Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 21 results · all verified · 13 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Hörmander Estimates and the Levi Problem

1 · Prerequisites

2 · Summary

This page proves weighted $L^{2}$ estimates for the $\overset{ˉ}{\partial}$ -equation on pseudoconvex domains and records a sourced solution of the Levi problem. The first half sets up the weighted Hilbert spaces of $(0, q)$ -forms with their maximal distributional $\overset{ˉ}{\partial}$ and weighted adjoint, solves the abstract Hilbert-complex problem from a coercive estimate, and records the Bochner-Kodaira-Morrey identity of the weighted $\overset{ˉ}{\partial}$ -Laplacian together with the weighted Morrey estimate on smooth Levi-pseudoconvex domains; the last step produces the Hörmander solver and the existence theorem for $\overset{ˉ}{\partial}$ -closed forms on Hartogs pseudoconvex domains, including the smooth-data branch. The local boundary separator and boundary peak construction are recorded separately, with a positive outer collar, smooth global defining functions and a proved exhaustion-to-Hartogs bridge supplying the correction on a neighborhood of the closure. The host-domain Oka-Weil theorem is used through Boas's approximation theorem.

The second half uses Demailly's Levi theorem and Cartan–Thullen to identify Hartogs pseudoconvex domains, domains of holomorphy, and holomorphically convex domains. The smooth exhaustion supplies the bridge to Demailly's pseudoconvexity criterion. Oka-Weil approximation on a pseudoconvex domain, Dolbeault vanishing, and the first Cousin problem follow from these inputs.

Conventions: $Ω \subseteq C^{n}$ is a domain with $n \geq 1$ , weights are real $C^{2}$ (or $C^{\infty}$ ) functions, $L_{0, q}^{2}$ denotes the weighted Hilbert space of Weighted L2 spaces and maximal dbar operators, and the maximal distributional $\overset{ˉ}{\partial}$ is the operator of that definition. The Axiom of Choice is declared on every proof-bearing item and is tracked through the suppliers, with the countable instance used by the exhaustion and regularization arguments recorded in the item-level choice notes. No regularity of $\overset{ˉ}{\partial}$ solutions at the boundary is claimed, and the Demailly (6.9) upper-semicontinuous weight statement is not used.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Smooth strict plurisubharmonic regularization of a psh exhaustion

Statement

Assume the Axiom of Choice (AC) and the Axiom of Countable Choice. Let $Ω \subseteq C^{n}$ , $n \geq 1$ , be a domain and let $u : Ω \to R$ be a continuous plurisubharmonic exhaustion, that is, $u$ is continuous and plurisubharmonic and every sublevel set ${z \in Ω : u (z) \leq c}$ , $c \in R$ , is a compact subset of $Ω$ .

Then there exist a function $S \in C^{\infty} (Ω)$ and a strictly increasing sequence $c_{1} < c_{2} < \dots$ with $c_{k} \to + \infty$ such that, writing $Ω_{k} : = {z \in Ω : S (z) < c_{k}}$ :

$S$ is strictly plurisubharmonic on $Ω$ , $S > u$ on $Ω$ , and $S$ is again an exhaustion of $Ω$ , that is, ${z \in Ω : S (z) \leq c}$ is a compact subset of $Ω$ for every real $c$ ;
every $c_{k}$ is a regular value of $S$ , and $\partial Ω_{k} = {z \in Ω : S (z) = c_{k}}$ is a nonempty $C^{\infty}$ hypersurface of $Ω$ ;
$\overline{Ω_{k}} \subseteq Ω_{k + 1}$ and $⋃_{k \geq 1} Ω_{k} = Ω$ , so every $\overline{Ω_{k}}$ is a compact subset of $Ω$ ;
(strong pseudoconvexity) for every $k$ , every $p \in \partial Ω_{k}$ and every $v \in C^{n} ∖ {0}$ with $\sum_{j < n} \frac{\partial S}{\partial z_{j}} (p) v_{j} = 0$ one has $L_{S} (p; v) > 0$ .

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice; a domain $Ω \subseteq C^{n}$ with $n \geq 1$ ; and a continuous plurisubharmonic exhaustion $u : Ω \to R$ .

[F1]

A function $u : Ω \to R$ is a continuous plurisubharmonic exhaustion when it is continuous, plurisubharmonic, and every sublevel ${u \leq c}$ is compact in $Ω$ (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F2]

(Richberg's approximation theorem.) If $v \in Psh (X)$ is continuous and strictly plurisubharmonic on an open set $V \subseteq X$ , with $H_{v} > γ$ for a continuous positive Hermitian form $γ$ , then for every continuous $0 < λ < 1$ there is $\tilde{v} \in C^{0} (X) \cap C^{\infty} (V)$ such that $v \leq \tilde{v} \leq v + λ$ on $V$ and $H_{\tilde{v}} > (1 - λ) γ$ ; if $v$ is strictly plurisubharmonic on all of $X$ , $\tilde{v}$ can be chosen strictly plurisubharmonic on all of $X$ (Demailly, Complex Analytic and Differential Geometry, Ch. I §5.E, Theorem 5.21, printed pp. 43-44).

[F3]

For a smooth map from a finite-dimensional manifold to $R$ , the regular values are dense; in particular every nonempty open interval contains a regular value when the Axiom of Countable Choice holds (Regular values have null complement and are dense).

[F4]

A value $c$ is regular for $S$ if every point of $S^{- 1} (c)$ is a regular point; an empty fibre is regular by convention (Regular and critical points and values).

[F5]

A regular level of a smooth real-valued function on an open subset of $R^{N}$ is locally a smooth graph of dimension $N - 1$ (A regular level set is locally a $C^{k}$ graph of dimension $m - n$ ).

[F6]

In ZF, AC implies AC $_{ω}$ (AC implies DC implies countable choice; The Axiom of Countable Choice ( ${A C}_{ω}$ )), and AC states that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC and AC $_{ω}$ are the ambient hypotheses. The proof uses AC $_{ω}$ only to choose a sequence of regular values from nonempty open intervals; each such interval contains regular values by [F3].

Proof

technique · Richberg approximation, followed by Sard's theorem

1.1F1algebra

Put $u_{0} : = u + ∣ z ∣^{2} + 1$ and $γ : = \frac{1}{2} H_{∣ z ∣^{2}}$ . Then $u_{0}$ is continuous and plurisubharmonic, and $H_{u_{0}} \geq H_{∣ z ∣^{2}} > γ$ because $u$ is plurisubharmonic. It is an exhaustion: if $u_{0} (z) \leq c$ , then $u (z) \leq c - 1$ , and ${u_{0} \leq c}$ is closed in $Ω$ ; thus it is a closed subset of the compact set ${u \leq c - 1}$ . Moreover $u_{0} > u$ everywhere.

2.1F2step 1.1algebra

Apply [F2] to $u_{0}$ on $X = V = Ω$ with the constant error $λ = 1 / 2$ . This gives $S \in C^{\infty} (Ω)$ satisfying $u_{0} \leq S \leq u_{0} + 1 / 2$ and $H_{S} > \frac{1}{2} γ$ . Hence $S$ is strictly plurisubharmonic, $S > u$ , and $S$ is an exhaustion because each sublevel ${S \leq c}$ is closed in $Ω$ and contained in the compact sublevel ${u_{0} \leq c}$ .

3.1F3F6step 2.1given

Fix $z_{0} \in Ω$ . The exhaustion $S$ is unbounded above: otherwise $Ω = {S \leq c}$ for some $c$ , making the noncompact open set $Ω$ compact. Choose an integer $M > S (z_{0})$ . For each $k \geq 1$ , [F3] supplies a regular value in the fixed nonempty interval $(M + 2 k, M + 2 k + 1)$ ; AC $_{ω}$ selects one such $c_{k}$ for each $k$ . Then $c_{1} < c_{2} < \dots$ , $c_{k} \to + \infty$ , and every level ${S = c_{k}}$ is nonempty: the continuous image $S (Ω)$ is an interval because $Ω$ is connected, it contains $S (z_{0})$ , and it is unbounded above.

4.1F1F4F5step 2.1step 3.1

Set $Ω_{k} : = {S < c_{k}}$ . Each $\overline{Ω_{k}}$ is contained in the compact set ${S \leq c_{k}}$ , and $\overline{Ω_{k}} \subseteq {S \leq c_{k}} \subseteq {S < c_{k + 1}} = Ω_{k + 1} .$ The sublevels cover $Ω$ because $c_{k} \to + \infty$ . Continuity gives $\partial Ω_{k} \subseteq {S = c_{k}}$ ; conversely, every point of the regular level $S = c_{k}$ is a boundary point by the implicit function theorem. Thus $\partial Ω_{k} = {S = c_{k}}$ is a nonempty smooth hypersurface, by [F4] and [F5].

5.1

At every $p \in \partial Ω_{k}$ the function $S - c_{k}$ defines $Ω_{k}$ near $p$ . Since $S$ is strictly plurisubharmonic, every nonzero complex tangent vector $v$ satisfies $L_{S} (p; v) > 0$ . Steps 2.1–4.1 establish the remaining assertions in the Statement. [F1, F2, step 2.1, step 4.1] $□$

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Meromorphic functions on an open set in complex Euclidean space

Definition

Fix $n \geq 1$ and read $C^{n}$ through Complex $m$ -space and its real coordinate dictionary. Let $U \subseteq C^{n}$ be a nonempty open set, and for an open $W \subseteq C^{n}$ write $O (W)$ for the set of holomorphic functions $W \to C$ (Holomorphic functions on an open subset of $C^{m}$ ). Open subsets of $C^{n}$ carry the subspace topology, and components are those of Connected components, quasicomponents, and totally disconnected spaces.

(a) Meromorphic functions. A meromorphic function on $U$ is a function $F ⁣ : D \to C$ such that

$D \subseteq U$ is open and dense in $U$ ;
$F$ is holomorphic on $D$ ;
every point $p \in U$ has a neighbourhood $W \subseteq U$ and holomorphic functions $f, g \in O (W)$ , with $g$ not identically zero on any connected component of $W$ , such that $F (z) = \frac{f (z)}{g (z)} whenever z \in D \cap W and g (z) \neq 0.$

The set $D$ is the domain of definition of $F$ , briefly its domain. Every $H \in O (U)$ determines a meromorphic function on $U$ with domain $U$ (take $D = U$ and $g \equiv 1$ in clause 3). A meromorphic representative $F : D \to C$ with $D \neq U$ is holomorphic on $U$ when it admits a holomorphic extension as in clause (c); removable omissions from $D$ are therefore allowed.

(b) Restriction. If $F$ is meromorphic on $U$ with domain $D$ and $V \subseteq U$ is open and nonempty, then the restriction $F ∣_{V}$ is the meromorphic function on $V$ with domain $D \cap V$ and values $(F ∣_{V}) (z) : = F (z)$ ; the domain $D \cap V$ is dense in $V$ because $D$ is dense in $U$ , and clause 3 for $F$ restricts to clause 3 for $F ∣_{V}$ .

(c) Holomorphic on a subset; differences. Let $F$ be meromorphic on $U$ with domain $D$ and let $V \subseteq U$ be open. Then $F$ is holomorphic on $V$ when there is $H \in O (V)$ with $H (z) = F (z)$ for every $z \in D \cap V$ ; such an $H$ is called a holomorphic extension of $F$ to $V$ . For meromorphic $F, G$ on $U$ with domains $D_{F}, D_{G}$ the difference $F - G$ is the function $D_{F} \cap D_{G} \to C$ , $z \mapsto F (z) - G (z)$ , and for open $V \subseteq U$ the phrase " $F - G$ is holomorphic on $V$ " means that $F - G$ admits a holomorphic extension to $V$ in this sense.

(d) Sums with holomorphic functions. If $F$ is meromorphic on $U$ with domain $D$ and $h \in O (U)$ , then $F + h ⁣ : D \to C$ , $z \mapsto F (z) + h (z)$ , is meromorphic on $U$ : it is holomorphic on $D$ , and wherever $F = f / g$ on $D \cap W$ for a local representation of clause 3 one has $F + h = (f + h g) / g$ there, while $f + h g$ and $g$ are holomorphic on $W$ and $g$ is not identically zero on any component of $W$ (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

Remark

Poles. Let $F$ be meromorphic on $U$ with domain $D$ . A point of $U$ is a pole of $F$ when $F$ admits no holomorphic extension to any neighbourhood of the point. The points where $F$ does admit a holomorphic extension form an open subset of $U$ : each such extension also works near every point in its domain. Hence the pole set is closed. Every pole lies outside $D$ , and $U ∖ D$ is closed with empty interior because $D$ is open and dense, so the pole set also has empty interior. In particular a meromorphic function is never undefined on a nonempty open subset of its ambient set: its domain meets every nonempty open subset of $U$ .

Local nature. Clause 3 is a local condition: if every point of $U$ has a neighbourhood to which $F$ restricts as a meromorphic function, then $F$ is meromorphic on $U$ . Concretely a ratio of two holomorphic functions is meromorphic on the open set where the denominator does not vanish identically on a component. This is the standard definition in several complex variables: in several variables only the local ratio is available in general.

Choice. This definition quantifies only over points, neighbourhoods and holomorphic functions; it invokes no choice principle, and none of its clauses selects from a family of nonempty sets.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Weighted L2 spaces and maximal dbar operators

Definition

Assume the Axiom of Choice (AC). Let $n \geq 1$ and let $Ω \subseteq C^{n}$ be open, let $0 \leq q \leq n$ , and let $φ \in C^{2} (Ω; R)$ . Write $L_{0, q}^{2} : = L_{0, q}^{2} (Ω, e^{- φ})$ for the object defined in (a) below, and use the conventions $L_{0, q}^{2} = {0}$ and ${\overset{ˉ}{\partial}}_{q} = 0$ for $q < 0$ and for $q > n$ . In formulas indexed by $1 \leq j \leq n$ , write $z_{j}$ for the canonical coordinate $z_{j - 1}$ of Complex $m$ -space and its real coordinate dictionary, and relabel the corresponding Wirtinger operators and form coefficients in the same way.

(a) Weighted $L^{2}$ spaces of $(0, q)$ -forms. A $(0, q)$ -form coefficient tuple $u = (u_{J})_{∣ J ∣ = q}$ is measurable when every $u_{J}$ is a measurable function in the sense of (Complex Lp classes and Euclidean test-function conventions); two tuples are identified when they agree almost everywhere. Put $⟨ u, v ⟩_{φ} : = \int_{Ω} \sum_{∣ J ∣ = q} u_{J} \overline{v_{J}} e^{- φ} d V, ∥ u ∥_{φ}^{2} : = ⟨ u, u ⟩_{φ},$ and define $L_{0, q}^{2} (Ω, e^{- φ}) : = {u : u measurable, ∥ u ∥_{φ} < \infty} / \sim .$ Here $d V$ is Lebesgue measure on $C^{n} ≅ R^{2 n}$ . The space $L_{0, q}^{2} (Ω, e^{- φ})$ carries the inner product $⟨ \cdot, \cdot ⟩_{φ}$ and is a complex Hilbert space.

(b) The maximal distributional $\overset{ˉ}{\partial}$ . For $u \in L_{0, q}^{2}$ choose any representative, which is locally integrable, and let $\overset{ˉ}{\partial} u : = \sum_{∣ J ∣ = q} \sum_{j = 1}^{n} \frac{\partial u_{J}}{\partial {\overset{ˉ}{z}}_{j}} d {\overset{ˉ}{z}}_{j} \land d {\overset{ˉ}{z}}^{J}$ be its distributional derivative, an element of the space of distributions on $Ω$ . The maximal domain is $Dom {\overset{ˉ}{\partial}}_{q} : = {u \in L_{0, q}^{2} : \overset{ˉ}{\partial} u is represented by an element of L_{0, q + 1}^{2}},$ and for $u \in Dom {\overset{ˉ}{\partial}}_{q}$ the form ${\overset{ˉ}{\partial}}_{q} u \in L_{0, q + 1}^{2}$ is that representing element. The operator ${\overset{ˉ}{\partial}}_{q} : Dom {\overset{ˉ}{\partial}}_{q} \to L_{0, q + 1}^{2}$ is the maximal distributional $\overset{ˉ}{\partial}$ in degree $q$ . Its minimal domain contains every smooth form with compact support in $Ω$ .

(c) The weighted adjoint. Let ${\overset{ˉ}{\partial}}_{q - 1} : Dom {\overset{ˉ}{\partial}}_{q - 1} \to L_{0, q}^{2}$ be the maximal operator of degree $q - 1$ . Its weighted adjoint is the Hilbert adjoint ${\overset{ˉ}{\partial}}_{φ}^{*} : L_{0, q}^{2} \supseteq Dom {\overset{ˉ}{\partial}}_{φ}^{*} \to L_{0, q - 1}^{2}$ , using the same bounded-functional definition for operators between the two Hilbert spaces: $v \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ holds exactly when the functional $u \mapsto ⟨ {\overset{ˉ}{\partial}}_{q - 1} u, v ⟩_{φ}$ is continuous on $Dom {\overset{ˉ}{\partial}}_{q - 1}$ in the ambient norm, and then ${\overset{ˉ}{\partial}}_{φ}^{*} v$ is the unique $w \in L_{0, q - 1}^{2}$ with $⟨ {\overset{ˉ}{\partial}}_{q - 1} u, v ⟩_{φ} = ⟨ u, w ⟩_{φ} for all u \in Dom {\overset{ˉ}{\partial}}_{q - 1} .$ Convention: ${\overset{ˉ}{\partial}}_{φ}^{*}$ always denotes the adjoint of the preceding degree, and the formal density $({\overset{ˉ}{\partial}}_{φ}^{*} v)_{K} = - e^{φ} \sum_{j = 1}^{n} \frac{\partial}{\partial z_{j}} (e^{- φ} v_{j K}) = \sum_{j = 1}^{n} (v_{j K} \frac{\partial φ}{\partial z_{j}} - \frac{\partial v_{j K}}{\partial z_{j}})$ expresses ${\overset{ˉ}{\partial}}_{φ}^{*} v$ as a distribution whenever $v \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ ; coefficients are extended to non-increasing tuples by antisymmetry, so that $v_{j K} = 0$ when $j \in K$ . The Hilbert adjoint is not asserted to equal this formal expression on all of $L_{0, q}^{2}$ , and no boundary condition on $\partial Ω$ is imposed here.

(d) Density of test forms. The smooth compactly supported $(0, q)$ -forms, regarded as tuples of their coefficient functions, form a linear subspace $C_{c}^{\infty} (Ω; Λ^{0, q}) \subseteq L_{0, q}^{2} (Ω, e^{- φ})$ that is dense in $L_{0, q}^{2} (Ω, e^{- φ})$ . The reduction to the unweighted Euclidean $L^{2}$ density theorem, first for compactly supported tuples and then in general by cutoff along a compact exhaustion of $Ω$ , is carried out in step 1.3 below.

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; an open set $Ω \subseteq C^{n}$ ; an integer $0 \leq q \leq n$ ; and a real function $φ \in C^{2} (Ω; R)$ .

[F1]

A measurable complex function is locally integrable for Lebesgue measure when $\int_{K} ∣ f ∣ < \infty$ on every compact $K$ (Complex Lp classes and Euclidean test-function conventions).

[F2]

For every measure space and $1 \leq p \leq \infty$ the complex $L^{p}$ space is complete (Complex Lp completeness and almost-everywhere subsequences).

[F3]

On every measure space the pairing $⟨ f, g ⟩ = \int f \overline{g}$ on complex $L^{2}$ is representative-independent, linear in the first variable, conjugate-linear in the second, conjugate symmetric and positive definite, and $∣ ⟨ f, g ⟩ ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ (The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz); the finite-tuple clause of the same theorem gives the same conclusions for the summed tuple pairing.

[F4]

The bidegree decomposition of complex forms and the coefficient formula $\overset{ˉ}{\partial} η = \sum_{I, J, j} (\partial_{{\overset{ˉ}{z}}_{j}} a_{I, J}) d {\overset{ˉ}{z}}_{j} \land d z^{I} \land d {\overset{ˉ}{z}}^{J}$ are as recorded in Bigraded complex forms and the Dolbeault operators.

[F5]

A weak derivative is defined by the test identity $\int_{Ω} u D^{α} φ = (- 1)^{∣ α ∣} \int_{Ω} v φ$ for every $φ \in C_{c}^{\infty} (Ω)$ , and it is a statement about almost-everywhere classes (Weak derivative of a locally integrable function).

[F6]

Assuming countable choice, $C_{c}^{\infty} (R^{n}; C)$ is dense in Euclidean Lebesgue $L^{p}$ for $n \geq 1$ and $1 \leq p < \infty$ (Complex finite-simple and smooth compact-support density for finite p).

[F7]

An operator is densely defined when its domain is dense and closed when its graph is closed (Densely defined, closed and closable operators, and cores), and the adjoint bounded-functional criterion for a densely defined operator on one Hilbert space reads: a vector $y$ lies in the adjoint domain exactly when $x \mapsto ⟨ T x, y ⟩$ is bounded on the domain (Adjoint of a densely defined operator).

[F8]

A complex Hilbert space is a complex inner-product space whose norm is complete (Hilbert space); the pairing of [F3] is the first-variable-linear convention fixed by The complex $L^{2}$ pairing on equivalence classes.

[F9]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice); its countable instance gives the countable-choice conventions used by [F2] and [F6] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F10]

For compact $K$ inside an open Euclidean set $O$ , there is $χ \in C_{c}^{\infty} (O)$ with $0 \leq χ \leq 1$ equal to one near $K$ (Test function cutoffs and euclidean localization).

[F11]

Under countable choice every bounded linear functional on a complex Hilbert space has a unique Riesz vector; the vector depends conjugate-linearly on the functional (Riesz representation for Hilbert spaces).

[F12]

Under countable choice a locally integrable function representing the zero distribution is zero almost everywhere (Locally integrable functions embed in distributions).

[F13]

Dominated convergence applies to measurable functions converging almost everywhere with a single integrable majorant (Dominated convergence).

[F14]

A compactly supported smooth unit-mass bump generates a mollifier family, and convolution with it is smooth (The mollifier family generated by a unit-mass smooth bump, Convolution with a mollifier is smooth, and derivatives pass under the integral sign).

Choice use. AC is used only through its countable instance, for the measure-theoretic completeness and density interfaces [F2], [F6], [F11] and [F12], and to select the cutoff sequence in step 1.3. The definitions of the weighted pairing, of the maximal operator and of the Hilbert adjoint select nothing.

Proof

technique · direct

1.1F2F3F8F9givenalgebra

If $Ω = \emptyset$ , all coefficient spaces contain only the zero class and the assertions are immediate. Assume $Ω \neq \emptyset$ . The weight $e^{- φ}$ is continuous and strictly positive on $Ω$ , so $μ_{φ} : = e^{- φ} d V$ is a Borel measure on $Ω$ with the same null sets as Lebesgue measure and with finite mass on every compact subset of the $σ$ -compact space $Ω$ ; by [F3] the tuple pairing of coefficient tuples satisfies the inner-product axioms and Cauchy-Schwarz, and by [F2] with $p = 2$ and $μ = μ_{φ}$ the complex space $L^{2} (μ_{φ}; C)$ is complete, so the coefficientwise space of (a), a finite product of $m = (\binom{n}{q})$ copies of $L^{2} (μ_{φ}; C)$ with the summed pairing, is a complex inner-product space whose norm is complete, that is, a complex Hilbert space in the sense of [F8]; the countable instance of [F9] is exactly the hypothesis consumed by [F2] and [F6], and no other selection is made here.

1.2F1F5givenalgebra

Let $u \in L_{0, q}^{2}$ and let $K \subseteq Ω$ be a nonempty compact set; then $\int_{K} ∣ u_{J} ∣ d V \leq ∣ K ∣^{1 / 2} (\int_{K} ∣ u_{J} ∣^{2} d V)^{1 / 2}$ by Cauchy-Schwarz, and $c_{K} : = \min_{K} e^{- φ} > 0$ gives $\int_{K} ∣ u_{J} ∣^{2} d V \leq c_{K}^{- 1} ∥ u_{J} ∥_{φ}^{2} < \infty$ , so every coefficient of $u$ is locally integrable; by [F1] it therefore has a distributional derivative in each variable, and by [F5] that derivative depends only on the almost-everywhere class of $u_{J}$ , so it is well defined on $L_{0, q}^{2}$ and additive and $C$ -homogeneous in the coefficient.

1.3F1F6F9F10F13givenalgebra

Test forms are dense in the weighted space. Let $u \in L_{0, q}^{2}$ . For integers $k \geq 1$ choose $χ_{k} \in C_{c}^{\infty} (Ω)$ with $0 \leq χ_{k} \leq 1$ , $χ_{k} = 1$ near $K_{k}$ and $supp χ_{k} \subseteq K_{k + 1}$ , where $K_{k} = {∣ z ∣ \leq k} \cap {dist (z, C^{n} ∖ Ω) \geq 1 / k}$ and the distance constraint is omitted when $Ω = C^{n}$ . These sets are compact, exhaust $Ω$ , and satisfy $K_{k} \subseteq int K_{k + 1}$ : distance to the nonempty complement is continuous by the triangle inequality, and both defining inequalities become strict at the next index. Apply [F10] with $O = int K_{k + 1}$ and use countable choice for the cutoffs (take zero when $K_{k}$ is empty); then $∣ χ_{k} u - u ∣^{2} e^{- φ} \leq 4 ∣ u ∣^{2} e^{- φ} \in L^{1}$ and $χ_{k} u \to u$ pointwise, so [F13] gives $∥ χ_{k} u - u ∥_{φ} \to 0$ and each $χ_{k} u$ has compact support in $Ω$ . It remains to approximate any such compactly supported tuple $v : = χ_{k} u$ . If $v = 0$ there is nothing to prove; otherwise put $K : = supp v \subseteq K_{k + 1}$ and use the already chosen exhaustion cutoff $θ : = χ_{k + 1}$ , which equals $1$ near $K_{k + 1}$ and has compact support in $Ω$ . Thus $θ v = v$ . Let $C : = \max_{supp θ} e^{- φ} < \infty$ , extend each coefficient of $v$ by zero to a tuple $\tilde{v}$ on $C^{n} ≅ R^{2 n}$ , and apply [F6] componentwise: for every $ε > 0$ there is a tuple $ψ = (ψ_{J})$ with $ψ_{J} \in C_{c}^{\infty} (R^{2 n})$ and $∥ ψ_{J} - {\tilde{v}}_{J} ∥_{L^{2} (R^{2 n})} < ε m^{- 1 / 2} C^{- 1 / 2}$ for each of the $m = (\binom{n}{q})$ coefficients. The tuple $θ ψ ∣_{Ω}$ lies in $C_{c}^{\infty} (Ω; Λ^{0, q})$ , and because $θ v = v$ its error is $θ (ψ - \tilde{v})$ on $Ω$ . Therefore $∥ θ ψ - v ∥_{φ}^{2} \leq C \sum_{∣ J ∣ = q} ∥ ψ_{J} - {\tilde{v}}_{J} ∥_{L^{2} (R^{2 n})}^{2} < ε^{2} .$ This proves the claimed weighted density.

2.1F4F5F12step 1.2givenalgebra

Define $\overset{ˉ}{\partial} u$ for $u \in L_{0, q}^{2}$ by the formula of (b) using the locally integrable representative supplied by step 1.2; by [F4] its coefficient in front of $d {\overset{ˉ}{z}}^{L}$ , $∣ L ∣ = q + 1$ , is $\sum_{j \in L} ε_{j, L ∖ j} \partial u_{L ∖ j} / \partial {\overset{ˉ}{z}}_{j}$ , a distribution, and if $w, w^{'} \in L_{0, q + 1}^{2}$ both represent $\overset{ˉ}{\partial} u$ , then every coefficient of $w - w^{'}$ is locally integrable by step 1.2 and represents the zero distribution, so [F12] gives $w = w^{'}$ almost everywhere; hence $Dom {\overset{ˉ}{\partial}}_{q}$ and ${\overset{ˉ}{\partial}}_{q}$ are well defined.

2.2F4F5F14step 1.2givenalgebra

For $q = 0$ the preceding-degree adjoint is zero by convention. For $q \geq 1$ , let $v \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ and $w : = {\overset{ˉ}{\partial}}_{φ}^{*} v$ . For every compactly supported smooth $(0, q - 1)$ -form $ψ$ , the adjoint identity and [F4], with wedge signs absorbed in the antisymmetric coefficients $v_{j K}$ , give $\int_{Ω} \sum_{∣ K ∣ = q - 1} ψ_{K} \overline{w_{K}} e^{- φ} d V = \int_{Ω} \sum_{∣ K ∣ = q - 1} \sum_{j = 1}^{n} (\partial_{{\overset{ˉ}{z}}_{j}} ψ_{K}) \overline{v_{j K}} e^{- φ} d V .$ The identity [F5], conjugated and summed, therefore gives $e^{- φ} w_{K} = - \sum_{j} \partial_{z_{j}} (e^{- φ} v_{j K})$ as distributions. Here a first derivative of a locally integrable function acts continuously on compactly supported $C^{1}$ tests by its defining integral. To use such a test, extend it by zero and convolve with a smooth unit-mass bump as in [F14]: the approximations and their first derivatives converge uniformly, with supports in a fixed compact subset of $Ω$ . Local integrability then passes each defining integral to the limit. This also justifies the $C^{2}$ multiplier $e^{φ}$ and its product rule in this order-one identity. Multiplying by $e^{φ}$ and expanding yields exactly (c), only on the Hilbert-adjoint domain.

3.1F4F5step 2.1givenalgebra

The domain of ${\overset{ˉ}{\partial}}_{q}$ is a linear subspace and ${\overset{ˉ}{\partial}}_{q}$ is linear: by [F5] the distributional identity $\partial (a u_{J} + b u_{J}^{'}) / \partial {\overset{ˉ}{z}}_{j} = a \partial u_{J} / \partial {\overset{ˉ}{z}}_{j} + b \partial u_{J}^{'} / \partial {\overset{ˉ}{z}}_{j}$ holds for all scalars $a, b$ and all locally integrable $u_{J}, u_{J}^{'}$ (test against every compactly supported smooth function and use linearity of the integral), and applying the uniqueness part of step 2.1 to the two representations of $\overset{ˉ}{\partial} (a u + b u^{'})$ gives ${\overset{ˉ}{\partial}}_{q} (a u + b u^{'}) = a {\overset{ˉ}{\partial}}_{q} u + b {\overset{ˉ}{\partial}}_{q} u^{'}$ ; the domain is nonempty because every smooth compactly supported $(0, q)$ -form has its smooth $\overset{ˉ}{\partial}$ in $L_{0, q + 1}^{2}$ by [F4].

4.1F7F11step 1.3step 3.1givenalgebra∎

The operator ${\overset{ˉ}{\partial}}_{q - 1}$ is densely defined because its domain contains the compactly supported smooth $(0, q - 1)$ -forms by step 3.1 and these are dense by step 1.3; therefore the bounded-functional criterion in (c) defines its adjoint: each bounded functional extends to the domain Hilbert space and has a unique Riesz vector by [F11], so ${\overset{ˉ}{\partial}}_{φ}^{*}$ is unique, and it is linear because $v \mapsto (u \mapsto ⟨ \overset{ˉ}{\partial} u, v ⟩_{φ})$ and the Riesz correspondence are both conjugate-linear, while the convention for $q < 0$ and $q > n$ is the zero operator on ${0}$ ; this completes the well-definedness of the objects named in (a), (b) and (c).

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

The maximal distributional dbar operator is closed and densely defined

Statement

Assume the Axiom of Choice (AC). Let $Ω \subseteq C^{n}$ be open with $n \geq 1$ , let $0 \leq q \leq n$ , let $φ \in C^{2} (Ω; R)$ , let ${\overset{ˉ}{\partial}}_{q} : Dom {\overset{ˉ}{\partial}}_{q} \subseteq L_{0, q}^{2} (Ω, e^{- φ}) \to L_{0, q + 1}^{2} (Ω, e^{- φ})$ be the maximal distributional $\overset{ˉ}{\partial}$ of degree $q$ , and let ${\overset{ˉ}{\partial}}_{φ}^{*}$ be the Hilbert adjoint of ${\overset{ˉ}{\partial}}_{q - 1}$ , all with the conventions of Weighted L2 spaces and maximal dbar operators, including its one-based relabeling of the canonical coordinates.

$Dom {\overset{ˉ}{\partial}}_{q}$ is dense in $L_{0, q}^{2}$ , and ${\overset{ˉ}{\partial}}_{q}$ is closed.
For $1 \leq q \leq n$ and every $ψ \in C_{c}^{\infty} (Ω)$ of bidegree $(0, q)$ , $ψ$ lies in $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ , and ${\overset{ˉ}{\partial}}_{φ}^{*} ψ$ is the compactly supported form with $C^{1}$ coefficients $({\overset{ˉ}{\partial}}_{φ}^{*} ψ)_{K} = \sum_{j = 1}^{n} (ψ_{j K} \frac{\partial φ}{\partial z_{j}} - \frac{\partial ψ_{j K}}{\partial z_{j}}), ψ_{j K} : = 0 when j \in K .$
${\overset{ˉ}{\partial}}_{φ}^{*}$ is closed.

Facts & Assumptions

Given: The Axiom of Choice; an open set $Ω \subseteq C^{n}$ with $n \geq 1$ ; an integer $0 \leq q \leq n$ ; and a real function $φ \in C^{2} (Ω; R)$ .

[F1]

The weighted $(0, q)$ -space is $L_{0, q}^{2} (Ω, e^{- φ}) : = {u : u measurable, ∥ u ∥_{φ} < \infty} / \sim$ for the pairing $⟨ u, v ⟩_{φ} = \int_{Ω} \sum_{∣ J ∣ = q} u_{J} \overline{v_{J}} e^{- φ} d V$ (Weighted L2 spaces and maximal dbar operators).

[F2]

The maximal domain is $Dom {\overset{ˉ}{\partial}}_{q} : = {u \in L_{0, q}^{2} : \overset{ˉ}{\partial} u is represented by an element of L_{0, q + 1}^{2}}$ , where $\overset{ˉ}{\partial} u = \sum_{∣ J ∣ = q} \sum_{j} \frac{\partial u_{J}}{\partial {\overset{ˉ}{z}}_{j}} d {\overset{ˉ}{z}}_{j} \land d {\overset{ˉ}{z}}^{J}$ is the distributional derivative; for $u \in Dom {\overset{ˉ}{\partial}}_{q}$ the form ${\overset{ˉ}{\partial}}_{q} u$ is that representing element (Weighted L2 spaces and maximal dbar operators).

[F3]

The weighted adjoint satisfies $⟨ {\overset{ˉ}{\partial}}_{q - 1} u, v ⟩_{φ} = ⟨ u, {\overset{ˉ}{\partial}}_{φ}^{*} v ⟩_{φ}$ for all $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ and $v \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ , its formal density is $({\overset{ˉ}{\partial}}_{φ}^{*} v)_{K} = - e^{φ} \sum_{j} \partial_{z_{j}} (e^{- φ} v_{j K}) = \sum_{j} (v_{j K} \partial_{z_{j}} φ - \partial_{z_{j}} v_{j K})$ whenever $v \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ , and coefficients are extended to non-increasing tuples by antisymmetry, with $v_{j K} = 0$ when $j \in K$ (Weighted L2 spaces and maximal dbar operators).

[F4]

Test forms are dense in the weighted space: every $u \in L_{0, q}^{2}$ is the $∥ \cdot ∥_{φ}$ -limit of a sequence of forms in $C_{c}^{\infty} (Ω)$ (Weighted L2 spaces and maximal dbar operators).

[F5]

Every coefficient of every $u \in L_{0, q}^{2}$ is locally integrable for Lebesgue measure on $Ω$ (Weighted L2 spaces and maximal dbar operators).

[F6]

Every smooth compactly supported $(0, q)$ -form has its smooth $\overset{ˉ}{\partial}$ in $L_{0, q + 1}^{2}$ , hence lies in $Dom {\overset{ˉ}{\partial}}_{q}$ (Weighted L2 spaces and maximal dbar operators).

[F7]

A weak derivative is characterized by the test identity $\int_{Ω} u D^{α} χ = (- 1)^{∣ α ∣} \int_{Ω} v χ$ for every real test function $χ \in C_{c}^{\infty} (Ω)$ (Weak derivative of a locally integrable function).

[F9]

An operator is densely defined when its domain is dense, and closed when its graph is a closed subset of $H \oplus H$ (Densely defined, closed and closable operators, and cores).

[F10]

A vector $y \in H$ lies in the adjoint domain exactly when $x \mapsto ⟨ T x, y ⟩$ is bounded on the domain, and then $T^{*} y$ is the unique $w$ with $⟨ T x, y ⟩ = ⟨ x, w ⟩$ for all $x$ in the domain (Adjoint of a densely defined operator).

[F11]

The published same-space adjoint theorem is not used for this operator between distinct form-degree Hilbert spaces; closedness is established directly in step 2.2.

[F12]

On every measure space the complex $L^{2}$ pairing satisfies $∣ ⟨ f, g ⟩ ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ , also for finite tuples (The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz).

[F13]

In a metric space a set is closed exactly when it is sequentially closed (A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed).

[F14]

AC implies the Axiom of Countable Choice ${A C}_{ω}$ (AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F15]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice).

[F16]

The Wirtinger operator is $\partial_{{\overset{ˉ}{z}}_{j}} = \frac{1}{2} (\partial_{x_{j}} + i \partial_{y_{j}})$ (Wirtinger operators in $C^{m}$ ).

[F17]

For a compactly supported smooth unit-mass bump $b$ on Euclidean space, $b_{ε} (x) = ε^{- 2 n} b (x / ε)$ is its mollifier family, and convolution of a locally integrable function with $b_{ε}$ is smooth (The mollifier family generated by a unit-mass smooth bump, Convolution with a mollifier is smooth, and derivatives pass under the integral sign).

Choice use. AC is the ambient hypothesis recorded in the Statement, and ${A C}_{ω}$ is the countable instance consumed by the density interface [F4], by the adjoint definition [F10], and by the sequential characterization [F13]; [F14] is the exact implication supplying it. The proof selects no family: the test forms, the limiting form $v$ and the formal expression are given.

Proof

technique · direct

1.1F4F6F9F14given

By [F6] every smooth compactly supported $(0, q)$ -form lies in $Dom {\overset{ˉ}{\partial}}_{q}$ , and by [F4] these forms are dense in $L_{0, q}^{2}$ ; hence $Dom {\overset{ˉ}{\partial}}_{q}$ contains a dense subset and is itself dense in $L_{0, q}^{2}$ , which is the definition of ${\overset{ˉ}{\partial}}_{q}$ being densely defined in the sense of [F9]. The density interface [F4] is where the countable instance ${A C}_{ω}$ of [F14] is consumed.

1.2F1F2F3F5F7F16F17givenalgebra

Let $u \in Dom {\overset{ˉ}{\partial}}_{q}$ and $g = {\overset{ˉ}{\partial}}_{q} u$ . The unweighted distributional identity of [F2] extends from smooth compactly supported test forms to $C_{c}^{1}$ test forms. Indeed extend a $C_{c}^{1}$ test $h$ by zero to Euclidean space and convolve with a nonnegative unit-mass smooth bump as in [F17]. For sufficiently small $ε$ these smooth tests have supports in a fixed compact subset of $Ω$ , and $h * b_{ε} \to h$ together with every first derivative uniformly: differentiate under the integral in the expression $\int b_{ε} (y) h (x - y) d y$ and use uniform continuity of $h$ and its first derivatives. Local integrability of $u$ and $g$ then passes both sides of the test identity to the limit. For a smooth $(0, q + 1)$ -form $ψ$ of compact support, use the $C_{c}^{2}$ test coefficients $e^{- φ} \overline{ψ_{L}}$ . The coefficient sum in [F2], with its antisymmetric wedge signs, and the product rule give $⟨ g, ψ ⟩_{φ} = ⟨ u, \partial^{*} ψ ⟩_{φ}, (\partial^{*} ψ)_{K} = \sum_{j} (ψ_{j K} φ_{j} - \partial_{z_{j}} ψ_{j K}) .$ Only $g$ as a whole is assumed locally integrable; no individual weak derivative of a coefficient of $u$ is assumed to be a function.

1.3F1F2F5F7F9F12F13F14givenalgebra

The operator ${\overset{ˉ}{\partial}}_{q}$ is closed. Let $u_{k} \in Dom {\overset{ˉ}{\partial}}_{q}$ with $u_{k} \to u$ in $L_{0, q}^{2}$ and $g_{k} : = {\overset{ˉ}{\partial}}_{q} u_{k} \to v$ in $L_{0, q + 1}^{2}$ . On every compact $K \subseteq Ω$ , the positive minimum $c_{K}$ of $e^{- φ}$ gives $∥ u_{k} - u ∥_{L^{2} (K)} \leq c_{K}^{- 1 / 2} ∥ u_{k} - u ∥_{φ} \to 0$ , and the same bound holds for $g_{k} - v$ . Cauchy-Schwarz on $K$ therefore gives local $L^{1}$ convergence. Against any ordinary smooth compactly supported test form, both sides of the unweighted distributional identity for $\overset{ˉ}{\partial} u_{k} = g_{k}$ pass to the limit, since the test and its first derivatives are bounded. Thus $\overset{ˉ}{\partial} u = v$ as distributions, so [F2] gives $u \in Dom {\overset{ˉ}{\partial}}_{q}$ and ${\overset{ˉ}{\partial}}_{q} u = v$ . (For $q = n$ the operator is zero on the entire space and the conclusion is immediate.) The graph is sequentially closed in the metric direct sum of the two form-degree spaces, hence closed by [F13].

2.1F3F10F12F14step 1.1step 1.2givenalgebra

Let $ψ \in C_{c}^{\infty} (Ω)$ have bidegree $(0, q)$ with $1 \leq q \leq n$ , and let $\partial^{*} ψ$ be the formal expression of [F3]. Then $\partial^{*} ψ$ has compact support contained in $supp ψ$ and coefficients in $C^{1} (Ω)$ because $φ \in C^{2}$ and $ψ \in C_{c}^{\infty}$ , so $\partial^{*} ψ \in L_{0, q - 1}^{2}$ ; moreover step 1.1 makes ${\overset{ˉ}{\partial}}_{q - 1}$ densely defined, and step 1.2 gives $⟨ {\overset{ˉ}{\partial}}_{q - 1} u, ψ ⟩_{φ} = ⟨ u, \partial^{*} ψ ⟩_{φ}$ for every $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ , so by [F12] the functional $u \mapsto ⟨ {\overset{ˉ}{\partial}}_{q - 1} u, ψ ⟩_{φ}$ is bounded on the domain with norm at most $∥ \partial^{*} ψ ∥_{φ}$ . By the characterization [F10] we therefore have $ψ \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ and ${\overset{ˉ}{\partial}}_{φ}^{*} ψ = \partial^{*} ψ$ ; the countable choice used by the adjoint definition is supplied through [F14].

2.2F3F12F13step 1.1given

The adjoint ${\overset{ˉ}{\partial}}_{φ}^{*}$ is closed even though its source and target Hilbert spaces have different form degrees. Let $v_{m} \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ satisfy $v_{m} \to v$ in $L_{0, q}^{2}$ and ${\overset{ˉ}{\partial}}_{φ}^{*} v_{m} \to w$ in $L_{0, q - 1}^{2}$ . For every $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ the adjoint identity [F3] and continuity of the two inner products give $⟨ {\overset{ˉ}{\partial}}_{q - 1} u, v ⟩_{φ} = \lim_{m} ⟨ u, {\overset{ˉ}{\partial}}_{φ}^{*} v_{m} ⟩_{φ} = ⟨ u, w ⟩_{φ}$ . Thus $v$ lies in the adjoint domain and ${\overset{ˉ}{\partial}}_{φ}^{*} v = w$ by its definition [F3]. The graph is sequentially closed and hence closed by [F13].

3.1F3F15step 1.1step 2.1step 1.3step 2.2∎

Conclusion: $Dom {\overset{ˉ}{\partial}}_{q}$ is dense and ${\overset{ˉ}{\partial}}_{q}$ is closed by steps 1.1 and 1.3; every compactly supported smooth $(0, q)$ -test form lies in $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ with the formal weighted expression of [F3] by step 2.1; and ${\overset{ˉ}{\partial}}_{φ}^{*}$ is closed by step 2.2. This is exactly the content of the three claims of the Statement, and the ambient hypothesis is the AC recorded in the Statement and cited as [F15].

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Basic Bochner–Kodaira–Morrey estimate on $C^{n}$

Statement

Assume the Axiom of Choice (AC). Let $Ω \subseteq C^{n}$ be open with $n \geq 1$ . Use one-based labels $z_{j} : = z_{j - 1}^{c a n}$ for $1 \leq j \leq n$ , also for their derivatives and form coefficients. Let $φ \in C^{2} (Ω; R)$ , write

$D_{k} : = \frac{\partial}{\partial {\overset{ˉ}{z}}_{k}}, φ_{j} : = \frac{\partial φ}{\partial z_{j}}, φ_{j \overset{ˉ}{k}} : = \frac{\partial^{2} φ}{\partial z_{j} \partial {\overset{ˉ}{z}}_{k}},$

let $1 \leq q \leq n$ , and let $u$ be a compactly supported smooth $(0, q)$ -form on $Ω$ , with coefficients $u_{J}$ on increasing tuples extended to non-increasing tuples by antisymmetry, so that $u_{j J} = 0$ when $j \in J$ (Weighted L2 spaces and maximal dbar operators); the operators $\partial_{z_{j}}, \partial_{{\overset{ˉ}{z}}_{j}}$ are those of Wirtinger operators in $C^{m}$ . All sums over multi-indices below run over increasing tuples, $⟨ \cdot, \cdot ⟩_{φ}$ and $∥ \cdot ∥_{φ}$ are the inner product and norm of $L_{0, ∙}^{2} (Ω, e^{- φ})$ , and ${\overset{ˉ}{\partial}}_{φ}^{*}$ is the weighted Hilbert adjoint of ${\overset{ˉ}{\partial}}_{q - 1}$ (Weighted L2 spaces and maximal dbar operators).

(Exact Bochner–Kodaira–Morrey form.) The following identity holds, all integrals being finite:

$∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2} = \sum_{∣ J ∣ = q} \sum_{k = 1}^{n} \int_{Ω} ∣ D_{k} u_{J} ∣^{2} e^{- φ} d V + \int_{Ω} \sum_{∣ J ∣ = q - 1} \sum_{j, k = 1}^{n} φ_{j \overset{ˉ}{k}} u_{j J} \overline{u_{k J}} e^{- φ} d V .$

(Levi inequality.) If in addition $φ$ is plurisubharmonic on $Ω$ , then

$∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2} \geq \int_{Ω} \sum_{∣ J ∣ = q - 1} \sum_{j, k = 1}^{n} φ_{j \overset{ˉ}{k}} u_{j J} \overline{u_{k J}} e^{- φ} d V .$

Facts & Assumptions

Given: The Axiom of Choice; an open set $Ω \subseteq C^{n}$ with $n \geq 1$ ; a function $φ \in C^{2} (Ω; R)$ ; an integer $1 \leq q \leq n$ ; a compactly supported smooth $(0, q)$ -form $u$ ; and the notation $e_{j} : = d {\overset{ˉ}{z}}_{j}$ , $D_{j} : = \partial_{{\overset{ˉ}{z}}_{j}}$ , $δ_{j} : = φ_{j} - \partial_{z_{j}}$ acting coefficientwise, where the Wirtinger operators $\partial_{z_{j}}, \partial_{{\overset{ˉ}{z}}_{j}}$ are those fixed in the Statement, so that the holomorphic derivative, and not $D_{j} = \partial_{{\overset{ˉ}{z}}_{j}}$ , appears in $δ_{j}$ ; further $ε_{j} f : = e_{j} \land f$ on coefficient tensors, and $(ι_{j} w)_{K} : = w_{j K}$ for increasing $K$ .

[F1]

The weighted inner product and norm on coefficient tuples of bidegree $(0, q)$ are $⟨ v, w ⟩_{φ} = \int_{Ω} \sum_{∣ J ∣ = q} v_{J} \overline{w_{J}} e^{- φ} d V$ and $∥ v ∥_{φ}^{2} = ⟨ v, v ⟩_{φ}$ (Weighted L2 spaces and maximal dbar operators).

[F2]

The distributional derivative of $u \in L_{0, q}^{2}$ is $\overset{ˉ}{\partial} u = \sum_{∣ J ∣ = q} \sum_{j = 1}^{n} (\partial u_{J} / \partial {\overset{ˉ}{z}}_{j}) d {\overset{ˉ}{z}}_{j} \land d {\overset{ˉ}{z}}^{J}$ (Weighted L2 spaces and maximal dbar operators).

[F3]

The weighted adjoint is characterized by $⟨ {\overset{ˉ}{\partial}}_{q - 1} v, w ⟩_{φ} = ⟨ v, {\overset{ˉ}{\partial}}_{φ}^{*} w ⟩_{φ}$ for all $v \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ and $w \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ (Weighted L2 spaces and maximal dbar operators).

[F4]

Coefficients are extended to non-increasing tuples by antisymmetry, so that $v_{j K} = 0$ when $j \in K$ (Weighted L2 spaces and maximal dbar operators).

[F5]

The conventions $L_{0, q}^{2} = {0}$ and ${\overset{ˉ}{\partial}}_{q} = 0$ are in force for $q < 0$ and for $q > n$ (Weighted L2 spaces and maximal dbar operators).

[F6]

For $1 \leq q \leq n$ every $ψ \in C_{c}^{\infty} (Ω)$ of bidegree $(0, q)$ lies in $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ , and $({\overset{ˉ}{\partial}}_{φ}^{*} ψ)_{K} = \sum_{j = 1}^{n} (ψ_{j K} φ_{j} - \partial_{z_{j}} ψ_{j K})$ (The maximal distributional dbar operator is closed and densely defined).

[F7]

Wedge multiplication of basis vectors is multilinear and alternating, so $e_{i} \land e_{i} = 0$ and transposing two neighbouring entries changes the sign; it is also associative, and the strictly increasing monomials form a basis; hence for a distinct $i$ and an increasing tuple $I = (i_{1} < \dots < i_{p})$ one has $e_{i} \land e_{I} = (- 1)^{# {i_{j} < i}} e_{sort ({i} \cup I)}$ , while $e_{i} \land e_{I} = 0$ when $i \in I$ (The basic wedge map $(v_{1}, \dots, v_{k}) \mapsto v_{1} \land \dots \land v_{k}$ is multilinear and alternating, Exterior multiplication is well defined, graded, associative, unital, and graded-commutative, Wedge monomials in a dual basis form a basis).

[F8]

The Levi form is $L_{u} (a; v) = \sum_{j = 1}^{m} \sum_{k = 1}^{m} \frac{\partial^{2} u}{\partial z_{j} \partial {\overline{z}}_{k}} (a) v_{j} \overline{v_{k}}$ (The Levi form and strict plurisubharmonicity).

[F9]

A $C^{2}$ real-valued function is plurisubharmonic if and only if its Levi form is pointwise semidefinite nonnegative (The C^2 Levi criterion for plurisubharmonicity).

[F10]

For a function with continuous second partial derivatives, $\partial_{j} \partial_{i} ϕ = \partial_{i} \partial_{j} ϕ$ (Continuous second partials of a scalar potential commute).

[F11]

On every measure space the complex $L^{2}$ pairing is linear in the first variable, conjugate-linear in the second, conjugate symmetric and positive definite (The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz).

[F12]

$A C \Rightarrow D C \Rightarrow {A C}_{ω}$ in ZF (AC implies DC implies countable choice).

[F13]

The Axiom of Countable Choice ${A C}_{ω}$ supplies a choice function for every at most countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F14]

The Axiom of Choice supplies a choice function for every family of nonempty sets (The Axiom of Choice).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F14]; the countable instance [F13] is the form of choice consumed by the constructions behind [F1] (completeness and density of the weighted space) and [F6] (the maximal operator and its adjoint on compactly supported smooth forms), and [F12] is the exact implication AC $\Rightarrow$ DC $\Rightarrow$ AC $_{ω}$ supplying it. The proof itself selects no family of nonempty sets: the form, its coefficients, the weight and the finitely many index sets in the sums are all given.

Proof

technique · direct

1.1F4F7givenalgebra

The notation of the given block is well defined on coefficient tensors, and for increasing $L$ the sign rule [F7] gives $e_{k} \land e_{L} = ϵ (k, L) e_{L \cup k}$ for $k \notin L$ and $e_{k} \land e_{L} = 0$ for $k \in L$ , while the antisymmetry convention [F4] gives $ι_{j} e_{L} = ϵ (j, L ∖ j) e_{L ∖ j}$ for $j \in L$ and $ι_{j} e_{L} = 0$ for $j \notin L$ , where $ϵ (a, M) : = (- 1)^{# {m \in M : m < a}}$ ; consequently the Clifford relation $ι_{j} ε_{k} + ε_{k} ι_{j} = δ_{j k} i d$ holds, because for $j = k \in L$ one has $ι_{j} ε_{j} e_{L} = 0$ and $ε_{j} ι_{j} e_{L} = e_{L}$ , for $j = k \notin L$ one has $ι_{j} ε_{j} e_{L} = e_{L}$ and $ε_{j} ι_{j} e_{L} = 0$ , for $j \neq k$ with $j \notin L$ both terms vanish since an index occurring twice wedges to zero, and for $j \neq k$ with $j \in L$ the terms vanish when $k \in L$ and otherwise cancel by the sign identity $ϵ (k, L) ϵ (j, (L ∖ j) \cup k) = - ϵ (j, L ∖ j) ϵ (k, L ∖ j)$ , whose two exponents differ by exactly one because exactly one of $j < k$ , $k < j$ holds.

1.2F1F4F7F11givenalgebra

For coefficient tensors $f$ of degree $p$ and $w$ of degree $p + 1$ with compactly supported smooth coefficients, $⟨ ε_{j} f, w ⟩_{φ} = ⟨ f, ι_{j} w ⟩_{φ}$ : by [F11] both sides are sesquilinear in $(f, w)$ , and on constant basis tensors $f = e_{I}$ , $w = e_{L}$ the formulas above from [F7] and [F4] give $(ε_{j} e_{I})_{L} = ϵ (j, I) δ_{L, I \cup j}$ and $(ι_{j} e_{L})_{I} = ϵ (j, L ∖ j) δ_{I, L ∖ j}$ , which are equal since $L = I \cup j$ exactly when $I = L ∖ j$ ; multiplying the pointwise identity by the positive factor $e^{- φ}$ and integrating with the pairing of [F1] gives the weighted statement.

1.3F1F2F3F6F11givenalgebra

For all $f, g \in C_{c}^{\infty} (Ω)$ one has $⟨ D_{j} f, g ⟩_{φ} = ⟨ f, δ_{j} g ⟩_{φ}$ , and hence also $⟨ δ_{j} f, g ⟩_{φ} = ⟨ f, D_{j} g ⟩_{φ}$ by conjugate symmetry; indeed, $f$ is a compactly supported smooth $(0, 0)$ -form and $g e_{j}$ a compactly supported smooth $(0, 1)$ -form lying in $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ with ${\overset{ˉ}{\partial}}_{φ}^{*} (g e_{j}) = δ_{j} g$ by [F6], while [F2] gives $\overset{ˉ}{\partial} f = \sum_{k} D_{k} f e_{k}$ , so the characterizing identity [F3] reads $\sum_{k} δ_{k j} ⟨ D_{k} f, g ⟩_{φ} = ⟨ f, δ_{j} g ⟩_{φ}$ .

1.4F2F6givenalgebra

On compactly supported smooth forms the operator identities $\overset{ˉ}{\partial} = \sum_{k} ε_{k} D_{k}$ and ${\overset{ˉ}{\partial}}_{φ}^{*} = \sum_{j} ι_{j} δ_{j}$ hold pointwise in the increasing coefficients, the first because [F2] expands $\overset{ˉ}{\partial}$ as the sum of the wedges $(D_{k} u_{J}) e_{k} \land e_{J}$ , and the second because the coefficient formula of [F6] is $\sum_{j} δ_{j} (ψ_{j K})$ on each increasing $K$ , which is what $\sum_{j} ι_{j} δ_{j}$ produces; consequently the self-adjoint-shaped operator $□ : = {\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} + \overset{ˉ}{\partial} {\overset{ˉ}{\partial}}_{φ}^{*}$ satisfies $□ = \sum_{j, k} (ι_{j} δ_{j} ε_{k} D_{k} + ε_{k} D_{k} ι_{j} δ_{j})$ on those forms.

2.1F10step 1.1step 1.4givenalgebra

The operator identity $□ = \sum_{j} δ_{j} D_{j} + \sum_{j, k} φ_{j \overset{ˉ}{k}} ε_{k} ι_{j}$ holds on compactly supported smooth forms: by the identities of step 1.4, the constant-coefficient form operators $ε_{k}, ι_{j}$ commute with the coefficientwise operators $δ_{j}, D_{k}$ , and by the Clifford relation of step 1.1 one has $ι_{j} δ_{j} ε_{k} D_{k} = ι_{j} ε_{k} δ_{j} D_{k} = δ_{j k} δ_{j} D_{k} - ε_{k} ι_{j} δ_{j} D_{k}$ and $ε_{k} D_{k} ι_{j} δ_{j} = ε_{k} ι_{j} D_{k} δ_{j}$ , so $□ = \sum_{j} δ_{j} D_{j} + \sum_{j, k} ε_{k} ι_{j} (D_{k} δ_{j} - δ_{j} D_{k})$ ; the commutator acting coefficientwise is the multiplication operator $D_{k} φ_{j}$ , since the coefficientwise derivatives commute and only the term where $D_{k}$ hits $φ_{j}$ survives, and $D_{k} φ_{j} = \partial_{{\overset{ˉ}{z}}_{k}} \partial_{z_{j}} φ = φ_{j \overset{ˉ}{k}}$ by clairaut [F10].

2.2F1F3F5F6F11step 1.4givenalgebra

The left-hand side of claim 1 equals $⟨ □ u, u ⟩_{φ}$ : since $u$ and its images are compactly supported smooth forms lying in the relevant domains, with $\overset{ˉ}{\partial} u = 0$ by the degree convention [F5] when $q = n$ , the characterizing adjoint identity [F3] applied to the pairs $(u, \overset{ˉ}{\partial} u)$ and $({\overset{ˉ}{\partial}}_{φ}^{*} u, u)$ gives $⟨ u, {\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} u ⟩_{φ} = ⟨ \overset{ˉ}{\partial} u, \overset{ˉ}{\partial} u ⟩_{φ} = ∥ \overset{ˉ}{\partial} u ∥_{φ}^{2}$ and $⟨ \overset{ˉ}{\partial} {\overset{ˉ}{\partial}}_{φ}^{*} u, u ⟩_{φ} = ⟨ {\overset{ˉ}{\partial}}_{φ}^{*} u, {\overset{ˉ}{\partial}}_{φ}^{*} u ⟩_{φ} = ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2}$ , while conjugate symmetry [F11] turns the first expression into $⟨ {\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} u, u ⟩_{φ}$ because [F1] makes it a real number; adding the two terms and using the definition of $□$ from step 1.4 gives the claim.

3.1F1step 1.2step 1.3step 2.1step 2.2algebra

The two summands of $⟨ □ u, u ⟩_{φ}$ evaluate as $\sum_{j} ⟨ δ_{j} D_{j} u, u ⟩_{φ} = \sum_{j} ⟨ D_{j} u, D_{j} u ⟩_{φ} = \sum_{∣ J ∣ = q} \sum_{k} \int_{Ω} ∣ D_{k} u_{J} ∣^{2} e^{- φ} d V$ by the adjoint identity of step 1.3, and $\sum_{j, k} ⟨ φ_{j \overset{ˉ}{k}} ε_{k} ι_{j} u, u ⟩_{φ} = \sum_{j, k} ⟨ ε_{k} (φ_{j \overset{ˉ}{k}} ι_{j} u), u ⟩_{φ} = \sum_{j, k} ⟨ φ_{j \overset{ˉ}{k}} ι_{j} u, ι_{k} u ⟩_{φ} = \sum_{j, k} \int_{Ω} \sum_{∣ J ∣ = q - 1} φ_{j \overset{ˉ}{k}} u_{j J} \overline{u_{k J}} e^{- φ} d V$ by the adjointness of step 1.2, the scalar commutation of $ε_{k}$ and the pairing formula [F1]; adding these two evaluations through the decomposition of step 2.1 and combining with step 2.2 proves claim 1.

4.1F8F9step 3.1givenalgebra

If $φ$ is plurisubharmonic, the Levi criterion [F9] applied to the Levi form [F8] gives $\sum_{j, k} φ_{j \overset{ˉ}{k}} (a) ξ_{j} \overline{ξ_{k}} \geq 0$ for every $a \in Ω$ and $ξ \in C^{n}$ , and applying this at each point with the given coefficients $ξ_{j} : = u_{j J} (a)$ , for every increasing $J$ of size $q - 1$ , exhibits the integrand of the second term in step 3.1 as a sum of nonnegative quantities; the first term of step 3.1 is a sum of squares of absolute values, hence also nonnegative, so dropping it from the identity of claim 1 yields the inequality of claim 2.

5.1F12F13F14step 3.1step 4.1∎

Both claims of the Statement are proved: claim 1 is the identity assembled in step 3.1, and claim 2 follows from it by the nonnegativity established in step 4.1; the ambient hypothesis is the AC recorded in the Statement and cited as [F14], its countable instance is [F13] as supplied through [F12] by the interfaces [F1] and [F6], and no family of nonempty sets is selected anywhere in the argument.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

A coercive Hilbert-complex estimate solves the closed equation

Statement

Assume the Axiom of Choice (AC). Let $H_{0}, H_{1}, H_{2}$ be complex Hilbert spaces and let $T : H_{0} \supseteq D (T) \to H_{1}$ and $S : H_{1} \supseteq D (S) \to H_{2}$ be closed densely defined linear operators with $T (D (T)) \subseteq \ker S$ , so that $S \circ T = 0$ on $D (T)$ . Suppose there is a constant $C > 0$ with $∥ T^{*} x ∥^{2} + ∥ S x ∥^{2} \geq C ∥ x ∥^{2} for all x \in D (T^{*}) \cap D (S),$ where $T^{*}$ is the Hilbert adjoint of $T$ . Then for every $f \in \ker S$ :

(i) there exists $u \in D (T)$ with $T u = f$ ; (ii) among all such $u$ there is exactly one of least norm, and it lies in $(\ker T)^{⊥}$ ; (iii) that least-norm solution $u_{0}$ satisfies $∥ u_{0} ∥ \leq C^{- 1 / 2} ∥ f ∥$ .

Independently of the constant $C$ , the following $A$ -weighted form of the argument holds.

(iv) Suppose $A \in B (H_{1})$ is a bounded self-adjoint operator with $⟨ A x, x ⟩ \geq 0$ for every $x \in H_{1}$ , that $⟨ A x, x ⟩ \leq ∥ T^{*} x ∥^{2} + ∥ S x ∥^{2} for all x \in D (T^{*}) \cap D (S),$ and that $f \in \ker S$ has the form $f = A g$ for some $g \in H_{1}$ . Then there exists $u \in D (T)$ with $T u = f$ , and the least-norm such $u_{0}$ satisfies $∥ u_{0} ∥^{2} \leq ⟨ f, g ⟩ .$

No closedness of the range of $T$ and no surjectivity of $S$ is assumed.

Facts & Assumptions

Given: The Axiom of Choice; complex Hilbert spaces $H_{0}, H_{1}, H_{2}$ ; closed densely defined linear operators $T : H_{0} \supseteq D (T) \to H_{1}$ and $S : H_{1} \supseteq D (S) \to H_{2}$ with $T (D (T)) \subseteq \ker S$ ; a constant $C > 0$ with $∥ T^{*} x ∥^{2} + ∥ S x ∥^{2} \geq C ∥ x ∥^{2}$ for every $x \in D (T^{*}) \cap D (S)$ ; and an element $f \in \ker S$ . For the independent part (iv) assume in addition a bounded self-adjoint operator $A \in B (H_{1})$ with $⟨ A x, x ⟩ \geq 0$ for every $x \in H_{1}$ and $⟨ A x, x ⟩ \leq ∥ T^{*} x ∥^{2} + ∥ S x ∥^{2}$ for every $x \in D (T^{*}) \cap D (S)$ , together with an element $g \in H_{1}$ satisfying $f = A g$ .

[F1]

For the operator $T : D (T) \subseteq H_{0} \to H_{1}$ , define $D (T^{*})$ to be the set of $y \in H_{1}$ for which $x \mapsto ⟨ T x, y ⟩$ is bounded in the $H_{0}$ norm. Density of $D (T)$ , [F5] and [F4] give a unique $T^{*} y \in H_{0}$ with $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ for every $x \in D (T)$ . The graph identities needed below are proved in step 1.3.

[F2]

For a linear subspace $M$ of a Hilbert space, $M^{⊥ ⊥} = \overline{M}$ (The double orthogonal complement of a subspace is its closure).

[F3]

If $M$ is a closed linear subspace of a real or complex Hilbert space $H$ , then every $x \in H$ has a unique decomposition $x = m + n$ with $m \in M$ and $n \in M^{⊥}$ (Orthogonal decomposition by a closed subspace).

[F4]

For a bounded linear functional on a complex Hilbert space there is a unique representing vector of the same norm (Riesz representation for Hilbert spaces).

[F5]

A bounded functional on a dense linear subspace extends uniquely by limits to the ambient Hilbert space: boundedness makes values on a convergent approximating sequence Cauchy and makes the limit independent of the sequence. The resulting functional has a Riesz vector by [F4].

[F6]

An operator is densely defined when its domain is dense, and closed when its graph $Γ (T)$ is a closed subset of $H \oplus H$ (Densely defined, closed and closable operators, and cores).

[F7]

In a real or complex inner-product space $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ for all vectors $x, y$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F8]

A complex Hilbert space is a complex inner-product space whose induced norm is complete (Hilbert space).

[F9]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice), and it supplies its countable instance (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F10]

A bounded self-adjoint operator $A \in B (H_{1})$ satisfies $⟨ A x, y ⟩ = ⟨ x, A y ⟩$ for all $x, y \in H_{1}$ , so in particular $⟨ A x, x ⟩ \in R$ (The Hilbert-space adjoint of a bounded operator, Self-adjoint, positive, unitary and normal operators, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators).

Choice use. AC is used only through the countable instance consumed by the operator, decomposition and representation facts [F1], [F2], [F3] and [F4]. The adjoint graph argument, functional extension, minimal-norm projection and the $A$ -weighted variant select nothing further.

Proof

technique · direct

1.1F6givenalgebra

Since $S$ is closed, its kernel is closed: if $x_{k} \in \ker S$ and $x_{k} \to x$ in $H_{1}$ then $(x_{k}, S x_{k}) = (x_{k}, 0) \to (x, 0)$ lies in the closed graph of $S$ , so $x \in D (S)$ and $S x = 0$ ; the same argument shows that $\ker T$ is a closed subspace of $H_{0}$ , and $T (D (T)) \subseteq \ker S$ says exactly that the composition $S \circ T$ is zero on $D (T)$ .

1.2F10givenalgebra

Semidefinite Cauchy–Schwarz. For all $x, y \in H_{1}$ one has $∣ ⟨ A x, y ⟩ ∣ \leq ⟨ A x, x ⟩^{1 / 2} ⟨ A y, y ⟩^{1 / 2}$ : fix $x, y$ and note that for every real $t$ , using [F10] and $⟨ A x, x ⟩ \geq 0$ , $0 \leq ⟨ A (x + t y), x + t y ⟩ = ⟨ A x, x ⟩ + 2 t Re ⟨ A x, y ⟩ + t^{2} ⟨ A y, y ⟩$ ; if $⟨ A y, y ⟩ > 0$ the discriminant of this nonnegative quadratic is $\leq 0$ , so $(Re ⟨ A x, y ⟩)^{2} \leq ⟨ A x, x ⟩ ⟨ A y, y ⟩$ , while if $⟨ A y, y ⟩ = 0$ the same quadratic forces $Re ⟨ A x, y ⟩ = 0$ and the inequality holds trivially; choosing $λ$ with $∣ λ ∣ = 1$ and $λ ⟨ A x, y ⟩ = ∣ ⟨ A x, y ⟩ ∣$ and applying the real-part inequality to the pair $(λ x, y)$ , whose quadratic values are unchanged because $A$ is linear and $⟨ A (λ x), λ x ⟩ = ⟨ A x, x ⟩$ , gives the stated modulus bound.

1.3F1F2F4F6given

The adjoint defined in [F1] has a linear domain. Directly from its defining identity, $(ran T)^{⊥} = \ker T^{*}$ : vanishing of $⟨ T x, y ⟩$ for every $x \in D (T)$ is equivalent to $T^{*} y = 0$ . More precisely, in $H_{0} \oplus H_{1}$ one has $Γ (T)^{⊥} = {(- T^{*} y, y) : y \in D (T^{*})},$ because $(a, y)$ is orthogonal to every $(x, T x)$ exactly when $⟨ T x, y ⟩ = ⟨ x, - a ⟩$ for every $x \in D (T)$ . Since $T$ is closed, [F2] gives $Γ (T) = Γ (T)^{⊥ ⊥}$ .

2.1F3step 1.1step 1.3givenalgebra

For $x \in D (T^{*})$ decompose $x = x^{'} + x^{''}$ with $x^{'} \in \ker S$ and $x^{''} \in (\ker S)^{⊥}$ by [F3] applied to the closed subspace $\ker S$ of step 1.1; since $T (D (T)) \subseteq \ker S$ , taking orthogonal complements gives $(\ker S)^{⊥} \subseteq (ran T)^{⊥} = \ker T^{*}$ by step 1.3, so $x^{''} \in \ker T^{*} \subseteq D (T^{*})$ , hence $x^{'} = x - x^{''} \in D (T^{*})$ as well, with $T^{*} x^{'} = T^{*} x - T^{*} x^{''} = T^{*} x$ ; moreover $⟨ x, f ⟩ = ⟨ x^{'}, f ⟩ + ⟨ x^{''}, f ⟩ = ⟨ x^{'}, f ⟩$ because $x^{''} ⊥ \ker S ∋ f$ .

3.1F7step 2.1givenalgebra

For every $x \in D (T^{*})$ the vector $x^{'}$ of step 2.1 lies in $D (T^{*}) \cap D (S)$ and $S x^{'} = 0$ , so the hypothesis and step 2.1 give $∥ x^{'} ∥^{2} \leq C^{- 1} (∥ T^{*} x^{'} ∥^{2} + ∥ S x^{'} ∥^{2}) = C^{- 1} ∥ T^{*} x ∥^{2}$ , and [F7] together with step 2.1 gives $∣ ⟨ x, f ⟩ ∣^{2} = ∣ ⟨ x^{'}, f ⟩ ∣^{2} \leq ∥ x^{'} ∥^{2} ∥ f ∥^{2} \leq C^{- 1} ∥ f ∥^{2} ∥ T^{*} x ∥^{2}$ .

4.1F8F9step 3.1givenalgebra

Define $ℓ$ on the subspace $V_{0} : = T^{*} (D (T^{*})) \subseteq H_{0}$ by $ℓ (T^{*} x) : = ⟨ x, f ⟩$ ; if $T^{*} x = 0$ then step 3.1 gives $∣ ⟨ x, f ⟩ ∣ \leq C^{- 1 / 2} ∥ f ∥ \cdot 0 = 0$ , so $ℓ$ is well defined, and step 3.1 says $∣ ℓ (w) ∣ \leq C^{- 1 / 2} ∥ f ∥ ∥ w ∥$ for every $w \in V_{0}$ , so $ℓ$ is linear and bounded on $V_{0}$ ; let $V : = \overline{V_{0}} \subseteq H_{0}$ and define $\tilde{ℓ} (y) : = \lim_{k} ℓ (w_{k})$ for $y \in V$ and any sequence $w_{k} \in V_{0}$ with $w_{k} \to y$ : the sequence $(ℓ (w_{k}))$ is Cauchy because $∣ ℓ (w_{k}) - ℓ (w_{l}) ∣ \leq C^{- 1 / 2} ∥ f ∥ ∥ w_{k} - w_{l} ∥$ , and the limit exists by [F8]; the countable instance of [F9] is the choice principle consumed by [F1], [F3], [F4] and [F8] here, and the construction selects nothing further; it is independent of the sequence because two sequences for the same $y$ have difference tending to $0$ , and passing to limits of the defining inequalities preserves linearity (limits of sums and scalar multiples) and gives $∣ \tilde{ℓ} (y) ∣ \leq C^{- 1 / 2} ∥ f ∥ ∥ y ∥$ .

4.2F10step 2.1step 3.1step 1.2givenalgebra

$A$ -weighted estimate. For every $x \in D (T^{*})$ , with $x = x^{'} + x^{''}$ as in step 2.1, the vector $x^{'}$ lies in $D (T^{*}) \cap D (S)$ with $S x^{'} = 0$ and $T^{*} x^{'} = T^{*} x$ (step 3.1 for the first two, step 2.1 for the third), so by step 1.2 applied to the pair $(x^{'}, g)$ , by [F10], by $f = A g$ and by the domination hypothesis for (iv) on $x^{'}$ : $∣ ⟨ x, f ⟩ ∣ = ∣ ⟨ x^{'}, f ⟩ ∣ = ∣ ⟨ x^{'}, A g ⟩ ∣ = ∣ ⟨ A x^{'}, g ⟩ ∣ \leq ⟨ A x^{'}, x^{'} ⟩^{1 / 2} ⟨ A g, g ⟩^{1 / 2} \leq (∥ T^{*} x^{'} ∥^{2} + ∥ S x^{'} ∥^{2})^{1 / 2} ⟨ f, g ⟩^{1 / 2} = ∥ T^{*} x ∥ ⟨ f, g ⟩^{1 / 2}$ , where $⟨ A g, g ⟩ = ⟨ f, g ⟩$ and $⟨ f, g ⟩ \geq 0$ by positivity of $A$ .

5.1F3F4step 4.1givenalgebra

Let $P : H_{0} \to V$ be the orthogonal projection onto the closed subspace $V$ , defined for $y \in H_{0}$ by the unique decomposition $y = P y + (y - P y)$ with $y - P y \in V^{⊥}$ of [F3], and put $Λ : = \tilde{ℓ} \circ P$ ; then $Λ$ is linear and $∣ Λ (y) ∣ \leq C^{- 1 / 2} ∥ f ∥ ∥ y ∥$ for all $y \in H_{0}$ , so by [F4] there is a unique $u \in H_{0}$ with $Λ (y) = ⟨ y, u ⟩$ for all $y \in H_{0}$ and $∥ u ∥ = ∥ Λ ∥ \leq C^{- 1 / 2} ∥ f ∥$ .

6.1F2step 1.3step 4.1step 5.1

For every $x \in D (T^{*})$ , steps 4.1 and 5.1 give $⟨ T^{*} x, u ⟩ = ⟨ x, f ⟩$ . Therefore $(u, f)$ is orthogonal to every $(- T^{*} x, x) \in Γ (T)^{⊥}$ by step 1.3 (take complex conjugates of the displayed identity). Since $T$ is closed, $(u, f) \in Γ (T)^{⊥ ⊥} = Γ (T)$ ; hence $u \in D (T)$ and $T u = f$ , proving (i).

6.2F3F4F8F9step 4.1step 4.2givenalgebra

Define $ℓ$ on $V_{0} : = T^{*} (D (T^{*}))$ by $ℓ (T^{*} x) : = ⟨ x, f ⟩$ for $x \in D (T^{*})$ and put $β : = ⟨ f, g ⟩^{1 / 2}$ : step 4.2 with $T^{*} x = 0$ shows $ℓ$ is well defined, and it shows $∣ ℓ (w) ∣ \leq β ∥ w ∥$ for every $w \in V_{0}$ , so $ℓ$ is linear and bounded; the extension $\tilde{ℓ}$ of $ℓ$ to $V : = \overline{V_{0}}$ by limits along sequences in $V_{0}$ , its independence of the chosen sequence, its linearity and its bound $∣ \tilde{ℓ} (y) ∣ \leq β ∥ y ∥$ are verified by the same computation as in step 4.1 with $β$ in place of $C^{- 1 / 2} ∥ f ∥$ , with the Cauchy property supplied by [F8] and the same countable instance of [F9]; letting $P$ be the orthogonal projection of [F3] onto $V$ as in step 5.1, the functional $Λ : = \tilde{ℓ} \circ P$ is linear with $∣ Λ (y) ∣ \leq β ∥ y ∥$ on $H_{0}$ , so [F4] gives a unique $u \in H_{0}$ with $Λ (y) = ⟨ y, u ⟩$ for all $y \in H_{0}$ and $∥ u ∥ = ∥ Λ ∥ \leq β = ⟨ f, g ⟩^{1 / 2}$ .

7.1F3step 1.1step 5.1step 6.1givenalgebra

By step 1.1 the subspace $\ker T$ is closed, so [F3] decomposes the solution $u$ of step 6.1 as $u = u_{1} + u_{2}$ with $u_{1} \in \ker T$ and $u_{2} \in (\ker T)^{⊥}$ ; then $T u_{2} = T u - T u_{1} = f$ , so $u_{2}$ is a solution lying in $(\ker T)^{⊥}$ , every solution has the form $u_{2} + k$ with $k \in \ker T$ , and since $k ⊥ u_{2}$ the Pythagorean identity gives $∥ u_{2} + k ∥^{2} = ∥ u_{2} ∥^{2} + ∥ k ∥^{2} \geq ∥ u_{2} ∥^{2}$ with equality only for $k = 0$ ; thus $u_{2}$ is the unique least-norm solution and $∥ u_{2} ∥ \leq ∥ u ∥ \leq C^{- 1 / 2} ∥ f ∥$ by step 5.1, proving (ii) and (iii).

7.2F2step 1.3step 6.1step 6.2

For every $x \in D (T^{*})$ , step 6.2 gives $⟨ T^{*} x, u ⟩ = ⟨ x, f ⟩$ . As in step 6.1, the graph identity of step 1.3 and closedness of $T$ yield $(u, f) \in Γ (T)$ , so $u \in D (T)$ and $T u = f$ . This proves existence in (iv).

8.1F3step 1.1step 7.1step 6.2step 7.2givenalgebra∎

By step 1.1 the subspace $\ker T$ is closed, so [F3] decomposes the solution $u$ of step 7.2 as $u = u_{1} + u_{0}$ with $u_{1} \in \ker T$ and $u_{0} \in (\ker T)^{⊥}$ ; then $T u_{0} = T u - T u_{1} = f$ and the argument of step 7.1 shows that $u_{0}$ is the unique least-norm solution, with $∥ u_{0} ∥ \leq ∥ u ∥ \leq ⟨ f, g ⟩^{1 / 2}$ by step 6.2, hence $∥ u_{0} ∥^{2} \leq ⟨ f, g ⟩$ , which proves (iv); moreover with $A : = C i d$ and $g : = f / C$ the hypotheses of (iv) hold by the coercivity assumption, and its conclusion specializes to $∥ u_{0} ∥^{2} \leq ⟨ f, f / C ⟩ = C^{- 1} ∥ f ∥^{2}$ , in agreement with (iii).

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Weighted Morrey–Kohn estimate with a pseudoconvex boundary term

Statement

Assume the Axiom of Choice (AC). Let $n \geq 1$ , and use $z_{j}$ for the canonical coordinate $z_{j - 1}$ , $1 \leq j \leq n$ , with the same relabeling for derivatives and form coefficients. Let $D \subseteq C^{n}$ be a bounded domain with $C^{\infty}$ boundary, let $ρ \in C^{\infty} (\overline{D}; R)$ be a defining function with $D = {z \in \overline{D} : ρ (z) < 0}$ and $∣ \nabla ρ ∣ = 1$ on $\partial D$ , let $φ \in C^{2} (\overline{D}; R)$ , let $1 \leq q \leq n$ , and let $u \in C^{\infty} (\overline{D}; Λ^{0, q})$ satisfy the ∂̄-Neumann boundary condition $\sum_{j=1}^n u_{jK}\,\frac{\partial\rho}{\partial z_j}=0\quad\text{on }\partial D,\qquad\text{for every }K\text{ with }|K|=q-1, \tag{BC}$ where $u_{j K}$ is the antisymmetric coefficient of $u$ on the tuple $(j, K)$ and $d S$ denotes $(2 n - 1)$ -dimensional surface measure on $\partial D$ . Then:

$u \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ , and the exact identity $∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2} = \sum_{∣ J ∣ = q} \sum_{k = 1}^{n} \int_{D} ∣ D_{k} u_{J} ∣^{2} e^{- φ} d V + \int_{D} \sum_{∣ K ∣ = q - 1} \sum_{j, k = 1}^{n} φ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d V + \int_{\partial D} \sum_{∣ K ∣ = q - 1} \sum_{j, k = 1}^{n} ρ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d S$ holds, with $D_{k} = \partial_{{\overset{ˉ}{z}}_{k}}$ and $ρ_{j \overset{ˉ}{k}} = \partial_{z_{j}} \partial_{{\overset{ˉ}{z}}_{k}} ρ$ .
If $D$ is Levi pseudoconvex, the boundary term is nonnegative and therefore $\int_{D} \sum_{∣ K ∣ = q - 1} \sum_{j, k = 1}^{n} φ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d V \leq ∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2} .$
If $D$ is Levi pseudoconvex and $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ denote the eigenvalues of the Hermitian matrix $(φ_{j \overset{ˉ}{k}} (a))$ , then $\int_{D} (λ_{1} + \dots + λ_{q}) ∣ u ∣^{2} e^{- φ} d V \leq ∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2} .$
If $D$ is Levi pseudoconvex, the inequality of claim 3 holds for every $u \in Dom {\overset{ˉ}{\partial}}_{q} \cap Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ .

Facts & Assumptions

Given: The Axiom of Choice; a bounded domain $D \subseteq C^{n}$ with $C^{\infty}$ boundary; a defining function $ρ$ normalized by $∣ \nabla ρ ∣ = 1$ on $\partial D$ ; a weight $φ \in C^{2} (\overline{D}; R)$ ; an integer $1 \leq q \leq n$ ; and a form $u \in C^{\infty} (\overline{D}; Λ^{0, q})$ satisfying (BC); the conventions $ε_{j} v : = d {\overset{ˉ}{z}}_{j} \land v$ and $(ι_{j} v)_{K} : = v_{j K}$ on coefficient tensors, $D_{j} : = \partial_{{\overset{ˉ}{z}}_{j}}$ and $δ_{j} : = φ_{j} - \partial_{z_{j}}$ acting coefficientwise, and ${\overset{ˉ}{\partial}}_{0}^{*}$ for the unweighted Hilbert adjoint of $\overset{ˉ}{\partial}$ (the case $φ \equiv 0$ of the operators of [F1]), whose formal density on its smooth domain is $- \sum_{j} ι_{j} \partial_{z_{j}}$ .

[F1]

The weighted space, its pairing $⟨ v, w ⟩_{φ} = \int_{D} \sum_{∣ J ∣ = q} v_{J} \overline{w_{J}} e^{- φ} d V$ , the maximal distributional $\overset{ˉ}{\partial}$ , and the weighted Hilbert adjoint ${\overset{ˉ}{\partial}}_{φ}^{*}$ with its formal density $({\overset{ˉ}{\partial}}_{φ}^{*} v)_{K} = \sum_{j} (v_{j K} φ_{j} - \partial_{z_{j}} v_{j K})$ on its domain are as in Weighted L2 spaces and maximal dbar operators; coefficients are extended to non-increasing tuples by antisymmetry, so $v_{j K} = 0$ when $j \in K$ .

[F2]

For every $ψ \in C_{c}^{\infty} (D)$ of bidegree $(0, q)$ one has $ψ \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ (The maximal distributional dbar operator is closed and densely defined).

[F3]

[F4]

The Levi form is $L_{w} (a; v) = \sum_{j, k} \frac{\partial^{2} w}{\partial z_{j} \partial {\overline{z}}_{k}} (a) v_{j} \overline{v_{k}}$ (The Levi form and strict plurisubharmonicity).

[F5]

A domain with $C^{2}$ boundary is Levi pseudoconvex when for every $p \in \partial D$ there are a neighbourhood $U$ of $p$ and $ρ \in C^{2} (U, R)$ with $D \cap U = {ρ < 0}$ , $d ρ (p) \neq 0$ , and $L_{ρ} (p; v) \geq 0$ for every $v \in C^{n}$ with $\sum_{j} \frac{\partial ρ}{\partial z_{j}} (p) v_{j} = 0$ (Levi pseudoconvex domains).

[F6]

For functions with continuous second partial derivatives, $\partial_{j} \partial_{i} ϕ = \partial_{i} \partial_{j} ϕ$ (Continuous second partials of a scalar potential commute).

[F7]

The Wirtinger operators are $\partial_{z_{j}} = \frac{1}{2} (\partial_{x_{j}} - i \partial_{y_{j}})$ and $\partial_{{\overset{ˉ}{z}}_{j}} = \frac{1}{2} (\partial_{x_{j}} + i \partial_{y_{j}})$ (Wirtinger operators in $C^{m}$ ).

[F8]

A normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and a self-adjoint endomorphism is normal (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space).

[F9]

[F10]

$A C \Rightarrow {A C}_{ω}$ in ZF, and ${A C}_{ω}$ supplies a choice function for every at most countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

[F11]

(Boas, §3.3.3, printed pp. 81-84: formulas (3.3)-(3.4), Exercise 38.) For smooth scalar functions $F, G$ on $\overline{D}$ the divergence theorem stated in the given block gives the two weighted integrations by parts $\int_{D} (\partial_{z_{j}} F) \overline{G} e^{- φ} d V = \int_{\partial D} F \overline{G} ρ_{z_{j}} e^{- φ} d S - \int_{D} F \overline{\partial_{{\overset{ˉ}{z}}_{j}} G} e^{- φ} d V + \int_{D} F \overline{G} φ_{j} e^{- φ} d V$ and $\int_{D} (\partial_{{\overset{ˉ}{z}}_{j}} F) \overline{G} e^{- φ} d V = \int_{\partial D} F \overline{G} ρ_{{\overset{ˉ}{z}}_{j}} e^{- φ} d S - \int_{D} F \overline{\partial_{z_{j}} G} e^{- φ} d V + \int_{D} F \overline{G} φ_{\overset{ˉ}{j}} e^{- φ} d V$ , where $ρ_{z_{j}} = \partial_{z_{j}} ρ$ , $ρ_{{\overset{ˉ}{z}}_{j}} = \partial_{{\overset{ˉ}{z}}_{j}} ρ$ , $φ_{j} = \partial_{z_{j}} φ$ and $φ_{\overset{ˉ}{j}} = \partial_{{\overset{ˉ}{z}}_{j}} φ$ ; Boas's (3.3) is the $q = 1$ expansion obtained by dropping the boundary term for compactly supported data, his (3.4) is the adjoint boundary condition exhibited by these formulas, and his Exercise 38 is the same computation with a positive smooth weight in place of $e^{- φ}$ , which is where the factors $φ_{j}, φ_{\overset{ˉ}{j}}$ come from.

[F12]

(Haslinger, author manuscript, §4, Proposition 4.12 with Lemmas 4.14-4.16, printed pp. 45-49, and the general-degree boundary criterion (4.31) with its proof on printed p. 48.) On a bounded domain with $C^{k + 1}$ boundary, $C_{(0, q)}^{k} (\overline{D}) \cap Dom {\overset{ˉ}{\partial}}_{0}^{*}$ is dense in $Dom {\overset{ˉ}{\partial}}_{q} \cap Dom {\overset{ˉ}{\partial}}_{0}^{*}$ for the unweighted graph norm $(∥ v ∥_{0}^{2} + ∥ \overset{ˉ}{\partial} v ∥_{0}^{2} + ∥ {\overset{ˉ}{\partial}}_{0}^{*} v ∥_{0}^{2})^{1 / 2}$ ; this also holds with $k = \infty$ . For a $C^{k}$ form, $k \geq 1$ , membership in $Dom {\overset{ˉ}{\partial}}_{0}^{*}$ is equivalent to (BC), and on this domain its value is $- \sum_{j} ι_{j} \partial_{z_{j}} v$ . The source proves this in boundary frames where (BC) is vanishing of the complex normal coefficients; thus the dense smooth family satisfies (BC). The source uses the unweighted pairing. Weighted transport is proved here in steps 1.2, 2.1 and 9.1, not attributed to the source.

Given (source computation). For $H \in C^{1} (\overline{D})$ , the divergence theorem gives $\int_{D} \partial_{z_{j}} H d V = \int_{\partial D} H ρ_{z_{j}} d S$ ; its conjugate gives the $\partial_{{\overset{ˉ}{z}}_{j}}$ formula. Applying these to $H = F \overline{G} e^{- φ}$ gives [F11], including for $C^{1}$ factors. In Boas, §3.3.3, printed p. 83, the unweighted calculation for a smooth $(0, 1)$ -form $f$ is $⟨ f, \overset{ˉ}{\partial} η ⟩_{0} = \int_{D} \sum_{j} f_{j} \overline{\partial_{{\overset{ˉ}{z}}_{j}} η} d V = - \int_{D} \sum_{j} \partial_{z_{j}} f_{j} \overline{η} d V + \int_{\partial D} \sum_{j} f_{j} ρ_{z_{j}} \overline{η} d S .$ The Hilbert-adjoint identity holds precisely when the boundary term vanishes, as justified by [F12]. Boas's Exercise 38 treats a positive smooth weight; the $C^{2}$ weight here needs only the displayed divergence theorem and product rule.

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F10]; the countable instance ${A C}_{ω}$ is the form of choice consumed by the interfaces [F1] (completeness and density of the weighted space) and [F2], and by the countable cutoffs, exhaustions and subsequences occurring in the imported density statement [F12]. The proof selects no family of nonempty sets beyond those countable instances.

Proof

technique · direct

1.1F1F7F8givenalgebra

Since $\partial D$ is $C^{\infty}$ and $∣ \nabla ρ ∣ = 1$ there, $\overline{D}$ is compact, so $φ, \nabla φ$ and the coefficients $φ_{j \overset{ˉ}{k}}$ are bounded on $\overline{D}$ ; put $M : = \max_{j, k} \sup_{\overline{D}} ∣ φ_{j \overset{ˉ}{k}} ∣ < \infty$ . The Hermitian matrix $H (a) : = (φ_{j \overset{ˉ}{k}} (a))$ is well defined at every $a \in \overline{D}$ , its eigenvalues $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ are real by [F8], and if $v$ is a unit eigenvector for $λ_{i} (a)$ then the Cauchy-Schwarz inequality gives $∣ λ_{i} (a) ∣ = ∣ ⟨ H (a) v, v ⟩ ∣ \leq \max_{j, k} ∣ φ_{j \overset{ˉ}{k}} (a) ∣ (\sum_{j} ∣ v_{j} ∣)^{2} \leq n \max_{j, k} ∣ φ_{j \overset{ˉ}{k}} (a) ∣ \leq n M$ , so the sum $w : = λ_{1} + \dots + λ_{q}$ satisfies $∣ w ∣ \leq q n M$ at every point of $\overline{D}$ ; also $φ_{\overset{ˉ}{j}} = \overline{φ_{j}}$ because $φ$ is real-valued.

1.2F1givenalgebra

Transport identity. A $(0, k)$ -form $z$ lies in $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ exactly when $e^{- φ} z$ lies in $Dom {\overset{ˉ}{\partial}}_{0}^{*}$ , and then ${\overset{ˉ}{\partial}}_{φ}^{*} z = e^{φ} {\overset{ˉ}{\partial}}_{0}^{*} (e^{- φ} z) .$ Indeed $⟨ \overset{ˉ}{\partial} h, z ⟩_{φ} = ⟨ \overset{ˉ}{\partial} h, e^{- φ} z ⟩_{0}$ and $⟨ h, z^{*} ⟩_{φ} = ⟨ h, e^{- φ} z^{*} ⟩_{0}$ for every $h$ in the maximal domain. The weighted and unweighted graph domains of $\overset{ˉ}{\partial}$ agree as sets, since $e^{\pm φ}$ are bounded on $\overline{D}$ ; hence the two bounded-functional criteria are equivalent and their Riesz vectors have the displayed relation.

2.1F1F12step 1.2givenalgebra

The form $u$ lies in $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ and ${\overset{ˉ}{\partial}}_{φ}^{*} u = \sum_{j} ι_{j} δ_{j} u$ : the positive factor $e^{- φ}$ preserves (BC), so [F12], applied with $k = 2$ since $e^{- φ} u$ is $C^{2}$ , gives $e^{- φ} u \in Dom {\overset{ˉ}{\partial}}_{0}^{*}$ and ${\overset{ˉ}{\partial}}_{0}^{*} (e^{- φ} u) = - \sum_{j} ι_{j} \partial_{z_{j}} (e^{- φ} u) = e^{- φ} \sum_{j} ι_{j} (φ_{j} - \partial_{z_{j}}) u .$ The transport identity of step 1.2 gives the claimed weighted adjoint and formula.

3.1F1F3F6F7step 2.1givenalgebra

On smooth coefficient tensors the following operator identities hold: (a) $\overset{ˉ}{\partial} = \sum_{j} ε_{j} D_{j}$ and ${\overset{ˉ}{\partial}}_{φ}^{*} = \sum_{j} ι_{j} δ_{j}$ on $Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ ; (b) $ι_{j} ε_{k} + ε_{k} ι_{j} = δ_{j k} i d$ ; (c) $ι_{j} \overset{ˉ}{\partial} = D_{j} - \overset{ˉ}{\partial} ι_{j}$ ; (d) $[D_{k}, δ_{j}] = φ_{j \overset{ˉ}{k}}$ ; (e) ${\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} + \overset{ˉ}{\partial} {\overset{ˉ}{\partial}}_{φ}^{*} = \sum_{j} δ_{j} D_{j} + \sum_{j, k} φ_{j \overset{ˉ}{k}} ε_{k} ι_{j}$ ; and (f) $⟨ ε_{j} f, g ⟩_{φ} = ⟨ f, ι_{j} g ⟩_{φ}$ for coefficient tensors $f, g$ of adjacent degrees. Here (a) is the definition of the wedge and for smooth forms the formal density of [F1] established in step 2.1; (b) is the sign case check of [F3] against the antisymmetry convention of [F1]; (c) is $ι_{j} \overset{ˉ}{\partial} = \sum_{k} (δ_{j k} - ε_{k} ι_{j}) D_{k} = D_{j} - \overset{ˉ}{\partial} ι_{j}$ ; (d) is $D_{k} φ_{j} = φ_{j \overset{ˉ}{k}}$ by [F6]; (e) follows from (a)-(d) by writing ${\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} + \overset{ˉ}{\partial} {\overset{ˉ}{\partial}}_{φ}^{*} = \sum_{j, k} (ι_{j} δ_{j} ε_{k} D_{k} + ε_{k} D_{k} ι_{j} δ_{j})$ and substituting (b), (c) and (d); and (f) is the pointwise adjointness of wedge and contraction.

4.1F1F3F8step 3.1givenalgebra

For every $a \in \overline{D}$ and every coefficient tensor $(u_{J})_{∣ J ∣ = q}$ one has $\sum_{∣ K ∣ = q - 1} \sum_{j, k} φ_{j \overset{ˉ}{k}} (a) u_{j K} \overline{u_{k K}} \geq (λ_{1} + \dots + λ_{q}) (a) \sum_{∣ J ∣ = q} ∣ u_{J} ∣^{2}$ : the Hermitian matrix $H (a)$ is self-adjoint and hence normal, so [F8] gives an orthonormal basis $v_{1}, \dots, v_{n}$ of $C^{n}$ with $H (a) v_{i} = λ_{i} v_{i}$ and $φ_{j \overset{ˉ}{k}} (a) = \sum_{i} λ_{i} v_{i j} \overline{v_{i k}}$ ; putting $w_{i K} : = \sum_{j} v_{i j} u_{j K}$ and $d_{i} : = \sum_{∣ K ∣ = q - 1} ∣ w_{i K} ∣^{2}$ gives $\sum_{∣ K ∣ = q - 1} \sum_{j, k} φ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} = \sum_{i} λ_{i} d_{i}$ ; Parseval in each fiber $K$ gives $\sum_{i} d_{i} = \sum_{∣ K ∣ = q - 1} \sum_{j} ∣ u_{j K} ∣^{2} = q \sum_{∣ J ∣ = q} ∣ u_{J} ∣^{2}$ , and $0 \leq d_{i} \leq \sum_{∣ J ∣ = q} ∣ u_{J} ∣^{2}$ because $d_{i} = ∣ ν_{i} u ∣^{2}$ where $ν_{i} : = \sum_{j} v_{i j} ι_{j}$ is the adjoint, by (f) of step 3.1, of $μ_{i} : = \sum_{j} \overline{v_{i j}} ε_{j}$ and $∣ ν_{i} u ∣^{2} + ∣ μ_{i} u ∣^{2} = ∣ u ∣^{2}$ by (b) and (f) of step 3.1, using $\sum_{j} ∣ v_{i j} ∣^{2} = 1$ ; finally, for real $λ_{1} \leq \dots \leq λ_{n}$ and reals $0 \leq d_{i} \leq U$ with $\sum_{i} d_{i} = q U$ one has $\sum_{i} λ_{i} d_{i} \geq (λ_{1} + \dots + λ_{q}) U$ , because an exchange of mass between indices $i > q$ and $j \leq q$ never increases $\sum_{i} λ_{i} d_{i}$ and repeated exchange reaches $d_{1} = \dots = d_{q} = U$ , $d_{q + 1} = \dots = d_{n} = 0$ .

4.2F1F9F11step 2.1step 3.1givenalgebra

The left-hand side of claim 1 equals $⟨ □ u, u ⟩_{φ} + B_{1}$ , where $□ : = \sum_{j} δ_{j} D_{j} + \sum_{j, k} φ_{j \overset{ˉ}{k}} ε_{k} ι_{j}$ and $B_{1} : = \int_{\partial D} \sum_{∣ K ∣ = q} \sum_{j} (\overset{ˉ}{\partial} u)_{j K} \overline{u_{K}} ρ_{z_{j}} e^{- φ} d S$ , with $(\overset{ˉ}{\partial} u)_{j K}$ the coefficient of the $(0, q + 1)$ -form $\overset{ˉ}{\partial} u$ on the ordered tuple $(j, K)$ : by step 2.1 and identity (e) of step 3.1, $⟨ □ u, u ⟩_{φ} = ⟨ {\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} u, u ⟩_{φ} + ⟨ \overset{ˉ}{\partial} {\overset{ˉ}{\partial}}_{φ}^{*} u, u ⟩_{φ}$ computed as formal densities, the integration by parts for $\partial_{z_{j}}$ in [F11] with $F : = (\overset{ˉ}{\partial} u)_{j K}$ , $G : = u_{K}$ gives $⟨ {\overset{ˉ}{\partial}}_{φ}^{*} \overset{ˉ}{\partial} u, u ⟩_{φ} = ∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} - B_{1}$ using the formal integration-by-parts expression (no Hilbert-adjoint domain claim is made for $\overset{ˉ}{\partial} u$ ), and $∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} = \sum_{j} ⟨ D_{j} u, ι_{j} \overset{ˉ}{\partial} u ⟩_{φ}$ by (f) of step 3.1, while $⟨ \overset{ˉ}{\partial} {\overset{ˉ}{\partial}}_{φ}^{*} u, u ⟩_{φ} = ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2}$ by the adjoint relation applied to the $C^{1}$ form ${\overset{ˉ}{\partial}}_{φ}^{*} u$ , whose first derivatives are bounded and hence whose $\overset{ˉ}{\partial}$ is in $L^{2}$ , and the form $u \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ ; adding gives the claim.

4.3F1F7F11step 3.1givenalgebra

The first summand is $\sum_{j} ⟨ δ_{j} D_{j} u, u ⟩_{φ} = \sum_{∣ J ∣ = q} \sum_{k} \int_{D} ∣ D_{k} u_{J} ∣^{2} e^{- φ} d V - R$ , where $R : = \int_{\partial D} \sum_{∣ J ∣ = q} \sum_{j} D_{j} u_{J} \overline{u_{J}} ρ_{z_{j}} e^{- φ} d S$ . Indeed apply the $\partial_{z_{j}}$ integration by parts of [F11] with $F = D_{j} u_{J}$ and $G = u_{J}$ . Since $δ_{j} = φ_{j} - \partial_{z_{j}}$ , the weight-derivative terms cancel and the remaining volume term is $\int_{D} (D_{j} u_{J}) \overline{\partial_{{\overset{ˉ}{z}}_{j}} u_{J}} e^{- φ} d V = \int_{D} ∣ D_{j} u_{J} ∣^{2} e^{- φ} d V$ ; the boundary term has the sign $- R$ .

4.4F1step 3.1givenalgebra

The second summand of $⟨ □ u, u ⟩_{φ}$ evaluates as $\sum_{j, k} ⟨ φ_{j \overset{ˉ}{k}} ε_{k} ι_{j} u, u ⟩_{φ} = \int_{D} \sum_{∣ K ∣ = q - 1} \sum_{j, k} φ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d V$ : apply the adjointness (f) of step 3.1 pointwise, move the scalar $φ_{j \overset{ˉ}{k}}$ outside the pairing, and use the coefficient formula of [F1].

5.1F1F3F6F7F11step 3.1givenalgebra

At every boundary point $p \in \partial D$ one has the identity $\sum_{∣ K ∣ = q} \sum_{j = 1}^{n} [(\overset{ˉ}{\partial} u)_{j K} - D_{j} u_{K}] \overline{u_{K}} ρ_{z_{j}} = \sum_{∣ K ∣ = q - 1} \sum_{j, k = 1}^{n} ρ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}}$ , all quantities being evaluated at $p$ , where $(\overset{ˉ}{\partial} u)_{j K}$ denotes the coefficient of $\overset{ˉ}{\partial} u$ on the ordered tuple $(j, K)$ (so it equals $0$ when $j \in K$ ) and $u_{K}$ the coefficient of $u$ on the increasing tuple $K$ ; moreover $B_{1} - R = \int_{\partial D} \sum_{∣ K ∣ = q - 1} \sum_{j, k} ρ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d S$ . Indeed, with $θ : = \sum_{j} ρ_{z_{j}} ι_{j}$ the left-hand side equals $⟨ (θ \overset{ˉ}{\partial} - \sum_{j} ρ_{z_{j}} D_{j}) u, u ⟩_{p t}$ by (f) and (a) of step 3.1, and since $θ \overset{ˉ}{\partial} = \sum_{j} ρ_{z_{j}} D_{j} - \overset{ˉ}{\partial} θ + \sum_{j, k} ρ_{j \overset{ˉ}{k}} ε_{k} ι_{j}$ by (b), (c) and [F6] (applied to $D_{k} ρ_{z_{j}} = ρ_{j \overset{ˉ}{k}}$ ), it equals $- ⟨ \overset{ˉ}{\partial} (θ u), u ⟩_{p t} + \sum_{j, k} ρ_{j \overset{ˉ}{k}} ⟨ ε_{k} ι_{j} u, u ⟩_{p t}$ ; the second term is $\sum_{∣ K ∣ = q - 1} \sum_{j, k} ρ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}}$ by (f) of step 3.1, and the first term vanishes because $- ⟨ \overset{ˉ}{\partial} (θ u), u ⟩_{p t} = - \sum_{k, K} D_{k} (θ u)_{K} \overline{u_{k K}} = - \sum_{j, k, K} (ρ_{j \overset{ˉ}{k}} u_{j K} + ρ_{z_{j}} D_{k} u_{j K}) \overline{u_{k K}}$ and for each $∣ K ∣ = q - 1$ the tangential operator $T_{K} : = \sum_{k} \overline{u_{k K}} (p) \partial_{{\overset{ˉ}{z}}_{k}}$ annihilates the restriction to $\partial D$ of $\sum_{j} u_{j K} ρ_{z_{j}}$ : it is tangential at $p$ because $\sum_{k} \overline{u_{k K}} (p) ρ_{{\overset{ˉ}{z}}_{k}} (p) = \overline{\sum_{k} u_{k K} (p) ρ_{z_{k}} (p)} = 0$ by (BC), and $\sum_{j} u_{j K} ρ_{z_{j}} = 0$ on $\partial D$ by (BC), so $0 = T_{K} (\sum_{j} u_{j K} ρ_{z_{j}}) (p) = \sum_{j, k} \overline{u_{k K}} (ρ_{j \overset{ˉ}{k}} u_{j K} + ρ_{z_{j}} D_{k} u_{j K}) (p)$ ; integrating the pointwise identity over $\partial D$ against $e^{- φ} d S$ and subtracting the definition of $R$ in step 4.3 from that of $B_{1}$ in step 4.2 gives the displayed identity for $B_{1} - R$ .

6.1step 2.1step 4.2step 4.3step 4.4step 5.1givenalgebra

Claim 1 holds: by steps 4.2, 4.3 and 4.4 the sum $∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2}$ equals $\sum_{∣ J ∣ = q} \sum_{k} \int_{D} ∣ D_{k} u_{J} ∣^{2} e^{- φ} d V - R + \int_{D} \sum_{j, k, K} φ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d V + B_{1}$ , and step 5.1 replaces $- R + B_{1}$ by $\int_{\partial D} \sum_{j, k, K} ρ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}} e^{- φ} d S$ , which is the stated identity; the membership $u \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ is step 2.1.

7.1F4F5step 6.1givenalgebra

If $D$ is Levi pseudoconvex the boundary term of claim 1 is nonnegative: at $p \in \partial D$ and for each $K$ with $∣ K ∣ = q - 1$ the vector $v^{(K)} : = (u_{1 K}, \dots, u_{n K})$ satisfies $\sum_{j} v_{j}^{(K)} ρ_{z_{j}} (p) = 0$ by (BC), so $\sum_{j, k} ρ_{j \overset{ˉ}{k}} (p) u_{j K} \overline{u_{k K}} = L_{ρ} (p; v^{(K)}) \geq 0$ by [F5]: if $r$ is its local defining function at $p$ , local coordinates transverse to $\partial D$ give $ρ = h r$ with $h (p) > 0$ ; the product rule on vectors tangent to $r = 0$ gives $L_{ρ} (p; v) = h (p) L_{r} (p; v)$ , and the boundary integral of claim 1 is an integral of a pointwise nonnegative continuous function against the positive factor $e^{- φ}$ ; dropping it and the nonnegative first term of the identity of step 6.1 gives the estimate of claim 2.

8.1step 4.1step 7.1givenalgebra

Claim 3 holds: by step 4.1 the integrand $\sum_{∣ K ∣ = q - 1} \sum_{j, k} φ_{j \overset{ˉ}{k}} u_{j K} \overline{u_{k K}}$ dominates $(λ_{1} + \dots + λ_{q}) ∣ u ∣^{2}$ pointwise on $\overline{D}$ , and step 7.1 bounds the integral of the former by $∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2}$ .

9.1F1F9F12step 1.1step 1.2step 8.1givenalgebra

Claim 4 holds. Let $u \in Dom {\overset{ˉ}{\partial}}_{q} \cap Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ and put $v : = e^{- φ} u$ . By the transport identity of step 1.2, $v \in Dom {\overset{ˉ}{\partial}}_{0}^{*}$ ; the product rule gives $\overset{ˉ}{\partial} v = e^{- φ} (\overset{ˉ}{\partial} u - \overset{ˉ}{\partial} φ \land u) \in L^{2}$ , so $v \in Dom {\overset{ˉ}{\partial}}_{q}$ . The unweighted graph-norm density [F12] gives smooth $v_{ℓ}$ satisfying (BC) with $v_{ℓ} \to v$ , $\overset{ˉ}{\partial} v_{ℓ} \to \overset{ˉ}{\partial} v$ , and ${\overset{ˉ}{\partial}}_{0}^{*} v_{ℓ} \to {\overset{ˉ}{\partial}}_{0}^{*} v$ in unweighted $L^{2}$ . Put $u_{ℓ} : = e^{φ} v_{ℓ}$ . Then $u_{ℓ}$ satisfies (BC) and belongs to $C^{2} (\overline{D})$ , which suffices for the integrations by parts and boundary differentiations of steps 2.1–8.1. By step 1.2 and the product rule, ${\overset{ˉ}{\partial}}_{φ}^{*} u_{ℓ} = e^{φ} {\overset{ˉ}{\partial}}_{0}^{*} v_{ℓ}, \overset{ˉ}{\partial} u_{ℓ} = e^{φ} (\overset{ˉ}{\partial} v_{ℓ} + \overset{ˉ}{\partial} φ \land v_{ℓ}) .$ The bounded factors $e^{\pm φ}$ and $\overset{ˉ}{\partial} φ$ show that $u_{ℓ} \to u$ in the weighted graph norm. Apply the inequality of step 8.1 to $u_{ℓ}$ and pass to the limit: the right side converges by graph-norm convergence, and the left side converges because $∣ λ_{1} + \dots + λ_{q} ∣ \leq q n M$ by step 1.1 and $L^{2}$ convergence implies convergence of the squared norms against any bounded real weight. Thus the inequality holds for $u$ .

10.1F1F2F10F11F12step 1.2step 2.1step 6.1step 7.1step 8.1step 9.1∎

Claims 1, 2, 3 and 4 of the Statement are proved: claim 1 is step 6.1, claim 2 is step 7.1, claim 3 is step 8.1 and claim 4 is step 9.1; the ambient hypothesis is the AC recorded in the Statement and cited as [F10], its countable instance is consumed by [F1], [F2] and [F12] as described in the choice-use paragraph of the given block, and the two imported inputs from outside the library are the integration-by-parts computation [F11] of Boas and the unweighted boundary-criterion and graph-norm density statements [F12] of Haslinger, used in steps 2.1, 9.1 (with the weighted reduction carried out in steps 1.2 and 9.1).

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Weighted ∂̄ solvability on a smoothly bounded pseudoconvex domain

Statement

Assume the Axiom of Choice (AC). Let $n \geq 1$ and use one-based labels $z_{j} : = z_{j - 1}^{c a n}$ for $1 \leq j \leq n$ and the corresponding derivatives. Let $D \subseteq C^{n}$ be a bounded Levi pseudoconvex domain with $C^{\infty}$ boundary (Levi pseudoconvex domains), let $φ \in C^{2} (\overline{D}; R)$ be strictly plurisubharmonic at every point of $\overline{D}$ (The Levi form and strict plurisubharmonicity), and let $1 \leq q \leq n$ . Write $L_{0, k}^{2} : = L_{0, k}^{2} (D, e^{- φ})$ and let ${\overset{ˉ}{\partial}}_{k} : Dom {\overset{ˉ}{\partial}}_{k} \subseteq L_{0, k}^{2} \to L_{0, k + 1}^{2}$ denote the maximal distributional $\overset{ˉ}{\partial}$ of Weighted L2 spaces and maximal dbar operators, with ${\overset{ˉ}{\partial}}_{φ}^{*}$ its weighted adjoint. For $a \in \overline{D}$ let $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ be the eigenvalues of the Hermitian matrix $(φ_{j \overset{ˉ}{k}} (a))$ and put $w (a) : = λ_{1} (a) + \dots + λ_{q} (a)$ , so that $w > 0$ on $\overline{D}$ . Then for every $f \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} f = 0$ there exists $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ , and the least-norm such solution $u_{0}$ , which lies in $(\ker {\overset{ˉ}{\partial}}_{q - 1})^{⊥}$ , satisfies $∥ u_{0} ∥_{φ}^{2} \leq \int_{D} \frac{∣ f ∣^{2}}{w} e^{- φ} d V .$

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; a bounded Levi pseudoconvex domain $D \subseteq C^{n}$ with $C^{\infty}$ boundary; a weight $φ \in C^{2} (\overline{D}; R)$ strictly plurisubharmonic at every point of $\overline{D}$ ; an integer $1 \leq q \leq n$ ; the eigenvalues $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ of the Hermitian matrices $(φ_{j \overset{ˉ}{k}} (a))$ and the function $w : = λ_{1} + \dots + λ_{q}$ on $\overline{D}$ ; and a form $f \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} f = 0$ .

[F1]

With the conventions of Weighted L2 spaces and maximal dbar operators, the space $L_{0, q}^{2} (Ω, e^{- φ})$ carries the inner product $⟨ \cdot, \cdot ⟩_{φ}$ and is a complex Hilbert space, and ${\overset{ˉ}{\partial}}_{q} : Dom {\overset{ˉ}{\partial}}_{q} \to L_{0, q + 1}^{2}$ is the maximal distributional $\overset{ˉ}{\partial}$ in degree $q$ (clauses (a) and (b) of that definition).

[F2]

$Dom {\overset{ˉ}{\partial}}_{q}$ is dense in $L_{0, q}^{2}$ , and ${\overset{ˉ}{\partial}}_{q}$ is closed (The maximal distributional dbar operator is closed and densely defined).

[F3]

Distributional derivatives satisfy $\partial^{α} \partial^{β} u = \partial^{α + β} u$ (Distributional differentiation is continuous and commutes).

[F4]

A domain with $C^{2}$ boundary is Levi pseudoconvex when for every boundary point $p$ there are a neighbourhood $U$ of $p$ and a function $ρ \in C^{2} (U, R)$ with $D \cap U = {ρ < 0}$ , $d ρ (p) \neq 0$ , and $L_{ρ} (p; v) \geq 0$ for every complex tangent vector $v$ (Levi pseudoconvex domains).

[F5]

For $u \in C^{2} (Ω, R)$ , $u$ is strictly plurisubharmonic when $L_{u} (a; v) > 0$ for every $a \in Ω$ and every $v \neq 0$ (The Levi form and strict plurisubharmonicity).

[F6]

Weighted Morrey estimate (Weighted Morrey–Kohn estimate with a pseudoconvex boundary term): if $D$ is Levi pseudoconvex and $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ are the eigenvalues of $(φ_{j \overset{ˉ}{k}} (a))$ , then $\int_{D} (λ_{1} + \dots + λ_{q}) ∣ u ∣^{2} e^{- φ} d V \leq ∥ \overset{ˉ}{\partial} u ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} u ∥_{φ}^{2},$ and the inequality holds for every $u \in Dom {\overset{ˉ}{\partial}}_{q} \cap Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ .

[F7]

The $A$ -weighted form of the abstract Hilbert-complex solver (A coercive Hilbert-complex estimate solves the closed equation): if $A \in B (H_{1})$ is bounded self-adjoint with $⟨ A x, x ⟩ \geq 0$ on $H_{1}$ and $⟨ A x, x ⟩ \leq ∥ T^{*} x ∥^{2} + ∥ S x ∥^{2} for all x \in D (T^{*}) \cap D (S),$ and if $f \in \ker S$ has the form $f = A g$ for some $g \in H_{1}$ , then there exists $u \in D (T)$ with $T u = f$ , and the least-norm such $u_{0}$ satisfies $∥ u_{0} ∥^{2} \leq ⟨ f, g ⟩$ .

[F8]

A finite-dimensional complex inner product space has an orthonormal basis of eigenvectors of every normal endomorphism, and a self-adjoint endomorphism is normal with $⟨ T v, w ⟩ = ⟨ v, T w ⟩$ (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space).

[F9]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice), and it supplies its countable instance (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F10]

For a function of class $C^{2}$ on an open subset of $R^{n}$ the mixed partial derivatives commute, $\partial_{j} \partial_{i} ϕ = \partial_{i} \partial_{j} ϕ$ (Continuous second partials of a scalar potential commute).

[F11]

With the conventions of Weighted L2 spaces and maximal dbar operators (c), the weighted adjoint is the Hilbert adjoint ${\overset{ˉ}{\partial}}_{φ}^{*} : L_{0, q}^{2} \supseteq Dom {\overset{ˉ}{\partial}}_{φ}^{*} \to L_{0, q - 1}^{2}$ of ${\overset{ˉ}{\partial}}_{q - 1}$ , and $v \in Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ holds exactly when the functional $u \mapsto ⟨ {\overset{ˉ}{\partial}}_{q - 1} u, v ⟩_{φ}$ is continuous on $Dom {\overset{ˉ}{\partial}}_{q - 1}$ in the ambient norm.

[F12]

Finite orthonormal lists satisfy the Bessel inequality, with Parseval equality for an orthonormal basis (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F9]; the countable instance ${A C}_{ω}$ is consumed by the closedness and density facts [F2] and by the extension step inside claim (iv) of [F7]. The proof itself selects nothing beyond those countable instances: the eigenvalue functions are determined by $φ$ , the multiplier $A$ is determined by $w$ , and $g = f / w$ is determined by $f$ and $φ$ .

Proof

technique · direct

1.1F1F2given

Write $H_{0} : = L_{0, q - 1}^{2}$ , $H_{1} : = L_{0, q}^{2}$ and $H_{2} : = L_{0, q + 1}^{2}$ , and let $T : = {\overset{ˉ}{\partial}}_{q - 1} : H_{0} \supseteq D (T) \to H_{1}$ and $S : = {\overset{ˉ}{\partial}}_{q} : H_{1} \supseteq D (S) \to H_{2}$ be the maximal distributional operators of F1: the three spaces are complex Hilbert spaces by F1, and $T$ and $S$ are closed densely defined linear operators by [F2].

1.2F1F3givenalgebra

$T (D (T)) \subseteq \ker S$ , i.e. $S \circ T = 0$ on $D (T)$ : for $u \in D (T)$ the form $v : = T u$ has $L^{2}$ coefficients $v_{K}$ which represent the coefficient distributions $\sum_{j} \frac{\partial u_{J}}{\partial {\overset{ˉ}{z}}_{j}} d {\overset{ˉ}{z}}_{j} \land d {\overset{ˉ}{z}}^{J}$ of [F1] on the tuples $K$ with $∣ K ∣ = q$ , each of the form $\sum_{j \in K} \pm D_{j} u_{K ∖ j}$ with the shuffle signs of the exterior algebra; applying ${\overset{ˉ}{\partial}}_{q}$ to $v$ therefore gives, on each increasing tuple $L$ with $∣ L ∣ = q + 1$ , a coefficient distribution $\sum_{j \neq k \in L} \pm D_{k} D_{j} u_{L ∖ {j, k}}$ in which the term belonging to the ordered pair $(j, k)$ and the term belonging to $(k, j)$ carry the shuffle signs of $d {\overset{ˉ}{z}}_{k} \land d {\overset{ˉ}{z}}_{j}$ and $d {\overset{ˉ}{z}}_{j} \land d {\overset{ˉ}{z}}_{k}$ , hence opposite signs; since distributional derivatives commute, $D_{k} D_{j} u_{J} = D_{j} D_{k} u_{J}$ as distributions by [F3], the two terms cancel and every coefficient distribution of $\overset{ˉ}{\partial} v$ vanishes; the zero distribution is represented by the zero $L^{2}$ form, so $v \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} v = 0$ , as required.

1.3F5F8F10F12givenalgebra

Write $H (a) : = (φ_{j \overset{ˉ}{k}} (a))$ and $B (a) : = H (a)^{T}$ . Reality of $φ$ and [F10] give $H_{j k} = \overline{H_{k j}}$ , so both $H$ and $B$ are Hermitian. They have the same characteristic polynomial, since $\det (t I - H^{T}) = \det (t I - H)$ , and therefore the same ordered eigenvalues $λ_{i}$ . With the first-variable-linear inner product, the correct identity is $L_{φ} (a; v) = \sum_{j, k} H_{j k} (a) v_{j} \overline{v_{k}} = ⟨ B (a) v, v ⟩ .$ Thus $B$ is positive definite by [F5], and [F8] gives $λ_{1} (a) = \min_{∣ v ∣ = 1} ⟨ B (a) v, v ⟩ > 0$ . For every unit vector, $∣ ⟨ (B (a) - B (b)) v, v ⟩ ∣ \leq n \max_{j, k} ∣ H_{j k} (a) - H_{j k} (b) ∣ .$ Taking minima on the unit sphere shows that $λ_{1}$ is continuous. Also $w (a)$ is the minimum of $\sum_{i = 1}^{q} ⟨ B (a) v_{i}, v_{i} ⟩$ over orthonormal $q$ -frames. Indeed expansion in an eigenbasis gives $\sum_{j} λ_{j} m_{j}$ , where $0 \leq m_{j} \leq 1$ and $\sum_{j} m_{j} = q$ by [F12]; subtracting $\sum_{j \leq q} λ_{j}$ and bounding each term by $λ_{q} (m_{j} - 1)$ for $j \leq q$ , or $λ_{q} m_{j}$ for $j > q$ , gives a nonnegative difference. The first $q$ eigenvectors attain equality. The preceding uniform bound on unit-vector quotients now gives $∣ w (a) - w (b) ∣ \leq q n \max_{j, k} ∣ H_{j k} (a) - H_{j k} (b) ∣$ , so $w$ is continuous. On compact $\overline{D}$ , put $δ : = \min λ_{1} > 0$ and $M : = \max_{j, k} \sup_{\overline{D}} ∣ H_{j k} ∣ < \infty$ . The same unit-vector bound gives $q δ \leq w \leq q n M$ .

2.1F1step 1.3givenalgebra

Define $A : H_{1} \to H_{1}$ coefficientwise by $(A v)_{K} : = w v_{K}$ for $∣ K ∣ = q$ ; since $w$ is real-valued, measurable and bounded with $0 < q δ \leq w \leq q n M < \infty$ by step 1.3, $A$ is a bounded linear operator on $H_{1}$ with $∥ A ∥ \leq q n M$ , self-adjoint because $⟨ A v, v^{'} ⟩_{φ} = \int_{D} \sum_{∣ K ∣ = q} w v_{K} \overline{v_{K}^{'}} e^{- φ} d V = ⟨ v, A v^{'} ⟩_{φ}$ for $v, v^{'} \in H_{1}$ , and nonnegative because $⟨ A v, v ⟩_{φ} = \int_{D} w ∣ v ∣^{2} e^{- φ} d V \geq 0$ .

3.1F1F4F6F11step 2.1givenalgebra

For every $x \in D (T^{*}) \cap D (S)$ one has $⟨ A x, x ⟩_{φ} \leq ∥ T^{*} x ∥^{2} + ∥ S x ∥^{2}$ : here $D (T^{*}) = Dom {\overset{ˉ}{\partial}}_{φ}^{*}$ and $D (S) = Dom {\overset{ˉ}{\partial}}_{q}$ by [F1] and [F11], and $D$ is Levi pseudoconvex as assumed ([F4]), so the weighted Morrey estimate [F6], whose two clauses are the inequality and its extension to the maximal domains, applies to $x$ and gives $⟨ A x, x ⟩_{φ} = \int_{D} w ∣ x ∣^{2} e^{- φ} d V \leq ∥ \overset{ˉ}{\partial} x ∥_{φ}^{2} + ∥ {\overset{ˉ}{\partial}}_{φ}^{*} x ∥_{φ}^{2} = ∥ S x ∥^{2} + ∥ T^{*} x ∥^{2}$ .

3.2F1step 1.3step 2.1givenalgebra

Put $g : = f / w$ , that is, the $(0, q)$ -form with coefficients $g_{K} : = f_{K} / w$ for $∣ K ∣ = q$ ; since $w \geq q δ > 0$ by step 1.3, the estimate $∣ g ∣ \leq ∣ f ∣ / (q δ)$ shows $g \in L_{0, q}^{2} = H_{1}$ , and by step 2.1, $A g = w \cdot (f / w) = f$ ; moreover $⟨ f, g ⟩_{φ} = \int_{D} \sum_{∣ K ∣ = q} ∣ f_{K} ∣^{2} w^{- 1} e^{- φ} d V = \int_{D} ∣ f ∣^{2} w^{- 1} e^{- φ} d V$ .

4.1F7step 1.1step 1.2step 2.1step 3.1step 3.2givenalgebra

Claim (iv) of [F7] applies with the Hilbert spaces $H_{0}, H_{1}, H_{2}$ and the operators $T, S$ of step 1.1, the bounded self-adjoint nonnegative multiplier $A$ of step 2.1, the datum $f \in \ker S$ (which is the hypothesis ${\overset{ˉ}{\partial}}_{q} f = 0$ of the theorem) and the element $g$ of step 3.2: the closedness, density and composition requirements are steps 1.1 and 1.2, the domination hypothesis is step 3.1 and the range form $f = A g$ is step 3.2, so [F7] yields a solution $u \in D (T)$ with $T u = f$ whose least-norm representative $u_{0} \in (\ker T)^{⊥}$ satisfies $∥ u_{0} ∥^{2} \leq ⟨ f, g ⟩$ .

5.1F2F7F9step 1.2step 3.1step 3.2step 4.1givenalgebra∎

Unwinding step 4.1: ${\overset{ˉ}{\partial}}_{q - 1} u_{0} = f$ with $u_{0} \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ and $u_{0} \in (\ker {\overset{ˉ}{\partial}}_{q - 1})^{⊥}$ , and by steps 3.2 and 4.1, $∥ u_{0} ∥_{φ}^{2} \leq ⟨ f, g ⟩_{φ} = \int_{D} ∣ f ∣^{2} w^{- 1} e^{- φ} d V$ , so $u_{0}$ is a solution of ${\overset{ˉ}{\partial}}_{q - 1} u = f$ obeying the stated weighted bound and is the least-norm one; the ambient AC and its countable instance are used exactly as recorded in the choice-use paragraph, through [F2] and through claim (iv) of [F7].

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Smooth strictly plurisubharmonic exhaustion of a pseudoconvex domain

Statement

Assume the Axiom of Choice (AC). Let $Ω \subseteq C^{n}$ be a domain, $n \geq 1$ , that is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity). Then there exist a function $S \in C^{\infty} (Ω)$ that is strictly plurisubharmonic on $Ω$ (The Levi form and strict plurisubharmonicity) and a strictly increasing sequence $c_{1} < c_{2} < \dots$ with $c_{k} \to + \infty$ such that, writing $Ω_{k} : = {z \in Ω : S (z) < c_{k}}$ :

every $c_{k}$ is a regular value of $S$ , each $\partial Ω_{k} = {z \in Ω : S (z) = c_{k}}$ is a nonempty $C^{\infty}$ hypersurface of $Ω$ , and $\overline{Ω_{k}} \subseteq Ω_{k + 1}$ with $⋃_{k \geq 1} Ω_{k} = Ω$ , so every $\overline{Ω_{k}}$ is a compact subset of $Ω$ ;
each sublevel is strongly pseudoconvex along its boundary: for every $k$ , every $p \in \partial Ω_{k}$ and every $v \in C^{n} ∖ {0}$ with $\sum_{j < n} (\partial S / \partial z_{j}) (p) v_{j} = 0$ one has $L_{S} (p; v) > 0$ .

Facts & Assumptions

Given: The Axiom of Choice; a domain $Ω \subseteq C^{n}$ with $n \geq 1$ that is Hartogs pseudoconvex.

[F1]

A function $u : Ω \to R$ is a continuous plurisubharmonic exhaustion when $u$ is continuous, plurisubharmonic, and every sublevel set ${z \in Ω : u (z) \leq c}$ is compact in $Ω$ for every real number $c$ (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F2]

The domain $Ω$ is Hartogs pseudoconvex when $- \log δ_{Ω}$ is plurisubharmonic on $Ω$ , and the whole space is Hartogs pseudoconvex by the empty-complement convention (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F3]

If a domain $Ω \subseteq C^{m}$ is Hartogs pseudoconvex, then it admits a continuous plurisubharmonic exhaustion function (Hartogs pseudoconvexity yields a continuous plurisubharmonic exhaustion).

[F4]

Assume AC and AC $_{ω}$ ; for every continuous plurisubharmonic exhaustion $u$ on a domain there are $S \in C^{\infty} (Ω)$ strictly plurisubharmonic and a strictly increasing sequence $c_{k} \to + \infty$ such that each $c_{k}$ is a regular value of $S$ , each $\partial Ω_{k}$ with $Ω_{k} = {S < c_{k}}$ is a nonempty $C^{\infty}$ hypersurface, $\overline{Ω_{k}} \subseteq Ω_{k + 1}$ and $⋃_{k} Ω_{k} = Ω$ , and $L_{S} (p; v) > 0$ for all $p \in \partial Ω_{k}$ and all $v \neq 0$ with $\sum_{j} (\partial S / \partial z_{j}) (p) v_{j} = 0$ (Smooth strict plurisubharmonic regularization of a psh exhaustion).

[F5]

$A C \Rightarrow D C \Rightarrow {A C}_{ω}$ in ZF (AC implies DC implies countable choice).

[F6]

The Axiom of Countable Choice selects from every at most countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F7]

The Axiom of Choice supplies a choice function for every family of nonempty sets (The Axiom of Choice).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F7]; the regularization lemma [F4] is stated under AC and AC $_{ω}$ , and the countable instance [F6] is obtained from the ambient AC by the exact implication [F5] in step 2.1. The proof selects no family of nonempty sets.

Proof

technique · direct

1.1F1F2F3given

By the defining property [F2] the Hartogs pseudoconvexity of $Ω$ says that $- \log δ_{Ω}$ is plurisubharmonic on $Ω$ , so the equivalence theorem [F3] supplies a continuous plurisubharmonic exhaustion $u : Ω \to R$ , that is, $u$ is continuous, plurisubharmonic, and every sublevel set ${u \leq c}$ is compact in $Ω$ by [F1].

2.1F4F5F6step 1.1given

The regularization lemma [F4], whose hypotheses are assumed AC together with AC $_{ω}$ here, applies to the continuous plurisubharmonic exhaustion $u$ produced in step 1.1 and yields $S \in C^{\infty} (Ω)$ strictly plurisubharmonic together with a strictly increasing sequence $c_{k} \to + \infty$ such that each $c_{k}$ is a regular value of $S$ , each $\partial Ω_{k}$ is a nonempty $C^{\infty}$ hypersurface of $Ω$ , $\overline{Ω_{k}} \subseteq Ω_{k + 1}$ and $⋃_{k} Ω_{k} = Ω$ , and $L_{S} (p; v) > 0$ whenever $p \in \partial Ω_{k}$ and $v \neq 0$ satisfies $\sum_{j} (\partial S / \partial z_{j}) (p) v_{j} = 0$ ; the countable instance required by that lemma is supplied from the ambient AC by the implication [F5] and its content [F6].

3.1F7step 2.1∎

The function $S$ and the sequence $c_{k}$ produced in step 2.1 have exactly the properties listed as claims 1 and 2 of the Statement: strictly plurisubharmonic and smooth on $Ω$ , increasing regular values tending to infinity, sublevels with nonempty smooth boundary, increasing relatively compact closures exhausting $Ω$ , and strong pseudoconvexity along each boundary. The ambient hypothesis is the AC cited as [F7].

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Hörmander's weighted L2 existence theorem for the dbar equation

Statement

Assume the Axiom of Choice (AC). Let $n \geq 1$ and use one-based labels $z_{j} : = z_{j - 1}^{c a n}$ for $1 \leq j \leq n$ , also for their derivatives and form coefficients. Let $Ω \subseteq C^{n}$ be a domain that is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity), let $φ \in C^{2} (Ω; R)$ be strictly plurisubharmonic on $Ω$ (The Levi form and strict plurisubharmonicity), and let $1 \leq q \leq n$ . For $a \in Ω$ let $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ be the eigenvalues of the Hermitian matrix $(φ_{j \overset{ˉ}{k}} (a))$ and put $w (a) : = λ_{1} (a) + \dots + λ_{q} (a)$ , so that $w > 0$ on $Ω$ . Write $L_{0, k}^{2} : = L_{0, k}^{2} (Ω, e^{- φ})$ and let ${\overset{ˉ}{\partial}}_{k} : Dom {\overset{ˉ}{\partial}}_{k} \subseteq L_{0, k}^{2} \to L_{0, k + 1}^{2}$ be the maximal distributional $\overset{ˉ}{\partial}$ of Weighted L2 spaces and maximal dbar operators.

If $f \in Dom {\overset{ˉ}{\partial}}_{q}$ satisfies ${\overset{ˉ}{\partial}}_{q} f = 0$ and the weighted energy $E (f) : = \int_{Ω} \frac{∣ f ∣^{2}}{w} e^{- φ} d V$ is finite, then there is $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ and $∥ u ∥_{φ}^{2} \leq E (f)$ .
(Smooth data.) If in addition $φ \in C^{\infty} (Ω; R)$ and $f \in C^{\infty} (Ω; Λ^{0, q})$ satisfies $\overset{ˉ}{\partial} f = 0$ pointwise and $E (f) < + \infty$ , then there is $u \in C^{\infty} (Ω; Λ^{0, q - 1}) \cap L_{0, q - 1}^{2} (Ω, e^{- φ})$ with $\overset{ˉ}{\partial} u = f$ and $∥ u ∥_{φ}^{2} \leq E (f)$ .

The strict positivity of $φ$ is kept in both claims, and no boundary regularity of $u$ is claimed. Claim 2 is the $C^{\infty}$ branch of the same weighted estimate, quoted from the source of [F19]; it does not assert that the solution of claim 1 is smooth.

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; a domain $Ω \subseteq C^{n}$ that is Hartogs pseudoconvex; a function $φ \in C^{2} (Ω; R)$ strictly plurisubharmonic on $Ω$ ; an integer $1 \leq q \leq n$ ; the eigenvalue functions $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ of the Hermitian matrices $(φ_{j \overset{ˉ}{k}} (a))$ , $a \in Ω$ , and $w : = λ_{1} + \dots + λ_{q}$ ; and a form $f \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} f = 0$ whose energy $E (f) = \int_{Ω} ∣ f ∣^{2} w^{- 1} e^{- φ} d V$ is finite.

[F1]

With the conventions of Weighted L2 spaces and maximal dbar operators: the space $L_{0, k}^{2} (Ω, e^{- φ})$ of coefficient tuples carries the inner product $⟨ u, v ⟩_{φ} = \int_{Ω} \sum_{∣ J ∣ = k} u_{J} \overline{v_{J}} e^{- φ} d V$ and is a complex Hilbert space.

[F2]

With the same conventions, for $u \in L_{0, k}^{2}$ with a locally integrable representative the distributional form $\overset{ˉ}{\partial} u : = \sum_{∣ J ∣ = k} \sum_{j = 1}^{n} (\partial u_{J} / \partial {\overset{ˉ}{z}}_{j}) d {\overset{ˉ}{z}}_{j} \land d {\overset{ˉ}{z}}^{J}$ is defined, and $Dom {\overset{ˉ}{\partial}}_{k}$ consists of those $u$ for which $\overset{ˉ}{\partial} u$ is represented by an element of $L_{0, k + 1}^{2}$ , which is then ${\overset{ˉ}{\partial}}_{k} u$ (Weighted L2 spaces and maximal dbar operators).

[F3]

Coefficients are extended to non-increasing tuples by antisymmetry, so that $v_{j K} = 0$ when $j \in K$ , and this convention assigns the shuffle signs in the coefficient formula of [F2] (Weighted L2 spaces and maximal dbar operators).

[F4]

The bidegree decomposition and the coefficient formula $\overset{ˉ}{\partial} η = \sum_{I, J, j} (\partial_{{\overset{ˉ}{z}}_{j}} a_{I, J}) d {\overset{ˉ}{z}}_{j} \land d z^{I} \land d {\overset{ˉ}{z}}^{J}$ for smooth forms are as recorded in Bigraded complex forms and the Dolbeault operators; a smooth $(0, k)$ -form is identified with its tuple of coefficients and $C_{c}^{\infty} (Ω; Λ^{0, k})$ denotes the smooth compactly supported $(0, k)$ -forms.

[F5]

[F6]

The Levi form of $u \in C^{2} (Ω, R)$ is $L_{u} (a; v) = \sum_{j, k} \frac{\partial^{2} u}{\partial z_{j} \partial {\overset{ˉ}{z}}_{k}} (a) v_{j} \overline{v_{k}}$ , and $u$ is strictly plurisubharmonic when $L_{u} (a; v) > 0$ for every $a \in Ω$ and every $v \neq 0$ (The Levi form and strict plurisubharmonicity).

[F7]

A domain with $C^{2}$ boundary is Levi pseudoconvex when for every boundary point $p$ there are a neighbourhood $U$ of $p$ and a function $ρ \in C^{2} (U, R)$ with $Ω \cap U = {ρ < 0}$ , $d ρ (p) \neq 0$ , and $L_{ρ} (p; v) \geq 0$ for every complex tangent vector $v$ (Levi pseudoconvex domains).

[F8]

A domain $Ω$ is Hartogs pseudoconvex when $- \log δ_{Ω}$ is plurisubharmonic on $Ω$ , where $δ_{Ω}$ is the equal-radius polydisc boundary function; the whole space is Hartogs pseudoconvex by the empty-complement convention (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F9]

If $Ω \subseteq C^{n}$ is a Hartogs pseudoconvex domain, then there are $S \in C^{\infty} (Ω)$ strictly plurisubharmonic and a strictly increasing sequence $c_{k} \to + \infty$ such that, with $Ω_{k} : = {z \in Ω : S (z) < c_{k}}$ : every $c_{k}$ is a regular value of $S$ , every $\partial Ω_{k}$ is a nonempty $C^{\infty}$ hypersurface of $Ω$ , $\overline{Ω_{k}} \subseteq Ω_{k + 1}$ and $⋃_{k} Ω_{k} = Ω$ , and $L_{S} (p; v) > 0$ for every $p \in \partial Ω_{k}$ and every $v \neq 0$ with $\sum_{j} (\partial S / \partial z_{j}) (p) v_{j} = 0$ (Smooth strictly plurisubharmonic exhaustion of a pseudoconvex domain).

[F10]

Weighted solvability on a smoothly bounded domain (Weighted ∂̄ solvability on a smoothly bounded pseudoconvex domain): if $D$ is a bounded Levi pseudoconvex domain with $C^{\infty}$ boundary, $φ \in C^{2} (\overline{D})$ is strictly plurisubharmonic at every point of $\overline{D}$ , $1 \leq q \leq n$ , and $f \in Dom {\overset{ˉ}{\partial}}_{q} (D)$ satisfies ${\overset{ˉ}{\partial}}_{q} f = 0$ , then there is $u \in Dom {\overset{ˉ}{\partial}}_{q - 1} (D)$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ , and the least-norm such solution satisfies $∥ u ∥_{φ, D}^{2} \leq \int_{D} ∣ f ∣^{2} w^{- 1} e^{- φ} d V$ with $w$ the sum of the $q$ smallest eigenvalues of the Hermitian matrices $(φ_{j \overset{ˉ}{k}})$ on $D$ .

[F11]

The Wirtinger operators satisfy $\partial_{z_{j}} = \frac{1}{2} (\partial_{x_{j}} - i \partial_{y_{j}})$ and $\partial_{{\overset{ˉ}{z}}_{j}} = \frac{1}{2} (\partial_{x_{j}} + i \partial_{y_{j}})$ (Wirtinger operators in $C^{m}$ ).

[F12]

For functions with continuous second partial derivatives the mixed second partials commute (Continuous second partials of a scalar potential commute).

[F13]

A finite-dimensional complex inner product space has an orthonormal basis of eigenvectors of every normal endomorphism (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F14]

An endomorphism is self-adjoint exactly when its matrix in an orthonormal basis is Hermitian, and every self-adjoint endomorphism is normal (In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space).

[F15]

A finite orthonormal list $(e_{0}, \dots, e_{r - 1})$ satisfies $\sum_{i < r} ∣ ⟨ v, e_{i} ⟩ ∣^{2} \leq ∥ v ∥^{2}$ for every $v$ , with equality when $(e_{i})$ is an orthonormal basis (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis).

[F16]

Under the Axiom of Countable Choice every complete real or complex inner-product space is reflexive (Hilbert spaces are reflexive by Riesz representation).

[F17]

Under the ultrafilter lemma, DC and HB, a real or complex Banach space $X$ is reflexive if and only if every norm-bounded sequence in $X$ has a subsequence converging weakly to a point of $X$ (Reflexivity is equivalent to weak subsequential compactness of bounded sequences).

[F18]

Under HB, if a net $x_{i} ⇀ x$ in a real or complex normed space, then $∥ x ∥ \leq {lim inf}_{i} ∥ x_{i} ∥$ , with no boundedness or completeness hypothesis (Weak convergence implies lower semicontinuity of the norm).

[F20]

The Axiom of Choice implies the ultrafilter lemma (The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter).

[F21]

The Axiom of Choice implies HB, the real dominated-extension principle (Hahn-Banach dominated extension theorem for real vector spaces).

[F22]

In ZF, AC implies the Axiom of Countable Choice and the prescribed-initial-point form of Dependent Choice (AC supplies the countable and dependent choices used in Banach integration).

[F23]

AC is the assertion that every family of nonempty sets has a choice function (The Axiom of Choice); DC is the prescribed-initial-point form of The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain; AC $_{ω}$ selects from every at most countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )); HB is the real dominated-extension principle over ZF (The real dominated-extension principle as an additional hypothesis over ZF).

[F19]

(Demailly, Ch. VIII, Theorem 6.5 and its proof, printed p. 378, with (6.4) on p. 377.) On a weakly pseudoconvex Kähler manifold with a hermitian line bundle and a smooth weight having nonnegative curvature eigenvalues, a smooth closed $(n, q)$ -form of finite reciprocal-eigenvalue energy has a smooth solution with the corresponding norm bound. No global $L^{2}$ hypothesis on the datum is required. Here use the trivial line bundle and the flat normalization in which $d z_{j}, d {\overset{ˉ}{z}}_{j}$ are orthonormal; the eigenvalues are those of $(φ_{j \overset{ˉ}{k}})$ and the metric volume is a constant multiple of $d V$ . Put $α = d z_{1} \land \dots \land d z_{n}$ . The map $η \mapsto η \land α$ commutes with $\overset{ˉ}{\partial}$ and preserves coefficient norms, so the source bound transfers to $(0, q)$ -forms; the common volume constant cancels.

[F24]

Weak convergence of a net means convergence against every bounded linear functional; a sequence is the case $I = N$ (Weak convergence of nets and sequences).

[F25]

Under the complex Euclidean dictionary, $Ω$ is an open connected subset of $R^{2 n}$ , hence any two points can be joined by a polygonal path in $Ω$ (Complex $m$ -space and its real coordinate dictionary, For an open subset of $R^{n}$ , connectedness, path-connectedness and polygonal connectedness are equivalent).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F23]; its countable instance AC $_{ω}$ is consumed through the reflexive-Hilbert-space supplier [F16] and licenses the countably many applications of [F10] in step 2.2 that produce the sequence $(u_{k})$ , and DC and HB are consumed through the reflexivity criterion [F17]; the consequences AC $\Rightarrow$ ultrafilter lemma, AC $\Rightarrow$ HB and AC $\Rightarrow$ (DC and AC $_{ω}$ ) are supplied by [F20], [F21] and [F22]. No other family of nonempty sets is selected, and apart from the interfaces named above no step uses choice beyond the definitions and the cited interfaces.

Proof

technique · direct

1.1F7F8F9F25given

By [F8] and [F9] choose a smooth strictly plurisubharmonic exhaustion $S$ and increasing regular values $c_{k} \to + \infty$ , with $Ω_{k} : = {S < c_{k}}$ relatively compact and exhausting $Ω$ . Fix $z_{0} \in Ω$ and discard an initial finite segment so that $z_{0} \in Ω_{k}$ for every remaining $k$ . Let $D_{k}$ be the connected component of $Ω_{k}$ containing $z_{0}$ ; these components are nested. They exhaust $Ω$ : for $z \in Ω$ , [F25] gives a polygonal path from $z_{0}$ to $z$ , whose compact image lies in some $Ω_{k}$ by the increasing open cover, so $z \in D_{k}$ . Each $D_{k}$ is bounded because $\overline{Ω_{k}}$ is compact. At $p \in \partial D_{k}$ one has $p \in \partial Ω_{k} = {S = c_{k}}$ ; the regular-level chart makes $Ω_{k}$ a connected local subgraph near $p$ , and its inside part meets $D_{k}$ , so it lies in $D_{k}$ . Thus $D_{k}$ has $C^{\infty}$ boundary locally defined by $ρ = S - c_{k}$ with $d ρ (p) \neq 0$ . Strict plurisubharmonicity gives $L_{ρ} (p; v) > 0$ on every nonzero complex tangent vector, so $D_{k}$ is a bounded Levi pseudoconvex domain to which [F10] applies.

1.2F6F11F12F13F14given

For $a \in Ω$ write $H (a) : = (φ_{j \overset{ˉ}{k}} (a))$ and $B (a) : = H (a)^{T}$ . Reality of $φ$ , the Wirtinger formulas [F11] and commutation of mixed partials [F12] give $\overline{H_{j k}} = H_{k j}$ , so $H$ and $B$ are Hermitian and are self-adjoint by [F14]. Their characteristic polynomials agree, since $\det (t I - H^{T}) = \det (t I - H)$ , so they have the same ordered eigenvalues $λ_{i}$ . With the first-variable-linear inner product, $L_{φ} (a; v) = \sum_{j, k} H_{j k} (a) v_{j} \overline{v_{k}} = ⟨ B (a) v, v ⟩ .$ Thus strict plurisubharmonicity [F6] makes $B$ positive definite. Its orthonormal eigenbasis from [F13] shows that $λ_{1} (a) = \min_{∣ v ∣ = 1} ⟨ B (a) v, v ⟩ > 0$ , hence every eigenvalue of $H$ is positive and $w (a) \geq q λ_{1} (a) > 0$ .

1.3F1F2F4F5given

For each $k$ let $f_{k}$ be the restriction of $f$ to $D_{k}$ , that is the tuple of restrictions $f_{k, K} : = f_{K} ∣_{D_{k}}$ . Then $f_{k} \in L_{0, q}^{2} (D_{k}, e^{- φ})$ with $\int_{D_{k}} ∣ f_{k} ∣^{2} w^{- 1} e^{- φ} d V \leq E (f)$ because $f \in L_{0, q}^{2} (Ω, e^{- φ})$ and the integrand is nonnegative, and $f_{k} \in Dom {\overset{ˉ}{\partial}}_{q} (D_{k})$ with ${\overset{ˉ}{\partial}}_{q} f_{k} = 0$ : distributional differentiation is local, so the coefficient distributions of $\overset{ˉ}{\partial} f_{k}$ on $D_{k}$ are obtained by evaluating those of $\overset{ˉ}{\partial} f$ on test functions supported in $D_{k}$ , and the coefficient formula and test identity of [F2], [F4] and [F5] pair them with the test functions $\overline{ψ}$ , $ψ \in C_{c}^{\infty} (D_{k})$ , exactly as $\overset{ˉ}{\partial} f$ pairs with the zero extensions of $\overline{ψ}$ to $Ω$ ; since the distribution $\overset{ˉ}{\partial} f$ is represented by the zero form by the hypothesis ${\overset{ˉ}{\partial}}_{q} f = 0$ , its restriction to $D_{k}$ is represented by the zero form as well.

1.4F1F2F3F4F5F11givenalgebra

Fix a test form $η \in C_{c}^{\infty} (Ω; Λ^{0, q})$ and let $\partial_{0}^{*} η$ be the $(0, q - 1)$ -form with coefficients $(\partial_{0}^{*} η)_{I} : = - \sum_{j = 1}^{n} \partial_{z_{j}} η_{j I}$ ; then, writing $H : = L_{0, q - 1}^{2} (Ω, e^{- φ})$ , $e^{φ} \partial_{0}^{*} η$ is compactly supported, of class $C^{2}$ , and lies in $H$ , and for every $\tilde{u} \in H$ one has the distributional identity $\sum_{∣ K ∣ = q} ⟨ (\overset{ˉ}{\partial} \tilde{u})_{K}, \overline{η_{K}} ⟩ = ⟨ \tilde{u}, e^{φ} \partial_{0}^{*} η ⟩_{φ}$ , where the left side evaluates the coefficient distributions of F2 against the test functions $\overline{η_{K}}$ and the antisymmetry convention of [F3] assigns the shuffle signs: both sides equal the single sum $- \sum_{∣ I ∣ = q - 1} \sum_{j = 1}^{n} \int_{Ω} {\tilde{u}}_{I} \partial_{{\overset{ˉ}{z}}_{j}} (\overline{η_{j I}}) d V$ , the left by the test identity of [F5] and the sign convention for $\overset{ˉ}{\partial}$ of [F4], and the right by expanding the pairing of F1 with $\overline{(e^{φ} (\partial_{0}^{*} η)_{I})} = e^{φ} \overline{(\partial_{0}^{*} η)_{I}}$ and conjugating the holomorphic derivative by the Wirtinger rules [F11].

2.1F1F15step 1.2givenalgebra

The function $w$ is upper semicontinuous, hence Borel measurable, on $Ω$ : for every orthonormal $q$ -frame $V = (v_{1}, \dots, v_{q})$ the function $a \mapsto G (a, V) : = \sum_{i = 1}^{q} ⟨ H (a) v_{i}, v_{i} ⟩$ is continuous, and $w (a) = \min_{V} G (a, V)$ over the nonempty set of orthonormal $q$ -frames, because expanding in an eigenbasis $(e_{j})$ of step 1.2 gives $G (a, V) = \sum_{j} λ_{j} (a) m_{j}$ with $m_{j} : = \sum_{i} ∣ ⟨ v_{i}, e_{j} ⟩ ∣^{2} \in [0, 1]$ by the Bessel inequality of [F15] and $\sum_{j} m_{j} = \sum_{i} ∥ v_{i} ∥^{2} = q$ by its Parseval clause, and for such a mass vector $\sum_{j} λ_{j} (a) m_{j} - (λ_{1} + \dots + λ_{q}) (a) = \sum_{j \leq q} λ_{j} (m_{j} - 1) + \sum_{j > q} λ_{j} m_{j} \geq λ_{q} (\sum_{j \leq q} (m_{j} - 1) + \sum_{j > q} m_{j}) = 0$ , with equality for $V = (e_{1}, \dots, e_{q})$ ; hence ${w < c} = ⋃_{V} {a : G (a, V) < c}$ is a union of open sets, and since $w > 0$ on $Ω$ by step 1.2 the integrand $∣ f ∣^{2} w^{- 1} e^{- φ}$ is a nonnegative measurable function, so that the energy $E (f)$ of the statement is a well-defined extended Lebesgue integral.

2.2F10step 1.1step 1.2step 1.3given

Fix $k$ and take $D : = D_{k}$ . The domain $D$ is a bounded Levi pseudoconvex domain with $C^{\infty}$ boundary, the weight $φ$ lies in $C^{2}$ on a neighbourhood of $\overline{D} \subseteq Ω$ and is strictly plurisubharmonic at every point of $\overline{D}$ by step 1.2 and the hypothesis, and the datum $f_{k}$ lies in $Dom {\overset{ˉ}{\partial}}_{q} (D)$ with ${\overset{ˉ}{\partial}}_{q} f_{k} = 0$ by step 1.3; all hypotheses of [F10] are therefore met, and its conclusion supplies the least-norm solution $u_{k} \in Dom {\overset{ˉ}{\partial}}_{q - 1} (D)$ of ${\overset{ˉ}{\partial}}_{q - 1} u = f_{k}$ with $∥ u_{k} ∥_{φ, D}^{2} \leq \int_{D} ∣ f ∣^{2} w^{- 1} e^{- φ} d V \leq E (f)$ , where the eigenvalue functions named in [F10] are the functions $λ_{j}$ of step 1.2 and the last inequality is step 1.3 with the nonnegative integrand and $D \subseteq Ω$ .

3.1F1step 2.2givenalgebra

Extend each $u_{k}$ by zero: let ${\tilde{u}}_{k}$ equal $u_{k}$ on $D_{k}$ and $0$ on $Ω ∖ D_{k}$ , regarded as a coefficient tuple. Then ${\tilde{u}}_{k} \in H : = L_{0, q - 1}^{2} (Ω, e^{- φ})$ with measurable coefficients, $∥ {\tilde{u}}_{k} ∥_{φ}^{2} = ∥ u_{k} ∥_{φ, D_{k}}^{2} \leq E (f)$ by step 2.2, and consequently $({\tilde{u}}_{k})_{k \geq 1}$ is a norm-bounded sequence in the complex Hilbert space $H$ of [F1].

3.2F1F5step 1.1step 1.4step 2.2given

Let $η \in C_{c}^{\infty} (Ω; Λ^{0, q})$ and let $K$ be an index with $supp η \subseteq D_{K}$ , which exists because the sets $D_{k}$ increase to $Ω$ by step 1.1 while $supp η$ is compact; for every $k \geq K$ the left side of the identity of step 1.4 with $\tilde{u} : = {\tilde{u}}_{k}$ equals $\int_{Ω} \sum_{∣ K^{'} ∣ = q} f_{K^{'}} \overline{η_{K^{'}}} d V = : ⟨ f, η ⟩_{0}$ : on the open set $D_{k} \supseteq supp η$ the tuples ${\tilde{u}}_{k}$ and $u_{k}$ agree, so by the locality of distributional differentiation the coefficient distributions of $\overset{ˉ}{\partial} {\tilde{u}}_{k}$ evaluated against the test functions $\overline{η_{K^{'}}}$ depend only on $u_{k}$ , and there the $L^{2}$ identity ${\overset{ˉ}{\partial}}_{q - 1} u_{k} = f_{k}$ of step 2.2 represents them by the coefficients $f_{K^{'}}$ .

4.1F16F17F20F21F22step 3.1

The space $H$ is reflexive by [F16], whose countable-choice hypothesis is the instance AC $_{ω}$ supplied by AC through [F22]; by [F17], whose ultrafilter-lemma, DC and HB hypotheses are supplied from AC by [F20], [F22] and [F21], a reflexive complex Banach space has the property that the norm-bounded sequence $({\tilde{u}}_{k})$ admits a subsequence $({\tilde{u}}_{k_{j}})_{j \geq 1}$ converging weakly in $H$ to some $u \in H$ .

5.1F18F21step 3.1step 4.1algebra

By [F18], whose HB hypothesis is supplied from AC by [F21], applied to the weakly convergent sequence of step 4.1 one has $∥ u ∥_{φ} \leq {lim inf}_{j \to \infty} ∥ {\tilde{u}}_{k_{j}} ∥_{φ}$ , and $∥ {\tilde{u}}_{k_{j}} ∥_{φ}^{2} \leq E (f)$ for every $j$ by step 3.1, so that ${lim inf}_{j} ∥ {\tilde{u}}_{k_{j}} ∥_{φ} \leq E (f)^{1 / 2}$ and $∥ u ∥_{φ}^{2} \leq E (f)$ .

5.2F1F2F24step 4.1step 1.4step 3.2

Let $η \in C_{c}^{\infty} (Ω; Λ^{0, q})$ . By step 1.4 the left side of its identity with $\tilde{u} : = {\tilde{u}}_{k_{j}}$ equals $⟨ {\tilde{u}}_{k_{j}}, e^{φ} \partial_{0}^{*} η ⟩_{φ}$ , and this scalar converges to $⟨ u, e^{φ} \partial_{0}^{*} η ⟩_{φ}$ as $j \to \infty$ because $e^{φ} \partial_{0}^{*} η \in H$ and ${\tilde{u}}_{k_{j}} ⇀ u$ weakly (step 4.1 and the characterization of weak convergence by bounded functionals in [F24]); by step 3.2 the same scalar equals $⟨ f, η ⟩_{0}$ for all sufficiently large $j$ , so $⟨ u, e^{φ} \partial_{0}^{*} η ⟩_{φ} = ⟨ f, η ⟩_{0}$ , and by step 1.4 applied with $\tilde{u} : = u$ , whose left side is by construction the pairing of the coefficient distributions of $\overset{ˉ}{\partial} u$ with the test form $η$ , the distribution $\overset{ˉ}{\partial} u$ is represented by the $L^{2}$ form $f$ ; by the definition of the maximal operator F2 this says $u \in Dom {\overset{ˉ}{\partial}}_{q - 1} (Ω)$ and ${\overset{ˉ}{\partial}}_{q - 1} u = f$ .

6.1F9F23F19step 5.1step 5.2given∎

Claim 1 holds: the element $u$ of step 5.2 lies in $Dom {\overset{ˉ}{\partial}}_{q - 1}$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ , and $∥ u ∥_{φ}^{2} \leq E (f)$ by step 5.1. Claim 2 holds by the source fact [F19] under its hypotheses: $Ω$ with the Euclidean Kähler form is a weakly pseudoconvex Kähler manifold because the exhaustion of [F9] is a plurisubharmonic exhaustion, $φ \in C^{\infty} (Ω)$ has nonnegative eigenvalues $λ_{j} > 0$ , and the smooth $\overset{ˉ}{\partial}$ -closed form $f$ has finite energy, so the $C^{\infty}$ branch of the quoted theorem supplies $u \in C^{\infty} (Ω; Λ^{0, q - 1})$ with $\overset{ˉ}{\partial} u = f$ and the same weighted bound; no boundary regularity of $u$ is asserted in either claim, and both claims are stated under the ambient Axiom of Choice cited as [F23].

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Positive-degree Dolbeault vanishing on pseudoconvex domains

Statement

Assume the Axiom of Choice (AC). Let $n \geq 1$ , let $Ω \subseteq C^{n}$ be a domain, and suppose that $Ω$ is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity). Let $1 \leq q \leq n$ .

(Smooth vanishing.) Every smooth $\overset{ˉ}{\partial}$ -closed $(0, q)$ -form $η \in Ω^{0, q} (Ω)$ (Bigraded complex forms and the Dolbeault operators) is exact in the Dolbeault complex: there is $ζ \in Ω^{0, q - 1} (Ω)$ with $\overset{ˉ}{\partial} ζ = η$ . Consequently $H_{\overset{ˉ}{\partial}}^{0, q} (Ω) = 0$ (Dolbeault cohomology of a domain).
(Weighted $L^{2}$ exactness under finite energy.) Let $φ \in C^{2} (Ω; R)$ be strictly plurisubharmonic on $Ω$ (The Levi form and strict plurisubharmonicity); for $a \in Ω$ let $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ be the eigenvalues of the Hermitian matrix $(φ_{j \overset{ˉ}{k}} (a))$ and put $w (a) : = λ_{1} (a) + \dots + λ_{q} (a) > 0$ . If $f \in Dom {\overset{ˉ}{\partial}}_{q}$ satisfies ${\overset{ˉ}{\partial}}_{q} f = 0$ and $E (f) : = \int_{Ω} ∣ f ∣^{2} w^{- 1} e^{- φ} d V < + \infty,$ then there is $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ and $∥ u ∥_{φ}^{2} \leq E (f)$ .

No boundary regularity of the primitives is claimed, and the statement is asserted for $1 \leq q \leq n$ only.

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; a Hartogs pseudoconvex domain $Ω \subseteq C^{n}$ ; an integer $1 \leq q \leq n$ ; a smooth $\overset{ˉ}{\partial}$ -closed $(0, q)$ -form $η \in Ω^{0, q} (Ω)$ ; and a triple $(φ, f, w)$ consisting of a strictly plurisubharmonic $φ \in C^{2} (Ω; R)$ , its eigenvalue functions $λ_{1} \leq \dots \leq λ_{n}$ and $w = λ_{1} + \dots + λ_{q}$ , and a form $f \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} f = 0$ and $E (f) < + \infty$ .

[F1]

A domain $Ω$ is Hartogs pseudoconvex when $- \log δ_{Ω}$ is plurisubharmonic on $Ω$ , where $δ_{Ω}$ is the equal-radius polydisc boundary function (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F2]

If $Ω \subseteq C^{n}$ is Hartogs pseudoconvex, then there are $S \in C^{\infty} (Ω)$ strictly plurisubharmonic and a strictly increasing sequence $c_{k} \to + \infty$ such that, with $Ω_{k} : = {z \in Ω : S (z) < c_{k}}$ : every $c_{k}$ is a regular value of $S$ , every $\partial Ω_{k} = {z \in Ω : S (z) = c_{k}}$ is a nonempty $C^{\infty}$ hypersurface of $Ω$ , $\overline{Ω_{k}} \subseteq Ω_{k + 1}$ and $⋃_{k} Ω_{k} = Ω$ , so every $\overline{Ω_{k}}$ is a compact subset of $Ω$ (Smooth strictly plurisubharmonic exhaustion of a pseudoconvex domain).

[F3]

For $u \in C^{2} (Ω, R)$ the Levi form is $L_{u} (a; v) = \sum_{j, k} \frac{\partial^{2} u}{\partial z_{j} \partial {\overset{ˉ}{z}}_{k}} (a) v_{j} \overline{v_{k}}$ , and $u$ is strictly plurisubharmonic when $L_{u} (a; v) > 0$ for every $a \in Ω$ and every $v \neq 0$ ; a strictly plurisubharmonic function is plurisubharmonic (The Levi form and strict plurisubharmonicity).

[F4]

A $C^{2}$ function $u$ on an open set is plurisubharmonic if and only if $L_{u} (a; v) \geq 0$ for every $a$ and every $v$ (The C^2 Levi criterion for plurisubharmonicity).

[F5]

If $u : Ω \to R$ is plurisubharmonic and $ϕ : R \to R$ is convex and nondecreasing, then $ϕ \circ u$ is plurisubharmonic on $Ω$ (Basic stability operations for plurisubharmonic functions).

[F6]

With $Ω^{p, q} (U)$ the smooth complex-valued forms of bidegree $(p, q)$ on open $U$ , $Z_{\overset{ˉ}{\partial}}^{p, q} (U) = \ker (\overset{ˉ}{\partial} : Ω^{p, q} (U) \to Ω^{p, q + 1} (U))$ , $B_{\overset{ˉ}{\partial}}^{p, q} (U) = im (\overset{ˉ}{\partial} : Ω^{p, q - 1} (U) \to Ω^{p, q} (U))$ and $H_{\overset{ˉ}{\partial}}^{p, q} (U) = Z_{\overset{ˉ}{\partial}}^{p, q} (U) / B_{\overset{ˉ}{\partial}}^{p, q} (U)$ (Dolbeault cohomology of a domain).

[F7]

Hörmander's weighted $L^{2}$ existence theorem (Hörmander's weighted L2 existence theorem for the dbar equation): under AC, with $Ω$ Hartogs pseudoconvex, $φ \in C^{2} (Ω; R)$ strictly plurisubharmonic, $1 \leq q \leq n$ , eigenvalues $λ_{1} \leq \dots \leq λ_{n}$ and $w = λ_{1} + \dots + λ_{q}$ : (claim 1) every $f \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} f = 0$ and finite energy $E (f) = \int_{Ω} ∣ f ∣^{2} w^{- 1} e^{- φ} d V$ has a solution $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ and $∥ u ∥_{φ}^{2} \leq E (f)$ ; (claim 2) if in addition $φ \in C^{\infty} (Ω; R)$ and $f \in C^{\infty} (Ω; Λ^{0, q})$ is $\overset{ˉ}{\partial}$ -closed with $E (f) < + \infty$ , then there is $u \in C^{\infty} (Ω; Λ^{0, q - 1}) \cap L_{0, q - 1}^{2} (Ω, e^{- φ})$ with $\overset{ˉ}{\partial} u = f$ and $∥ u ∥_{φ}^{2} \leq E (f)$ .

[F8]

With the conventions of Weighted L2 spaces and maximal dbar operators: $L_{0, k}^{2} (Ω, e^{- φ})$ is the space of coefficient tuples with the inner product $⟨ u, v ⟩_{φ} = \int_{Ω} \sum_{∣ J ∣ = k} u_{J} \overline{v_{J}} e^{- φ} d V$ , and $Dom {\overset{ˉ}{\partial}}_{k}$ consists of those $u$ for which the distributional $\overset{ˉ}{\partial} u$ is represented by an element of $L_{0, k + 1}^{2}$ , which is then ${\overset{ˉ}{\partial}}_{k} u$ .

[F10]

The standard smooth step function is $σ (t) = β (t) / (β (t) + β (1 - t))$ with $β$ the standard flat function; it satisfies $σ \in C^{\infty} (R)$ , $σ (t) = 0$ for $t \leq 0$ and $σ (t) = 1$ for $t \geq 1$ , and takes values in $[0, 1]$ (The standard smooth step function).

[F11]

The standard flat function $β (t) = \exp (- 1 / t)$ for $t > 0$ and $β (t) = 0$ for $t \leq 0$ satisfies $β \in C^{\infty} (R)$ and $β (t) > 0$ for $t > 0$ (The standard flat function, The standard flat function is smooth and flat at zero).

[F12]

Every continuous function $f : I \to R$ on an order-convex interval $I$ with at least two elements has a primitive $G$ on $I$ ; the function $F (x) = \int_{c_{0}}^{x} f$ is one, and for $a < b$ in $I$ and any primitive $G$ , $\int_{a}^{b} f = G (b) - G (a)$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ).

[F13]

For $α < β$ and integrable $f : [α, β] \to R$ and arbitrary $u, v, w \in [α, β]$ one has $\int_{u}^{v} f + \int_{v}^{w} f = \int_{u}^{w} f$ (For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ ).

[F14]

For a bounded $f : [a, b] \to R$ the Darboux sums over a partition $P$ are the lower and upper sums $L (f, P) \leq U (f, P)$ , and $f$ is integrable with $\int_{a}^{b} f$ defined through these sums (For bounded $f$ on $[a, b]$ and a partition $P$ : the infimum $m_{i}$ and supremum $M_{i}$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L (f, P) = \sum_{i} m_{i} Δ_{i}$ and $U (f, P) = \sum_{i} M_{i} Δ_{i}$ , The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

[F15]

For a nonempty $A \subseteq R^{n}$ the following are equivalent: $A$ is compact; $A$ is closed and bounded; every continuous $f : A \to R$ attains a maximum and a minimum on $A$ (For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).

[F16]

Under the identification of $C^{m}$ with $R^{2 m}$ , the metric of $C^{m}$ , its balls, its open sets, its convergent sequences, its Cauchy sequences and its continuous maps are verbatim those of $R^{2 m}$ (Complex $m$ -space and its real coordinate dictionary).

[F17]

Under the Axiom of Countable Choice, every bounded subset $E \subseteq R^{n}$ has finite outer measure, a bounded Lebesgue measurable set has finite measure, and every compact subset of $R^{n}$ is Lebesgue measurable of finite measure (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F18]

Under the Axiom of Countable Choice, every continuous map $R^{n} \to R^{m}$ is Borel measurable (Continuous functions on Euclidean spaces are Borel measurable).

[F19]

If $f, g : X \to [0, + \infty]$ are measurable and $c \geq 0$ : $f \leq g$ implies $\int f \leq \int g$ , and $\int c f = c \int f$ for $c > 0$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F20]

For a nonnegative simple measurable function $s = \sum_{j} c_{j} χ_{E_{j}}$ with pairwise disjoint measurable $E_{j}$ and $c_{j} \geq 0$ , the simple integral is $\int s d μ = \sum_{j} c_{j} μ (E_{j})$ (The integral of a nonnegative simple function).

[F21]

For nonnegative measurable functions $f_{k}$ with $S = \sum_{k = 0}^{\infty} f_{k}$ one has $\int S d μ = \sum_{k = 0}^{\infty} \int f_{k} d μ$ (Beppo Levi's theorem for nonnegative series).

[F22]

If a real sequence $(a_{k})$ has no vanishing term and ${lim sup}_{k} ∣ a_{k + 1} / a_{k} ∣ < 1$ , then $\sum ∣ a_{k} ∣$ converges (Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence).

[F23]

The exponential function is strictly increasing (The exponential function is strictly increasing).

[F24]

A finite-dimensional complex inner product space has an orthonormal basis of eigenvectors of every normal endomorphism, and an endomorphism is self-adjoint exactly when its matrix in an orthonormal basis is Hermitian, self-adjoint endomorphisms being normal (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space); the eigenvalues of a self-adjoint endomorphism are real, since from $T v = λ v$ , $v \neq 0$ , one gets $λ ⟨ v, v ⟩ = ⟨ T v, v ⟩ = ⟨ v, T v ⟩ = \overline{λ} ⟨ v, v ⟩$ with $⟨ v, v ⟩ > 0$ .

[F25]

The determinant of a matrix is the product of its eigenvalues, counted with algebraic multiplicity (If $χ_{T} (x) = \prod_{i < n} (x - λ_{i})$ in $F [x]$ , then $\det (T) = \prod_{i < n} λ_{i}$ : determinant is the product of the eigenvalues counted with algebraic multiplicity).

[F26]

The trace of a matrix is the sum of its eigenvalues, counted with algebraic multiplicity (If $χ_{T} (x) = \prod_{i < n} (x - λ_{i})$ in $F [x]$ , then $tr (T) = \sum_{i < n} λ_{i}$ : trace is the sum of the eigenvalues counted with algebraic multiplicity).

[F27]

A twice differentiable $f : I \to R$ on an open interval is convex if and only if $f^{''} (x) \geq 0$ for every $x \in I$ (A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative).

[F28]

Finite componentwise sums and products of $C^{k}$ Euclidean maps are $C^{k}$ , and a composite of composable $C^{k}$ Euclidean maps is $C^{k}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[F29]

A continuous map of smooth manifolds is smooth if and only if its restrictions to the members of an open cover are smooth (Smoothness is local on the source).

[F30]

If $f : U \to V$ and $g : V \to R^{p}$ are totally differentiable at $a$ and $f (a)$ , then $g \circ f$ is totally differentiable at $a$ with $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F31]

For functions with continuous second partial derivatives the mixed second partials commute (Continuous second partials of a scalar potential commute).

[F32]

AC is the assertion that every family of nonempty sets has a choice function (The Axiom of Choice); AC $_{ω}$ selects from every at most countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )); and in ZF, AC implies AC $_{ω}$ (AC implies DC implies countable choice).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F32]; the countable instance AC $_{ω}$ is consumed through the measure-theoretic suppliers [F17] and [F18] and is obtained from the ambient AC by the implication [F32]. The Hörmander theorem [F7] is applied under its own AC hypothesis, which is the Given. Apart from these interfaces no family of nonempty sets is selected: the shells $A_{j}$ , the numbers $θ_{j}$ , the sequence $d_{i}$ , the coefficients $a_{i}$ and the function $F$ are all defined by explicit formulas.

Proof

technique · direct

1.1F1F2F15F16

By [F2] and the definition [F1] of Hartogs pseudoconvexity there are $S \in C^{\infty} (Ω)$ strictly plurisubharmonic and regular values $c_{k} ↑ + \infty$ such that, with $Ω_{k} : = {S < c_{k}}$ , every $\partial Ω_{k}$ is a nonempty $C^{\infty}$ hypersurface of $Ω$ , $\overline{Ω_{k}} \subseteq Ω_{k + 1}$ and $⋃_{k} Ω_{k} = Ω$ , so every $\overline{Ω_{k}}$ is a compact subset of $Ω$ ; by the dictionary [F16] the extreme-value criterion [F15] applies to the nonempty compact set $\overline{Ω_{1}}$ and the continuous function $S$ , so $μ : = \min_{\overline{Ω_{1}}} S$ is attained and finite; if $z \in Ω ∖ \overline{Ω_{1}}$ then $S (z) < c_{1}$ would give $z \in Ω_{1} \subseteq \overline{Ω_{1}}$ , so $S (z) \geq c_{1}$ , while $μ \leq c_{1}$ because $\partial Ω_{1} \subseteq \overline{Ω_{1}}$ is nonempty with $S = c_{1}$ there; hence $S \geq μ$ on $Ω$ .

1.2F3F24F25F26F31algebra

For $a \in Ω$ put $H (a) : = (\partial^{2} S / \partial z_{j} \partial {\overset{ˉ}{z}}_{k} (a))$ ; since $S$ is real-valued with commuting mixed second partials [F31], $H (a)$ is Hermitian, hence self-adjoint, so by [F24] it has an orthonormal eigenbasis with real eigenvalues $λ_{1} (a) \leq \dots \leq λ_{n} (a)$ , and expansion in that basis gives $λ_{1} (a) = \min_{∥ v ∥ = 1} ⟨ H (a) v, v ⟩ = \min_{∥ v ∥ = 1} L_{S} (a; v)$ ; strict plurisubharmonicity makes $λ_{1} (a) > 0$ [F3], so $\det H (a) = λ_{1} (a) \dots λ_{n} (a) > 0$ and $tr H (a) = λ_{1} (a) + \dots + λ_{n} (a) \geq λ_{1} (a)$ by [F25] and [F26], and since $0 < λ_{j} (a) \leq tr H (a)$ for every $j$ , one has $λ_{1} (a) \geq \det H (a) / (tr H (a))^{n - 1}$ ; the functions $\det H$ and $tr H$ are continuous on $Ω$ because the entries $S_{j \overset{ˉ}{k}}$ are, so $g : = \det H / (tr H)^{n - 1}$ is a continuous positive function on $Ω$ with $λ_{1} (a) \geq g (a) > 0$ for all $a \in Ω$ .

1.3F10F11

Let $χ$ be the standard smooth step function [F10]; its defining formula is $χ (t) = β (t) / (β (t) + β (1 - t))$ with $β$ the standard flat function, and the flat function vanishes on $(- \infty, 0]$ and is positive on $(0, \infty)$ [F11], so $χ \in C^{\infty} (R)$ with $0 \leq χ \leq 1$ on all of $R$ , $χ = 0$ on $(- \infty, 0]$ and $χ = 1$ on $[1, \infty)$ .

1.4F7F8given

Claim 2 is the instance of [F7] claim 1 with the given strictly plurisubharmonic $φ \in C^{2}$ , the given $f \in Dom {\overset{ˉ}{\partial}}_{q}$ with ${\overset{ˉ}{\partial}}_{q} f = 0$ and $E (f) < + \infty$ : it yields $u \in Dom {\overset{ˉ}{\partial}}_{q - 1}$ with ${\overset{ˉ}{\partial}}_{q - 1} u = f$ and $∥ u ∥_{φ}^{2} \leq E (f)$ , which is exactly the assertion of claim 2.

2.1F2F3step 1.1algebra

Put $S_{1} : = S + (1 - μ)$ . Then $S_{1} \in C^{\infty} (Ω)$ , $S_{1} \geq 1$ on $Ω$ , the complex Hessian of $S_{1}$ equals that of $S$ , and adding the constant changes neither the Levi form nor strict plurisubharmonicity [F3], so $S_{1}$ is strictly plurisubharmonic; every sublevel set ${S_{1} \leq t} = {S \leq t - 1 + μ}$ is closed in $Ω$ and contained in ${S < c_{k}}$ for every $k$ with $c_{k} > t - 1 + μ$ , hence is compactly contained in $Ω$ by [F2].

2.2F10F12step 1.3

Define $κ (t) : = \int_{0}^{t} χ (s) d s$ for $t \in R$ . By [F12] the function $κ$ is a primitive of the continuous function $χ$ on $R$ , hence is differentiable with $κ^{'} = χ$ ; since $χ \in C^{\infty}$ (step 1.3), $κ \in C^{\infty}$ as well.

3.1F18step 2.1

For every integer $j \geq 2$ put $A_{j} : = {j - 1 \leq S_{1} < j}$ . Each $A_{j}$ is a Borel subset of $Ω$ , being the preimage under the continuous $S_{1}$ of a Borel subset of $R$ [F18]; the $A_{j}$ are pairwise disjoint and $⋃_{j \geq 2} A_{j} = Ω$ because $S_{1} \geq 1$ on $Ω$ by step 2.1, so the indicator functions $1_{A_{j}}$ sum pointwise to $1_{Ω}$ .

3.2F12F13F14step 1.3step 2.2

One has $κ (0) = 0$ and $κ = 0$ on $(- \infty, 0]$ because $χ$ vanishes there (step 1.3); $κ \geq 0$ on $[0, \infty)$ because $χ \geq 0$ : for every partition of $[0, t]$ all lower and upper Darboux sums of $χ$ are $\geq$ those of the zero function, so the integral is $\geq 0$ by the definition of the Darboux integral [F13, F14]; and for $t \geq 1$ additivity over subintervals gives $κ (t) = \int_{0}^{1} χ + \int_{1}^{t} χ \geq \int_{1}^{t} χ = \int_{1}^{t} 1 = t - 1$ , the last equality because $u \mapsto u$ is a primitive of the constant function $1$ and $\int_{1}^{t} 1 = [u]_{1}^{t} = t - 1$ by the evaluation clause of [F12], while the first summand is $\geq 0$ .

4.1F15F16F17F19F20step 1.2step 3.1

The function $\tilde{H} : = ∣ η ∣^{2} / g$ is continuous and nonnegative on $Ω$ by step 1.2, and for every $j \geq 2$ one has $\int_{A_{j}} \tilde{H} d V \leq θ_{j}$ for a finite number $θ_{j} \geq 0$ : if $A_{j} = \emptyset$ take $θ_{j} : = 0$ ; otherwise $\overline{A_{j}} \subseteq {S_{1} \leq j}$ is a nonempty compact subset of $Ω$ by step 2.1, so by [F16] and [F15] the continuous function $1 + \tilde{H}$ attains on it a finite maximum $M_{j} : = \max_{\overline{A_{j}}} (1 + \tilde{H})$ , while $λ (\overline{A_{j}}) < + \infty$ because $\overline{A_{j}}$ is bounded and Lebesgue measurable [F15, F17]; then $M_{j} λ (\overline{A_{j}}) < + \infty$ and with $θ_{j} : = M_{j} (1 + λ (\overline{A_{j}}))$ one gets $\int_{A_{j}} \tilde{H} d V \leq \int_{\overline{A_{j}}} (1 + \tilde{H}) d V \leq \int_{\overline{A_{j}}} M_{j} d V = M_{j} λ (\overline{A_{j}}) \leq θ_{j}$ by monotonicity of the nonnegative integral [F19] and the simple-integral formula $\int c χ_{E} d μ = c μ (E)$ [F20].

4.2F12F13F14step 3.2

Define $β (t) : = \int_{0}^{t} κ (s) d s$ . Then $β$ is a primitive of the continuous $κ$ [F12], so $β \in C^{\infty}$ with $β^{'} = κ$ and $β^{''} = κ^{'} = χ$ by step 2.2; $β = 0$ on $(- \infty, 0]$ since $κ$ vanishes there, and $β \geq 0$ on $[0, \infty)$ by monotonicity of the integral and $κ \geq 0$ (step 3.2); finally $β (2) = \int_{0}^{2} κ \geq \int_{3 / 2}^{2} κ \geq \int_{3 / 2}^{2} \frac{1}{2} d s = \frac{1}{2} (2 - \frac{3}{2}) = \frac{1}{4}$ , using $κ (s) \geq s - 1 \geq \frac{1}{2}$ for $s \geq \frac{3}{2}$ (step 3.2), additivity and monotonicity of the integral [F13, F14], and the primitive evaluation [F12].

5.1F28F29F30step 4.1step 4.2

With the numbers $θ_{j} \geq 0$ of step 4.1 define $d_{i} : = i + \log (1 + θ_{i + 1})$ for $i \geq 1$ , $C_{0} : = \max (1, d_{1}, d_{2}) \geq 1$ , $a_{i} : = \max (0, 4 (d_{i + 2} - C_{0} (i + 2))) \geq 0$ for $i \geq 1$ , and $F (t) : = C_{0} t + \sum_{i = 1}^{\infty} a_{i} β (t - i)$ for $t \in R$ . At each $t$ only the finitely many indices with $i \leq t$ contribute a nonzero term because $β (t - i) = 0$ for $t - i \leq 0$ (step 4.2), so near $t$ the function $F$ agrees with a finite sum of $C^{\infty}$ functions and is $C^{\infty}$ by the algebra and locality properties of smooth maps [F28, F29]; differentiating that finite sum termwise by the chain rule [F30] gives $F^{'} (t) = C_{0} + \sum_{i \leq t} a_{i} κ (t - i)$ and $F^{''} (t) = \sum_{i \leq t} a_{i} χ (t - i)$ .

6.1F12F13F14step 1.3step 3.2step 4.2step 5.1

By step 5.1 and steps 1.3, 3.2, 4.2 the coefficients are nonnegative and $κ \geq 0$ , $χ \geq 0$ , so $F^{'} (t) \geq C_{0} \geq 1$ and $F^{''} (t) \geq 0$ for every $t$ ; $F (0) = 0$ ; and since $F$ is a primitive of the continuous $F^{'}$ , the evaluation clause of [F12] gives $F (t) - F (s) = \int_{s}^{t} F^{'} \geq t - s > 0$ for $s < t$ , so $F$ is strictly increasing, and $F (t) \geq F (0) + t = t \geq 0$ for $t \geq 0$ ; moreover for $j \geq 3$ one has $F (j) = C_{0} j + \sum_{i \leq j} a_{i} β (j - i) \geq C_{0} j + a_{j - 2} β (2) \geq C_{0} j + a_{j - 2} / 4 \geq d_{j}$ by step 4.2 and the definition of $a_{j - 2}$ , while $F (1) \geq C_{0} \geq d_{1}$ and $F (2) \geq 2 C_{0} \geq d_{2}$ because $C_{0} \geq d_{1}, d_{2}$ ; hence $F (j) \geq d_{j}$ for every integer $j \geq 1$ .

7.1F4F5F12F27step 2.1step 6.1

Let $i d (t) : = t$ and put $ψ : = (F - i d) \circ S_{1}$ on $Ω$ . On $R$ one has $(F - i d)^{'} = F^{'} - 1 \geq 0$ and $(F - i d)^{''} = F^{''} \geq 0$ by step 6.1, so $F - i d$ is convex by [F27] and nondecreasing because $(F - i d) (t) - (F - i d) (s) = \int_{s}^{t} (F^{'} - 1) \geq 0$ for $s < t$ by [F12] and $F^{'} \geq 1$ ; the function $S_{1}$ is real-valued with $L_{S_{1}} (a; v) > 0$ for $v \neq 0$ by step 2.1, hence $L_{S_{1}} \geq 0$ everywhere on the real vector space $C^{n}$ , and the $C^{2}$ Levi criterion [F4] makes $S_{1}$ plurisubharmonic on $Ω$ ; therefore the composition $ψ = (F - i d) \circ S_{1}$ is plurisubharmonic on $Ω$ by [F5].

8.1F3F4F28F30step 2.1step 7.1

Put $Φ : = F \circ S_{1} = S_{1} + ψ$ . Then $Φ \in C^{\infty} (Ω)$ by the chain rule and the algebra of smooth maps [F28, F30], and for every $a \in Ω$ and $v \in C^{n}$ the Levi form is additive, $L_{Φ} (a; v) = L_{S_{1}} (a; v) + L_{ψ} (a; v) \geq L_{S_{1}} (a; v) > 0$ : the middle inequality holds because $ψ \in C^{2}$ is plurisubharmonic, so $L_{ψ} \geq 0$ by the $C^{2}$ Levi criterion [F4], while $L_{S_{1}} (a; v) > 0$ for $v \neq 0$ by step 2.1 and [F3]; hence $Φ$ is strictly plurisubharmonic on $Ω$ and $Φ \in C^{\infty}$ .

9.1F24step 2.1step 1.2step 8.1algebra

Let $H_{Φ} : = (\partial^{2} Φ / \partial z_{j} \partial {\overset{ˉ}{z}}_{k})$ and let $w_{Φ}$ be the sum of its $q$ smallest eigenvalues. By steps 8.1 and 1.2 and the Rayleigh characterization recorded in step 1.2, $λ_{1} (H_{Φ}) (a) = \min_{∥ v ∥ = 1} L_{Φ} (a; v) \geq \min_{∥ v ∥ = 1} L_{S_{1}} (a; v) = λ_{1} (H_{S_{1}}) (a) = λ_{1} (H) (a) \geq g (a) > 0$ , the last equality because $S_{1}$ and $S$ have the same complex Hessian (step 2.1) and the last inequality by step 1.2; hence $w_{Φ} (a) \geq λ_{1} (H_{Φ}) (a) \geq g (a) > 0$ on $Ω$ .

10.1F18F19F21F22F23step 3.1step 4.1step 5.1step 6.1step 9.1

The function $a \mapsto \tilde{H} (a) e^{- Φ (a)}$ is continuous, hence Borel measurable, on $Ω$ [F18]; since the $A_{j}$ partition $Ω$ (step 3.1), Beppo Levi's theorem [F21] gives $\int_{Ω} \tilde{H} e^{- Φ} d V = \sum_{j \geq 2} \int_{A_{j}} \tilde{H} e^{- Φ} d V$ ; on $A_{j}$ one has $S_{1} \geq j - 1$ , hence $Φ = F (S_{1}) \geq F (j - 1)$ because $F$ is strictly increasing (step 6.1), and $\int_{A_{j}} \tilde{H} d V \leq θ_{j}$ (step 4.1), so the scalar rule and monotonicity [F19] give $\int_{A_{j}} \tilde{H} e^{- Φ} d V \leq θ_{j} e^{- F (j - 1)} \leq θ_{j} (1 + θ_{j})^{- 1} e^{- (j - 1)} \leq e^{- (j - 1)}$ , the middle inequality because $F (j - 1) \geq d_{j - 1} = (j - 1) + \log (1 + θ_{j})$ (steps 5.1 and 6.1); the series $\sum_{j \geq 2} e^{- (j - 1)} = \sum_{k \geq 1} (e^{- 1})^{k}$ converges by the ratio test [F22] since $e^{- 1} < 1$ by strict increase of the exponential [F23]; therefore the energy $E_{Φ} (η) : = \int_{Ω} ∣ η ∣^{2} w_{Φ}^{- 1} e^{- Φ} d V$ of $η$ with respect to $Φ$ satisfies $E_{Φ} (η) \leq \int_{Ω} \tilde{H} e^{- Φ} d V < + \infty$ by step 9.1.

11.1F6F7step 8.1step 10.1

Since $η$ is a smooth $\overset{ˉ}{\partial}$ -closed $(0, q)$ -form and $Φ \in C^{\infty} (Ω; R)$ is strictly plurisubharmonic with $E_{Φ} (η) < + \infty$ (steps 8.1 and 10.1), the $C^{\infty}$ branch of the Hörmander theorem [F7] (claim 2) supplies $ζ \in C^{\infty} (Ω; Λ^{0, q - 1})$ with $\overset{ˉ}{\partial} ζ = η$ ; thus $η \in B_{\overset{ˉ}{\partial}}^{0, q} (Ω)$ , its class in $H_{\overset{ˉ}{\partial}}^{0, q} (Ω) = Z_{\overset{ˉ}{\partial}}^{0, q} (Ω) / B_{\overset{ˉ}{\partial}}^{0, q} (Ω)$ is zero by [F6], and since $η$ was an arbitrary smooth $\overset{ˉ}{\partial}$ -closed $(0, q)$ -form one has $H_{\overset{ˉ}{\partial}}^{0, q} (Ω) = 0$ .

12.1F7F17F18F32step 11.1step 1.4∎

Claim 1 of the statement is proved by steps 10.1 and 11.1, and claim 2 by step 1.4; both are stated under the ambient Axiom of Choice recorded in the Given and cited as [F32], consumed in this proof only through the AC $_{ω}$ instances of the measure-theoretic suppliers [F17] and [F18] and through the AC hypothesis of [F7], and the weight $Φ = F \circ S_{1}$ produced in step 8.1 is smooth and strictly plurisubharmonic with no boundary regularity claimed.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

A strictly pseudoconvex boundary point has a local holomorphic separator

Statement

Assume the Axiom of Choice (AC). Let $D \subseteq C^{n}$ , $n \geq 1$ , be a domain and let $p \in \partial D$ be a boundary point such that $\partial D$ is of class $C^{2}$ near $p$ and strongly pseudoconvex at $p$ . In this item, $z_{j}$ denotes the canonical coordinate $z_{j - 1}$ for $1 \leq j \leq n$ , with the same relabeling for derivatives and form coefficients. There are a neighbourhood $U$ of $p$ and a $C^{2}$ function $ρ : U \to R$ with $D \cap U = {z \in U : ρ (z) < 0}, d ρ (p) \neq 0,$ and $L_{ρ} (p; v) > 0 for every v \neq 0 with \sum_{j = 1}^{n} \frac{\partial ρ}{\partial z_{j}} (p) v_{j} = 0.$

Then there are a neighbourhood $V \subseteq U$ of $p$ and a holomorphic function $h : V \to C$ such that $h (p) = 0 and Re h (z) < 0 for every z \in (D \cap V) ∖ {p} .$

In particular $h$ has no zero on $(D \cap V) ∖ {p}$ . Moreover the separator is quantitative in suitable coordinates: there are a holomorphic chart $w$ centred at $p$ , a radius $δ > 0$ and a constant $c > 0$ such that $h = w_{n}$ in this chart and $Re w_{n} \leq - c ∣ w ∣^{2}$ at every point of $D$ corresponding to $∣ w ∣ < δ$ .

Facts & Assumptions

Given: The Axiom of Choice; a domain $D \subseteq C^{n}$ ; a boundary point $p \in \partial D$ with $C^{2}$ boundary near $p$ ; a $C^{2}$ defining function $ρ$ on a neighbourhood $U$ of $p$ with $D \cap U = {ρ < 0}$ , $d ρ (p) \neq 0$ , and $L_{ρ} (p; v) > 0$ for every nonzero complex tangent vector $v$ at $p$ .

[F1]

For $u \in C^{2}$ open set and $a$ , $v$ as above, the Levi form is $L_{u} (a; v) : = \sum_{j, k} \frac{\partial^{2} u}{\partial z_{j} \partial {\overset{ˉ}{z}}_{k}} (a) v_{j} \overline{v_{k}},$ and $u$ is strictly plurisubharmonic when $L_{u} (a; v) > 0$ for all $a$ and all $v \neq 0$ (The Levi form and strict plurisubharmonicity).

[F2]

With $ρ$ a $C^{2}$ defining function of a domain on a neighbourhood of a boundary point $p$ , the complex tangent vectors at $p$ are those with $\sum_{j} ρ_{z_{j}} (p) v_{j} = 0$ , and the Levi form is evaluated on that subspace (Levi pseudoconvex domains).

[F3]

Two $C^{2}$ defining functions near the same boundary point have Levi forms on complex tangent vectors differing by a positive scalar factor; in particular the strict positivity demanded at $p$ does not depend on the choice of $ρ$ (Levi pseudoconvexity does not depend on the defining function).

[F4]

For a $C^{2}$ scalar field near $a$ , $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} ⟨ H_{f} (a) h, h ⟩ + o (∥ h ∥^{2})$ (Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ ).

[F5]

The Wirtinger operators in several variables satisfy $D f (a) h = \sum_{k} ((\partial_{z_{k}} f (a)) h_{k} + (\partial_{{\overset{ˉ}{z}}_{k}} f (a)) \overline{h_{k}})$ for real totally differentiable $f$ , and $ρ$ real-valued is recovered from its Wirtinger partials by this identity (Wirtinger operators in $C^{m}$ ).

[F6]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC is the ambient hypothesis stated in the lemma. The normalization, the multiplication by the positive function $g$ , the choice of the polynomial $q$ and the final pullback are all explicit formulas, so neither [F6] nor any weaker selection principle is consumed by the construction itself; the cited suppliers are used as stated.

Proof

technique · direct

1.1F2F5F6given

With $U$ and $ρ$ as given, the only rephrasing needed is the description of the complex tangent space: by [F2] the complex tangent vectors at $p$ form the kernel of the $C$ -linear form $ℓ (v) : = \sum_{a} ρ_{z_{a}} (p) v_{a}$ , and $ℓ \neq 0$ because $d ρ (p) \neq 0$ (if all Wirtinger partials of $ρ$ vanished at $p$ , then $D ρ (p) = 0$ by [F5]); relabel the indices so that $ρ_{z_{n}} (p) \neq 0$ , so that $\ker ℓ$ has complex dimension $n - 1$ and the hypothesis of the statement says that $\sum_{a, b} ρ_{z_{a} {\overset{ˉ}{z}}_{b}} (p) v_{a} \overline{v_{b}} > 0$ for every $0 \neq v \in \ker ℓ$ . The Axiom of Choice [F6] is the ambient hypothesis of the statement, and this step selects nothing.

2.1F4F5step 1.1algebra

Expansion of $ρ$ at $p$ in complex notation: applying the second-order Taylor expansion [F4] to the $C^{2}$ function $ρ$ and rewriting its linear and quadratic terms with the differential identity of [F5] (for the linear term $D ρ (p) h = \sum_{a} (ρ_{z_{a}} (p) h_{a} + ρ_{{\overset{ˉ}{z}}_{a}} (p) \overline{h_{a}}) = 2 Re ℓ (h)$ , because $ρ$ is real; for the quadratic term, substituting the real coordinates $ξ_{a} = (h_{a} + \overline{h_{a}}) / 2$ and $η_{a} = (h_{a} - \overline{h_{a}}) / (2 i)$ into the real Hessian form and collecting the $h_{a} h_{b}$ , $h_{a} \overline{h_{b}}$ , $\overline{h_{a}} \overline{h_{b}}$ terms), one obtains with $ℓ (h) : = \sum_{a} ρ_{z_{a}} (p) h_{a}$ , $A (h) : = \sum_{a, b} ρ_{z_{a} z_{b}} (p) h_{a} h_{b}$ and $Q_{0} (h) : = \sum_{a, b} ρ_{z_{a} {\overset{ˉ}{z}}_{b}} (p) h_{a} \overline{h_{b}}$ the expansion $ρ (p + h) = 2 Re ℓ (h) + Re A (h) + Q_{0} (h) + o (∣ h ∣^{2})$ as $h \to 0$ .

3.1F1F2F3step 1.1step 2.1algebra

First normalization: define the holomorphic affine map $Φ$ by $Φ (ζ)_{a} : = p_{a} + ζ_{a}$ for $1 \leq a < n$ and $Φ (ζ)_{n} : = p_{n} + c (ζ_{n} - 2 \sum_{1 \leq a < n} ρ_{z_{a}} (p) ζ_{a})$ with $c : = (2 ρ_{z_{n}} (p))^{- 1}$ , so that $ℓ (Φ (ζ) - p) = ζ_{n} / 2$ and the Jacobian of $Φ$ is triangular with diagonal entries $1$ and $c \neq 0$ ; shrinking $U$ makes $Φ$ a biholomorphism onto a neighbourhood of $0$ , and $ρ_{1} : = ρ \circ Φ$ is a $C^{2}$ defining function of $D_{1} : = Φ^{- 1} (D \cap U)$ near $0$ with expansion $ρ_{1} (ζ) = Re ζ_{n} + Q_{1} (ζ) + Re A_{1} (ζ) + o (∣ ζ ∣^{2})$ from step 2.1, where $Q_{1} (ζ) = \sum_{a, b} ρ_{1, ζ_{a} {\overset{ˉ}{ζ}}_{b}} (0) ζ_{a} \overline{ζ_{b}}$ and $A_{1} (ζ) = \sum_{a, b} ρ_{1, ζ_{a} ζ_{b}} (0) ζ_{a} ζ_{b}$ . The hypothesis survives: $ℓ_{1} (w) : = \sum_{a} ρ_{1, ζ_{a}} (0) w_{a}$ equals $ℓ (L w) = w_{n} / 2$ for the linear part $L$ of $Φ$ , and for $v \neq 0$ with $ℓ_{1} (v) = 0$ the chain rule gives $L v \in \ker ℓ ∖ {0}$ together with $L_{ρ_{1}} (0; v) = L_{ρ} (p; L v) > 0$ by step 1.1.

4.1F1step 3.1algebra

Multiplication by a positive function: for $t > 0$ put $g_{t} (ζ) : = 1 + t Re ζ_{n}$ and $ρ_{t} : = g_{t} ρ_{1}$ , a $C^{2}$ defining function of the same domain near $0$ with $g_{t} (0) = 1$ and $d ρ_{t} (0) = d ρ_{1} (0) \neq 0$ . Writing $ℓ_{1} (ζ) = ζ_{n} / 2$ and $L_{t} : = t ℓ_{1}$ and using the identity $2 Re (X) 2 Re (Y) = 2 Re (X \overline{Y}) + 2 Re (X Y)$ , multiplication of the expansion of step 3.1 by $g_{t} = 1 + 2 Re L_{t}$ gives $ρ_{t} (ζ) = 2 Re [ℓ_{1} (ζ) + \frac{1}{2} A_{1} (ζ) + L_{t} (ζ) ℓ_{1} (ζ)] + [Q_{1} (ζ) + 2 Re (L_{t} (ζ) \overline{ℓ_{1} (ζ)})] + o (∣ ζ ∣^{2})$ , the $O (∣ ζ ∣^{3})$ terms of $2 Re (L_{t}) (Re A_{1} + Q_{1})$ having been absorbed into $o (∣ ζ ∣^{2})$ ; thus the holomorphic quadratic part of $ρ_{t}$ is $B_{t} : = ζ_{n} / 2 + A_{1} / 2 + t ζ_{n}^{2} / 4$ and its Hermitian quadratic part is the Hermitian form $H_{t} (v) : = Q_{1} (v) + (t / 2) ∣ v_{n} ∣^{2}$ in the variable $v$ .

5.1step 1.1step 4.1algebra

$H_{t}$ is positive definite for all large $t$ : the hypothesis of step 1.1 says $Q_{1} (v) > 0$ for $0 \neq v$ with $v_{n} = 0$ , so on the hyperplane $E : = {v_{n} = 0}$ there is $δ > 0$ with $Q_{1} (v) \geq δ ∣ v ∣^{2}$ , while the Hermitian form $Q_{1}$ satisfies $∣ Q_{1} (v, w) ∣ \leq C ∣ v ∣ ∣ w ∣$ for $v, w \in E$ with a constant $C$ independent of $t$ . Decomposing $v = v^{'} + λ e_{n}$ with $v^{'} \in E$ and $λ = v_{n}$ gives $H_{t} (v) = Q_{1} (v^{'}) + 2 Re Q_{1} (v^{'}, λ e_{n}) + ∣ λ ∣^{2} Q_{1} (e_{n}) + (t / 2) ∣ λ ∣^{2} \geq δ ∣ v^{'} ∣^{2} - 2 C ∣ v^{'} ∣ ∣ λ ∣ + (Q_{1} (e_{n}) + t / 2) ∣ λ ∣^{2}$ , and $2 C ∣ v^{'} ∣ ∣ λ ∣ \leq (δ / 2) ∣ v^{'} ∣^{2} + (2 C^{2} / δ) ∣ λ ∣^{2}$ because $(\sqrt{δ / 2} ∣ v^{'} ∣ - \sqrt{2 / δ} C ∣ λ ∣)^{2} \geq 0$ . Choosing $t > 0$ with $Q_{1} (e_{n}) + t / 2 \geq 1 + 2 C^{2} / δ$ therefore gives $H_{t} (v) \geq (δ / 2) ∣ v^{'} ∣^{2} + ∣ λ ∣^{2} \geq c ∣ v ∣^{2}$ for all $v$ , where $c : = \min (δ / 2, 1) > 0$ and $∣ v ∣^{2} = ∣ v^{'} ∣^{2} + ∣ λ ∣^{2}$ ; fix such a $t$ and write $ρ_{t}$ and $H$ for $ρ_{t}$ and $H_{t}$ .

6.1step 4.1step 5.1algebra

Killing the holomorphic quadratic part: let $q (ζ) : = - A_{1} (ζ) - (t / 2) ζ_{n}^{2}$ and define the holomorphic polynomial map $Φ_{2} (w) : = w + q (w) e_{n}$ , which fixes the first $n - 1$ coordinates and sends $w_{n}$ to $w_{n} + q (w)$ ; since $D Φ_{2} (0) = I$ it is a local biholomorphism fixing $0$ , and with $\hat{ρ} : = ρ_{t} \circ Φ_{2}$ one computes for $z = Φ_{2} (w)$ that $ℓ_{1} (w + q (w) e_{n}) = w_{n} / 2 + q (w) / 2$ and $A_{1} (w + q (w) e_{n}) = A_{1} (w) + O (∣ w ∣^{3})$ , hence $2 Re B_{t} (Φ_{2} (w)) = Re w_{n} + Re [q (w) + A_{1} (w) + (t / 2) w_{n}^{2}] + O (∣ w ∣^{3}) = Re w_{n} + O (∣ w ∣^{3})$ , while $H (Φ_{2} (w)) = H (w) + O (∣ w ∣^{3})$ because $H$ is quadratic; therefore $\hat{ρ} (w) = Re w_{n} + H (w) + o (∣ w ∣^{2})$ , and $\hat{ρ}$ is a $C^{2}$ defining function near $0$ of the image of $D$ under the change of coordinates.

7.1step 5.1step 6.1algebra∎

Conclusion: since $\hat{ρ} (w) - Re w_{n} - H (w) = o (∣ w ∣^{2})$ , after shrinking the ball to a radius $δ > 0$ on which $Φ_{2}$ is biholomorphic and $∣ \hat{ρ} (w) - Re w_{n} - H (w) ∣ \leq (c / 2) ∣ w ∣^{2}$ , every $w$ with $∣ w ∣ < δ$ , $w \neq 0$ and $\hat{ρ} (w) < 0$ satisfies $Re w_{n} = \hat{ρ} (w) - H (w) - (\hat{ρ} (w) - Re w_{n} - H (w)) \leq 0 - c ∣ w ∣^{2} + (c / 2) ∣ w ∣^{2} = - (c / 2) ∣ w ∣^{2} < 0$ ; define $V : = Φ (Φ_{2} (B (0, δ))) \subseteq U$ and $h (z) : = w_{n} (z)$ where $w = Φ_{2}^{- 1} (Φ^{- 1} (z))$ is the inverse chart, so that $h$ is holomorphic on $V$ , $h (p) = 0$ , and $Re h < 0$ on $(D \cap V) ∖ {p}$ because those points correspond exactly to the parameters $∣ w ∣ < δ$ , $w \neq 0$ , $\hat{ρ} (w) < 0$ , and the displayed inequality is the quantitative bound $Re w_{n} \leq - (c / 2) ∣ w ∣^{2}$ in the chart $w$ with the constant $c / 2 > 0$ of step 5.1.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Positive smooth collars for strictly plurisubharmonic negative sets

Statement

Assume the Axiom of Choice. Let $D \subset C^{n}$ , $n \geq 1$ , be bounded and open, with nonempty boundary. Suppose $U \supset \overline{D}$ is open, $ρ \in C^{\infty} (U, R)$ , $D = {ρ < 0}$ in $U$ , and $ρ$ is strictly plurisubharmonic near $\partial D$ . No nonvanishing-gradient condition is imposed.

For every open $O$ with $\overline{D} \subset O \subset U$ , there are finitely many pairwise disjoint bounded domains $G_{1}, \dots, G_{N}$ such that $\overline{D} \subset G : = ⋃_{i = 1}^{N} G_{i}, \overline{G} \subset O,$ each $G_{i}$ has $C^{\infty}$ strongly pseudoconvex boundary, and each $G_{i}$ admits a continuous plurisubharmonic exhaustion.

Facts & Assumptions

Given: AC; the data of the Statement; and the prescribed neighborhood $O$ .

[F1]

A fixed smooth Euclidean bump $b$ is nonnegative, equals $1$ on the closed unit ball and has support in the radius-two ball (Explicit compactly supported smooth cutoffs).

[F2]

Uniform convergence of continuously differentiable functions and their derivatives on a closed interval permits termwise differentiation of the limit (If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit).

[F3]

Smooth real-valued maps have dense regular values; a regular level is locally a smooth graph (Regular values have null complement and are dense, A regular level set is locally a $C^{k}$ graph of dimension $m - n$ ).

[F4]

For smooth functions the nonnegative Levi form characterizes plurisubharmonicity; strict positivity is the positive-definite Levi form condition (The C^2 Levi criterion for plurisubharmonicity, The Levi form and strict plurisubharmonicity).

[F5]

Nonnegative sums, finite maxima and convex nondecreasing composition preserve plurisubharmonicity (Basic stability operations for plurisubharmonic functions). An exhaustion has compact sublevel sets in its domain (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

Choice use. AC supplies the choice hypotheses of [F3]. The bump sequence, its coefficients and their sum below are explicit after fixing an enumeration of rational balls; the remaining selections are finite.

Proof

1.1F1F2givenconstruct

Put $F = \overline{D}$ . Enumerate all rational centers $q_{j} \in Q^{2 n}$ and positive rational radii $r_{j}$ for which $\overline{B} (q_{j}, 2 r_{j}) \cap F = \emptyset$ . Their inner balls cover $C^{n} ∖ F$ : openness of the complement supplies a sufficiently small ball, then a rational center and radius. Set $b_{j} (x) = b ((x - q_{j}) / r_{j})$ , $M_{j} = \max_{∣ α ∣ \leq j} \sup_{R^{2 n}} ∣ D^{α} b_{j} ∣, a_{j} = \frac{2^{- j}}{1 + M_{j}}, β = \sum_{j \geq 1} a_{j} b_{j} .$ Each $M_{j}$ is finite since the derivatives have compact support. For every fixed multi-index $α$ , the tail with $j \geq ∣ α ∣$ is bounded termwise by $2^{- j}$ , so the series of derivatives converges uniformly. Apply [F2] on coordinate segments in closed boxes, successively to every derivative: the limit is $C^{\infty}$ with $D^{α} β = \sum_{j} a_{j} D^{α} b_{j}$ . Every derivative vanishes on $F$ , while at any point outside $F$ some $b_{j}$ is $1$ ; hence $β \geq 0$ and $β^{- 1} (0) = F$ . This argument proves smoothness across $F$ , without assuming local finiteness of the bumps there.

2.1F4step 1.1given

Choose a compact neighborhood of $\partial D$ contained in the strict Levi collar of $ρ$ and in $U$ . Compactness of this neighborhood times the unit sphere gives a uniform positive lower Levi bound for $ρ$ and a finite upper absolute Levi bound for $β$ . Thus for some $t > 0$ , $ψ : = ρ + t β$ is strictly plurisubharmonic on an open neighborhood $V$ of $\partial D$ . On $F$ it agrees with $ρ$ , so it is negative on $D$ and zero on $\partial D$ ; on $U ∖ F$ both $ρ \geq 0$ and $β > 0$ , so $ψ > 0$ . In particular ${ψ < 0} = D$ and the zero set of $ψ$ in $U$ is exactly $\partial D$ .

3.1F3F4step 2.1

Choose a bounded open $W$ with $F \subset W$ and $\overline{W} \subset U$ . Then $\partial W$ is compact and disjoint from $F$ , so $\min_{\partial W} ψ > 0$ . On the compact set $\overline{W} ∖ O$ the function $ψ$ is positive whenever that set is nonempty. On $\overline{W} ∖ V$ the compact subset where $ψ \geq 0$ also has a positive minimum if nonempty, since its zero set would lie in $\partial D \subset V$ . Choose a positive regular value $ε$ smaller than all these positive minima, using [F3]. Then $S = {x \in W : ψ (x) < ε}$ contains $F$ , has $\overline{S} \subset O \cap W$ , and its boundary lies in $V \cap {ψ = ε}$ . Regular-level charts show that $\partial S$ is smooth and that the inside half of each such chart is connected. Consequently every connected component of $S$ has smooth boundary locally defined by $ψ - ε$ , with strictly positive tangential Levi form by step 2.1.

4.1F4F5step 3.1given

Components of the open set $S$ are open and cover the compact set $F$ , so finitely many distinct components $G_{i}$ cover $F$ . Their union $G$ satisfies the required compact containment. In each $G_{i}$ choose an open neighborhood $N_{i}$ of $\partial G_{i}$ with $\overline{N_{i}} \subset V$ . Put $f = - \log (ε - ψ)$ on $G_{i}$ . It is smooth and plurisubharmonic near $\partial G_{i}$ by [F5], and tends to $+ \infty$ there. The set $\overline{G_{i}} ∖ N_{i}$ is a compact subset of $G_{i}$ , so choose $A_{i}$ greater than its maximum of $f$ . The function $E_{i} = \max {f, A_{i}} + ∣ z ∣^{2}$ is continuous and plurisubharmonic: near the boundary both terms in the maximum are psh; near every point outside $N_{i}$ the maximum is the constant $A_{i}$ ; these descriptions agree on their overlap. Its sublevels are closed in $G_{i}$ and stay away from the boundary, hence are compact in the bounded $G_{i}$ . Thus it is an exhaustion.

5.1step 2.1step 3.1step 4.1∎

The domains $G_{i}$ constructed in steps 3.1–4.1 have all the properties in the Statement, including when the original boundary has critical points or the original $ρ$ has zeros outside $\overline{D}$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Smooth global defining functions for strongly pseudoconvex boundaries

Statement

Assume the Axiom of Choice. Let $D \subset C^{n}$ , $n \geq 1$ , be a bounded domain with $C^{\infty}$ boundary, strongly pseudoconvex at every boundary point. Then there are a neighborhood $U$ of $\overline{D}$ and $ρ \in C^{\infty} (U, R)$ with $D = {ρ < 0}$ in $U$ , $d ρ \neq 0$ on $\partial D$ , and $ρ$ strictly plurisubharmonic near $\partial D$ .

Facts & Assumptions

Given: AC; $D$ and its local smooth strongly pseudoconvex boundary data.

[F1]

A local defining function $r_{i}$ is smooth, defines the negative side $D$ , has nonzero differential on the boundary, and has positive Levi form on every nonzero complex tangent vector (Levi pseudoconvex domains).

[F2]

For compact $K$ in an open Euclidean set $V$ there is a smooth cutoff in $[0, 1]$ , equal to $1$ near $K$ and compactly supported in $V$ (Test function cutoffs and euclidean localization).

[F3]

Strict plurisubharmonicity is positive definiteness of the Levi form (The Levi form and strict plurisubharmonicity).

Choice use. AC licenses the stated ambient hypotheses; the compact-boundary cover and cutoffs use finitely many selections.

Proof

1.1F1F2givenconstruct

Compactness of $\partial D$ supplies finitely many local defining charts and smaller relatively compact neighborhoods covering $\partial D$ . Shrink them so each local differential stays nonzero on the boundary in its chart and each tangential Levi form stays positive there. By [F2] take nonnegative smooth bumps $b_{i}$ supported in the charts and equal to $1$ on the smaller neighborhoods. On a neighborhood $T$ of $\partial D$ where $B = \sum_{i} b_{i} > 0$ , put $λ_{i} = b_{i} / B$ and $r_{0} = \sum_{i} λ_{i} r_{i}$ , extending each supported product by zero outside its chart. This is smooth and has the same negative, zero and positive sides as the local defining functions. At a boundary point, all active differentials are positive multiples of one outward conormal: they annihilate the common real tangent hyperplane and evaluate positively on an outward vector. Thus $d r_{0} = \sum_{i} λ_{i} d r_{i} \neq 0$ .

2.1F1F2step 1.1algebra

If $v$ is complex tangent at a boundary point, then $\partial r_{i} (v) = 0$ for all active charts and $r_{i} = 0$ . The product rule therefore gives $L_{r_{0}} (v) = \sum_{i} λ_{i} L_{r_{i}} (v) > 0 (v \neq 0) .$ Terms involving derivatives of the weights vanish because they contain either $r_{i}$ or a tangential first derivative of $r_{i}$ . By [F2] choose $η \in C_{c}^{\infty} (T, [0, 1])$ equal to $1$ near $\partial D$ . Define $s = - 1$ on $D$ and $s = 1$ on $C^{n} ∖ \overline{D}$ , and define $r = η r_{0} + (1 - η) s$ off $\partial D$ , with $r = r_{0} = 0$ on the boundary. Here $η r_{0}$ is extended by zero off $T$ , and $(1 - η) s$ is zero near the boundary. Hence $r$ is globally smooth, negative exactly on $D$ , positive outside $\overline{D}$ , and agrees with $r_{0}$ near the boundary.

3.1F3step 1.1step 2.1algebra

On the compact boundary put $a = \partial r$ . Its norm has a positive lower bound $m$ , and the operator norm of the Levi matrix of $r$ has a finite bound $M$ . For $n > 1$ the tangential Levi form has a uniform positive bound $λ$ . Write any $v = t + w$ with $t \in \ker a$ and $w ⊥ \ker a$ . Then $∣ a (v) ∣ = ∣ a ∣ ∣ w ∣ \geq m ∣ w ∣$ and $L_{r} (v) \geq λ ∣ t ∣^{2} - 2 M ∣ t ∣ ∣ w ∣ - M ∣ w ∣^{2} \geq \frac{λ}{2} ∣ t ∣^{2} - (M + \frac{2 M^{2}}{λ}) ∣ w ∣^{2} .$ Choose $C$ with $C m^{2} > M + 2 M^{2} / λ$ . For $n = 1$ the tangent space is zero and one instead chooses $C m^{2} > M$ . In either case $L_{r} (v) + C ∣ a (v) ∣^{2} > 0$ for every nonzero $v$ on the boundary.

4.1F3step 2.1step 3.1algebra∎

Set $ρ = e^{C r} - 1$ . The chain rule gives $L_{ρ} (v) = C e^{C r} (L_{r} (v) + C ∣ \partial r (v) ∣^{2}) .$ Step 3.1 and compactness give strict positivity on a neighborhood of $\partial D$ . The function is $C^{\infty}$ because the constructed $r$ is $C^{\infty}$ ; its negative set is exactly $D$ and $d ρ = C d r \neq 0$ on the boundary. Restricting to any neighborhood $U$ of $\overline{D}$ gives the Statement.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

A smooth psh exhaustion gives Hartogs pseudoconvexity on bounded domains

Statement

Assume the Axiom of Choice. Let $Ω \subset C^{n}$ , $n \geq 1$ , be a bounded domain with a smooth plurisubharmonic exhaustion $S$ . Then $Ω$ is Hartogs pseudoconvex: $- \log δ_{Ω}$ is plurisubharmonic, where $δ_{Ω}$ is the equal-radius polydisc boundary function.

Facts & Assumptions

Given: AC; the bounded domain $Ω$ ; and $S \in C^{\infty} (Ω, R)$ plurisubharmonic, with compact sublevels in $Ω$ .

[F1]

Holomorphic pullback preserves plurisubharmonicity for a $C^{2}$ psh function (Holomorphic pullbacks of $C^{2}$ plurisubharmonic functions are plurisubharmonic). Psh is subharmonicity on affine complex lines (Plurisubharmonic functions).

[F2]

An upper semicontinuous, finite function on a plane domain is subharmonic if it satisfies harmonic comparison on every compactly contained closed disc (Subharmonicity is equivalent to harmonic comparison on compactly contained discs).

[F3]

On a disc a harmonic function is the real part of a holomorphic function, since a disc is homologically simply connected (Harmonic conjugates exist on homologically simply connected plane domains).

[F4]

A subharmonic function attaining a finite interior maximum is constant (A plane subharmonic function with an interior maximum is constant on its component); the submean convention is that of Subharmonic functions on plane domains.

[F5]

The equal-radius polydisc radius is the distance to the complement in the coordinate sup norm (The equal-radius polydisc boundary function), and its negative logarithm being psh is Hartogs pseudoconvexity (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

Choice use. AC is the ambient hypothesis. The argument makes only finitely many selections for each disc, direction and harmonic majorant.

Proof

1.1givenconstruct

For each fixed nonzero $ξ \in C^{n}$ define $d_{ξ} (z) = \sup {r > 0 : z + {t : ∣ t ∣ < r} ξ \subset Ω}, u_{ξ} = - \log d_{ξ} .$ These radii are positive and finite because $Ω$ is open and bounded. If $r < d_{ξ} (z)$ , the closed directional disc of radius $r$ is compact in $Ω$ , and sufficiently small translations stay in $Ω$ . Thus $d_{ξ}$ is lower semicontinuous, so $u_{ξ}$ is upper semicontinuous.

2.1F3step 1.1givenconstruct

Fix an affine base disc $z (ζ) = a + ζ v$ with $z (\overline{D}) \subset Ω$ , and a continuous real harmonic majorant $h$ on the closed unit disc with $h \geq u_{ξ} \circ z$ on its boundary. For $0 < s < 1$ the function $h_{s} (ζ) = h (s ζ)$ is harmonic on a disc of radius greater than $1$ , and $h_{s} \to h$ uniformly on the closed unit disc. Given $η > 0$ , choose $s$ so that $∣ h_{s} - h ∣ < η$ there; by [F3] choose a holomorphic $H$ on a disc of radius greater than $1$ with $Re H = h_{s} + η$ . Then $Φ (ζ, t) = z (ζ) + t e^{- H (ζ)} ξ$ is holomorphic near the closed base disc, and for $∣ ζ ∣ = 1$ , $∣ t ∣ < 1$ , its value belongs to $Ω$ , since $e^{- Re H} \leq d_{ξ} (z (ζ))$ . Compactness of the base disc also puts all its images in $Ω$ for sufficiently small $∣ t ∣$ .

3.1F1F4step 2.1givenassume-contradischarge-contradiction

Let $r_{*}$ be the supremum of radii $r \leq 1$ for which $Φ (\overline{D} \times {t : ∣ t ∣ < r}) \subset Ω$ . Suppose $r_{*} < 1$ , and fix $r_{*} < b < 1$ . The image of $\partial D \times {∣ t ∣ \leq b}$ is a compact subset of $Ω$ by step 2.1. Let $M$ be the maximum of $S$ on this image. For each $∣ t ∣ < r_{*}$ , [F1] makes $S \circ Φ (\cdot, t)$ subharmonic, continuous on the closed base disc; its boundary values are at most $M$ , so [F4] bounds it everywhere by $M$ . All these images therefore lie in the fixed compact sublevel $K = {S \leq M} \subset Ω$ . By continuity their limits with $∣ t ∣ \leq r_{*}$ also lie in $K$ . Uniform continuity on a slightly larger compact product then increases the admissible radius beyond $r_{*}$ , contradicting its definition. Thus $r_{*} = 1$ , and $d_{ξ} (z (ζ)) \geq e^{- h_{s} (ζ) - η}$ throughout the base disc.

4.1F1F2step 1.1step 2.1step 3.1

Step 3.1 gives $u_{ξ} \circ z \leq h_{s} + η$ . Choose $s \to 1$ and $η \to 0$ with the stated uniform error, to conclude $u_{ξ} \circ z \leq h$ . The affine-disc normalization covers every closed disc in every complex line in $Ω$ . Hence [F2], together with the upper semicontinuity of step 1.1, makes each $u_{ξ}$ plurisubharmonic. The dilation of the majorant in step 2.1 ensures that $H$ and $Φ$ are defined past the base boundary; no boundary continuity of an arbitrary harmonic conjugate is assumed.

5.1F1F4F5step 4.1∎

Write $∥ ξ ∥_{\infty} = \max_{j < n} ∣ ξ_{j} ∣$ . A sup-norm polydisc of radius $r$ consists exactly of all directional discs of radius $r$ with $∥ ξ ∥_{\infty} = 1$ , so $δ_{Ω} (z) = \inf_{∥ ξ ∥_{\infty} = 1} d_{ξ} (z), u (z) : = - \log δ_{Ω} (z) = \sup_{∥ ξ ∥_{\infty} = 1} u_{ξ} (z) .$ By [F5], $δ_{Ω}$ is a positive continuous distance function on $Ω$ , so $u$ is continuous. On any compactly contained affine circle, each $u_{ξ}$ satisfies its submean inequality and is at most $u$ on the circle. Therefore $u_{ξ}$ at the center is at most the circle average of $u$ ; taking the supremum gives that same bound for $u$ at the center. Its continuity and these submean inequalities make it psh by [F1] and [F4]. This proves the exact Hartogs convention of [F5].

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Peak functions at strongly pseudoconvex boundary points, by a dbar correction

Statement

Assume the Axiom of Choice (AC). Let $n \geq 1$ , let $D \subseteq C^{n}$ be a bounded open set and let $p \in \partial D$ . Suppose that there are a neighbourhood $U$ of $\overline{D}$ and a function $ρ \in C^{\infty} (U, R)$ such that $ρ (p) = 0, D = {z \in U : ρ (z) < 0},$ and such that $ρ$ is strictly plurisubharmonic on some neighbourhood of $\partial D$ (The Levi form and strict plurisubharmonicity).

Then there is a function $h$ , holomorphic on a neighbourhood of $\overline{D}$ , with $h (p) = 1 and ∣ h (z) ∣ < 1 for every z \in D ∖ {p} .$
(Strongly pseudoconvex boundaries.) The same conclusion holds when $D$ is a bounded domain whose boundary is of class $C^{\infty}$ and strongly pseudoconvex at every point, that is: for every $q \in \partial D$ there are a neighbourhood $V$ of $q$ and $ρ \in C^{\infty} (V, R)$ with $D \cap V = {ρ < 0}$ , $d ρ (q) \neq 0$ and $L_{ρ} (q; v) > 0$ for every nonzero complex tangent vector $v$ at $q$ (Levi pseudoconvex domains); namely, there is then $h$ holomorphic on a neighbourhood of $\overline{D}$ with $h (p) = 1$ and $∣ h ∣ < 1$ on $D ∖ {p}$ .

Facts & Assumptions

Given: AC; $n \geq 1$ ; a bounded open $D \subset C^{n}$ ; $p \in \partial D$ ; and the smooth negative-set defining data $(U, ρ)$ in branch 1. Branch 2 is reduced to this data in step 7.1. Coordinates are canonical $z_{0}, \dots, z_{n - 1}$ .

[F1]

Real second-order Taylor expansion, rewritten using Wirtinger derivatives, separates the real part of a holomorphic linear/quadratic polynomial from the Hermitian Levi quadratic form. Strict psh means the latter is positive definite (Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ , Wirtinger operators in $C^{m}$ , The Levi form and strict plurisubharmonicity).

[F2]

The negative set of smooth data strictly psh near its boundary has arbitrarily small outer neighborhoods consisting of finitely many bounded smooth strongly pseudoconvex domains, each with a continuous psh exhaustion; critical boundary points are allowed (Positive smooth collars for strictly plurisubharmonic negative sets).

[F3]

Smooth strongly pseudoconvex boundary data admit a global smooth defining function strictly psh near the boundary (Smooth global defining functions for strongly pseudoconvex boundaries), for the boundary convention of Levi pseudoconvex domains.

[F4]

Under AC and countable choice a continuous psh exhaustion has a smooth strictly psh exhaustive majorant (Smooth strict plurisubharmonic regularization of a psh exhaustion). On a bounded domain a smooth psh exhaustion implies Hartogs pseudoconvexity with the equal-radius polydisc convention (A smooth psh exhaustion gives Hartogs pseudoconvexity on bounded domains).

[F5]

On a Hartogs pseudoconvex domain, with smooth strictly psh weight $φ$ , every smooth closed $(0, 1)$ -form of finite weighted energy has a smooth scalar solution $v$ of $\overset{ˉ}{\partial} v = α$ (Hörmander's weighted L2 existence theorem for the dbar equation, Statement, smooth-data branch).

[F6]

There is a smooth cutoff in $[0, 1]$ , equal to $1$ on a smaller closed ball and supported in a larger open ball (A smooth bump between concentric Euclidean balls).

[F7]

Smooth forms satisfy ${\overset{ˉ}{\partial}}^{2} = 0$ (The d, partial and dbar identities). A smooth function with all $\overset{ˉ}{z}$ derivatives zero is holomorphic (For $C^{1}$ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree); polynomial algebra and reciprocals of nonzero holomorphic functions are holomorphic (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

[F8]

The complex exponential satisfies $∣ e^{w} ∣ = e^{Re w}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F9]

AC implies countable choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), AC implies DC implies countable choice).

Choice use. AC is inherited by the collar, regularization and Hörmander interfaces; it supplies countable choice for [F4]. Only finitely many outer components and corrections are selected. The polynomial, cutoff, corrected quotient and exponential below are explicit once those data are fixed.

Proof

1.1F1F7givenconstruct

Put $w = z - p$ and define the holomorphic Levi polynomial $f (z) = 2 \sum_{j < n} ρ_{z_{j}} (p) w_{j} + \sum_{j, k < n} ρ_{z_{j} z_{k}} (p) w_{j} w_{k} .$ By [F1], $ρ (p + w) = Re f (p + w) + \sum_{j, k < n} ρ_{z_{j} {\overset{ˉ}{z}}_{k}} (p) w_{j} {\overset{ˉ}{w}}_{k} + o (∣ w ∣^{2}) .$ The Hermitian form is bounded below by $λ ∣ w ∣^{2}$ for some $λ > 0$ . Choose $δ > 0$ with $B (p, δ) \subset U$ and remainder at most $λ ∣ w ∣^{2} / 2$ . Since $ρ \leq 0$ on $\overline{D}$ , $Re f (z) \leq - \frac{λ}{2} ∣ z - p ∣^{2} (z \in \overline{D} \cap B (p, δ)) .$ Thus $f (p) = 0$ and $f$ is zero-free on this part of $\overline{D} ∖ {p}$ . The argument includes $d ρ (p) = 0$ : the linear term then vanishes and the same quadratic estimate applies.

2.1F2F6F7step 1.1

Fix $0 < r < R < δ$ . By [F6] take $χ \in C_{c}^{\infty} (B (p, R), [0, 1])$ equal to $1$ on a neighborhood of $\overline{B} (p, r)$ . Its derivative support lies in the compact annulus $A = {r \leq ∣ z - p ∣ \leq R}$ . The compact set $Z = A \cap {f = 0}$ is disjoint from $\overline{D}$ by step 1.1. Apply [F2] with the prescribed open neighborhood $O = U ∖ Z$ to get $G = ⋃_{i = 1}^{N} G_{i} \supset \overline{D}$ , with $\overline{G} \subset O$ . Hence $f$ is bounded away from zero on $A \cap \overline{G}$ when this set is nonempty. On $G$ define $α = (\overset{ˉ}{\partial} χ) / f$ in the annulus and zero outside it. More precisely, use the quotient on the open zero-free neighborhood of $supp (\overset{ˉ}{\partial} χ) \cap \overline{G}$ and zero wherever $χ$ is locally constant. These definitions agree, so $α$ is smooth, bounded on $G$ , and $\overset{ˉ}{\partial} α = 0$ by [F7]. It vanishes near $p$ and satisfies $f α = \overset{ˉ}{\partial} χ$ everywhere on $G$ . No sublevel of $∣ f ∣$ is asserted to lie in a ball.

3.1F2F4F5F7F9step 2.1

Each $G_{i}$ has a continuous psh exhaustion by [F2]. Using [F9], apply [F4] to regularize it and then conclude that $G_{i}$ is Hartogs pseudoconvex in the actual polydisc-radius convention. Take $φ (z) = ∣ z ∣^{2}$ : its Levi eigenvalues are all $1$ . The energy $\int_{G_{i}} ∣ α ∣^{2} e^{- ∣ z ∣^{2}}$ is finite because $G_{i}$ is bounded and $α$ is bounded. Thus [F5] gives a smooth scalar $v_{i}$ on $G_{i}$ with $\overset{ˉ}{\partial} v_{i} = α$ . Define $v = v_{i}$ on each of the finitely many disjoint components. Then $v \in C^{\infty} (G)$ solves $\overset{ˉ}{\partial} v = α$ , is holomorphic near $p$ , and is bounded on the compact set $\overline{D} \subset G$ . The data need not have compact support in each $G_{i}$ : boundedness on the bounded domain proves the required finite energy.

4.1F7step 2.1step 3.1construct

Choose $c > 1 + \max_{\overline{D}} ∣ v ∣$ and put $Q = (c + v) f - χ$ on $G$ . Since $f α = \overset{ˉ}{\partial} χ$ , $\overset{ˉ}{\partial} Q = f \overset{ˉ}{\partial} v - \overset{ˉ}{\partial} χ = 0,$ so $Q$ is holomorphic by [F7]. Where $χ = 0$ in a neighborhood, $v$ is holomorphic and the expression $g = 1 / (c + v)$ is holomorphic wherever $c + v \neq 0$ . Where $Q \neq 0$ , the expression $g = f / Q$ is holomorphic. The two expressions agree on their common domain where $χ$ is locally zero: $Q = (c + v) f$ and $Q \neq 0$ there forces $f \neq 0$ . They therefore glue on the union of these open sets.

5.1F7step 1.1step 3.1step 4.1algebra

This union contains $\overline{D}$ . At a point of $\overline{D}$ outside $supp χ$ , $χ$ is locally zero and $Re (c + v) > 0$ . At a point $z \in \overline{D} \cap supp χ$ other than $p$ , step 1.1 gives $f (z) \neq 0$ and $Re (1 / f (z)) < 0$ , whence $Re (c + v (z) - \frac{χ (z)}{f (z)}) \geq c - ∣ v (z) ∣ > 0.$ Thus $Q (z) \neq 0$ . At $p$ one has $Q (p) = - 1$ , so $f / Q$ is holomorphic on a full neighborhood of $p$ and $g (p) = 0$ . Every point of $D$ has $Re g > 0$ : where $χ = 0$ this follows from $g = 1 / (c + v)$ , and where $χ \neq 0$ from the same displayed inequality and $g = 1 / (c + v - χ / f)$ . Therefore $g$ is holomorphic on an open neighborhood of $\overline{D}$ , vanishes at $p$ , and has positive real part on $D$ .

6.1F7F8step 5.1

Set $h = e^{- g}$ on this neighborhood. It is holomorphic, $h (p) = 1$ , and $∣ h (z) ∣ = e^{- Re g (z)} < 1$ for every $z \in D$ . This proves branch 1 under exactly its stated smooth negative-set hypotheses, including critical boundary points and disconnected $D$ .

7.1F3step 6.1given

Under the smooth strongly pseudoconvex boundary hypotheses of branch 2, [F3] constructs a $C^{\infty}$ defining function on a neighborhood of $\overline{D}$ , strictly psh near $\partial D$ . Its proof glues the given smooth local defining functions with a finite partition near the compact boundary, extends with a sign-constant interior/exterior term, and applies $e^{C r} - 1$ after a tangent/normal Levi estimate. Thus it supplies the smooth data required by branch 1 without upgrading a merely $C^{2}$ function. Step 6.1 now gives the same $h$ for branch 2.

8.1F9step 6.1step 7.1∎

Both branches of the Statement hold with all their original hypotheses, under the ambient AC.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

Oka-Weil approximation on a domain of holomorphy (host-domain lemma)

Statement

Assume the Axiom of Choice (AC). Let $D \subseteq C^{n}$ be a domain of holomorphy (Holomorphic extension and domains of holomorphy in several variables), let $K ⋐ D$ be compact and convex with respect to the holomorphic functions on $D$ , i.e. ${\hat{K}}_{D} = K$ (Holomorphic hulls and holomorphic convexity), and let $f$ be holomorphic in an open neighbourhood of $K$ . Then for every $ε > 0$ there is $F \in O (D)$ with $\sup_{z \in K} ∣ F (z) - f (z) ∣ < ε .$

Facts & Assumptions

Given: The Axiom of Choice; a domain $D \subseteq C^{n}$ of holomorphy; a compact $K ⋐ D$ with ${\hat{K}}_{D} = K$ ; a function $f$ holomorphic on an open neighbourhood of $K$ ; and $ε > 0$ .

[F1]

The holomorphic hull is ${\hat{E}}_{D} : = {a \in D : ∣ g (a) ∣ \leq \sup_{z \in E} ∣ g (z) ∣ for every g \in O (D)}$ (Holomorphic hulls and holomorphic convexity). Thus ${\hat{K}}_{D} = K$ is exactly the compact $O (D)$ -convexity hypothesis.

[F2]

(Harold P. Boas, Lecture Notes on Multidimensional Complex Analysis, §3.3.2, Theorem 21, printed p. 79, with proof on printed pp. 79–80.) If $G$ is a domain of holomorphy in $C^{n}$ and $K_{0} ⋐ G$ is compact and convex with respect to $O (G)$ , then every function holomorphic in a neighbourhood of $K_{0}$ is uniformly approximable on $K_{0}$ by functions in $O (G)$ . The proof uses finite analytic-polyhedron reduction, Oka's graph lift and a $\overset{ˉ}{\partial}$ correction, a power-series approximation, and a telescoping exhaustion.

[F3]

The Axiom of Choice supplies a choice function for every family of nonempty sets (The Axiom of Choice).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F3]. No additional choice is made in applying [F2].

Proof

technique · direct application of the cited approximation theorem

1.1F1F2given

If $K = \emptyset$ , take $F = 0$ , since the supremum of the nonnegative empty family is $0$ in the convention of [F1]. Otherwise the hypotheses of Boas's Theorem 21 [F2] hold with $G : = D$ and $K_{0} : = K$ : $D$ is a domain of holomorphy, and ${\hat{K}}_{D} = K$ is the required $O (D)$ -convexity condition by [F1]. The given $f$ is holomorphic in a neighbourhood of $K$ .

2.1F2F3step 1.1given∎

For nonempty $K$ , apply [F2] with approximation tolerance $ε$ . It gives $F \in O (D)$ with $\sup_{z \in K} ∣ F (z) - f (z) ∣ < ε$ , which is the Statement under the ambient AC assumption [F3].

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

The Levi problem: pseudoconvexity, domains of holomorphy, and holomorphic convexity

Statement

Assume the Axiom of Choice (AC). Let $Ω \subseteq C^{n}$ be a domain, $n \geq 1$ . Then the following three conditions are equivalent:

$Ω$ is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity);
$Ω$ is a domain of holomorphy (Holomorphic extension and domains of holomorphy in several variables);
$Ω$ is holomorphically convex, that is, ${\hat{K}}_{Ω} ⋐ Ω$ for every compact $K ⋐ Ω$ (Holomorphic hulls and holomorphic convexity).

Facts & Assumptions

Given: The Axiom of Choice and a domain $Ω \subseteq C^{n}$ , $n \geq 1$ .

[F1]

Hartogs pseudoconvexity gives a continuous plurisubharmonic exhaustion on $Ω$ (Hartogs pseudoconvexity yields a continuous plurisubharmonic exhaustion).

[F2]

In Demailly, Complex Analytic and Differential Geometry, Ch. I §7.A, Theorem 7.2(c) implies (e): a plurisubharmonic exhaustion on a proper open subset of $C^{n}$ makes $- \log d (z, C^{n} ∖ Ω)$ plurisubharmonic, which is his pseudoconvexity criterion. Ch. VIII §9, Theorem 9.11(a), printed pp. 392–393, says that an open subset of $C^{n}$ is a domain of holomorphy if and only if it is pseudoconvex. Its proof of the forward implication used here applies Skoda's Theorem 9.10 to the coordinate functions $z_{j} - a_{j}$ at a boundary point $a$ with a plurisubharmonic distance weight; the resulting identity $\sum_{j} (z_{j} - a_{j}) h_{j} = 1$ prevents common holomorphic continuation across $a$ .

[F3]

For a domain in $C^{n}$ , being a domain of holomorphy is equivalent to holomorphic convexity (Cartan-Thullen theorem).

[F4]

Every domain of holomorphy in $C^{n}$ is Hartogs pseudoconvex (Domains of holomorphy are Hartogs pseudoconvex).

[F5]

The holomorphic hull of a compact set is closed in its ambient domain and bounded in each coordinate (Basic properties of the holomorphic hull). A closed bounded subset of $C^{n} ≅ R^{2 n}$ is compact (Complex $m$ -space and its real coordinate dictionary, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F6]

AC supplies the ambient choice assumptions of the source theorem and the cited library interfaces (The Axiom of Choice).

Choice use. AC is the stated ambient hypothesis. The proof itself makes no new arbitrary selection; the nontrivial existence theorem imported in [F2] is used under its classical choice setting.

Proof

technique · direct application of the cited Levi theorem and Cartan–Thullen

1.1F1F2F6given

Suppose $Ω$ is Hartogs pseudoconvex and proper in $C^{n}$ . By [F1] it has a continuous plurisubharmonic exhaustion. Demailly's pseudoconvexity equivalence in [F2] makes it pseudoconvex in his sense; his Levi theorem in [F2] then gives that $Ω$ is a domain of holomorphy.

1.2F3F5

If $Ω = C^{n}$ , then for every compact $K \subseteq Ω$ its holomorphic hull is closed in $C^{n}$ and coordinate-bounded by [F5], hence compact. Thus $C^{n}$ is holomorphically convex and therefore a domain of holomorphy by [F3]. This covers the whole-space convention separately.

2.1F3F4step 1.1step 1.2∎

By [F3], the domain-of-holomorphy conclusion of steps 1.1–1.2 is equivalent to holomorphic convexity. Conversely, either of those conditions gives Hartogs pseudoconvexity by [F4]. Hence all three conditions in the Statement are equivalent.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Behnke-Stein: increasing unions of pseudoconvex domains

Statement

Assume the Axiom of Choice (AC). Let $Ω_{1} \subseteq Ω_{2} \subseteq \dots$ be an increasing sequence of Hartogs pseudoconvex domains in $C^{n}$ , $n \geq 1$ , whose union $Ω : = ⋃_{j \geq 1} Ω_{j}$ is a domain. Then $Ω$ is Hartogs pseudoconvex: when $Ω = C^{n}$ this is the whole-space convention, and otherwise, for every $J \geq 1$ , the decreasing tail $(- \log δ_{Ω_{j}} ∣_{Ω_{J}})_{j \geq J}$ consists of plurisubharmonic functions on $Ω_{J}$ and converges pointwise there to $- \log δ_{Ω} ∣_{Ω_{J}}$ .

Facts & Assumptions

Given: The Axiom of Choice; an increasing sequence of domains $Ω_{1} \subseteq Ω_{2} \subseteq \dots$ in $C^{n}$ , $n \geq 1$ , each Hartogs pseudoconvex, with union $Ω = ⋃_{j \geq 1} Ω_{j}$ a domain.

[F1]

A domain $Ω$ is Hartogs pseudoconvex when the function $z \mapsto - \log δ_{Ω} (z)$ is plurisubharmonic on $Ω$ , where $δ_{Ω}$ is the equal-radius polydisc boundary function (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F2]

When $Ω = C^{m}$ one has $δ_{Ω} \equiv + \infty$ and the boundary function is by convention the constant function $0$ ; thus the whole space is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F3]

The equal-radius polydisc boundary function is $δ_{Ω} (a) : = \sup {r > 0 : Δ_{r} (a) \subseteq Ω} \in (0, + \infty],$ for $a \in Ω$ , where $Δ_{r} (a)$ is the open polydisc of constant polyradius $r$ (The equal-radius polydisc boundary function).

[F4]

The closed polydisc is ${\overline{Δ}}_{r} (a) : = {z : ∣ z_{k} - a_{k} ∣ \leq r_{k} for every k < m},$ and the open polydisc $Δ_{r} (a)$ is defined by the strict inequalities $∣ z_{k} - a_{k} ∣ < r_{k}$ (Balls, polydiscs and the distinguished boundary in $C^{m}$ ).

[F5]

A subset of $R^{N}$ is compact if and only if it is closed and bounded (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F6]

Under the identification of $C^{m}$ with $R^{2 m}$ the metric, the balls, the open sets, the convergent sequences and the continuous maps of $C^{m}$ are verbatim those of $R^{2 m}$ (Complex $m$ -space and its real coordinate dictionary).

[F7]

If $u_{1} \geq u_{2} \geq \dots$ is a decreasing sequence of plurisubharmonic functions on a domain and $u = \lim_{n} u_{n}$ pointwise, then either $u \equiv - \infty$ on a connected component, or $u$ is plurisubharmonic (Decreasing limits of plurisubharmonic functions).

[F8]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC is the ambient hypothesis recorded in the Statement. The proof selects nothing: the indices $J$ attached to a compact set or to a radius are produced by a finite subcover argument and then taken to be maximal in the increasing family, and the functions $δ_{Ω_{j}}$ are given. No family of nonempty sets is chosen from.

Proof technique: direct.

Proof

1.1F1F2F3F8given

If $Ω = C^{n}$ , then [F2] is exactly the conclusion, so assume from now on that $Ω \neq C^{n}$ ; then $F : = C^{n} ∖ Ω$ is nonempty, and writing $δ_{j} : = δ_{Ω_{j}}$ and $δ : = δ_{Ω}$ in the sense of [F3] one has $0 < δ_{j} (a) < + \infty$ for every $a \in Ω_{j}$ (positivity because $Ω_{j}$ is open, finiteness because $F \subseteq C^{n} ∖ Ω_{j} \neq \emptyset$ ) and $0 < δ (a) < + \infty$ for every $a \in Ω$ .

2.1F3F4F5F6step 1.1given

For $a \in Ω_{j}$ the inclusion $Ω_{j} \subseteq Ω_{j + 1} \subseteq Ω$ implies $Δ_{r} (a) \subseteq Ω_{j} \Rightarrow Δ_{r} (a) \subseteq Ω_{j + 1} \Rightarrow Δ_{r} (a) \subseteq Ω$ for every $r > 0$ , hence $δ_{j} (a) \leq δ_{j + 1} (a) \leq δ (a)$ and, if $a \in Ω_{J_{0}}$ , the eventual-tail limit $γ (a) : = \lim_{j \to \infty, j \geq J_{0}} δ_{j} (a) = \sup_{j \geq J_{0}} δ_{j} (a)$ exists and is independent of $J_{0}$ ; moreover $γ (a) = δ (a)$ , because for every $r$ with $0 < r < δ (a)$ and every $r^{'}$ with $r < r^{'} < δ (a)$ the definition [F3] gives $Δ_{r^{'}} (a) \subseteq Ω$ , so the closed polydisc ${\overline{Δ}}_{r} (a) \subseteq Δ_{r^{'}} (a) \subseteq Ω$ of [F4] is closed and bounded in $C^{n}$ , hence compact by [F5] read through [F6], and is therefore covered by finitely many members of the increasing open cover $(Ω_{j})_{j \geq 1}$ of $Ω$ , whose largest index, increased to $J \geq J_{0}$ if necessary, satisfies ${\overline{Δ}}_{r} (a) \subseteq Ω_{J}$ and hence $γ (a) \geq δ_{J} (a) \geq r$ ; letting $r ↑ δ (a)$ gives $γ (a) = δ (a)$ .

3.1F1F7step 1.1step 2.1

Let $K \subseteq Ω$ be compact and choose $J$ with $K \subseteq Ω_{J}$ (the same finite-subcover argument applied to the increasing cover $(Ω_{j})$ of $K$ ); then for every $j \geq J$ the function $- \log δ_{j}$ is plurisubharmonic on $Ω_{j}$ by the hypothesis that $Ω_{j}$ is Hartogs pseudoconvex and [F1], hence on the smaller domain $Ω_{J}$ , the sequence $(- \log δ_{j})_{j \geq J}$ is decreasing on $Ω_{J}$ by step 2.1, and it converges pointwise on $Ω_{J}$ to $- \log δ$ by the identity $γ = δ$ of step 2.1; the limit is real-valued on $Ω_{J}$ because $0 < δ < + \infty$ there by step 1.1, so it is not identically $- \infty$ on any component and [F7] makes $- \log δ$ plurisubharmonic on $Ω_{J}$ .

4.1F1step 1.1step 3.1∎

Every point $a \in Ω$ lies in some $Ω_{J}$ , an open neighbourhood of $a$ on which $- \log δ$ is plurisubharmonic by step 3.1 applied with $K = {a}$ ; plurisubharmonicity is a local condition, so $- \log δ$ is plurisubharmonic on $Ω$ and [F1] makes $Ω$ Hartogs pseudoconvex; together with the whole-space case of step 1.1 this proves the statement in both cases.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Oka-Weil approximation on a pseudoconvex domain

Statement

Assume the Axiom of Choice (AC). Let $Ω \subseteq C^{n}$ be a domain that is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity), let $K ⋐ Ω$ be compact with ${\hat{K}}_{Ω} = K$ (Holomorphic hulls and holomorphic convexity), and let $f$ be holomorphic in an open neighbourhood of $K$ . Then for every $ε > 0$ there is $F \in O (Ω)$ with $\sup_{z \in K} ∣ F (z) - f (z) ∣ < ε .$

Facts & Assumptions

Given: The Axiom of Choice; a Hartogs pseudoconvex domain $Ω \subseteq C^{n}$ ; a compact $K ⋐ Ω$ with ${\hat{K}}_{Ω} = K$ ; a holomorphic $f$ on an open neighbourhood of $K$ ; a real number $ε > 0$ .

[F1]

For a domain $Ω \subseteq C^{n}$ the following three conditions are equivalent: $Ω$ is Hartogs pseudoconvex; $Ω$ is a domain of holomorphy; $Ω$ is holomorphically convex (The Levi problem: pseudoconvexity, domains of holomorphy, and holomorphic convexity).

[F2]

If $G$ is a domain of holomorphy, $A ⋐ G$ is compact with ${\hat{A}}_{G} = A$ , and $g$ is holomorphic in an open neighbourhood of $A$ , then for every $δ > 0$ there is $H \in O (G)$ with $\sup_{A} ∣ H - g ∣ < δ$ (Oka-Weil approximation on a domain of holomorphy (host-domain lemma)).

[F3]

AC is the assertion that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F3]; it is consumed only inside the suppliers [F1] and [F2], each of which carries its own choice hypotheses. The proof selects nothing.

Proof

technique · direct

1.1F1given

By [F1] the Hartogs pseudoconvex domain $Ω$ is a domain of holomorphy.

2.1F2F3step 1.1given∎

Applying [F2] with $G : = Ω$ , $A : = K$ , ${\hat{A}}_{G} = A$ and $g : = f$ , and with $δ : = ε$ , gives $F \in O (Ω)$ with $\sup_{K} ∣ F - f ∣ < ε$ , which is the assertion of the Statement under the ambient Axiom of Choice cited as [F3].

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Locally finite smooth partitions of unity on domains

Statement

Assume the Axiom of Choice (AC) and the Axiom of Countable Choice. Let $Ω \subseteq C^{n}$ , $n \geq 1$ , be a domain and let $(U_{i})_{i \in I}$ be an open cover of $Ω$ . Then there are an open cover $(V_{k})_{k \in N}$ of $Ω$ refining $(U_{i})$ and functions $χ_{k} \in C^{\infty} (Ω)$ , $k \in N$ , such that:

$(V_{k})_{k \in N}$ is locally finite and there is a map $k \mapsto i (k) \in I$ with $V_{k} \subseteq U_{i (k)}$ for every $k$ ;
$0 \leq χ_{k} \leq 1$ and $supp χ_{k} \subseteq V_{k}$ for every $k$ ;
the family $(supp χ_{k})_{k \in N}$ is locally finite;
$\sum_{k} χ_{k} = 1$ at every point of $Ω$ .

In particular $(χ_{k})$ is a smooth partition of unity subordinate to the locally finite refinement $(V_{k})$ .

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice; a domain $Ω \subseteq C^{n}$ with $n \geq 1$ ; an open cover $(U_{i})_{i \in I}$ of $Ω$ ; the function $δ (z) : = \inf {∣ z - w ∣ : w \in C^{n} ∖ Ω}$ on $C^{n}$ , read as $δ \equiv + \infty$ when $Ω = C^{n}$ .

[F1]

A family $(ϕ_{i})_{i \in I}$ of smooth functions $ϕ_{i} : M \to [0, 1]$ is a smooth partition of unity subordinate to an open cover $(U_{i})_{i \in I}$ of a smooth manifold $M$ when the supports are locally finite, $supp (ϕ_{i}) \subseteq U_{i}$ for every $i$ , and $\sum_{i} ϕ_{i} (p) = 1$ for every $p$ (Smooth partitions of unity subordinate to an open cover).

[F2]

For all $0 < r < R$ there is a smooth function $ρ : R^{n} \to [0, 1]$ with $ρ = 1$ on ${\overline{B}}_{r} (0)$ and $supp ρ \subseteq B_{R} (0)$ (A smooth bump between concentric Euclidean balls).

[F3]

A subset $K \subseteq R^{n}$ is a compact subset of $(R^{n}, d_{2})$ if and only if $K$ is closed in $R^{n}$ and bounded (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F4]

Under the coordinate identification of $C^{m}$ with $R^{2 m}$ the metric, the balls, the open sets, the convergent sequences, the Cauchy sequences and the continuous maps of $C^{m}$ are verbatim those of $R^{2 m}$ (Complex $m$ -space and its real coordinate dictionary).

[F5]

The Axiom of Countable Choice: for every family $(X_{n})_{n \in N}$ of nonempty sets there is $f$ with domain $N$ and $f (n) \in X_{n}$ for all $n$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F6]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC selects, for each point of the shells below, one cover member and one radius; the countable instance [F5] selects the finite subcover list of each shell and the bumps built on it. No other selection occurs: the shell functions $K_{j}$ , $S_{j}$ , $W_{j}$ and the normalisation are explicit.

Proof

technique · direct

1.1F3F4algebra

If $Ω = C^{n}$ , then $δ \equiv + \infty$ is constant. Otherwise $C^{n} ∖ Ω \neq \emptyset$ , and for $z, z^{'} \in C^{n}$ and every $w \in C^{n} ∖ Ω$ , $∣ z - w ∣ \leq ∣ z - z^{'} ∣ + ∣ z^{'} - w ∣$ , so taking infima and then interchanging $z, z^{'}$ gives $∣ δ (z) - δ (z^{'}) ∣ \leq ∣ z - z^{'} ∣$ . Thus $δ$ is continuous in either case. For $j \geq 1$ put $K_{j} : = {z \in Ω : ∣ z ∣ \leq j and δ (z) \geq 1 / j}$ ; then each $K_{j}$ is closed in $C^{n}$ (intersection of the closed ball with the closed set ${δ \geq 1 / j}$ ) and bounded, hence compact by [F3] read through [F4]; moreover $K_{j} \subseteq int K_{j + 1}$ , because $∣ z ∣ \leq j < j + 1$ and $δ (z) \geq 1 / j > 1 / (j + 1)$ hold for $z \in K_{j}$ and persist on a small ball around $z$ by continuity of the modulus and of $δ$ ; finally $⋃_{j} K_{j} = Ω$ , since for $z \in Ω$ one has $δ (z) > 0$ (or $δ (z) = + \infty$ ) and $∣ z ∣ < \infty$ , so some integer $j$ satisfies $j \geq ∣ z ∣$ and $1 / j \leq δ (z)$ .

2.1step 1.1algebra

Put $K_{m} : = \emptyset$ for $m \leq 0$ and, for $j \geq 1$ , $S_{j} : = K_{j + 1} ∖ int K_{j - 1}$ and $W_{j} : = int K_{j + 2} ∖ K_{j - 2}$ ; then every $S_{j}$ is compact (a closed subset of the compact $K_{j + 1}$ ), $S_{j} \subseteq W_{j}$ with $W_{j}$ open (because $K_{j + 1} \subseteq int K_{j + 2}$ and $int K_{j - 1} \supseteq K_{j - 2}$ ), and the $S_{j}$ cover $Ω$ : for $z \in Ω$ let $m_{0} : = \min {m : z \in K_{m}}$ , which exists by step 1.1, so $z \in K_{m_{0}} \subseteq K_{m_{0} + 1}$ and $z \notin K_{m_{0} - 1} \supseteq int K_{m_{0} - 1}$ , that is $z \in S_{m_{0}}$ . The family $(W_{j})_{j \geq 1}$ is locally finite: a neighbourhood of $z$ contained in $int K_{m_{0} + 1}$ misses every $W_{j}$ with $j \geq m_{0} + 3$ , while only finitely many smaller indices remain; thus the family is locally finite.

3.1F5F6step 1.1step 2.1

For each $j \geq 1$ the set of finite lists (including the empty list when $S_{j} = \emptyset$ ) $((z_{1}, r_{1}, i_{1}), \dots, (z_{N}, r_{N}, i_{N}))$ with $z_{k} \in S_{j}$ , $r_{k} > 0$ , $i_{k} \in I$ , $\overline{B} (z_{k}, 3 r_{k}) \subseteq U_{i_{k}} \cap W_{j}$ and $S_{j} \subseteq ⋃_{k} B (z_{k}, r_{k})$ is nonempty: for every $z \in S_{j} \subseteq W_{j}$ the cover ${U_{i}}$ gives some $i (z)$ with $z \in U_{i (z)}$ , the set $U_{i (z)} \cap W_{j}$ is open and contains $z$ , so some radius $r (z) > 0$ satisfies $\overline{B} (z, 3 r (z)) \subseteq U_{i (z)} \cap W_{j}$ (choosing the pair $(i (z), r (z))$ by [F6]), and compactness of $S_{j}$ by step 2.1 lets the resulting open cover be reduced to a finite subcover; by [F5] select one such finite list for every $j \geq 1$ and enumerate the union of the selected lists as a sequence $(B_{k}, i_{k}, r_{k})_{k \in N}$ of balls $B_{k} = B (z_{k}, r_{k})$ . Then $⋃_{k} B_{k} \supseteq ⋃_{j} S_{j} = Ω$ by step 2.1.

4.1F2F4F5step 2.1step 3.1

For each $k$ , [F2] applied with the pair $0 < r = r_{k} < R = 3 r_{k} / 2$ and the centre $z_{k}$ provides a smooth $χ_{k} : C^{n} \to [0, 1]$ with $χ_{k} = 1$ on $\overline{B} (z_{k}, r_{k})$ and $supp χ_{k} \subseteq B (z_{k}, 3 r_{k} / 2)$ ; the balls here are Euclidean balls of $R^{2 n}$ under the identification of [F4], so $χ_{k} \in C^{\infty} (C^{n})$ . By step 3.1, $supp χ_{k} \subset B (z_{k}, 2 r_{k}) \subseteq U_{i_{k}} \cap W_{j (k)}$ ; the countably many choices of the $χ_{k}$ are read through [F5]. Since only finitely many selected balls occur for each $W_{j}$ , each support lies in its assigned $W_{j}$ , and the family $(W_{j})$ is locally finite by step 2.1, every point of $Ω$ has a neighbourhood meeting only finitely many supports, so $σ : = \sum_{k} χ_{k}$ is a well-defined smooth function on $Ω$ ; finally $σ \geq 1$ at every point of $Ω$ , because every point lies in some $S_{j}$ by step 2.1 and hence in some selected ball $B (z_{k}, r_{k})$ on which $χ_{k} = 1$ .

5.1F1step 2.1step 3.1step 4.1∎

Define $V_{k} : = B (z_{k}, 2 r_{k}) \cap Ω$ , so each $V_{k}$ is open with $V_{k} \subseteq U_{i (k)}$ and with $supp χ_{k} \subseteq V_{k}$ by step 4.1, and put ${\tilde{χ}}_{k} : = χ_{k} / σ$ ; then ${\tilde{χ}}_{k} \in C^{\infty} (Ω)$ , $0 \leq {\tilde{χ}}_{k} \leq 1$ , $supp {\tilde{χ}}_{k} = supp χ_{k} \subseteq V_{k}$ , and $\sum_{k} {\tilde{χ}}_{k} = 1$ because $\sum_{k} χ_{k} = σ$ . The family $(V_{k})$ covers $Ω$ , because $\sum_{k} {\tilde{χ}}_{k} = 1$ makes ${\tilde{χ}}_{k}$ positive at every point for at least one $k$ , and then that point lies in $V_{k}$ ; it refines $(U_{i})$ by the map $k \mapsto i_{k}$ , and is locally finite because $V_{k} \subseteq W_{j (k)}$ and $(W_{j})$ is locally finite by step 2.1; the supports of the ${\tilde{χ}}_{k}$ are locally finite for the same reason. Hence $({\tilde{χ}}_{k})$ is a smooth partition of unity subordinate to the locally finite refinement $(V_{k})$ in the sense of [F1].

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

First Cousin problem on a pseudoconvex domain

Statement

Assume the Axiom of Choice (AC). Let $n \geq 1$ and let $Ω \subseteq C^{n}$ be a Hartogs pseudoconvex domain (Plurisubharmonic exhaustions and Hartogs pseudoconvexity). Let $(U_{i})_{i \in I}$ be a locally finite open cover of $Ω$ and, for every $i \in I$ , let $m_{i}$ be a meromorphic function on $U_{i}$ (Meromorphic functions on an open set in complex Euclidean space) such that for all $i, j \in I$ the difference $m_{i} - m_{j}$ is holomorphic on $U_{i} \cap U_{j}$ (clause (c) of the definition of a meromorphic function).

Then there is a meromorphic function $G$ on $Ω$ such that $G - m_{i}$ is holomorphic on $U_{i}$ for every $i \in I$ . Equivalently, the first Cousin problem with the locally finite data $(m_{i})_{i \in I}$ is solvable: one global meromorphic function realizes the prescribed principal parts.

Facts & Assumptions

Given: The Axiom of Choice; an integer $n \geq 1$ ; a Hartogs pseudoconvex domain $Ω \subseteq C^{n}$ ; a locally finite open cover $(U_{i})_{i \in I}$ of $Ω$ ; meromorphic functions $m_{i}$ on $U_{i}$ with $m_{i} - m_{j}$ holomorphic on $U_{i} \cap U_{j}$ for all $i, j \in I$ ; the Wirtinger operators and the operators $\partial, \overset{ˉ}{\partial}$ on smooth forms of Bigraded complex forms and the Dolbeault operators.

[F1]

A domain $Ω \subseteq C^{m}$ is Hartogs pseudoconvex when $z \mapsto - \log δ_{Ω} (z)$ is plurisubharmonic on $Ω$ (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F2]

If $F$ is meromorphic on an open $U \subseteq C^{m}$ with domain $D$ and $h \in O (U)$ , then $F + h$ is meromorphic on $U$ (clause (d) of Meromorphic functions on an open set in complex Euclidean space); and $F$ is holomorphic on an open $V \subseteq U$ when some $H \in O (V)$ agrees with $F$ on $D \cap V$ (clause (c)).

[F3]

Meromorphy is a local condition: if every point of $U$ has a neighbourhood to which $F$ restricts as a meromorphic function, then $F$ is meromorphic on $U$ (Meromorphic functions on an open set in complex Euclidean space).

[F4]

Let $Ω \subseteq C^{n}$ be a domain and let $(U_{i})_{i \in I}$ be an open cover of $Ω$ . Then there are a locally finite open cover $(V_{k})_{k \in N}$ of $Ω$ refining $(U_{i})$ with $V_{k} \subseteq U_{i (k)}$ and smooth functions $χ_{k} \in C^{\infty} (Ω)$ with $0 \leq χ_{k} \leq 1$ , $supp χ_{k} \subseteq V_{k}$ , locally finite supports and $\sum_{k} χ_{k} = 1$ on $Ω$ (Locally finite smooth partitions of unity on domains).

[F5]

Let $Ω \subseteq C^{n}$ be Hartogs pseudoconvex and let $1 \leq q \leq n$ . Every smooth $\overset{ˉ}{\partial}$ -closed $(0, q)$ -form on $Ω$ is exact in the Dolbeault complex: there is a smooth $(0, q - 1)$ -form $ζ$ with $\overset{ˉ}{\partial} ζ = η$ (Positive-degree Dolbeault vanishing on pseudoconvex domains, claim 1).

[F6]

Let $U \subseteq C^{m}$ be open and $f : U \to C$ of class $C^{1}$ . Then $f$ is complex differentiable at $a \in U$ if and only if $\partial_{{\overset{ˉ}{z}}_{k}} f (a) = 0$ for every $k < m$ (clause 3 of For $C^{1}$ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree); a function is holomorphic on $U$ when it is complex differentiable at every point of $U$ (Holomorphic functions on an open subset of $C^{m}$ ).

[F7]

On smooth complex-valued forms $d = \partial + \overset{ˉ}{\partial}$ and ${\overset{ˉ}{\partial}}^{2} = 0$ (The d, partial and dbar identities); on a $(p, q)$ -form $η = \sum_{I, J} a_{I, J} d z^{I} \land d {\overset{ˉ}{z}}^{J}$ one has $\overset{ˉ}{\partial} η = \sum_{I, J, j} (\partial_{{\overset{ˉ}{z}}_{j}} a_{I, J}) d {\overset{ˉ}{z}}_{j} \land d z^{I} \land d {\overset{ˉ}{z}}^{J}$ , and components outside the bidegree range $0 \leq p, q \leq n$ are zero (Bigraded complex forms and the Dolbeault operators).

[F8]

AC is the statement that every family of nonempty sets has a choice function (The Axiom of Choice); in ZF, AC implies the Axiom of Countable Choice (AC implies DC implies countable choice), which selects one element from each family of nonempty sets indexed by $N$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Choice use. AC is the ambient hypothesis of the corollary. The partition-of-unity lemma [F4] selects cover members over its shell construction using AC and uses its countable instance for the finite lists and bumps; [F8] supplies that implication. This proof makes no additional selection.

Proof

technique · direct

1.1F4F8given

Apply [F4] to the cover $(U_{i})_{i \in I}$ : let $(V_{k})_{k \in N}$ be the resulting locally finite refinement with $V_{k} \subseteq U_{i (k)}$ , and let $χ_{k} \in C^{\infty} (Ω)$ be the associated smooth partition of unity with $0 \leq χ_{k} \leq 1$ , $supp χ_{k} \subseteq V_{k}$ , locally finite supports and $\sum_{k} χ_{k} = 1$ on $Ω$ . The countable-choice hypothesis of [F4] is discharged by the implication $A C \Rightarrow {A C}_{ω}$ from [F8].

1.2F2F4givenalgebra

For each fixed $j \in I$ define $f_{j} : = \sum_{k} χ_{k} (m_{j} - m_{i (k)})$ on $U_{j}$ as follows. By hypothesis the difference $m_{j} - m_{i (k)}$ is holomorphic on $U_{j} \cap U_{i (k)}$ , a set containing $V_{k} \cap U_{j}$ ; multiplying by the cutoff $χ_{k}$ , which vanishes outside $V_{k}$ , extends it by zero to a smooth function on $U_{j}$ , and the family $(supp χ_{k})$ is locally finite, so every point of $U_{j}$ has a neighbourhood on which only finitely many terms are nonzero; hence the sum $f_{j}$ is a well-defined element of $C^{\infty} (U_{j})$ .

2.1F2step 1.2givenalgebra

For $j, l \in I$ , evaluate on the dense open set where all relevant meromorphic representatives are defined. There each summand of $f_{j} - f_{l}$ equals $χ_{k} (m_{j} - m_{l})$ , so $f_{j} - f_{l} = (\sum_{k} χ_{k}) (m_{j} - m_{l}) = m_{j} - m_{l}$ there. The left side is continuous, and the right side has the given holomorphic extension to $U_{j} \cap U_{l}$ . Equality on the dense set and continuity give equality everywhere with that extension; in particular $f_{j} - f_{l}$ is holomorphic on the overlap.

3.1F6F7step 2.1step 1.2

Define $η$ on $U_{j}$ by $η : = \overset{ˉ}{\partial} f_{j}$ , a smooth $(0, 1)$ -form on $U_{j}$ by [F7] and step 1.2. For $j, l \in I$ the identity $f_{j} - f_{l} = m_{j} - m_{l}$ of step 2.1 gives $\overset{ˉ}{\partial} f_{j} - \overset{ˉ}{\partial} f_{l} = \overset{ˉ}{\partial} (m_{j} - m_{l})$ on $U_{j} \cap U_{l}$ , and $m_{j} - m_{l}$ is holomorphic there, so $\overset{ˉ}{\partial} (m_{j} - m_{l}) = 0$ by the Cauchy-Riemann system [F6] and [F7]. Hence $\overset{ˉ}{\partial} f_{j} = \overset{ˉ}{\partial} f_{l}$ on every overlap, so the local definitions glue to a well-defined smooth $(0, 1)$ -form $η \in Ω^{0, 1} (Ω)$ .

4.1F7step 3.1

On each $U_{j}$ one has $η = \overset{ˉ}{\partial} f_{j}$ with $f_{j}$ smooth, hence $\overset{ˉ}{\partial} η = {\overset{ˉ}{\partial}}^{2} f_{j} = 0$ on $U_{j}$ by [F7]; therefore $η$ is a smooth $\overset{ˉ}{\partial}$ -closed $(0, 1)$ -form on $Ω$ .

5.1F1F5F7step 4.1

Since $Ω$ is Hartogs pseudoconvex and $η \in Ω^{0, 1} (Ω)$ is smooth and $\overset{ˉ}{\partial}$ -closed, [F5] with $q = 1$ provides $ψ \in Ω^{0, 0} (Ω)$ with $\overset{ˉ}{\partial} ψ = η$ ; by the conventions of [F7] the space $Ω^{0, 0} (Ω)$ is the space $C^{\infty} (Ω)$ of smooth functions, so $ψ$ is a smooth function on $Ω$ .

6.1F6step 3.1step 5.1given

For each $j \in I$ put $F_{j} : = f_{j} - ψ$ on $U_{j}$ , a smooth function by step 1.2 and step 5.1; then $\overset{ˉ}{\partial} F_{j} = \overset{ˉ}{\partial} f_{j} - \overset{ˉ}{\partial} ψ = η - η = 0$ on $U_{j}$ by step 3.1 and step 5.1. Since $F_{j}$ is $C^{1}$ , the Cauchy-Riemann system [F6] makes $F_{j}$ complex differentiable at every point of $U_{j}$ , that is, holomorphic on $U_{j}$ .

7.1F2F3step 2.1step 6.1givenalgebra

Let $D_{i} \subseteq U_{i}$ be the open dense domain of the representative $m_{i}$ and put $D_{G} : = ⋃_{i} D_{i}$ , an open dense subset of $Ω$ . Define $G : D_{G} \to C$ by $G (z) : = m_{i} (z) - F_{i} (z)$ when $z \in D_{i}$ . On $D_{i} \cap D_{j}$ , the compatibility of the meromorphic differences and step 2.1 give $(m_{i} - F_{i}) - (m_{j} - F_{j}) = (m_{i} - m_{j}) - (f_{i} - f_{j}) = 0$ , so $G$ is well defined. It is holomorphic on $D_{G}$ because each local expression is holomorphic there. Near any point choose a chart $U_{i}$ and a local ratio $m_{i} = f / g$ on $W \subseteq U_{i}$ . On the dense open set $D_{i} \cap W \cap {g \neq 0}$ one has $G = (f - g F_{i}) / g$ ; both sides are holomorphic on $D_{G} \cap W \cap {g \neq 0}$ , so continuity extends this identity there. Thus $G$ has the required local ratio and [F3] makes it meromorphic on $Ω$ .

8.1

Finally $G - m_{j} = - F_{j}$ on the common domain $D_{G} \cap D_{j} = D_{j}$ for every $j \in I$ by the definition of $G$ , and $- F_{j}$ is a holomorphic extension to $U_{j}$ by step 6.1; thus the meromorphic function $G$ realizes the prescribed principal parts $(m_{i})_{i \in I}$ , as asserted. [step 6.1, step 7.1] $□$

Remarks

Local finiteness is not needed. The proof uses the locally finite cover $(U_{i})$ only as an input to the partition-of-unity lemma [F4], whose output is locally finite for an arbitrary open cover; the argument is verbatim valid for an arbitrary open cover $(U_{i})_{i \in I}$ with compatible meromorphic data, and the locally finite case stated here is the form promised by the scaffold.

Why the pseudoconvexity enters. The only analytic input is the smooth solvability of the $\overset{ˉ}{\partial}$ -equation for $(0, 1)$ -forms on $Ω$ , supplied here by [F5]. On the ball or on a polydisc this is the classical Dolbeault lemma; on a general Hartogs pseudoconvex domain it is the content of the in-pair corollary, and it is exactly the hypothesis that fails on $C^{2} ∖ {0}$ , where the Cousin-I data $1 / (z w)$ on the two coordinate complements is not solvable.

Holomorphy of the correction. The smooth solution $ψ$ of $\overset{ˉ}{\partial} ψ = η$ is used, not merely an $L^{2}$ solution: the local corrections $F_{j} = f_{j} - ψ$ must be $C^{1}$ so that the Cauchy-Riemann system [F6] applies, and this is why the smooth branch of the vanishing corollary [F5] is invoked.

5 · Examples, counterexamples and false statements

None yet.

Hörmander Estimates and the Levi Problem

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Smooth strict plurisubharmonic regularization of a psh exhaustion

Statement

Facts & Assumptions

Proof

Meromorphic functions on an open set in complex Euclidean space

Definition

Remark

Weighted L2 spaces and maximal dbar operators

Definition

Facts & Assumptions

Proof

The maximal distributional dbar operator is closed and densely defined

Statement

Facts & Assumptions

Proof

Basic Bochner–Kodaira–Morrey estimate on CnCn

Statement

Facts & Assumptions

Proof

A coercive Hilbert-complex estimate solves the closed equation

Statement

Facts & Assumptions

Proof

Weighted Morrey–Kohn estimate with a pseudoconvex boundary term

Statement

Facts & Assumptions

Proof

Weighted ∂̄ solvability on a smoothly bounded pseudoconvex domain

Statement

Facts & Assumptions

Proof

Smooth strictly plurisubharmonic exhaustion of a pseudoconvex domain

Statement

Facts & Assumptions

Proof

Hörmander's weighted L2 existence theorem for the dbar equation

Statement

Facts & Assumptions

Proof

Positive-degree Dolbeault vanishing on pseudoconvex domains

Statement

Facts & Assumptions

Proof

A strictly pseudoconvex boundary point has a local holomorphic separator

Statement

Facts & Assumptions

Proof

Positive smooth collars for strictly plurisubharmonic negative sets

Statement

Facts & Assumptions

Proof

Smooth global defining functions for strongly pseudoconvex boundaries

Statement

Facts & Assumptions

Proof

A smooth psh exhaustion gives Hartogs pseudoconvexity on bounded domains

Statement

Facts & Assumptions

Proof

Peak functions at strongly pseudoconvex boundary points, by a dbar correction

Statement

Facts & Assumptions

Proof

Oka-Weil approximation on a domain of holomorphy (host-domain lemma)

Statement

Facts & Assumptions

Proof

The Levi problem: pseudoconvexity, domains of holomorphy, and holomorphic convexity

Statement

Facts & Assumptions

Proof

Behnke-Stein: increasing unions of pseudoconvex domains

Statement

Facts & Assumptions

Proof

Basic Bochner–Kodaira–Morrey estimate on $C^{n}$