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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].

Depends on

Used by

Hörmander's weighted L2 existence theorem for the dbar equation Theorem

Dependency tree · two levels

68 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean-Pierre Demailly, Complex Analytic and Differential Geometry (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables (standard reference, not scraped)
Mohammad Jabbari, Several Complex Variables course notes (standard reference, not scraped)