Hörmander's weighted L2 existence theorem for the dbar equation

By [F8] and [F9] choose a smooth strictly plurisubharmonic exhaustion $S$ and increasing regular values $c_k\to +\mathrm{\infty }$, with ${\mathrm{\Omega }}_k:=\{S<c_k\}$ relatively compact and exhausting $\mathrm{\Omega }$. Fix $z_0\in \mathrm{\Omega }$ and discard an initial finite segment so that $z_0\in {\mathrm{\Omega }}_k$ for every remaining $k$. Let $D_k$ be the connected component of ${\mathrm{\Omega }}_k$ containing $z_0$; these components are nested. They exhaust $\mathrm{\Omega }$: for $z\in \mathrm{\Omega }$, [F25] gives a polygonal path from $z_0$ to $z$, whose compact image lies in some ${\mathrm{\Omega }}_k$ by the increasing open cover, so $z\in D_k$. Each $D_k$ is bounded because $\overline{{\mathrm{\Omega }}_k}$ is compact. At $p\in \mathrm{\partial }{D}_k$ one has $p\in \mathrm{\partial }{\mathrm{\Omega }}_k=\{S=c_k\}$; the regular-level chart makes ${\mathrm{\Omega }}_k$ a connected local subgraph near $p$, and its inside part meets $D_k$, so it lies in $D_k$. Thus $D_k$ has $C^{\mathrm{\infty }}$ boundary locally defined by $\rho =S-c_k$ with $d\rho (p)\mathrm{\ne }0$. Strict plurisubharmonicity gives ${\mathcal{L}}_{\rho }(p;v)>0$ on every nonzero complex tangent vector, so $D_k$ is a bounded Levi pseudoconvex domain to which [F10] applies.

For $a\in \mathrm{\Omega }$ write $H(a):=({\phi }_{j\bar{k}}(a))$ and $B(a):=H(a{)}^T$. Reality of $\phi$, the Wirtinger formulas [F11] and commutation of mixed partials [F12] give $\overline{H_{jk}}=H_{kj}$, so $H$ and $B$ are Hermitian and are self-adjoint by [F14]. Their characteristic polynomials agree, since $\det (tI-H^T)=\det (tI-H)$, so they have the same ordered eigenvalues ${\lambda }_i$. With the first-variable-linear inner product, ${\mathcal{L}}_{\phi }(a;v)={\sum }_{j,k}{H}_{jk}(a)v_j\overline{v_k}=⟨B(a)v,v⟩\mathrm{.}$ Thus strict plurisubharmonicity [F6] makes $B$ positive definite. Its orthonormal eigenbasis from [F13] shows that ${\lambda }_1(a)={\min }_{\mathrm{\mid }v\mathrm{\mid }=1}⟨B(a)v,v⟩>0$, hence every eigenvalue of $H$ is positive and $w(a)\ge q{\lambda }_1(a)>0$.

For each $k$ let $f_k$ be the restriction of $f$ to $D_k$, that is the tuple of restrictions $f_{k,K}:=f_K{\mathrm{\mid }}_{D_k}$. Then $f_k\in L_{0,q}^2(D_k,e^{-\phi })$ with ${\int }_{D_k}\mathrm{\mid }{f}_k{\mathrm{\mid }}^2{w}^{-1}{e}^{-\phi }dV\le E(f)$ because $f\in L_{0,q}^2(\mathrm{\Omega },e^{-\phi })$ and the integrand is nonnegative, and $f_k\in \mathrm{Dom}{\bar{\mathrm{\partial }}}_q(D_k)$ with ${\bar{\mathrm{\partial }}}_q{f}_k=0$: distributional differentiation is local, so the coefficient distributions of $\bar{\mathrm{\partial }}{f}_k$ on $D_k$ are obtained by evaluating those of $\bar{\mathrm{\partial }}f$ 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 $\bar{\psi }$, $\psi \in C_c^{\mathrm{\infty }}(D_k)$, exactly as $\bar{\mathrm{\partial }}f$ pairs with the zero extensions of $\bar{\psi }$ to $\mathrm{\Omega }$; since the distribution $\bar{\mathrm{\partial }}f$ is represented by the zero form by the hypothesis ${\bar{\mathrm{\partial }}}_{q}f=0$, its restriction to $D_k$ is represented by the zero form as well.

Fix a test form $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathrm{\Lambda }}^{0,q})$ and let ${\mathrm{\partial }}_0^{\ast }\eta$ be the $(0,q-1)$-form with coefficients $({\mathrm{\partial }}_0^{\ast }\eta {)}_I:=-{\sum }_{j=1}^n{\mathrm{\partial }}_{z_j}{\eta }_{jI}$; then, writing $H:=L_{0,q-1}^2(\mathrm{\Omega },e^{-\phi })$, $e^{\phi }{\mathrm{\partial }}_0^{\ast }\eta$ is compactly supported, of class $C^2$, and lies in $H$, and for every $\tilde{u}\in H$ one has the distributional identity ${\sum }_{\mathrm{\mid }K\mathrm{\mid }=q}⟨(\bar{\mathrm{\partial }}\tilde{u}{)}_K,\overline{{\eta }_K}⟩=⟨\tilde{u},e^{\phi }{\mathrm{\partial }}_0^{\ast }\eta {⟩}_{\phi }$, where the left side evaluates the coefficient distributions of F2 against the test functions $\overline{{\eta }_K}$ and the antisymmetry convention of [F3] assigns the shuffle signs: both sides equal the single sum $-{\sum }_{\mathrm{\mid }I\mathrm{\mid }=q-1}{\sum }_{j=1}^n{\int }_{\mathrm{\Omega }}{\tilde{u}}_I{\mathrm{\partial }}_{{\bar{z}}_j}\left(\overline{{\eta }_{jI}}\right)dV$, the left by the test identity of [F5] and the sign convention for $\bar{\mathrm{\partial }}$ of [F4], and the right by expanding the pairing of F1 with $\overline{(e^{\phi }({\mathrm{\partial }}_0^{\ast }\eta {)}_I)}=e^{\phi }\overline{({\mathrm{\partial }}_0^{\ast }\eta {)}_I}$ and conjugating the holomorphic derivative by the Wirtinger rules [F11].

The function $w$ is upper semicontinuous, hence Borel measurable, on $\mathrm{\Omega }$: for every orthonormal $q$-frame $\mathcal{V}=(v_1,\dots ,v_q)$ the function $a\mapsto G(a,\mathcal{V}):={\sum }_{i=1}^q⟨H(a)v_i,v_i⟩$ is continuous, and $w(a)={\min }_{\mathcal{V}}G(a,\mathcal{V})$ over the nonempty set of orthonormal $q$-frames, because expanding in an eigenbasis $(e_j)$ of step 1.2 gives $G(a,\mathcal{V})={\sum }_j{\lambda }_j(a)m_j$ with $m_j:={\sum }_i\mathrm{\mid }⟨v_i,e_j⟩{\mathrm{\mid }}^2\in [0,1]$ by the Bessel inequality of [F15] and ${\sum }_j{m}_j={\sum }_i\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2=q$ by its Parseval clause, and for such a mass vector ${\sum }_j{\lambda }_j(a)m_j-({\lambda }_1+\cdots +{\lambda }_q)(a)={\sum }_{j\le q}{\lambda }_j(m_j-1)+{\sum }_{j>q}{\lambda }_j{m}_j\ge {\lambda }_q({\sum }_{j\le q}(m_j-1)+{\sum }_{j>q}{m}_j)=0$, with equality for $\mathcal{V}=(e_1,\dots ,e_q)$; hence $\{w<c\}={\bigcup }_{\mathcal{V}}\{a:G(a,\mathcal{V})<c\}$ is a union of open sets, and since $w>0$ on $\mathrm{\Omega }$ by step 1.2 the integrand $\mathrm{\mid }f{\mathrm{\mid }}^2{w}^{-1}{e}^{-\phi }$ is a nonnegative measurable function, so that the energy $E(f)$ of the statement is a well-defined extended Lebesgue integral.

Fix $k$ and take $D:=D_k$. The domain $D$ is a bounded Levi pseudoconvex domain with $C^{\mathrm{\infty }}$ boundary, the weight $\phi$ lies in $C^2$ on a neighbourhood of $\bar{D}\subseteq \mathrm{\Omega }$ and is strictly plurisubharmonic at every point of $\bar{D}$ by step 1.2 and the hypothesis, and the datum $f_k$ lies in $\mathrm{Dom}{\bar{\mathrm{\partial }}}_q(D)$ with ${\bar{\mathrm{\partial }}}_q{f}_k=0$ by step 1.3; all hypotheses of [F10] are therefore met, and its conclusion supplies the least-norm solution $u_k\in \mathrm{Dom}{\bar{\mathrm{\partial }}}_{q-1}(D)$ of ${\bar{\mathrm{\partial }}}_{q-1}u=f_k$ with $\mathrm{\parallel }{u}_k{\mathrm{\parallel }}_{\phi ,D}^2\le {\int }_D\mathrm{\mid }f{\mathrm{\mid }}^2{w}^{-1}{e}^{-\phi }dV\le E(f)$, where the eigenvalue functions named in [F10] are the functions ${\lambda }_j$ of step 1.2 and the last inequality is step 1.3 with the nonnegative integrand and $D\subseteq \mathrm{\Omega }$.

Extend each $u_k$ by zero: let ${\tilde{u}}_k$ equal $u_k$ on $D_k$ and $0$ on $\mathrm{\Omega }\setminus D_k$, regarded as a coefficient tuple. Then ${\tilde{u}}_k\in H:=L_{0,q-1}^2(\mathrm{\Omega },e^{-\phi })$ with measurable coefficients, $\mathrm{\parallel }{\tilde{u}}_k{\mathrm{\parallel }}_{\phi }^2=\mathrm{\parallel }{u}_k{\mathrm{\parallel }}_{\phi ,D_k}^2\le E(f)$ by step 2.2, and consequently $({\tilde{u}}_k{)}_{k\ge 1}$ is a norm-bounded sequence in the complex Hilbert space $H$ of [F1].

Let $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathrm{\Lambda }}^{0,q})$ and let $K$ be an index with $\mathrm{supp}\eta \subseteq D_K$, which exists because the sets $D_k$ increase to $\mathrm{\Omega }$ by step 1.1 while $\mathrm{supp}\eta$ is compact; for every $k\ge K$ the left side of the identity of step 1.4 with $\tilde{u}:={\tilde{u}}_k$ equals ${\int }_{\mathrm{\Omega }}{\sum }_{\mathrm{\mid }{K}^{\mathrm{\prime }}\mathrm{\mid }=q}{f}_{K^{\mathrm{\prime }}}\stackrel{‾}{{\eta }_{K^{\mathrm{\prime }}}}dV=:⟨f,\eta {⟩}_0$: on the open set $D_k\supseteq \mathrm{supp}\eta$ the tuples ${\tilde{u}}_k$ and $u_k$ agree, so by the locality of distributional differentiation the coefficient distributions of $\bar{\mathrm{\partial }}{\tilde{u}}_k$ evaluated against the test functions $\stackrel{‾}{{\eta }_{K^{\mathrm{\prime }}}}$ depend only on $u_k$, and there the $L^2$ identity ${\bar{\mathrm{\partial }}}_{q-1}{u}_k=f_k$ of step 2.2 represents them by the coefficients $f_{K^{\mathrm{\prime }}}$.

The space $H$ is reflexive by [F16], whose countable-choice hypothesis is the instance AC${}_{\omega }$ 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 $({\tilde{u}}_{k_j}{)}_{j\ge 1}$ converging weakly in $H$ to some $u\in H$.

By [F18], whose HB hypothesis is supplied from AC by [F21], applied to the weakly convergent sequence of step 4.1 one has $\mathrm{\parallel }u{\mathrm{\parallel }}_{\phi }\le {\mathrm{lim\hspace{0.17em}inf}}_{j\to \mathrm{\infty }}\mathrm{\parallel }{\tilde{u}}_{k_j}{\mathrm{\parallel }}_{\phi }$, and $\mathrm{\parallel }{\tilde{u}}_{k_j}{\mathrm{\parallel }}_{\phi }^2\le E(f)$ for every $j$ by step 3.1, so that ${\mathrm{lim\hspace{0.17em}inf}}_j\mathrm{\parallel }{\tilde{u}}_{k_j}{\mathrm{\parallel }}_{\phi }\le E(f{)}^{1\mathrm{/}2}$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\phi }^2\le E(f)$.

Let $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathrm{\Lambda }}^{0,q})$. By step 1.4 the left side of its identity with $\tilde{u}:={\tilde{u}}_{k_j}$ equals $⟨{\tilde{u}}_{k_j},e^{\phi }{\mathrm{\partial }}_0^{\ast }\eta {⟩}_{\phi }$, and this scalar converges to $⟨u,e^{\phi }{\mathrm{\partial }}_0^{\ast }\eta {⟩}_{\phi }$ as $j\to \mathrm{\infty }$ because $e^{\phi }{\mathrm{\partial }}_0^{\ast }\eta \in H$ and ${\tilde{u}}_{k_j}\rightharpoonup u$ weakly (step 4.1 and the characterization of weak convergence by bounded functionals in [F24]); by step 3.2 the same scalar equals $⟨f,\eta {⟩}_0$ for all sufficiently large $j$, so $⟨u,e^{\phi }{\mathrm{\partial }}_0^{\ast }\eta {⟩}_{\phi }=⟨f,\eta {⟩}_0$, and by step 1.4 applied with $\tilde{u}:=u$, whose left side is by construction the pairing of the coefficient distributions of $\bar{\mathrm{\partial }}u$ with the test form $\eta$, the distribution $\bar{\mathrm{\partial }}u$ is represented by the $L^2$ form $f$; by the definition of the maximal operator F2 this says $u\in \mathrm{Dom}{\bar{\mathrm{\partial }}}_{q-1}(\mathrm{\Omega })$ and ${\bar{\mathrm{\partial }}}_{q-1}u=f$.

Claim 1 holds: the element $u$ of step 5.2 lies in $\mathrm{Dom}{\bar{\mathrm{\partial }}}_{q-1}$ with ${\bar{\mathrm{\partial }}}_{q-1}u=f$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\phi }^2\le E(f)$ by step 5.1. Claim 2 holds by the source fact [F19] under its hypotheses: $\mathrm{\Omega }$ with the Euclidean Kähler form is a weakly pseudoconvex Kähler manifold because the exhaustion of [F9] is a plurisubharmonic exhaustion, $\phi \in C^{\mathrm{\infty }}(\mathrm{\Omega })$ has nonnegative eigenvalues ${\lambda }_j>0$, and the smooth $\bar{\mathrm{\partial }}$-closed form $f$ has finite energy, so the $C^{\mathrm{\infty }}$ branch of the quoted theorem supplies $u\in C^{\mathrm{\infty }}(\mathrm{\Omega };{\mathrm{\Lambda }}^{0,q-1})$ with $\bar{\mathrm{\partial }}u=f$ 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].