Weighted L2 spaces and maximal dbar operators

If $\mathrm{\Omega }=\mathrm{\varnothing }$, all coefficient spaces contain only the zero class and the assertions are immediate. Assume $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$. The weight $e^{-\phi }$ is continuous and strictly positive on $\mathrm{\Omega }$, so ${\mu }_{\phi }:=e^{-\phi }dV$ is a Borel measure on $\mathrm{\Omega }$ with the same null sets as Lebesgue measure and with finite mass on every compact subset of the $\sigma$-compact space $\mathrm{\Omega }$; by [F3] the tuple pairing of coefficient tuples satisfies the inner-product axioms and Cauchy-Schwarz, and by [F2] with $p=2$ and $\mu ={\mu }_{\phi }$ the complex space $L^2({\mu }_{\phi };\mathbb{C})$ is complete, so the coefficientwise space of (a), a finite product of $m=\left(\genfrac{}{}{0px}{}{n}{q}\right)$ copies of $L^2({\mu }_{\phi };\mathbb{C})$ 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.

Let $u\in L_{0,q}^2$ and let $K\subseteq \mathrm{\Omega }$ be a nonempty compact set; then ${\int }_K\mathrm{\mid }{u}_J\mathrm{\mid }dV\le \mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}2}({\int }_K\mathrm{\mid }{u}_J{\mathrm{\mid }}^{2}dV{)}^{1\mathrm{/}2}$ by Cauchy-Schwarz, and $c_K:={\min }_K{e}^{-\phi }>0$ gives ${\int }_K\mathrm{\mid }{u}_J{\mathrm{\mid }}^{2}dV\le c_K^{-1}\mathrm{\parallel }{u}_J{\mathrm{\parallel }}_{\phi }^2<\mathrm{\infty }$, 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 $\mathbb{C}$-homogeneous in the coefficient.

Test forms are dense in the weighted space. Let $u\in L_{0,q}^2$. For integers $k\ge 1$ choose ${\chi }_k\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $0\le {\chi }_k\le 1$, ${\chi }_k=1$ near $K_k$ and $\mathrm{supp}{\chi }_k\subseteq K_{k+1}$, where $K_k=\{\mathrm{\mid }z\mathrm{\mid }\le k\}\cap \{\mathrm{dist}(z,{\mathbb{C}}^n\setminus \mathrm{\Omega })\ge 1\mathrm{/}k\}$ and the distance constraint is omitted when $\mathrm{\Omega }={\mathbb{C}}^n$. These sets are compact, exhaust $\mathrm{\Omega }$, and satisfy $K_k\subseteq \mathrm{int}{K}_{k+1}$: distance to the nonempty complement is continuous by the triangle inequality, and both defining inequalities become strict at the next index. Apply [F10] with $O=\mathrm{int}{K}_{k+1}$ and use countable choice for the cutoffs (take zero when $K_k$ is empty); then $\mathrm{\mid }{\chi }_{k}u-u{\mathrm{\mid }}^2{e}^{-\phi }\le 4\mathrm{\mid }u{\mathrm{\mid }}^2{e}^{-\phi }\in L^1$ and ${\chi }_{k}u\to u$ pointwise, so [F13] gives $\mathrm{\parallel }{\chi }_{k}u-u{\mathrm{\parallel }}_{\phi }\to 0$ and each ${\chi }_{k}u$ has compact support in $\mathrm{\Omega }$. It remains to approximate any such compactly supported tuple $v:={\chi }_{k}u$. If $v=0$ there is nothing to prove; otherwise put $K:=\mathrm{supp}v\subseteq K_{k+1}$ and use the already chosen exhaustion cutoff $\theta :={\chi }_{k+1}$, which equals $1$ near $K_{k+1}$ and has compact support in $\mathrm{\Omega }$. Thus $\theta v=v$. Let $C:={\max }_{\mathrm{supp}\theta }{e}^{-\phi }<\mathrm{\infty }$, extend each coefficient of $v$ by zero to a tuple $\tilde{v}$ on ${\mathbb{C}}^n\cong {\mathbb{R}}^{2n}$, and apply [F6] componentwise: for every $\epsilon >0$ there is a tuple $\psi =({\psi }_J)$ with ${\psi }_J\in C_c^{\mathrm{\infty }}({\mathbb{R}}^{2n})$ and $\mathrm{\parallel }{\psi }_J-{\tilde{v}}_J{\mathrm{\parallel }}_{L^2({\mathbb{R}}^{2n})}<\epsilon m^{-1\mathrm{/}2}{C}^{-1\mathrm{/}2}$ for each of the $m=\left(\genfrac{}{}{0px}{}{n}{q}\right)$ coefficients. The tuple $\theta \psi {\mathrm{\mid }}_{\mathrm{\Omega }}$ lies in $C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathrm{\Lambda }}^{0,q})$, and because $\theta v=v$ its error is $\theta (\psi -\tilde{v})$ on $\mathrm{\Omega }$. Therefore $\mathrm{\parallel }\theta \psi -v{\mathrm{\parallel }}_{\phi }^2\le C{\sum }_{\mathrm{\mid }J\mathrm{\mid }=q}\mathrm{\parallel }{\psi }_J-{\tilde{v}}_J{\mathrm{\parallel }}_{L^2({\mathbb{R}}^{2n})}^2<{\epsilon }^2\mathrm{.}$ This proves the claimed weighted density.

Define $\bar{\mathrm{\partial }}u$ 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 $d{\bar{z}}^L$, $\mathrm{\mid }L\mathrm{\mid }=q+1$, is ${\sum }_{j\in L}{\epsilon }_{j,L\setminus j}\mathrm{\partial }{u}_{L\setminus j}\mathrm{/}\mathrm{\partial }{\bar{z}}_j$, a distribution, and if $w,w^{\mathrm{\prime }}\in L_{0,q+1}^2$ both represent $\bar{\mathrm{\partial }}u$, then every coefficient of $w-w^{\mathrm{\prime }}$ is locally integrable by step 1.2 and represents the zero distribution, so [F12] gives $w=w^{\mathrm{\prime }}$ almost everywhere; hence $\mathrm{Dom}{\bar{\mathrm{\partial }}}_q$ and ${\bar{\mathrm{\partial }}}_q$ are well defined.

For $q=0$ the preceding-degree adjoint is zero by convention. For $q\ge 1$, let $v\in \mathrm{Dom}{\bar{\mathrm{\partial }}}_{\phi }^{\ast }$ and $w:={\bar{\mathrm{\partial }}}_{\phi }^{\ast }v$. For every compactly supported smooth $(0,q-1)$-form $\psi$, the adjoint identity and [F4], with wedge signs absorbed in the antisymmetric coefficients $v_{jK}$, give ${\int }_{\mathrm{\Omega }}{\sum }_{\mathrm{\mid }K\mathrm{\mid }=q-1}{\psi }_K\overline{w_K}{e}^{-\phi }dV={\int }_{\mathrm{\Omega }}{\sum }_{\mathrm{\mid }K\mathrm{\mid }=q-1}{\sum }_{j=1}^n({\mathrm{\partial }}_{{\bar{z}}_j}{\psi }_K)\overline{v_{jK}}{e}^{-\phi }dV\mathrm{.}$ The identity [F5], conjugated and summed, therefore gives $e^{-\phi }{w}_K=-{\sum }_j{\mathrm{\partial }}_{z_j}(e^{-\phi }{v}_{jK})$ 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 $\mathrm{\Omega }$. Local integrability then passes each defining integral to the limit. This also justifies the $C^2$ multiplier $e^{\phi }$ and its product rule in this order-one identity. Multiplying by $e^{\phi }$ and expanding yields exactly (c), only on the Hilbert-adjoint domain.

The domain of ${\bar{\mathrm{\partial }}}_q$ is a linear subspace and ${\bar{\mathrm{\partial }}}_q$ is linear: by [F5] the distributional identity $\mathrm{\partial }(a{u}_J+b{u}_J^{\mathrm{\prime }})\mathrm{/}\mathrm{\partial }{\bar{z}}_j=a\mathrm{\partial }{u}_J\mathrm{/}\mathrm{\partial }{\bar{z}}_j+b\mathrm{\partial }{u}_J^{\mathrm{\prime }}\mathrm{/}\mathrm{\partial }{\bar{z}}_j$ holds for all scalars $a,b$ and all locally integrable $u_J,u_J^{\mathrm{\prime }}$ (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 $\bar{\mathrm{\partial }}(au+b{u}^{\mathrm{\prime }})$ gives ${\bar{\mathrm{\partial }}}_q(au+b{u}^{\mathrm{\prime }})=a{\bar{\mathrm{\partial }}}_{q}u+b{\bar{\mathrm{\partial }}}_q{u}^{\mathrm{\prime }}$; the domain is nonempty because every smooth compactly supported $(0,q)$-form has its smooth $\bar{\mathrm{\partial }}$ in $L_{0,q+1}^2$ by [F4].

The operator ${\bar{\mathrm{\partial }}}_{q-1}$ 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 ${\bar{\mathrm{\partial }}}_{\phi }^{\ast }$ is unique, and it is linear because $v\mapsto (u\mapsto ⟨\bar{\mathrm{\partial }}u,v{⟩}_{\phi })$ 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).