6.4 Interior density and the weighted estimate4E04198982E6

Proof. Step 1: cutting off. Let $f\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$, and write $\chi =\chi _\nu $. Since $\chi $ is real, $|\partial _{z_ j}\chi |=|\partial _{\bar z_ j}\chi |$, so the Cauchy–Schwarz inequality gives $|\sum _ j(\partial _{z_ j}\chi )f_ j|^2\le |\bar\partial \chi |^2|f|^2$. Lagrange's identity $\sum _{j<k}|a_ jb_ k-a_ kb_ j|^2=|a|^2|b|^2-|\sum _ ja_ j\bar b_ j|^2$ gives likewise $|\bar\partial \chi \wedge f|^2\le |\bar\partial \chi |^2|f|^2$. Since $\varphi _1-2\varphi _2=-\varphi =-\varphi _3$ and $|\bar\partial \chi |^2\mathrm e^{-\varphi }\le \mathrm e^{-\varphi _2}$, the form $\bar\partial \chi \wedge f$ lies in $\mathcal H_2$, the function $\mathrm e^{\varphi _1-\varphi _2}\sum _ j(\partial _{z_ j}\chi )f_ j$ lies in $\mathcal H_0$, and both of their squared norms are at most

Let $u\in \operatorname{Dom}T$. Then $\chi u\in \operatorname{Dom}T$ and $T(\chi u)=\chi Tu+u\, \bar\partial \chi $, because $u\, \bar\partial \chi \in \mathcal H_1$, the weights being bounded on the support of $\chi $. As $\chi $ is real and $\overline{\partial _{\bar z_ j}\chi }=\partial _{z_ j}\chi $, we get

Hence $\chi f\in \operatorname{Dom}T^*$, and the distributional product rule shows that $\chi f\in \operatorname{Dom}S$, with

By (6.4.3), the squared norms of both differences are bounded by $\int |f|^2|\bar\partial \chi _\nu |^2\mathrm e^{-\varphi }\, \mathrm dV$. As $\nu \to \infty $, $\bar\partial \chi _\nu $ tends to zero pointwise, since it vanishes near each point for large $\nu $, and the integrand is dominated by the integrable function $|f|^2\mathrm e^{-\varphi _2}$. By dominated convergence, both differences tend to zero. Again by dominated convergence, $\chi _\nu f\to f$, $\chi _\nu T^*f\to T^*f$ and $\chi _\nu Sf\to Sf$. Therefore $\chi _\nu f\to f$ in the graph norm.

Step 2: smoothing. It remains to approximate a form $f\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$ with compact support $K\subset \Omega $. By (6.4.1),

and $\bar\partial f=Sf\in L^2_{\mathrm{loc}}(\Omega )$. Let $f^\varepsilon =\sum _ j(f_ j*\rho _\varepsilon )\, \mathrm d\bar z_ j$, with $\rho _\varepsilon $ the mollifier of section 2.5; for small $\varepsilon $ these forms are smooth and supported in a fixed compact neighborhood $K'$ of $K$ in $\Omega $. Convolution commutes with derivatives with constant coefficients (proposition 2.5.3), so $\bar\partial f^\varepsilon =(\bar\partial f)*\rho _\varepsilon $ and $\sum _ j\partial _{z_ j}f^\varepsilon _ j=(\sum _ j\partial _{z_ j}f_ j)*\rho _\varepsilon $. By lemma 2.5.1, $f^\varepsilon $, $\sum _ j\partial _{z_ j}f^\varepsilon _ j$ and $\bar\partial f^\varepsilon $ converge in $L^2(K')$ to $f$, $\sum _ j\partial _{z_ j}f_ j$ and $\bar\partial f$. Since (6.4.1) applies to both $f^\varepsilon $ and $f$, and its remaining coefficients are smooth, $T^*f^\varepsilon \to T^*f$ in $L^2(K')$. The weights are bounded above and below on $K'$, so $f^\varepsilon \to f$ in the graph norm. $\square$

Proof. For smooth compactly supported functions $g$ and $w$, integration by parts (Fubini's theorem and the fundamental theorem of calculus on lines) gives

because $\varphi $ is real; thus $\delta _ j$ is the formal adjoint of $\partial _{\bar z_ j}$ for the inner product $\langle \cdot ,\cdot \rangle _\varphi $, and taking complex conjugates we also have $\langle \delta _ jw,g\rangle _\varphi =\langle w,\partial _{\bar z_ j}g\rangle _\varphi $. Since $\partial _{\bar z_ k}\bigl ((\partial _{z_ j}\varphi )g\bigr )=\varphi _{j\bar k}g+(\partial _{z_ j}\varphi )\partial _{\bar z_ k}g$ and the derivatives $\partial _{z_ j}$ and $\partial _{\bar z_ k}$ commute, we obtain the commutation relation

By the adjoint relation, then the commutation relation, and then the conjugate adjoint relation,

On the other hand, $\bar\partial f=\sum _{j<k}(\partial _{\bar z_ j}f_ k-\partial _{\bar z_ k}f_ j)\, \mathrm d\bar z_ j\wedge \mathrm d\bar z_ k$, and expanding the squares gives

When we add the two identities, the terms with $j\ne k$ in the last sums cancel, and the remaining terms $\| \partial _{\bar z_ j}f_ j\| _\varphi ^2$ combine with $\sum _{j\ne k}\| \partial _{\bar z_ k}f_ j\| _\varphi ^2$ to give $\sum _{j,k}\| \partial _{\bar z_ k}f_ j\| _\varphi ^2$. This proves the identity.

For the estimate, we have $\varphi _1-\varphi _2=-\psi $ and $\partial _{z_ j}\varphi _2=\partial _{z_ j}\varphi -\partial _{z_ j}\psi $, so (6.4.1) reads

By the elementary inequality $|x+y|^2\le (1+t^{-1})|x|^2+(1+t)|y|^2$ for $x,y\in \mathbb C$,

The first integral is $\| T^*f\| _{\mathcal H_0}^2$, because $\varphi _1=\varphi -2\psi $, and $|\sum _ j(\partial _{z_ j}\psi )f_ j|^2\le |\bar\partial \psi |^2|f|^2$ by the Cauchy–Schwarz inequality, since $|\partial _{z_ j}\psi |=|\partial _{\bar z_ j}\psi |$ for real $\psi $. Moreover $\| \bar\partial f\| _\varphi ^2=\| Sf\| _{\mathcal H_2}^2$, because $\varphi _3=\varphi $. Inserting these bounds into the identity and dropping its last, nonnegative, term, we obtain (6.4.5). $\square$

Proof. The idea is to modify the weight far out in $\Omega $ so that lemma 6.4.2 applies and the error term in (6.4.5) is absorbed by additional convexity, and then to let the modification recede to the boundary.

Step 1: auxiliary functions. The matrix $H_\varphi $ is positive semidefinite because $\varphi $ is plurisubharmonic, and by the discussion preceding the theorem, applied with $A=H_\varphi $, the finiteness of $M$ implies that $\alpha \in L^2_{\mathrm{loc},(0,1)}(\Omega )$. By theorem 6.3.4, $\Omega $ has a nonnegative smooth strictly plurisubharmonic exhaustion function $s$. Fix an integer $t\ge 1$. Let $\kappa _0$ be the function fixed at the beginning of this section, put $c_\kappa =\sup |\kappa _0'|$, and for integers $\nu \ge 1$ put $\chi _\nu =\kappa _0(s-t-1-\nu )$. Then $0\le \chi _\nu \le 1$, and $\chi _\nu \in C^\infty _ c(\Omega )$, because $\{ s\le t+2+\nu \} $ is compact; $\chi _\nu =1$ on $\{ s\le t+1+\nu \} $, hence on a neighborhood of any given compact set for large $\nu $; and $\bar\partial \chi _\nu =\kappa _0'(s-t-1-\nu )\, \bar\partial s$ vanishes on $\{ s<t+2\} $ and satisfies $|\bar\partial \chi _\nu |^2\le c_\kappa ^2|\bar\partial s|^2$. We put

Then $\psi $ is smooth and nonnegative; $\psi $ and $\bar\partial \psi $ vanish on $\{ s\le t+1\} $, since $1-\kappa _0(x-t-1)$ and its derivative vanish for $x\le t+1$; and $|\bar\partial \chi _\nu |^2\le \mathrm e^\psi $ for every $\nu $, because $\mathrm e^\psi =1+c_\kappa ^2|\bar\partial s|^2$ on $\{ s\ge t+2\} $ and $\bar\partial \chi _\nu =0$ on $\{ s<t+2\} $.

Next we choose a smooth convex nondecreasing function $\gamma \colon \mathbb R\to [0,\infty )$ with $\gamma =0$ on $(-\infty ,t]$ such that

For integers $l\ge 1$ the sets $K_ l=\{ t+l\le s\le t+l+1\} $ are compact. The smallest eigenvalue of $H_ s$ is a positive continuous function on $\Omega $ (see the discussion following proposition 5.9.2), so its minimum $\lambda _ l$ over $K_ l$ is positive (if $K_ l=\varnothing $, put $\lambda _ l=1$ and $m_ l=P_ l=I_ l=0$), and the numbers $m_ l=(1+t)\max _{K_ l}|\bar\partial \psi |^2$, $P_ l=2\max _{K_ l}\psi $ and $I_ l=\int _{K_ l}|\alpha |^2\mathrm e^{-\varphi +\psi }\, \mathrm dV$ are finite, since $\alpha \in L^2_{\mathrm{loc},(0,1)}(\Omega )$. Choose a smooth nondecreasing function $g\colon \mathbb R\to [0,\infty )$ that vanishes on $(-\infty ,t]$ and satisfies, for every $l\ge 1$,

For instance, let $\kappa _1(x)=1-\kappa _0(2x)$, a smooth nondecreasing function with $\kappa _1=0$ on $(-\infty ,0]$ and $\kappa _1=1$ on $[1/2,\infty )$, and put $g(x)=\sum _{l\ge 1}c_ l\kappa _1(x-t-l+1)$ with $c_ l=\max \{ m_ l/\lambda _ l,2P_ l,2\log ^+(2^ lI_ l)\} $. The sum is locally finite, since only the terms with $l<x-t+1$ are nonzero at $x$, and all its terms vanish for $x\le t$. The $l$-th term equals $c_ l$ on $[t+l-1/2,\infty )$, so $g\ge c_ l$ on $[t+l,t+l+1]$ and $\int _{t+l-1}^{t+l}g(x)\, \mathrm dx\ge c_ l/2$.

Put $\gamma (x)=\int _{-\infty }^ xg(y)\, \mathrm dy$. Then $\gamma $ is smooth, nondecreasing and convex, because $\gamma '=g$ is nondecreasing, and $\gamma =0$ on $(-\infty ,t]$. Since $\Omega $ is the union of $\{ s\le t+1\} $ and the sets $K_ l$, we may check the first two conditions in (6.4.8) on each of these sets. On $\{ s\le t+1\} $ they hold because $\psi $ and $\bar\partial \psi $ vanish there while $\gamma \ge 0$ and $H_{\gamma (s)}\ge 0$. On $K_ l$ we have $\gamma (s)\ge \gamma (t+l)\ge P_ l\ge 2\psi $, and the chain rule

gives $H_{\gamma (s)}\ge (m_ l/\lambda _ l)\lambda _ lI\ge (1+t)|\bar\partial \psi |^2I$. Finally, the integral in (6.4.8) over the compact set $\{ s\le t+1\} $ is finite, and its integral over $K_ l$ is at most $\mathrm e^{-\gamma (t+l)}I_ l\le 2^{-l}$, because $\gamma (s)\ge \gamma (t+l)\ge \log ^+(2^ lI_ l)$ there; summing over $l$, we obtain the third condition.

Step 2: the estimate. Put $\varphi _0=\varphi +\gamma (s)$ and $\varphi _ k=\varphi _0+(k-3)\psi $ for $k=1,2,3$, and let $T$ and $S$ be the corresponding operators. By the last condition in (6.4.8), $\alpha \in \mathcal H_1=L^2_{(0,1)}(\Omega ,\mathrm e^{-\varphi _2})$, and $\alpha \in \ker S$. Let $f$ be a smooth compactly supported $(0,1)$-form. We apply (6.4.5) with $\varphi _0$ in place of $\varphi $ and with the integer $t$ fixed in Step 1. Since $H_{\varphi _0}=H_\varphi +H_{\gamma (s)}$, the second condition in (6.4.8) absorbs the last term of (6.4.5), and we obtain

Moreover $\varphi -2\varphi _2=-\varphi _0-\gamma (s)+2\psi \le -\varphi _0$ by the first condition in (6.4.8). Since $M<\infty $, the dual energy of $\alpha $ is finite almost everywhere, and the inequality $|\sum _ j\alpha _ j\bar f_ j|^2\le |\alpha |_{H_\varphi ^{-1}}^2H_\varphi [f]$ together with the Cauchy–Schwarz inequality gives

Both sides are continuous for the graph norm of lemma 6.4.2, whose hypotheses hold by Step 1; hence the inequality extends to all $f\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$. By theorem 6.2.4 there is $u_ t\in \mathcal H_0$ with $\bar\partial u_ t=\alpha $ and $\| u_ t\| _{\mathcal H_0}^2\le M(1+t^{-1})$. On $\{ s<t\} $ both $\gamma (s)$ and $\psi $ vanish, so $\varphi _1=\varphi $ there and

Step 3: passage to the limit. For each integer $l\ge 1$ the functions $u_ t$ with $t\ge l$ have squared norms at most $2M$ in $L^2(\{ s<l\} ,\mathrm e^{-\varphi })$. By proposition 6.2.2 and a diagonal argument, there is a sequence $t_ i\to \infty $ such that $u_{t_ i}$ converges weakly in $L^2(\{ s<l\} ,\mathrm e^{-\varphi })$ for every $l$. The limits on different sublevel sets agree on their intersections, because restriction from $\{ s<l+1\} $ to $\{ s<l\} $ is a bounded linear map, and they define a function $u$ on $\Omega $. Weak convergence in these spaces implies convergence in the sense of distributions: a test function $\chi $ has support in some set $\{ s<l\} $, and $\int u_{t_ i}\chi \, \mathrm dV=\langle u_{t_ i},\bar\chi \mathrm e^\varphi \rangle _\varphi $ with $\bar\chi \mathrm e^\varphi \in L^2(\{ s<l\} ,\mathrm e^{-\varphi })$. Hence $\bar\partial u=\alpha $. By the lower semicontinuity of the norm in proposition 6.2.2, $\int _{\{ s<l\} }|u|^2\mathrm e^{-\varphi }\, \mathrm dV\le \varliminf _ iM(1+t_ i^{-1})=M$ for every $l$, and the monotone convergence theorem gives $\int _\Omega |u|^2\mathrm e^{-\varphi }\, \mathrm dV\le M$. $\square$

Proof. We regularize $p$ on relatively compact pseudoconvex subsets of $\Omega $, apply theorem 6.4.7 there, and pass to a weak limit. Since the weight $p+q$ is bounded above on compact sets, the discussion preceding theorem 6.4.7 shows that $\alpha \in L^2_{\mathrm{loc},(0,1)}(\Omega )$, so that $\bar\partial \alpha $ is defined. By corollary 6.3.5 there are pseudoconvex open sets $\Omega _1\subset \Omega _2\subset \cdots $ with $\Omega _ j\Subset \Omega $ and $\bigcup _ j\Omega _ j=\Omega $. Let $\rho _\varepsilon $ be the radial mollifier of section 2.5, and choose $\varepsilon _ j\downarrow 0$ with $\varepsilon _ j<\operatorname {dist}(\overline\Omega _ j,\mathbb C^ n\setminus \Omega )$. By proposition 5.9.2(2), $p_ j=p*\rho _{\varepsilon _ j}$ is defined, smooth and plurisubharmonic on the open neighborhood $\{ z\in \Omega :\operatorname {dist}(z,\mathbb C^ n\setminus \Omega )>\varepsilon _ j\} $ of $\overline\Omega _ j$, so its complex Hessian is positive semidefinite there, and $p_ j\downarrow p$ pointwise as $j\to \infty $. Put $\varphi _ j=p_ j+q$. Then $\varphi _ j$ is smooth with complex Hessian matrix $H_{\varphi _ j}\ge A\ge 0$ near $\overline\Omega _ j$, so it is plurisubharmonic there by proposition 5.9.2(1), and $\varphi _ j\ge p+q$. Since the dual energy decreases when the matrix increases,

By theorem 6.4.7 there is $u_ j$ on $\Omega _ j$ with $\bar\partial u_ j=\alpha $ and $\int _{\Omega _ j}|u_ j|^2\mathrm e^{-\varphi _ j}\, \mathrm dV\le M$.

Let $K\subset \Omega $ be compact. Then $K\subset \Omega _ k$ for some $k$, and for $j\ge k$ we have $\varphi _ j\le \varphi _ k$ on $K$, because $p_ j$ decreases. Hence $\int _ K|u_ j|^2\, \mathrm dV\le \mathrm e^{\max _ K\varphi _ k}M$ for $j\ge k$, so the sequence $(u_ j)$ is eventually bounded in $L^2(K)$ for every compact set $K$. Let $(K_ m)$ be a compact exhaustion of $\Omega $ (section 2.2). By proposition 6.2.2 and a diagonal argument, we find a subsequence that converges weakly in $L^2(K_ m)$ for every $m$. The weak limits are compatible, because restriction from $K_{m+1}$ to $K_ m$ is a bounded linear map, and they define a function $u\in L^2_{\mathrm{loc}}(\Omega )$; since every compact subset of $\Omega $ lies in some $K_ m$, the subsequence converges weakly to $u$ in $L^2(K)$ for every compact $K\subset \Omega $. Then $u_ j\to u$ in the sense of distributions, and $\bar\partial u=\alpha $.

Now let $K\subset \Omega $ be compact, and let $k$ be so large that $K\subset \Omega _ k$. On $L^2(K)$ the norm with weight $\mathrm e^{-\varphi _ k}$ is equivalent to the unweighted norm, so the subsequence also converges weakly for it, and the lower semicontinuity in proposition 6.2.2 gives

This holds for all large $k$, and $\mathrm e^{-\varphi _ k}$ increases to $\mathrm e^{-p-q}$ as $k\to \infty $; so the monotone convergence theorem gives $\int _ K|u|^2\mathrm e^{-p-q}\, \mathrm dV\le M$, and exhausting $\Omega $ by compact sets proves the estimate. The last assertion follows from the pointwise inequality $|\alpha |_{A^{-1}}^2\le \lambda ^{-1}|\alpha |^2$. $\square$

Proof. (1) The real and imaginary parts of $h$, given by $\operatorname{Re}\langle h,\chi \rangle $ and $\operatorname{Im}\langle h,\chi \rangle $ for real test functions $\chi $, are real distributions. Since derivatives of distributions commute, (5.2.9) shows that the Laplacian of $h$ is $4\sum _ j\partial _{z_ j}\partial _{\bar z_ j}h=0$. As the Laplacian has real coefficients, the real and imaginary parts of $h$ are harmonic distributions. By Weyl's lemma (theorem 3.1.11) they are given by smooth functions, and hence $h$ is given by a smooth function $g$. By integration by parts, the classical derivatives $\partial _{\bar z_ j}g$ represent the distributions $\partial _{\bar z_ j}h=0$, so they vanish almost everywhere by proposition 2.5.3(4), and hence everywhere, being continuous functions. Thus $g$ is holomorphic.

(2) Let $L\subset \Omega $ be compact, and let $m_ L$ be a real upper bound for $\varphi $ on $L$. For $u\in L^2(\Omega ,\mathrm e^{-\varphi })$ we have $\int _ L|u|^2\, \mathrm dV\le \mathrm e^{m_ L}\int _ L|u|^2\mathrm e^{-\varphi }\, \mathrm dV$. Hence $L^2(\Omega ,\mathrm e^{-\varphi })\subset L^2_{\mathrm{loc}}(\Omega )$, so that $\bar\partial u$ is defined in the sense of distributions, and convergence in $L^2(\Omega ,\mathrm e^{-\varphi })$ implies convergence in $L^2_{\mathrm{loc}}(\Omega )$, hence in the sense of distributions. The space $L^2(\Omega ,\mathrm e^{-\varphi })$ is a Hilbert space, by the completeness argument at the beginning of section 6.2, applied to the positive measure $\mathrm e^{-\varphi }\, \mathrm dV$. The set $\mathcal K$ is a linear subspace. It is closed: if $g_ j\in \mathcal K$ converge to $g$ in $L^2(\Omega ,\mathrm e^{-\varphi })$, then $g_ j\to g$ in the sense of distributions, so $\bar\partial g=\lim _ j\bar\partial g_ j=0$, and by (1) $g$ agrees almost everywhere with a holomorphic function, that is, $g\in \mathcal K$.

Let $u_0$ be a solution of $\bar\partial u=\alpha $ in $L^2(\Omega ,\mathrm e^{-\varphi })$. By (1), an element $u$ of this space is a solution if and only if $\bar\partial (u-u_0)=0$, that is, $u-u_0\in \mathcal K$. Thus the solutions form the nonempty closed convex set $u_0+\mathcal K$, which by proposition 6.2.1 contains a unique element $u_{\min }$ of smallest norm. If a solution $u$ is orthogonal to $\mathcal K$, then $\| u+k\| ^2=\| u\| ^2+\| k\| ^2\geq \| u\| ^2$ for every $k\in \mathcal K$, so $u$ has smallest norm, that is, $u=u_{\min }$. Conversely, if $P$ is the orthogonal projection onto $\mathcal K$ (proposition 6.2.1), then $u_{\min }-Pu_{\min }$ is a solution orthogonal to $\mathcal K$, hence equal to $u_{\min }$; so $u_{\min }$ is orthogonal to $\mathcal K$. $\square$