The Bergman Kernel on Forms: General Theory

TL;DR

This paper investigates the properties and construction of the Bergman kernel on differential forms, revealing fundamental differences from the function case and providing explicit computations for specific forms and domains.

Contribution

It demonstrates the failure of basic properties for the Bergman kernel on forms and offers a detailed construction and explicit formulas, especially for (0,n-1)-forms.

Findings

Pointwise evaluation fails for (0,q)-forms, q ≥ 1.

Explicit Bergman kernel for (0,n-1)-forms is derived.

Bergman kernel size on (0,1)-forms in the ball is not controlled by the metric.

Abstract

The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(\Omega)\cap\ker\bar\partial_q$, fail for $(0,q)$-forms, $q \geq 1$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to Bergman kernel on functions.

Equations

$$(f,{\varphi }_j)={\int }_{\Omega }⟨f,{\varphi }_j⟩dV={\int }_{\Omega }f\wedge \star {\varphi }_j.$$

$$B_{q}f(z)=\sum _{j=1}^{\infty }(f,{\varphi }_j){\varphi }_j(z)$$

$$B_{q}f(z)=\sum_{j=1}^{\infty}\Big{(}\int_{\Omega}f(w)\wedge\star\phi_{j}(w)\Big{)}\phi_{j}(z)=\int_{\Omega}f(w)\wedge\Big{(}\sum_{j=1}^{\infty}\star\phi_{j}(w)\wedge\phi_{j}(z)\Big{)}.$$

$$B_q(z,w)=\sum _{j=1}^{\infty }\star {\varphi }_j(w)\wedge {\varphi }_j(z)$$

$$\widetilde{d{\bar{z}}_j}=d{\bar{z}}_1\wedge \cdots \wedge \widehat{d{\bar{z}}_j}\wedge \cdots \wedge d{\bar{z}}_n$$

$$B_{q}f(z)={\int }_{\Omega }f(w)\wedge B_q(z,w)$$

$$B_{q}f(z)=\sum _{J,J^{\prime }\in {\mathcal{I}}_q}{B}_{J^{\prime }J}{f}_J(z)d{\bar{z}}^{J^{\prime }};$$

$$B_q(z,w)=\sum _{j=1}^{\infty }\star {\varphi }_j(w)\wedge {\varphi }_j(z)$$

$$B_{n-1}f(z)=f(z)-{\int }_{\Omega }f(w)\wedge {\vartheta }_{n-1,z}{\partial }_{n-1,w}^{\ast }{N}_n(z,w);$$

$$B_{[\hat{k}][\hat{j}]}(z,w)={\delta }_{jk}{\delta }_z(w)+(-1{)}^{n+j+k-1}4\frac{{\partial }^{2}G(z,w)}{\partial z_k\partial {\bar{w}}_j}$$

$$\displaystyle B_{[\hat{k}][\hat{j}]}(z,w)=\delta_{jk}\delta_{z}(w)+(-1)^{n+j+k-1}\frac{(n-1)!}{\pi^{n}}\bigg{[}\frac{\delta_{jk}}{|z-w|^{2n}}-n\frac{(z_{k}-w_{k})(\bar{z}_{j}-\bar{w}_{j})}{|z-w|^{2n+2}}$$

$$\displaystyle-\frac{\delta_{jk}-\bar{z}_{j}w_{k}}{(|z-w|^{2}+(1-|w|^{2})(1-|z|^{2}))^{n}}+n\frac{((z_{k}-w_{k})+w_{k}(1-|z|^{2}))((\bar{z}_{j}-\bar{w}_{j})-\bar{z}_{j}(1-|w|^{2}))}{(|z-w|^{2}+(1-|w|^{2})(1-|z|^{2}))^{n+1}}\bigg{]}$$

$$\displaystyle|K(z,z)|=\sum_{j=1}^{\infty}|\varphi_{j}(z)|^{2}\,dV(z)=\Big{(}\sup_{\genfrac{}{}{0.0pt}{2}{\{a\}\in\ell^{2}}{\|a\|_{\ell^{2}}=1}}\Big{|}\sum_{j=1}^{\infty}a_{j}\varphi_{j}(z)\Big{|}\Big{)}^{2}\,dV(z)=\sup_{\genfrac{}{}{0.0pt}{2}{f\in\ker\bar{\partial}}{\|f\|_{L^{2}}=1}}|f(z)|^{2}\,dV(z)$$

$$\displaystyle\big{(}F^{*}\phi,F^{*}\psi\big{)}=\int_{\Omega_{1}}F^{*}\phi(w)\wedge\star\big{(}F^{*}\psi(w)\big{)}$$

$$\displaystyle=\sum_{\genfrac{}{}{0.0pt}{2}{I,I^{\prime},K\in\mathcal{I}_{p}}{J,J^{\prime},L^{\prime}\in\mathcal{I}_{q}}}\int_{\Omega_{1}}\Big{(}\phi_{IJ}\circ F(w)\Big{)}\Big{(}\overline{\psi_{I^{\prime},J^{\prime}}\circ F(w)}\Big{)}\bigg{|}\frac{\partial F^{I}}{\partial w^{K}}\bigg{|}\bigg{|}\overline{\frac{\partial F^{J}}{\partial w^{L}}}\bigg{|}\bigg{|}\overline{\frac{\partial F^{[\hat{I}]}}{\partial w^{[\hat{K}]}}}\bigg{|}\bigg{|}\frac{\partial F^{[\hat{J}]}}{\partial w^{[\hat{L}]}}\bigg{|}\,dV(w)$$

$$\frac{\partial F^{I}}{\partial w^{K}}=\Big{(}\frac{\partial F_{I_{j}}}{\partial w_{K_{k}}}\Big{)}_{j,k=1}^{p}$$

$$f=\sum _{J\in {\mathcal{I}}_q}{f}_{J}d{\bar{z}}^J$$

$$B_q(f_{J}d{\bar{z}}^J)=\sum _{J^{\prime }\in {\mathcal{I}}_q}(B_q{f}_{J}d{\bar{z}}^J{)}_{J^{\prime }}d{\bar{z}}^{J^{\prime }}.$$

$$B_{J^{\prime }J}^{\prime }:L_{0,q}^2(\Omega )\to L_{0,q}^2(\Omega )$$

$$B_{J^{\prime }J}^{\prime }f=(B_q{f}_{J}d{\bar{z}}^J{)}_{J^{\prime }}d{\bar{z}}^{J^{\prime }}.$$

$$B_{q}f=\sum_{J,J^{\prime}\in\mathcal{I}_{q}}B_{J^{\prime}J}^{\prime}\big{(}f_{J}\,d\bar{z}^{J}\big{)}$$

$$B_{J^{\prime}J}f_{J}=\big{(}B^{\prime}_{J^{\prime}J}f_{J}\big{)}_{J^{\prime}}$$

$$\mathcal{K}\phi (x_1)={\int }_{\Omega }K(x_1,x_2)\phi (x_2)dV(x_2),\phi \in C_c(\Omega ),x_1\in \Omega .$$

$$⟨\mathcal{K}\phi ,\psi ⟩=K(\psi \otimes \phi )={\int }_{\Omega }{\int }_{\Omega }K(x_1,x_2)\phi (x_2)\psi (x_1)dV(x_2)dV(x_1).$$

$$⟨\mathcal{K}\phi ,\psi ⟩=K(\psi \otimes \phi );\psi ,\phi \in C_c^{\infty }(\Omega )$$

$$(B_{q}f,g)={\int }_{\Omega }{\int }_{\Omega }f(w)\wedge B_q(z,w)\wedge \ast g(z)dV(w)dV(z)=K_q(f\otimes g)$$

$$K_N(z,w)=\sum _{j=1}^N\ast {\varphi }_j(w)\wedge {\varphi }_j(z),$$

$$K_N(z,w)\to B_q(z,w)\text{ in }{L}_{(0,q),(n,n-q)}^2(\Omega ).$$

$$\Big{|}K_{M}(f\otimes g)-K_{N}(f\otimes g)\Big{|}=\Big{|}\int_{\Omega\times\Omega}\sum_{j=N+1}^{M}f(w)\wedge*\phi_{j}(w)\wedge\phi_{j}(z)\wedge*g(z)\Big{|}<\epsilon\|f\|_{L^{2}(\Omega)}\|g\|_{L^{2}(\Omega)}.$$

$$\sum _{j=1}^n{u}_{jK}\frac{\partial \rho }{\partial z_j}=0\text{ in bΩ for all }K\in {\mathcal{I}}_{q-1}$$

$$u_{jK}=\sum _{J\in {\mathcal{I}}_q}{ϵ}_J^{jK}{u}_J.$$

Full text

The author was partially supported by NSF grant DMS-1405100. I would also like to thank Phil Harrington for several helpful discussions on this project.

1. Introduction

On a domain $\Omega\subset\mathbb{C}^{n}$, the Bergman projection $B_{q}$ is the the orthogonal projection $B_{q}:L^{2}_{0,q}(\Omega)\to\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$. The basic theory of the classical Bergman projection $B_{0}$ is, well, classical and can be found in any several complex variables textbook, e.g., [Kra01]. The Bergman projection $B_{0}$ is one of the most basic objects in the analysis of both one and several variables, and its mapping properties have been exhaustively (though not conclusively) researched, as have formulas for its kernel. See, for example, [Cat83, Cat87, KN65, FK72, PS77, McN89, NRSW89, CD06, NS06, McN94, MS94, KR, Fef74, D’A78, D’A94] for just a small samplings of the results in the literature. Surprisingly, when $q\geq 1$, only mapping properties have been investigated – regularity properties for Bergman projects often follows from estimates of the $\bar{\partial}$-Neumann operator and Kohn’s formula (see, for example, [HR15, BS90]). There is essentially no literature about explicit construction of the kernels, pointwise size estimates, or geometry.

A standard discussion of $B_{0}$ includes a formal construction of the integral kernel, its transformation law under biholomorphic mappings, and a computation of the Bergman kernel on the ball (and perhaps the polydisk). One of the goals of this paper is to show that several of the main features of $B_{0}$ and its construction fail for $B_{q}$, $q\geq 1$. In particular, we show that:

1. (1)

Pointwise evaluation is not a bounded linear functional on $L^{2}_{0,q}(\Omega)\cap\ker(\bar{\partial}_{q})$; 2. (2)

It is unrealistic for a transformation formula to hold for $B_{p,q}(z,w)$ unless $p,q\in\{0,n\}$; 3. (3)

In $\mathbb{C}^{2}$, the Bergman kernel $B_{1}(z,w)$ on the ball does not behave according to the control geometry (in start constrast to $B_{0}(z,w)$).

There is no additional information to be gained by looking at the Bergman projection on $L^{2}_{p,q}(\Omega)$, so we focus on the $p=0$ case, except when we investigate the existence of transformation formulas because the $B_{p,0}$ behaves worse that $B_{0}$.

We start by carefully constructing $B_{q}$, which, while using well known Hilbert space and distribution theory, does not seem to appear in the literature. We then exploit Kohn’s formula and the knowledge of the $\bar{\partial}$-Neumann problem in the top degree to give a general formula for the Bergman projection $B_{n-1}$, and its associated integral kernel $B_{n-1}(z,w)$. We conclude the paper with a discussion on the ball. We compute $B_{n-1}$ explicitly and then restrict ourselves to the $\mathbb{C}^{2}$ case. There, we observe that the control geometry, which governs the size of $B_{0}(z,w)$, does not reflect the scaling present in the kernel $B_{1}(z,w)$. We conclude with a remark about future directions.

Fix $q\geq 1$. The kernel, $\ker\bar{\partial}_{q}$, is a closed subspace of $L^{2}_{0,q}(\Omega)$, so the projection $B_{q}$ onto $\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$ can be given as a Fourier series in terms of a basis. The construction of $B_{q}$ can proceeds as follows: suppose that $\{\phi_{j}\}_{j=1}^{\infty}$ is an orthonormal basis of $\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$. The vector projection of $f\in L^{2}_{0,q}(\Omega)$ onto $\operatorname{span}\phi_{j}$ is $(f,\phi_{j})\phi_{j}$ where the inner product

[TABLE]

where $\star$ is the Hodge-$\star$ operator (see, e.g., [CS01, p.208]) and $dV$ is Lebesgue measure. The orthogonal projection of $f$ on $\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$ is therefore given by the Fourier series

[TABLE]

where the sum converges in $L^{2}_{0,q}(\Omega)$.

Working formally, we see that

[TABLE]

This suggests that the Bergman kernel ought to be

[TABLE]

for any orthonormal basis $\{\phi_{j}\}$ of $\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$. For this formula to be rigorous, of course, the sum defining $B_{q}(\cdot,w)$ must converge in $L^{2}_{0,q}(\Omega)$, be independent of the orthonormal system $\{\phi_{j}\}$, and be the orthogonal projection onto $\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$. This is contain in Theorem 1.1, our structure theorem for the Bergman projection. To state our results, we need the following notation. Let $\mathcal{I}_{q}=\{J=(j_{1},\dots,j_{q})\in\mathbb{N}^{q}:1\leq j_{1}<\cdots<j_{q}\leq n\}$ be the set of increasing $q$-tuples and let

[TABLE]

where $\widehat{d\bar{z}_{j}}$ represents the omission of $d\bar{z}_{j}$ from the wedge product. We will also use the $[\hat{I}]$ to denote the $(n-|I|)$-tuple $\{1,\dots,n\}\setminus I$.

Theorem 1.1**.**

Let $\Omega\subset\mathbb{C}^{n}$ be a domain and $1\leq q\leq n-1$. Then:

1. (1)

There exists an integral kernel $B_{q}(z,w)$ so that the Bergman projection $B_{q}:L^{2}_{0,q}(\Omega)\to L^{2}_{0,q}(\Omega)\cap\ker\bar{\partial}_{q}$ is given by

[TABLE]

for any $f\in L^{2}_{0,q}(\Omega)$; 2. (2)

Moreover, there exist bounded operators $B_{J^{\prime}J}:L^{2}(\Omega)\to L^{2}(\Omega)$ so that if $f=\sum_{J\in\mathcal{I}_{q}}f_{J}\,d\bar{z}^{J}$, then

[TABLE] 3. (3)

Given any orthonormal basis $\{\phi_{j}\}\subset L^{2}_{0,q}(\Omega)\cap\ker\bar{\partial}$,

[TABLE]

where the sum converges in $L^{2}_{(0,q),(n,n-q)}(\Omega\times\Omega)$.

We have additional information about the operators $B_{J^{\prime}J}$ in the case that $q=n-1$.

Theorem 1.2**.**

Let $\Omega\subset\mathbb{C}^{n}$ be a domain and $G(z,w)$ be the Green’s function for the Laplacian $\triangle$. Then

1. (1)

[TABLE] 2. (2)

[TABLE]

where $\delta_{jk}$ is the Kronecker $\delta$ and $\delta_{z}(w)$ is the Dirac $\delta$. 3. (3)

In the case that $\Omega=B(0,1)$ is the unit ball then

[TABLE]

Our final result is the failure of the boundedness of pointwise evaluation in $L^{2}_{0,q}(\Omega)\cap\ker\bar{\partial}_{q}$, $q\geq 1$. This result stands in stark contract to $B_{0}$, and, in fact, boundedness pointwise evaluation in $L^{2}(\Omega)$ is a critical fact for $B_{0}$ and (more generally) one of the defining assumptions in the expansive theory of reproducing kernel Hilbert spaces, see, e.g., [BTA04]. To observe the first instance of the boundedness of pointwise evaluation in the theory of the Bergman project, we simply need to recall the standard construction for $B_{0}$. This construction works equally well for reproducing kernels in reproducing kernel Hilbert spaces. Suppose that the evaluation functional $e_{z}(\varphi)=\varphi(z)$ was a bounded, linear functional, i.e., $|e_{z}(\varphi)|\leq C\|\varphi\|_{L^{2}_{0,q}(\Omega)}$ for some constant $C$ that may depend on $z$ but not on $\varphi$. This would mean for any $f\in\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$, $|f(z)|\leq C\|f\|_{L^{2}_{0,q}(\Omega)}$ where $C=C(z)$ does not depend on $f$. This is critical for the following reason: for any $\{a_{j}\}\in\ell^{2}$, $f(z)=\sum_{j=1}^{\infty}a_{j}\varphi_{j}(z)\in\ker\bar{\partial}_{q}\cap L^{2}_{0,q}(\Omega)$, with the consequence that

[TABLE]

Consequently, boundedness on the diagonal implies finiteness of $\sup_{\genfrac{}{}{0.0pt}{2}{f\in\ker\bar{\partial}}{\|f\|_{L^{2}}=1}}|f(z)|$. From Theorem 1.2, it is immediate that $B_{n-1}(z,w)$ blows up as $w\to z$.

Theorem 1.3**.**

Let $\Omega\subset\mathbb{C}^{n}$ be a domain. If $1\leq q\leq n$, then pointwise evaluation is not a bounded, linear functional on $L^{2}_{0,q}(\Omega)\cap\ker\bar{\partial}_{q}$.

Proof.

Since forms are not functions, we consider pointwise evaluation to be the pointwise evaluation functionals $\varphi\mapsto\varphi_{J}$ for each $J\in\mathcal{I}_{q}$. Without loss of generality, we may suppose that $0\in\Omega$. Let $q\geq 1$, $J\in\mathcal{I}_{q}$, and $\psi\in(C^{\infty}_{c})_{0,q-1}(\Omega)$ so that $(\bar{\partial}\psi(0))_{J}\neq 0$. Set $\varphi(z)=\frac{\bar{\partial}\psi(z)}{\|(\bar{\partial}\psi)_{J}\|_{L^{2}(\Omega)}}$. Then $\varphi_{\epsilon}(z)=\epsilon^{-\frac{n}{2}}\varphi(z/\epsilon)\in(C^{\infty}_{c})_{0,q}(\Omega)\cap\ker\bar{\partial}_{q}$ since $\bar{\partial}^{2}=0$. Moreover, our normalization ensure $\|(\varphi_{\epsilon})_{J}\|_{L^{2}(\Omega)}=1$ for all $\epsilon>0$ but $|(\varphi_{\epsilon})_{J}(z)|\to\infty$ as $\epsilon\to 0$. ∎

Remark 1.4*.*

It is very unlikely that the Bergman kernel $B_{p,q}(z,w)$ satisfies a nice transformation formula under biholomorphisms unless $p,q\in\{0,n\}$. The transformation law for $B_{0}$ essentially follows from the pullback relationship $F^{*}\bar{\partial}=\bar{\partial}F^{*}$ and the fact that $J_{\mathbb{R}}F=|J_{\mathbb{C}}F|^{2}$ where $J_{\mathbb{R}}F$ is the determinant of the real Jacobian and $J_{\mathbb{C}}F$ is the determinant of the complex Jacobian. In general, while the pullback interacts nicely with $\bar{\partial}$, it behaves poorly with respect to $L^{2}$-inner products. In particular, if $F:\Omega_{1}\to\Omega_{2}$ is a biholomorphism and $\phi,\psi\in L^{2}_{p,q}(\Omega_{2})$, then

[TABLE]

where $\frac{\partial F^{I}}{\partial w^{K}}$ is the $p\times p$ minor of the complex Jacobian of the mapping $F=(F_{1},\dots,F_{n})$ given by

[TABLE]

where $I=(I_{1},\dots,I_{p})$ and $K=(K_{1},\dots,K_{p})$ and similarly for the other terms. The complicated product of determinants only simplifies dramatically in the cases $p,q\in\{0,n\}$ to $J_{\mathbb{R}}F$ and a change of variables may proceed as in the $B_{0}$ case.

1.1. Existence of the Bergman kernel and the proof of Theorem 1.1

We know that the Bergman projection is a bounded, linear operator. We now show that $B_{q}$ is an integral operator and that the Bergman kernel exists. Given $f\in L^{2}_{0,q}(\Omega)$, we can write

[TABLE]

The Bergman projection is a linear operator so that

[TABLE]

Define

[TABLE]

by the mapping

[TABLE]

It is easy to see that the operator norm $\|B_{J^{\prime}J}\|_{L^{2}\to L^{2}}\leq 1$ and

[TABLE]

For each operator $B_{J^{\prime}J}^{\prime}$, define an auxiliary operator $B_{J^{\prime}J}:L^{2}(\Omega)\to L^{2}(\Omega)$ that satisfies

[TABLE]

Essentially, $B_{J^{\prime}J}$ the operator that takes the coefficient of $f$ on $d\bar{z}^{J}$ and maps it to the $d\bar{z}^{J^{\prime}}$ coefficient of $B_{J^{\prime}J}^{\prime}f$.

Recall the Schwartz Kernel Theorem [Hör90, Theorem 5.2.1]. We state a version of it for our particular setup. Every function $K\in C(\Omega\times\Omega)$ defines an integral operator $\mathcal{K}$ from $C_{c}(\Omega)$ to $C(\Omega)$ by the formula

[TABLE]

The Schwartz Kernel Theorem extends this definition to arbitrary distributions $K$ if $\varphi$ is restricted to $C^{\infty}_{c}(\Omega)$ and $\mathcal{K}\varphi$ is allowed to be a distribution. The first observation is that if $K\in C(\Omega\times\Omega)$, then

[TABLE]

Theorem 1.5** (Schwartz Kernel Theorem).**

Every $K\in\mathcal{D}^{\prime}(\Omega\times\Omega)$ defines according to

[TABLE]

a linear map $\mathcal{K}$ from $C^{\infty}_{c}(\Omega)$ to $\mathcal{D}^{\prime}(\Omega)$ which is continuous in the sense that $\mathcal{K}\varphi_{j}\to 0$ in $\mathcal{D}^{\prime}(\Omega)$ if $\varphi_{j}\to 0$ in $C^{\infty}_{c}(\Omega)$. Conversely, to every such linear map $\mathcal{K}$ there is one and only one distribution $K$ such that (1.3) is valid. One calls $K$ the kernel of $\mathcal{K}$.

Since the maps $B_{J^{\prime}J}:L^{2}(\Omega)\to L^{2}(\Omega)$ boundedly, they certainly map from $C^{\infty}_{c}(\Omega)\to\mathcal{D}^{\prime}(\Omega)$. Consequently, the Schwartz Kernel Theorem applies to each $B_{J^{\prime}J}$. As a result, the Bergman kernel on $(0,q)$-forms exists as a distributional kernel, and we can write (for $f,g\in\mathcal{D}_{0,q}(\Omega)$)

[TABLE]

where the integral is understood in the distributional sense.

We now turn to establishing greater regularity for $B_{q}(z,w)$. Let $\{\phi_{j}\}$ be an orthonormal basis of $L^{2}_{0,q}(\Omega)\cap\ker\bar{\partial}_{q}$,

[TABLE]

and $\mathcal{K}_{N}$ as the operator with kernel $K_{N}$. We will show that

[TABLE]

Since $\mathcal{K}_{N}f\to Bf$ in $L^{2}_{0,q}(\Omega)$ and $\{\phi_{j}\}$ are orthogonal, there exists $N^{\prime}>0$ so that if $M\geq N\geq N^{\prime}$, then

[TABLE]

Consequently, the sequence of operators $\{\mathcal{K}_{N}\}$ with distributional kernels $\{K_{N}\}$ forms a Cauchy sequence acting on $L^{2}_{0,q}(\Omega)\times L^{2}_{0,q}(\Omega)$ and therefore converges to an operator $B^{\prime}$ acting on $L^{2}_{0,q}(\Omega)\times L^{2}_{0,q}(\Omega)$ and with distributional kernel $K(z,w)$. Moreover, since $K_{N}(z,w)$ forms a Cauchy sequence in $L^{2}_{0,q}(\Omega)\otimes L^{2}_{n,n-q}(\Omega)$, it follows that $B_{q}(z,w)\in L^{2}_{0,q}(\Omega)\otimes L^{2}_{n,n-q}(\Omega)\subset L^{2}_{(0,q),(n,n-q)}(\Omega\times\Omega)$. That this sum is independent of the basis is a standard Hilbert space fact. This concludes the proof of Theorem 1.1.

2. The Bergman projection $B_{n-1}$ and the proof of Theorem 1.2, parts (1) and (2)

Recall that the boundary condition for a form $u=\sum_{J\in\mathcal{I}_{q}}u_{J}\,d\bar{z}^{J}\in L^{2}_{0,q}(\Omega)$ to be an element of $\operatorname{Dom}(\bar{\partial}^{*})$ is that

[TABLE]

where

[TABLE]

If $q=n$ the boundary requirement is exactly that $u_{\{1,\dots,n\}}\frac{\partial\rho}{\partial z_{j}}=0$ for all $j=1,\dots,n$, i.e., $u=0$ on $\textrm{b}\Omega$. This is the Dirichlet boundary condition and the $\bar{\partial}$-Neumann problem reduces to the standard Dirichlet problem for the Laplacian. We normalize the Laplacian $\triangle$ so that $\triangle=-4\sum_{j=1}^{n}\frac{\partial^{2}}{\partial z_{j}\partial\bar{z}_{j}}$. Consequently, if $G(z,w)$ is the Green’s function for the Laplacian on $\Omega$, then the $\bar{\partial}$-Neumann operator on the top degree is

[TABLE]

with the notation $dw=dw_{1}\wedge\cdots\wedge dw_{n}$ and $d\bar{z}=d\bar{z}_{1}\wedge\cdots\wedge d\bar{z}_{n}$. The integral operator $N_{n}$ applied to a $(0,n)$-form $F=f\,d\bar{z}$ is then

[TABLE]

Thus we have an explicit integral kernel for $N_{n}$ for every case for which there is an explicit formula for $G(z,w)$.

Recall Kohn’s formula for the Bergman projection:

[TABLE]

We now compute $B_{n-1}$ and recall that $G(x,y)=0$ whenever either $x\in\textrm{b}\Omega$ or $y\in\textrm{b}\Omega$. Suppose $f\in L^{2}_{0,n-1}(\Omega)$. Then

[TABLE]

We would like to bring the operator $\vartheta_{n-1,z}$ inside the integral but this requires care because the Newtonian potential on $\mathbb{C}^{n}$ is

[TABLE]

and two derivatives means that the kernel would blow up like a singular integral. In point of fact, this will not cause a problem because derivatives of two derivatives of $\Phi(z)$ generate a Calderón-Zygmund singular integral. But care certainly must be taken! In particular, the Green’s function $G(z,w)$ is built from the Newtonian potential and a harmonic function. Therefore, the singularity of $\frac{\partial^{2}G(z,w)}{\partial z_{j}\partial\bar{w}_{k}}$ can only come from the $\frac{\partial^{2}}{\partial z_{j}\partial\bar{w}_{k}}\frac{1}{|w-z|^{2n-2}}$ which we now compute.

[TABLE]

The case $j\neq k$ yields the kernel $\frac{(w_{k}-z_{k})\overline{(w_{j}-z_{j})}}{|w-z|^{2(n+1)}}$ which is a classic Calderón-Zygmund convolution kernel – homogeneous of degree $-2n$ and integrates to [math] over any sphere centered around the origin. The case $j=k$ is only slightly more complicated. Observe that if $\sigma_{2n-1}$ is the surface area of the unit sphere in $\mathbb{C}^{n}$, then by symmetry

[TABLE]

By homogeneity, the integral is [math] around any sphere, thus we can write

[TABLE]

where the integral is taken in the sense of (tempered) distributions. A version of this formula (written directly in terms of the Green’s function) appears in [Bel92, Theorem 15.3] for domains in $\mathbb{C}$ and the Bergman projection $B_{0}$. Breaking down $B_{n-1}$ into its constituent parts, we compute

[TABLE]

from which (1.1) follows.

2.1. The proof of Theorem 1.2, parts (3) and (4)

We now restrict ourselves to the case $\Omega$ is the unit ball on which the Green’s function

[TABLE]

where $\tilde{w}=\frac{w}{|w|^{2}}$ is the reflection of $w$ across the unit sphere. Since

[TABLE]

it follows that

[TABLE]

In this case, note that

[TABLE]

and so $\frac{\partial G(z,w)}{\partial\bar{w}_{k}}\equiv 0$ whenever $w\in B(0,1)$ and $z\in\textrm{b}B(0,1)$ (reflecting the fact that $N_{n}\bar{\partial}_{n-1}\in\operatorname{Dom}(\bar{\partial}^{*}_{n-1})$). Also,

[TABLE]

from which part (3) of Theorem 1.2 follows.

3. Control geometry and the unit ball in $\mathbb{C}^{2}$

Observe that if $z\to\textrm{b}B(0,1)$, then

[TABLE]

as $z\to\textrm{b}B(0,1)$. Let $a_{jk}(z,w)=\frac{\bar{z}_{j}w_{k}}{|z-w|^{4}}-2\frac{\bar{z}_{j}(z_{k}-w_{k})(1-|w|^{2})}{|z-w|^{6}}$.

A defining function for $B(0,1)$ is $r(z)=|z|^{2}-1$. Consequently, the $(1,0)$ complex tangential vector field is $L=\bar{z}_{2}\frac{\partial}{\partial z_{1}}-\bar{z}_{1}\frac{\partial}{\partial z_{2}}$ and the complex normal is given by $S=2z_{1}\frac{\partial}{\partial z_{1}}+2z_{2}\frac{\partial}{\partial z_{2}}$. Observe that $[L,\bar{L}]=-\operatorname{Im}S$. If $z=(0,1)$ and $w=(w_{1},1-h)$, then $a_{1k}((0,1),w)=0$ and

[TABLE]

and

[TABLE]

while the Bergman kernel

[TABLE]

For the proper size estimate comparisons with $B_{0}(z,w)$, we recall the control metric from [NSW85] and the Bergman kernel estimates of [NRSW89, McN89]. At $(0,1)$, note that $L=\frac{\partial}{\partial z_{1}}$ and $S=2\frac{\partial}{\partial z_{2}}$ which means that the distance from $(0,1)$ in the $w_{1}$-direction is weighted by order 1 and in the $w_{2}$-direction by order 2. In other words, $d((0,1),(w_{1},1-h))\approx|w_{1}|+|h|^{1/2}$. It is clear that $a_{2k}(z,w)$ observes different scaling and size estimates that $B_{0}(z,w)$ as $|w_{1}|$ appears with the same weighting as $|h|$. Once again, $B_{1}$ behaves quite differently than $B_{0}$!

4. Conclusion

This paper checks the functional analysis to show that the Bergman projection has a well-defined integral kernel and that $B_{n-1}(z,w)$ is quite computable from the Green’s function $G(z,w)$. Of course, computing the Green’s function for domains of interest in several complex variables (and domains in general) is a complicated task. We will return to this topic in a future paper, in particular for $n=2$ case, as we can say much more there.