The closed range property for the $\overline{\partial}$-operator on planar domains

TL;DR

This paper characterizes when the ar operator has closed range on planar domains using potential-theoretic conditions, linking functional analysis with complex potential theory.

Contribution

It establishes necessary and sufficient potential-theoretic conditions for the ar operator to have closed range in L^2 spaces on planar domains.

Findings

ar has closed range iff the Poincare9-Dirichlet inequality holds.

Provides new potential-theoretic criteria for closed range of ar.

Characterizes when the Bergman space is infinite dimensional.

Abstract

Let $\Omega\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ if and only if the Poincar\'e-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for the $\overline{\partial}$-operator to have closed range in $L^{2}(\Omega)$. We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of $\Omega$ to be infinite dimensional.

Figures

Equations

$$\overline{\partial }f=\sum _{j=1}^n{\frac{\partial f}{\partial \bar{z}}}_{j}d{\bar{z}}_j$$

$$\Vert u{\Vert }_{L^2(\Omega )}\le C\Vert \overline{\partial }u{\Vert }_{L^2(\Omega )}$$

$$\Vert u{\Vert }_{L^2(\Omega )}\le C\Vert u_{\bar{z}}{\Vert }_{L^2(\Omega )}\forall u\perp A^2(\Omega )\text{ with }{u}_{\bar{z}}\in L^2(\Omega ).$$

$$\Vert v-v_{\Omega }{\Vert }_{L^2(\Omega )}\le C\Vert \nabla v{\Vert }_{L^2(\Omega )}$$

$$\Vert u{\Vert }_{L^2(\Omega )}\le C\Vert \nabla u{\Vert }_{L^2(\Omega )}\forall u\in H^1(\Omega )\cap {\left(\ker \nabla \right)}^{\perp }.$$

$$\Vert v{\Vert }_{L^2(\Omega )}\le C\Vert \nabla v{\Vert }_{L^2(\Omega )}\forall v\in H_0^1(\Omega )$$

$${\rho }_{\mathrm{cap}}(\Omega )=\sup \left\{R\ge 0:\forall \delta >0\exists z\in \mathbb{C}\text{ such that }\mathrm{cap}\left(\mathbb{D}(z,R)\cap {\Omega }^c\right)<\delta \right\},$$

$$\Vert f{\Vert }_{H^1(\Omega )}={\left(\Vert f{\Vert }_{L^2(\Omega )}^2+\Vert \nabla f{\Vert }_{L^2(\Omega )}^2\right)}^{\frac{1}{2}}$$

$$\mathrm{Dom}(\overline{\partial })=\left\{u\in L^2(\Omega ):\overline{\partial }u\text{ (in the sense of distributions) in }{L}^2(\Omega )\right\}.$$

$$|(\overline{\partial }u,v{)}_{L^2(\Omega )}|\le C\Vert u{\Vert }_{L^2(\Omega )}\forall u\in \mathrm{Dom}(\overline{\partial }),$$

$$(\overline{\partial }u,v{)}_{L^2(\Omega )}=(u,w{)}_{L^2(\Omega )}\forall u\in \mathrm{Dom}(\overline{\partial }).$$

$$v\mapsto {\left(\Vert v{\Vert }_{L^2(\Omega )}^2+\Vert {\overline{\partial }}^{\star }v{\Vert }_{L^2(\Omega )}^2\right)}^{1/2}.$$

$$(u,v{)}_{L^2(\Omega )}+({\overline{\partial }}^{\star }u,{\overline{\partial }}^{\star }v{)}_{L^2(\Omega )}=0\forall v\in {\mathcal{C}}_c^{\infty }(\Omega ).$$

$$\Vert u{\Vert }_{L^2(\Omega )}^2+\Vert {\overline{\partial }}^{\star }u{\Vert }_{L^2(\Omega )}^2=0.$$

$$\Vert v{\Vert }_{L^2(\Omega )}\le C\Vert f{\Vert }_{L^2(\Omega )}.$$

$$\Vert v{\Vert }_{L^2(\Omega )}\le C\Vert \overline{\partial }u{\Vert }_{L^2(\Omega )}.$$

$$\Vert u{\Vert }_{L^2(\Omega )}\le C\Vert \overline{\partial }u{\Vert }_{L^2(\Omega )}$$

$$\mathfrak{C}(\Omega )=\inf \left\{C:\Vert u{\Vert }_{L^2(\Omega )}\le C\Vert \overline{\partial }u{\Vert }_{L^2(\Omega )}\forall u\in \mathrm{Dom}(\overline{\partial })\cap A^2(\Omega {)}^{\perp }\right\}.$$

$$p_{\mu }(z)={\int }_{\mathbb{C}}\ln |z-w|d\mu (w).$$

$$I_{\mu }={\int }_{\mathbb{C}}{p}_{\mu (z)}d\mu (z)={\int }_{\mathbb{C}}{\int }_{\mathbb{C}}\ln |z-w|d\mu (w)d\mu (z).$$

$$I_{\nu }=\sup \{I_{\mu }:\mu \text{ finite Borel probability measure with support in }K\},$$

$${\rho }_{\mathrm{cap}}(\Omega )=\sup \left\{R\ge 0:\forall \delta >0\exists z\in \mathbb{C}\text{ such that }\mathrm{cap}\left(\mathbb{D}(z,R)\cap {\Omega }^c\right)<\delta \right\}.$$

$$\Vert v{\Vert }_{L^2(\Omega )}\le C\Vert \nabla v{\Vert }_{L^2(\Omega )}\forall v\in H_0^1(\Omega ).$$

$$\displaystyle\lambda_{1}(\Omega)=\min\Biggl{\{}\frac{\|\nabla\varphi\|^{2}_{L^{2}(\Omega)}}{\|\varphi\|^{2}_{L^{2}(\Omega)}}:\varphi\in H_{0}^{1}(\Omega),\;\varphi\not\equiv 0\Biggr{\}}.$$

$$f_{\phi }(t)=\frac{{\Vert \nabla \left(\psi +t\phi \right)\Vert }_{L^2(\Omega )}^2}{{\Vert \psi +t\phi \Vert }_{L^2(\Omega )}^2}$$

$$\{\begin{array}{l}△\psi +\lambda \psi =0\text{ on }\Omega \\ \psi {|}_{b\Omega }=0\end{array}$$

$${\lambda }_1(\Omega )\le \frac{\Vert \nabla \psi {\Vert }_{L^2(\Omega )}^2}{\Vert \psi {\Vert }_{L^2(\Omega )}^2}=\frac{(-△\psi ,\psi {)}_{L^2(\Omega )}}{\Vert \psi {\Vert }_{L^2(\Omega )}^2}\le \frac{\Vert △\psi {\Vert }_{L^2(\Omega )}}{\Vert \psi {\Vert }_{L^2(\Omega )}}.$$

$$\mathfrak{C}(\mathbb{C})=\mathfrak{C}(r\mathbb{C})=r\mathfrak{C}(\mathbb{C})$$

$$\Omega =\mathbb{C}\setminus \left(\mathbb{Z}+\sqrt{-1}\mathbb{Z}\right)$$

$$\mathfrak{C}(\Omega )=\frac{2}{\sqrt{{\lambda }_1(\Omega )}}.$$

Full text

1. Introduction

The purpose of this paper is to understand the closed range property for the $\overline{\partial}$-operator for open sets in the complex plane. That is, we study a concept, which is fundamental for the analysis of holomorphic functions in higher dimensions, in one complex variable. Because of the tight relationship of harmonic and holomorphic functions on open sets in $\mathbb{C}$, many phenomena of the latter may be explained and derived through potential theory in the complex plane. Our main result, Theorem 1.3, is another manifestation of this deep connection. In fact, we completely describe the closed range property for $\overline{\partial}$ for open sets in $\mathbb{C}$ through two different kinds of potential-theoretic conditions.

The $\overline{\partial}$-operator is initially defined as

[TABLE]

for any function $f$ which is differentiable on an open set in $\mathbb{C}^{n}$. We shall consider the maximal $L^{2}$-extension of $\overline{\partial}$ for the given open set. A reason for considering the $\overline{\partial}$-operator as an $L^{2}$-operator is that it allows one to employ Hilbert space methods to solve the inhomogeneous Cauchy–Riemann equations. This is of importance for the construction of holomorphic functions in higher dimension due to the lack of power series techniques which are available in one complex dimension. Note that for planar open sets, the $\overline{\partial}$-operator may be identified with an extension of the derivative operator $\frac{\partial}{\partial\bar{z}}$.

In this article, we give necessary and sufficient potential-theoretic conditions for the range of $\overline{\partial}$ on an open set $\Omega\subset\mathbb{C}$ to be closed in $L^{2}(\Omega)$. The closed range property is known to hold for $\overline{\partial}$ on $\Omega$ iff there exists a constant $C>0$ such that

[TABLE]

for all $u\in L^{2}(\Omega)$ with $\overline{\partial}u\in L^{2}(\Omega)$ and $u$ orthogonal to the kernel of $\overline{\partial}$, see [6, Theorem 1.1.1]. The kernel of $\overline{\partial}$ is the closed subspace of $L^{2}(\Omega)$ consisting of functions holomorphic on $\Omega$. This space is commonly called the Bergman space and denoted by $A^{2}(\Omega)$. Inequality (1.1) may be reformulated as

[TABLE]

The relevance of (1.2) (or (1.1)) lies in the fact that, if the $\overline{\partial}$-operator has closed range for an open set in dimension greater than $1$, on two consecutive form levels, then the $\overline{\partial}$-Neumann operator exists as a bounded $L^{2}$-operator. Characterizing such open sets in higher dimensions is an unresolved problem. A first step towards resolving this question is to establish necessary and sufficient conditions for the closed range property to hold on planar open sets.

Another point of interest of (1.2) is its formal similarity to the Poincaré–Wirtinger inequality. The latter is said to hold on a domain $\Omega\subset\mathbb{C}$, if there exists a constant $C>0$ such that

[TABLE]

for all $v$ in $H^{1}(\Omega)$, the $L^{2}$-Sobolev-$1$-space of $\Omega$. Here, $v_{\Omega}$ is the average value of $v$ on $\Omega$. Since the kernel, $\ker\nabla$, of $\nabla$ is either the set of constants or trivial, it follows that $v-v_{\Omega}$ is orthogonal to $\ker\nabla$. In fact, $u\in L^{2}(\Omega)\cap\left(\ker\nabla\right)^{\perp}$ iff $u_{\Omega}=0$. Thus, the Poincaré-Wirtinger inequality is

[TABLE]

Hence the closed range property of $\overline{\partial}$ may be considered a Poincaré–Wirtinger inequality for $\overline{\partial}$.

It turns out that the closed range property for $\overline{\partial}$ is more closely related to the Poincaré–Dirichlet inequality. That is, the inequality

[TABLE]

for some constant $C>0$; here $H_{0}^{1}(\Omega)$ is the completion of $\mathcal{C}^{\infty}_{c}(\Omega)$ with respect to the Sobolev-$1$-norm. At first, this might seem surprising as membership to the domain of $\overline{\partial}$ has no boundary condition folded in. However, the domain of the Hilbert space adjoint, $\overline{\partial}^{\star}$, of $\overline{\partial}$ is contained in $H_{0}^{1}(\Omega)$. Due to the Closed Range Theorem of Banach, the $\overline{\partial}$-operator has closed range if and only if its Hilbert space adjoint does. So it might be less surprising that the Poincaré–Dirichlet inequality is in fact equivalent to $\overline{\partial}$ having closed range in $L^{2}(\Omega)$, see Theorem 1.3 below.

To describe the closed range property for $\overline{\partial}$ on planar open sets in potential-theoretic terms, we use the notion of logarithmic capacity of a set in the complex plane. We denote the logarithmic capacity of a set $E\subset\mathbb{C}$ by $\operatorname{cap}{E}$; see Section 2.2 for the definition. Following nomenclature used in describing sufficiency conditions for the Poincaré–Dirichlet inequality, see, e.g., [8, §2],[9, Proposition 2.1], and references therein, we introduce the following terminology. For a set $\Omega\subset\mathbb{C}$, define the capacity inradius of $\Omega$ by

[TABLE]

see Section 2.2 for more details on this concept. Finiteness of the capacity inradius completely characterizes those open sets for which $\overline{\partial}$ has closed range:

Theorem 1.3**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Then the following are equivalent:

• (1)

$\overline{\partial}$* has closed range in $L^{2}(\Omega)$.*

• (2)

The Poincaré–Dirichlet inequality holds on $\Omega$.

• (3)

$\rho_{\operatorname{cap}}(\Omega)<\infty$.

• (4)

There exists a bounded function $\varphi\in\mathcal{C}^{\infty}(\Omega)$ and a constant $c>0$ such that $\triangle\varphi(z)>c$ holds for all $z\in\Omega$.

The implication “(4)$\Rightarrow$(1)” is a consequence of work by Hörmander in [6, Theorem 2.2.1*′* ], see also [5, Corollary 6.11]. Our proof of “(3)$\Rightarrow$(4)” is constructive. In fact, the function $\varphi$ in (4) is built from a sequence of potential functions associated to the equilibrium measures of certain compact sets in the complement of the open set.

The idea for the proof of “(3)$\Rightarrow$(4)” lead us to the completion of the characterization of planar open sets with infinite dimensional Bergman spaces in terms of the existence of bounded, strictly subharmonic functions, see (4) in the following theorem.

Theorem 1.4**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Then the following are equivalent:

• (1)

$A^{2}(\Omega)\neq\{0\}$.

• (2)

$\dim A^{2}(\Omega)=\infty$.

• (3)

$\operatorname{cap}(\Omega^{c})>0$.

• (4)

There exists a bounded function $\varphi\in\mathcal{C}^{\infty}(\Omega)$ such that $\triangle\varphi(z)>0$ for all $z\in\Omega$.

The equivalence of (1) and (2) is shown by Wiegerinck in [11], the equivalence of (1) and (3) by Carleson in [1, Theorem 1.a in §VI], the implication “(4)$\Rightarrow$(2)” by Harz, Herbort, and the first author in [4]. It follows from “(3)$\Rightarrow$(4)” and from the proof of “(4)$\Rightarrow$(2)” that $A^{2}(\Omega)$ is a separating set for $\Omega$ iff it is non-trivial.

The paper is structured as follows. We define basic notions of the $L^{2}$-theory for $\overline{\partial}$ and potential theory for open sets in the complex plane in Section 2. In this section, we also recall the connection between the best constant in the Poincaré–Dirichlet inequality and the lowest eigenvalue of the Dirichlet–Laplacian. Moreover, we derive basic characteristics of the closed range property of $\overline{\partial}$ and conclude the section with a proof of the equivalence of the closed range property for $\overline{\partial}$ and the Poincaré–Dirichlet inequality. Section 3 contains the proof of the implication “(1)$\Rightarrow$(3)” of Theorem 1.3. We first give a proof of this implication under an additional assumption, since it is based on standard $\overline{\partial}$-arguments that indicate how to approach the higher dimensional case. The general proof, also in Section 3, is based on the connection of the closed range property to the Poincaré–Dirichlet inequality for bounded open sets, a solution to the (lowest) eigenvalue problem for the Dirichlet–Laplacian on the unit disc, and so-called $r$-logarithmic potentials. The proofs of “(3)$\Rightarrow$(4)” of Theorem 1.3 and “(3)$\Rightarrow$(4)” of Theorem 1.4 are given in Sections 4 and 5, respectively. Both are constructive and based on using potential functions associated to certain compact sets in the complement of the open set in consideration.

2. Preliminaries

2.1. The $\overline{\partial}$-operator and its closed range property

on open sets in $\mathbb{C}$

For an open set $\Omega\subset\mathbb{C}$, we denote by $\mathcal{C}^{\infty}(\Omega)$ and $\mathcal{C}^{\infty}_{c}(\Omega)$ the family of smooth functions on $\Omega$ and the family of smooth functions on $\Omega$ whose (closed) support is compact in $\Omega$, respectively. As usual, $L^{2}(\Omega)$ is the space of square-integrable functions on $\Omega$, the associated norm and inner product are denoted by $\|.\|_{L^{2}(\Omega)}$ and $(.,.)_{L^{2}(\Omega)}$, respectively. The $L^{2}$-Sobolev-$1$-space, $H^{1}(\Omega)$, on $\Omega$ is the subspace of functions $f\in L^{2}(\Omega)$ for which the norm

[TABLE]

is finite. Here $\nabla f$ is meant in the sense of distributions. $H_{0}^{1}(\Omega)$ is the closure of $\mathcal{C}^{\infty}_{c}(\Omega)$ with respect to $\|.\|_{H^{1}(\Omega)}$.

The $\overline{\partial}$-operator on $\Omega$ is defined as $\overline{\partial}u=u_{\bar{z}}\,d\bar{z}$ for any $u\in\mathcal{C}^{\infty}(\Omega)$. Since $(0,1)$-forms on $\Omega$ may be identified with functions on $\Omega$, we henceforth identify $\overline{\partial}u$ with $u_{\bar{z}}$. The maximal extension of the $\overline{\partial}$-operator, still denoted by $\overline{\partial}$, is defined as follows: we first allow $\overline{\partial}$ to act on functions in $L^{2}(\Omega)$ in the sense of distributions and then restrict its domain to those functions whose image under $\overline{\partial}$ lies in $L^{2}(\Omega)$. That is,

[TABLE]

As $\mathcal{C}^{\infty}_{c}(\Omega)$ is dense in $L^{2}(\Omega)$ with respect to the $L^{2}(\Omega)$-norm, it follows that $\overline{\partial}$ is a densely defined operator on $L^{2}(\Omega)$; moreover, it is a closed operator. To define the Hilbert space adjoint, $\overline{\partial}^{\star}$, of $\overline{\partial}$ we first define its domain $\operatorname{Dom}(\overline{\partial}^{\star})$ to be the space of those $v\in L^{2}(\Omega)$ for which there exists a positive constant $C=C(v)$ such that

[TABLE]

i.e., for any $v\in\operatorname{Dom}(\overline{\partial}^{\star})$, the map $u\mapsto(\overline{\partial}u,v)_{L^{2}(\Omega)}$ is a bounded linear functional on $\operatorname{Dom}(\overline{\partial})$. Hence, by Hahn–Banach, the map extends to a bounded linear functional on $L^{2}(\Omega)$. It then follows from the Riesz Representation theorem that for any $v\in\operatorname{Dom}(\overline{\partial}^{\star})$ there exists a $w\in L^{2}(\Omega)$ such that

[TABLE]

Set $\overline{\partial}^{\star}v=w$. If $\Omega$ has smooth boundary, it follows from an integration by parts argument, that whenever $v\in\operatorname{Dom}(\overline{\partial}^{\star})\cap\mathcal{C}^{\infty}(\overline{\Omega})$, then $v|_{b\Omega}=0$ and $\overline{\partial}^{\star}v=-v_{z}$. Furthermore the following density result holds.

Lemma 2.1**.**

Let $\Omega\subset\mathbb{C}$ be an open set, then $\mathcal{C}^{\infty}_{c}(\Omega)$ is dense in $\operatorname{Dom}(\overline{\partial}^{\star})$ with respect to the graph norm

[TABLE]

This density result could be expected considering that elements of $\operatorname{Dom}(\overline{\partial}^{\star})$ in some sense vanish on the boundary while the above graph norm restricted to $\mathcal{C}^{\infty}_{c}(\Omega)$ is equivalent to $\|.\|_{H^{1}(\Omega)}$, see the argument in the proof of Lemma 2.10. A concise proof of Lemma 2.1 may be found in [10, Proposition 2.3]. For the convenience of the reader, we reproduce the proof here.

Proof.

Let $u\in\operatorname{Dom}(\overline{\partial}^{\star})$ be given. Suppose $u$ is orthogonal to all functions on $\mathcal{C}_{c}^{\infty}(\Omega)$ with respect to the inner product associated to the graph norm, i.e.,

[TABLE]

If this forces $u$ to be zero, then the claim follows. Note first that $(\overline{\partial}^{\star}u,\overline{\partial}^{\star}v)_{L^{2}(\Omega)}$, $v\in\mathcal{C}^{\infty}_{c}(\Omega)$, defines $\overline{\partial}\overline{\partial}^{\star}u$ in the sense of distributions. In particular, it follows that $u+\overline{\partial}\overline{\partial}^{\star}u$ is zero as a distribution. As $u\in L^{2}(\Omega)$, it then follows that $\overline{\partial}\overline{\partial}^{\star}u\in L^{2}(\Omega)$. Thus $(u+\overline{\partial}\overline{\partial}^{\star}u,u)_{L^{2}(\Omega)}=0$, hence

[TABLE]

Therefore $u=0$, which proves the claim. ∎

The next proposition gives basic, equivalent descriptions for the $\overline{\partial}$-operator to have closed range, which is the property that whenever $\{\overline{\partial}u_{n}\}_{n\in\mathbb{N}}$ converges in $L^{2}(\Omega)$ for $\{u_{n}\}_{n\in\mathbb{N}}\subset\operatorname{Dom}(\overline{\partial})$, then $\|\overline{\partial}u_{n}-\overline{\partial}u\|_{L^{2}(\Omega)}$ goes to [math] as $n\to\infty$ for some $u\in\operatorname{Dom}(\overline{\partial})$.

Proposition 2.2**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Then the following are equivalent:

• (i)

$\overline{\partial}$* has closed range in $L^{2}(\Omega)$.*

• (ii)

There exists a constant $C>0$ such that $\|u\|_{L^{2}(\Omega)}\leq C\|\overline{\partial}u\|_{L^{2}(\Omega)}$ holds for all $u\in\operatorname{Dom}(\overline{\partial})$ with $u\perp\ker\overline{\partial}$.

• (iii)

There exists a constant $C>0$ such that $\|v\|_{L^{2}(\Omega)}\leq C\|\overline{\partial}^{\star}v\|_{L^{2}(\Omega)}$ holds for all $v\in\operatorname{Dom}(\overline{\partial}^{\star})$ with $v\perp\ker\overline{\partial}^{\star}$.

• (iv)

There exists a constant $C>0$ such that for all $f$ in the range of $\overline{\partial}$ there exists a $v\in L^{2}(\Omega)$ such that $\overline{\partial}v=f$ holds in the distributional sense and

[TABLE]

These equivalences are well-known, and, in fact, higher dimensional analogs of (i)–(iv) are true. For the convenience of the reader, we give either references or short arguments for the proofs of Proposition 2.2.

Proof.

The equivalences of (i)–(iii) are proved in [6, Theorem 1.1.1].

To see that (iv) implies (ii), let $u\in\operatorname{Dom}(\overline{\partial})$ with $u\perp\ker\overline{\partial}$ be given. By (iv) there exists a $v\in L^{2}(\Omega)$ such that $\overline{\partial}v=\overline{\partial}u$ in the distributional sense and

[TABLE]

Note that for any $h\in A^{2}(\Omega)$, $w=v+h$ also satisfies $\overline{\partial}w=f$ in the distributional sense. Since $A^{2}(\Omega)$ is a closed subspace of $L^{2}(\Omega)$, it follows that there exists a $L^{2}(\Omega)$-minimal solution $w_{0}$ to the $\overline{\partial}$-equation with data $f$. It is easy to show that $w_{0}$ is orthogonal to $A^{2}(\Omega)$. Since $\overline{\partial}(u-w_{0})=0$ in the distributional sense, it follows from the ellipticity of $\overline{\partial}$ on functions that $u-w_{0}\in A^{2}(\Omega)$. Since $u-w_{0}$ is also in $(A^{2}(\Omega))^{\perp}$, it follows that $u=w_{0}$ so that (ii) holds.

The implication (iii)$\Rightarrow$(iv) follows from a standard duality argument, see Theorem 1.1.4 in [6, Theorem 1.1.4] with $A=\frac{1}{C}\text{Id}$, $T=\overline{\partial}$, $S=0$, $H_{1}=L^{2}(\Omega)=H_{2}$, $H_{3}=0$ and $F=(\ker\overline{\partial}^{\star})^{\perp}$. ∎

The constants in (ii)–(iv) of Proposition 2.2 may be chosen to be the same. For the best possible constant, we introduce the following notation.

Definition 2.3**.**

Let $\Omega$ be an open set in $\mathbb{C}$. Then $\overline{\partial}$ is said to have closed range in $L^{2}(\Omega)$ with constant $C$ if

[TABLE]

holds for all $u\in\operatorname{Dom}(\overline{\partial})\cap A^{2}(\Omega)^{\perp}$. In that case, set

[TABLE]

If the closed range property for $\overline{\partial}$ does not hold in $L^{2}(\Omega)$, we say that $\mathfrak{C}(\Omega)=\infty$.

2.2. Terminology from potential theory in the

plane

Let $\mu$ be a finite Borel measure with compact support in $\mathbb{C}$. The potential, $p_{\mu}$, associated to $\mu$ is defined by

[TABLE]

The energy, $I_{\mu}$, of $\mu$ is given by

[TABLE]

A set $E\subset\mathbb{C}$ is called polar if the energy of every non-trival, finite Borel measure with compact support in $E$ is $-\infty$. If for a compact set $K\subset\mathbb{C}$, there is a finite Borel probability measure $\nu$ with support in $K$ such that

[TABLE]

then $\nu$ is said to be an equilibrium measure for $K$. Any compact set has an equilibrium measure, see, e.g., [7, Theorem 3.3.2]. Moreover, this equilibrium measure is unique for any non-polar compact set, see [7, Theorem 3.7.6]. The logarithmic capacity of a compact, non-polar set $K$ is defined as $\operatorname{cap}{K}=e^{I_{\nu}}$, where $\nu$ is the equilibrium measure of $K$. If $K$ is compact and polar, then $\operatorname{cap}(K)=0$. For a general set $E\subset\mathbb{C}$, the logarithmic capacity, $\operatorname{cap}(E)$, of $E$ is defined as $\sup e^{I_{\mu}}$ for $\mu$ is a finite Borel probability measure with compact support in $E$. Note that a set $E$ is polar if and only if $\operatorname{cap}(E)=0$.

That the notion of positive logarithmic capacity comes into play for the description of the dimension of the Bergman space can be seen through the following observation. Let $K\subset\mathbb{C}$ be a compact, non-polar set, and $\mu$ the associated equilibrium measure. Then $p_{\mu}$ is a non-constant function, which is harmonic on $K^{c}$, and bounded from below by $\ln(\operatorname{cap}(K))$ by Frostman’s theorem. Thus $e^{-p_{\mu}}$ is a bounded, smooth, subharmonic, non-harmonic function on $K^{c}$. Hence it is a good candidate for the construction of subharmonic functions in part (4) of 1.4.

This construction is also used in the proof of the necessity of the existence of bounded, strictly subharmonic functions for the closed range property to hold for $\overline{\partial}$, see part (4) of Theorem 1.3. To achieve this strict subharmonicity we need compact sets, contained in the complement, and of sufficiently large logarithmic capacity, to be somewhat regularly distributed over the complex plane. This vague description can be made precise using the terminology of capacity inradius as introduced in the first section. Recall that for a set $\Omega\subset\mathbb{C}$, the capacity inradius of $\Omega$ is defined by

[TABLE]

Note that finiteness of the capacity inradius of $\Omega$ means that for any $M>\rho_{\operatorname{cap}}(\Omega)$ there is a $\delta>0$ such that for any point in $z\in\Omega$ there is a set in the complement of $\Omega$, whose logarithmic capacity is larger than $\delta$ while its distance to $z$ is less than $M$. For instance, both $\rho_{\operatorname{cap}}(\mathbb{C})$ and $\rho_{\operatorname{cap}}\left(\mathbb{C}\setminus\left(\mathbb{Z}+\sqrt{-1}\,\mathbb{Z})\right)\right)$ are infinite. However, if for given $\epsilon>0$, $K_{j,\ell}$ is the disc of radius $\epsilon$ centered at $j+\sqrt{-1}\,\ell$ or a line segment of length $\epsilon$ containing $j+\sqrt{-1}\,\ell$, then $\rho(\mathbb{C}\setminus\bigcup_{j,\ell\in\mathbb{Z}}K_{j,\ell})$ is finite. We note that in the case of the removed discs, the Poincaré–Dirichlet inequality is known to hold, see part (ii) in Proposition 2.1 in [9] and references therein. It appears to be new that, as a consequence of Theorem 1.3, the Poincaré–Dirichlet inequality is true in the removed line segments case as well.

2.3. The Poincaré–Dirichlet inequality

Let $\Omega\subset\mathbb{C}$ be an open set. The Poincaré–Dirichlet inequality is said to hold on $\Omega$, if there exists a constant $C>0$ such that

[TABLE]

Whenever (2.4) holds, it is customary to consider

[TABLE]

This notation stems from the fact that $\lambda_{1}(\Omega)$ is the smallest eigenvalue for the Dirichlet–Laplacian. In fact, if $\lambda_{1}(\Omega)$ in (2.5) is attained at $\psi\in H_{0}^{1}(\Omega)$, then, for fixed $\varphi\in H_{0}^{1}(\Omega)$, the function

[TABLE]

is differentiable near $t=0$ and has a critical point there. Unraveling the equation $f_{\varphi}^{\prime}(0)=0$ for all $\varphi\in H_{0}^{1}(\Omega)$ then leads to observing that $\psi$ is a distributional solution to the boundary value problem

[TABLE]

for $\lambda=\lambda_{1}(\Omega)$.

Furthermore, we note that, if $\Omega$ has smooth boundary and $\psi\in\mathcal{C}^{\infty}(\overline{\Omega})$ with $\psi=0$ on $bD$, then integration by parts yields

[TABLE]

2.4. Basic characteristics of the closed range property

for $\overline{\partial}$

Proposition 2.8**.**

Let $\Omega\subset\mathbb{C}$ be an open set.

• (a)

Invariance under rigid transformations: If $\Omega^{\prime}$ is obtained from $\Omega$ by translation, rotation, reflection, or a combination thereof, then $\mathfrak{C}(\Omega^{\prime})=\mathfrak{C}(\Omega)$.

• (b)

Linearity under scaling: $r\mathfrak{C}(\Omega)=\mathfrak{C}(r\Omega)$ for $r\Omega=\{rz:z\in\Omega\}$ and $r>0$.

• (c)

Invariance under polar sets: $\mathfrak{C}(\Omega)=\mathfrak{C}(\Omega^{\prime})$ for any open set $\Omega^{\prime}\subset\Omega$ such that $\Omega\setminus\Omega^{\prime}$ is polar.

Remark**.**

(i) If $\Omega\subset\mathbb{C}$ is a bounded open set, then $\mathfrak{C}(\Omega)<\infty$ by the implication “(4)$\Rightarrow$(1)” Theorem 1.3 with, say, $\varphi(z)=|z|^{2}$, see [6, Theorem 2.2.1′].

(ii) Note that (b) implies that

[TABLE]

for all $r>0$. Hence either $\mathfrak{C}(\mathbb{C})$ equals [math] or $\infty$. If $\mathfrak{C}(\mathbb{C})$ was [math], then by (ii) of Proposition 2.2 it would follow that $\|\nabla u\|_{L^{2}(\mathbb{C})}=2\|u_{z}\|_{L^{2}(\mathbb{C})}=0$ for all $u\in\mathcal{C}_{c}^{\infty}(\mathbb{C})$. As this is clearly not true, it follows that $\mathfrak{C}(\mathbb{C})=\infty$, i.e., $\overline{\partial}$ does not have closed range in $L^{2}(\mathbb{C})$.

Since $\mathbb{Z}+\sqrt{-1}\,\mathbb{Z}$ is a polar set, (c) implies that $\overline{\partial}$ for

[TABLE]

also does not have closed range in $L^{2}(\Omega)$.

(iii) An argument similar to the one given in (ii) yields that $\overline{\partial}$ for the upper half plane does not have closed range. Since $\mathfrak{C}(\mathbb{D}(0,1))$ is finite by (i), it follows that the closed range property is not invariant under biholomorphic equivalences.

Proof.

Translations and rotations are biholomorphic maps for which the absolute value of its Jacobian is $1$. Hence, invariance readily follows. Any reflection may be written as a composition of translations, rotations, and complex conjugation. So it remains to show the invariance under complex conjugation. Denote complex conjugation by $T$, i.e., $Tz=\bar{z}$ for $z\in\mathbb{C}$. Write $\Omega_{T}=\{z\in\mathbb{C}:Tz\in\Omega\}$. Observe that the map $u\mapsto T\circ u\circ T$ yields an isometry of $L^{2}(\Omega)$ and $L^{2}(\Omega_{T})$ as well as $A^{2}(\Omega)$ and $A^{2}(\Omega_{T})$. Further, one easily verifies that $\overline{\partial}(T\circ u\circ T)=T\circ\overline{\partial}u\circ T$. Hence $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ iff it has closed range in $L^{2}(\Omega_{T})$, and $\mathfrak{C}(\Omega)=\mathfrak{C}(\Omega_{T})$.

Part (b) follows straightforwardly from the fact that $\overline{\partial}(u(rz))=r(\overline{\partial}u)(rz)$ for any scalar $r$.

To prove part (c), note first that $u$, $\overline{\partial}u\in L^{2}(\Omega^{\prime})$ whenever $u$, $\overline{\partial}u\in L^{2}(\Omega)$, since $\Omega\setminus\Omega^{\prime}$ is of Lebesgue measure zero. Moreover, since $\Omega\setminus\Omega^{\prime}$ is polar, $A^{2}(\Omega)=A^{2}(\Omega^{\prime})$, see e.g., part (c) of Theorem 9.5 in [2]. Hence $A^{2}(\Omega)^{\perp}=A^{2}(\Omega^{\prime})^{\perp}$. Therefore, $\mathfrak{C}(\Omega)=\mathfrak{C}(\Omega^{\prime})$. ∎

Proposition 2.9**.**

Let $\Omega\subset\mathbb{C}$ be an open set. The Poincaré–Dirichlet inequality holds on $\Omega$ if and only if $\overline{\partial}$ has closed range in $L^{2}(\Omega)$. Moreover,

[TABLE]

In our proof of Proposition 2.9 the following two lemmata are essential.

Lemma 2.10**.**

Let $\Omega\subset\mathbb{C}$ be an open set. If the Poincaré–Dirichlet inequality holds, then

[TABLE]

Proof of Lemma 2.10.

Let $\Omega\subset\mathbb{C}$ be an open set. Then for $\varphi\in\mathcal{C}_{c}^{\infty}(\Omega)$, it follows from integration by parts that

[TABLE]

∎

Lemma 2.11**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Then non-trivial constant functions are not contained in $H_{0}^{1}(\Omega)$.

Proof.

For a function $f\in H_{0}^{1}(\Omega)$, write $\widehat{f}$ to denote the function obtained from extending $f$ to be [math] outside of $\Omega$. Note that $\widehat{f}\in H^{1}(\mathbb{R}^{2})$. In fact, since $f\in H_{0}^{1}(\Omega)$, there exists a $\{\phi_{n}\}_{n\in\mathbb{N}}\subset\mathcal{C}_{c}^{\infty}(\Omega)$ such that $\phi_{n}\longrightarrow f$ in the Sobolev-$1$-norm on $\Omega$ as $n\to\infty$. As $\{\phi_{n}\}_{n\in\mathbb{N}}$ may be considered a subset of $\mathcal{C}_{c}^{\infty}(\mathbb{R}^{2})$, we get that $\{\phi_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence in the Sobolev-$1$-norm on $\mathbb{R}^{2}$. It then follows that $\phi_{n}\longrightarrow\widehat{f}$ in the Sobolev-$1$-norm on $\mathbb{R}^{2}$ as $n\to\infty$.

Now suppose $f\equiv c\neq 0$. Then $\widehat{f}=c\chi_{\Omega}$, where $\chi_{\Omega}$ is the characteristic function of $\Omega$. Fubini’s theorem yields that there is a set $\mathcal{N}\subset\mathbb{R}$ of $1$-dimensional measure [math], such that $\widehat{f}(.,y_{0})\in H^{1}(\mathbb{R})$ for all $y_{0}\notin\mathcal{N}$. This implies that $f(.,y_{0})$ is equal to a continuous function almost everywhere in $\mathbb{R}$ for all $y_{0}\notin\mathcal{N}$. Since $\Omega$ is open, it follows that there exists a $y_{0}\notin\mathcal{N}$ such that the intersection of the line $y=y_{0}$ with $\Omega$ is of positive one-dimensional Lebesgues measure. Thus $\widehat{f}(.,y_{0})=c$ almost everywhere. This is a contradiction as $c\notin L^{2}(\mathbb{R})$. ∎

We are now set to prove Proposition 2.9.

Proof of Proposition 2.9.

Suppose the Poincaré–Dirichlet inequality holds. The proof for $\overline{\partial}$ having closed range is based on Lemmata 2.10 and 2.1. To wit, let $u\in\operatorname{Dom}(\overline{\partial}^{\star})\cap(\ker\overline{\partial}^{\star})^{\perp}$ and $\epsilon>0$ be given. Then by the density result, there exists a $\varphi\in\mathcal{C}^{\infty}_{c}(\Omega)$ such that

[TABLE]

Hence using Lemma 2.10, after an application of the triangle inequality, yields

[TABLE]

It then follows from the density result that

[TABLE]

Since $\epsilon$ was chosen arbitrarily, it follows from Proposition 2.2, that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ and $\mathfrak{C}(\Omega)\leq\frac{2}{\sqrt{\lambda_{1}(\Omega)}}$.

Now suppose that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ with constant $C$. Note first that the adjoint operator $\overline{\partial}^{\star}$ is closed because $\overline{\partial}$ is. Hence $\ker\overline{\partial}^{\star}$ is closed in $L^{2}(\Omega)$, and we obtain the orthogonal decomposition:

[TABLE]

Now let $\phi\in\mathcal{C}^{\infty}_{c}(\Omega)$. By the above decomposition of $L^{2}(\Omega)$, $\phi$ may be written as $\phi_{1}+\phi_{2}$ for some $\phi_{1}\in(\ker\overline{\partial}^{\star})^{\perp}$ and $\phi_{2}\in\ker\overline{\partial}^{\star}$. Note that both $\phi$ and $\phi_{2}$ are in $\operatorname{Dom}(\overline{\partial}^{\star})$. Since the latter is a vector space, it follows that $\phi_{1}\in\operatorname{Dom}(\overline{\partial}^{\star})$ as well. Next, by assumption $\overline{\partial}$ has closed range in $L^{2}(\Omega)$. Therefore

[TABLE]

We shall employ Lemmata 2.1 and 2.11 to show that $\phi_{2}=0$ almost everywhere. Note first that by Lemma 2.1, there exists a sequence $\{f_{n}\}\subset\mathcal{C}_{c}^{\infty}(\Omega)$ such that

[TABLE]

Therefore $\|\overline{\partial}^{\star}f_{n}\|_{L^{2}(\Omega)}$ goes to [math] as $n$ goes to $\infty$. However, since $f_{n}\in\mathcal{C}_{c}^{\infty}(\Omega)$ integrating by parts twice yields

[TABLE]

for all $n\in\mathbb{N}$, i.e., $\nabla f_{n}$ converges to [math] in $L^{2}(\Omega)$. Hence $\{f_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $H_{0}^{1}(\Omega)$. Therefore, there exists a function $f\in H_{0}^{1}(\Omega)$ such that $f_{n}\longrightarrow f$ in $L^{2}(\Omega)$ and $\nabla f=0$ almost everywhere. However, using a mollification argument, one can show that $f$ is constant almost everywhere on each connected component of $\Omega$. This is a contradiction to Lemma 2.11 unless $f=0$ almost everywhere. Since $f$ and $\phi_{2}$ are both the $L^{2}(\Omega)$-limit of $f_{n}$, it follows that $\phi_{2}=0$ almost everywhere. Thus

[TABLE]

holds. That is, the Poincaré–Dirichlet inequality holds and $\mathfrak{C}(\Omega)\geq\frac{2}{\sqrt{\lambda_{1}(\Omega)}}.$ ∎

Corollary 2.13**.**

Let $\Omega^{\prime}\subset\Omega\subset\mathbb{C}$ be open sets. Then $\mathfrak{C}(\Omega^{\prime})\leq\mathfrak{C}(\Omega)$.

Proof.

If $\mathfrak{C}(\Omega)=\infty$, the claim is trivially true. Otherwise, this follows from Proposition 2.9 and the easily verified fact that $\lambda_{1}(\Omega)\leq\lambda_{1}(\Omega^{\prime})$. ∎

It also follows from Proposition 2.9 that $\ker\overline{\partial}^{\star}=\{0\}$ which yields a somewhat stronger version of the equivalence of (i) and (iv) in Proposition 2.2.

Corollary 2.14**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Then the following are equivalent.

• (i)

$\overline{\partial}$* has closed range in $L^{2}(\Omega)$.*

• (iv*′*)

There exists a constant $C>0$ such that for all $f\in L^{2}(\Omega)$ there exists a $v\in L^{2}(\Omega)$ such that $\overline{\partial}v=f$ holds in the distributional sense and

[TABLE]

Proof.

Only the implication “(i)$\Rightarrow$(iv’)” needs to be proved. Note first that Proposition 2.9 together with the density result Lemma 2.1 implies that $\ker\overline{\partial}^{\star}=\{0\}$. Thus $(\ker\overline{\partial}^{\star})^{\perp}=L^{2}(\Omega)$. However, $(\ker\overline{\partial}^{\star})^{\perp}$ equals the closure of the range of $\overline{\partial}$, which is closed by assumption. Thus $L^{2}(\Omega)$ is the range of $\overline{\partial}$. The implication “(i)$\Rightarrow$(iv’)” now follows from “(i)$\Rightarrow$(iv)” in Proposition 2.2. ∎

In the following, we observe that the best closed range constant $\mathfrak{C}(.)$ satisfies a continuity from below property.

Proposition 2.15**.**

Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be an increasing sequence of open sets, set $\Omega=\cup_{j\in\mathbb{N}}\Omega_{j}$. If $\overline{\partial}$ has closed range in $L^{2}(\Omega_{j})$ with constant $C$ for all $j\in\mathbb{N}$, then $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ with constant $C$. In particular, $\mathfrak{C}(\Omega)=\lim_{j\to\infty}\mathfrak{C}(\Omega_{j})$.

For the proof of Proposition 2.15 we shall use the Bergman projection. Recall that for $\Omega\subset\mathbb{C}$, the Bergman projection is the orthogonal projection $B^{\Omega}$ of $L^{2}(\Omega)$ onto its closed subspace $A^{2}(\Omega)$. That is, $B^{\Omega}$ satisfies

• (a)

$B^{\Omega}\circ B^{\Omega}=B^{\Omega}$,

• (b)

$B^{\Omega}h=h$ for all $h\in A^{2}(\Omega)$,

• (c)

$B^{\Omega}f-f$ is orthogonal to $A^{2}(\Omega)$.

Proof.

Let $u\in\operatorname{Dom}(\overline{\partial})\cap\left(A^{2}(\Omega)\right)^{\perp}$ be given. Throughout, write $B$ for $B^{\Omega}$, $B^{j}$ for $B^{\Omega_{j}}$ and $\|.\|_{\Omega}$ for $\|.\|_{L^{2}(\Omega)}$, $\|.\|_{\Omega_{j}}$ for $\|.\|_{L^{2}(\Omega_{j})}$. Then for $j\in\mathbb{N}$,

[TABLE]

Let $\chi_{j}$ be the characteristic function of $\Omega_{j}$, and set $f_{j}=(1-\chi_{j})u^{2}$. Then $f_{j}$ converges to [math] almost everywhere in $\Omega$. Moreover, $|f_{j}|\leq|u|^{2}$ for all $j\in\mathbb{N}$ and $|u|^{2}$ is in $L^{1}(\Omega)$. It follows from the dominated convergence theorem, that $\lim_{j\to\infty}\int_{\Omega}f_{j}dV=\int_{\Omega}\lim_{j\to\infty}f_{j}<\infty$, i.e.,

[TABLE]

Therefore, for a given $\epsilon>0$ there exists a $j_{0}\in\mathbb{N}$, such that

[TABLE]

holds for all $j\geq j_{0}$. Hence

[TABLE]

for all $j\geq j_{0}$. Note that $u-B^{j}u$ is orthogonal to $A^{2}(\Omega_{j})$. Since $u,B^{j}u\in\operatorname{Dom}(\overline{\partial})$, so is $u-B^{j}u$. As $\overline{\partial}$ has closed range in $L^{2}(\Omega_{j})$ with constant $C$, it now follows that

[TABLE]

which yields

[TABLE]

for all $j\geq j_{0}$.

It remains to estimate the term $\left\|B^{j}u\right\|_{\Omega_{j}}$. To that end, notice first that the sequence $\{\chi_{j}B^{j}u\}_{j\in\mathbb{N}}$ is uniformly bounded in $L^{2}(\Omega)$ by $\|u\|_{L^{2}(\Omega)}$. Thus it has a weakly convergent subsequence, say, $\{\chi_{j_{k}}B^{j_{k}}u\}_{k\in\mathbb{N}}$. That is, there exists a $g\in L^{2}(\Omega)$ such that

[TABLE]

We shall first show that $g$ is holomorphic, and then use this fact to derive that $\|B^{j_{k}}u\|_{\Omega_{j_{k}}}$ converges to [math] as $k$ tends to $\infty$. It follows from (2.17) that

[TABLE]

Since $\{\Omega_{j_{k}}\}_{k}$ is an increasing sequence of open sets, it follows that for each $\varphi\in\mathcal{C}^{\infty}_{c}(\Omega)$ there exists a $j_{k_{0}}\in\mathbb{N}$ such that $\operatorname{supp}\varphi\Subset\Omega_{j_{k}}$ for all $j_{k}\geq j_{k_{0}}$. Let $S_{\varphi}$ be a smoothly bounded open set such that $\operatorname{supp}\varphi\Subset S_{\varphi}\Subset\Omega_{j_{k}}$ for all $j_{k}\geq j_{k_{0}}$. Then it follows from integration by parts that

[TABLE]

for all $j_{k}\geq j_{k_{0}}$. This, together with (2.18), implies that $(g,\varphi_{z})_{\Omega}=0$ for all $\varphi\in\mathcal{C}_{c}^{\infty}(\Omega)$. Thus, $g\in A^{2}(\Omega)$, in particular $(g,u)_{\Omega}=0$. The weak convergence (2.17) yields

[TABLE]

Since $u-B^{j_{k}}u$ is orthogonal to $A^{2}(\Omega_{j_{k}})$ while $B^{j_{k}}u\in A^{2}(\Omega_{j_{k}})$, it follows that

[TABLE]

Now repeat all arguments leading up to the estimate (2.16) with $B^{j_{k}}$ instead of $B^{j}$. Since $\left\|B^{j_{k}}u\right\|_{\Omega_{j_{k}}}$ tends to [math] as $k\to\infty$, it follows that $\|u\|_{\Omega}\leq C\|\overline{\partial}u\|_{\Omega}$.

To show that $\mathfrak{C}(\Omega_{j})$ converges to $\mathfrak{C}(\Omega)$ as $j\to\infty$, set $C:=\sup\{\mathfrak{C}(\Omega_{j}):j\in\mathbb{N}\}$. By hypothesis, $C<\infty$. The above argument then yields $\mathfrak{C}(\Omega)\leq C$. However, the monotonicity property in Corollary 2.13, yields $\mathfrak{C}(\Omega_{j})\leq\mathfrak{C}(\Omega)$. Hence $C\leq\mathfrak{C}(\Omega)$. Thus $C=\mathfrak{C}(\Omega)$ holds, which completes the proof. ∎

3. Proof of “(1)$\Rightarrow$(3)” of Theorem

1.3

3.1. Special case

The proof of “(1)$\Rightarrow$(3)” is done by contraposition, i.e., we assume that (1) holds while (3) does not. We shall first consider cases of open sets for which (3) does not hold in a particular manner, see hypothesis of Lemma 3.1 and the example below. The proof of “(1)$\Rightarrow$(3)” for these open sets is a straightforward consequence of Proposition 2.15.

Lemma 3.1**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Suppose that for all $M>0$, there exist a sequence $\{\delta_{j}\}_{j\in\mathbb{N}}\subset\mathbb{R}^{+}$ with $\lim_{j\to\infty}\delta_{j}=0$ and $\{z_{M,j}\}_{j}\subset\mathbb{C}$ such that

• (i)

$\operatorname{cap}\left(\Omega^{c}\cap\mathbb{D}(z_{M,j},M)\right)<\delta_{j}$,

• (ii)

the sequence of open sets given by $\left(\Omega\cap\mathbb{D}(z_{M,j},M)\right)-z_{M,j}\subset\mathbb{D}(0,M)$ is increasing in $j$.

Then $\overline{\partial}$ does not have closed range in $L^{2}(\Omega)$.

Example 1**.**

For each $(j,\ell)\in\mathbb{Z}\times\mathbb{Z}$, let $K_{j,\ell}$ be the closed horizontal line segment of length $\frac{\arctan(j)}{\pi}+\frac{1}{2}$ centered at $j+\sqrt{-1}\,\ell$. Let

[TABLE]

Then $\overline{\partial}$ does not have closed range in $L^{2}(\Omega)$ by Lemma 3.1. In fact, for any $m\in\mathbb{N}$,

[TABLE]

Moreover, for any $j\in\mathbb{N}$ there exists an $m_{j}\in\mathbb{N}$ such that

[TABLE]

Hence, conditions (i) and (ii) of Lemma 3.1 are satisfied with $z_{M,j}=-m_{j}$ and $\delta_{j}=\frac{1}{j}$.

That $\overline{\partial}$ does not have closed range in $L^{2}(\Omega)$ can also be seen more directly from Propositions 2.8 and 2.15. To wit, set $\Omega_{m}$ to be the shift of $\Omega$ to the right by $m$ units. It follows from part (a) of Proposition 2.8 that $\mathfrak{C}(\Omega)=\mathfrak{C}(\Omega_{m})$. Further, $\Omega_{m}\subset\Omega_{m+1}$ for any $m\in\mathbb{Z}$. By Proposition 2.15, it now follows that

[TABLE]

However, the complement of $\bigcup_{m\in\mathbb{Z}}\Omega_{m}$ is the lattice $\mathbb{Z}+\sqrt{-1}\,\mathbb{Z}$, which is a polar set. By part (c) of Proposition 2.8, it follows that

[TABLE]

Since $\mathfrak{C}(\mathbb{C})=\infty$, we obtain from (3.2) that $\mathfrak{C}(\Omega)=\infty$.

Proof of Lemma 3.1.

We suppose that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ with constant $C$. Choose $M>0$ such that $\frac{C}{M}<\mathfrak{C}(\mathbb{D}(0,1))$. By hypothesis, there exist a sequence of positive scalars $\{\delta_{j}\}_{j\in\mathbb{N}}$ with $\lim_{j\to\infty}\delta_{j}=0$ and $\{z_{j}\}_{j\in\mathbb{N}}\in\mathbb{C}$ such that

[TABLE]

Define $D_{j}$ to be the set obtained from translating $\Omega\cap\mathbb{D}(z_{j},M)$ by $-z_{j}$ and then scaling it by a factor of $\frac{1}{M}$, i.e.,

[TABLE]

Then by properties (a),(b) and (c) of Proposition 2.8,

[TABLE]

As the logarithmic capacity satisfies analogous properties, see part (c) of Theorem 5.1.2 in [7], we also have $\operatorname{cap}(\mathbb{D}(0,1)\setminus D_{j})\leq\frac{\delta_{j}}{M}$. Since $D_{j}\subset D_{j+1}$, it follows from monotonicity, see Theorem 5.1.2 (a) in [7], that

[TABLE]

By part (c) of Proposition 2.8 we then get that

[TABLE]

Moreover, Proposition 2.15 yields for any given $\epsilon>0$ a $j\in\mathbb{N}$ such that

[TABLE]

By (3.3), it then follows that

[TABLE]

This is a contradiction to the choice of $M$ for $\epsilon>0$ sufficiently small. ∎

3.2. General case

Proposition 3.4**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Suppose $\overline{\partial}$ has closed range in $L^{2}(\Omega)$. Then $\rho_{\operatorname{cap}}(\Omega)<\infty$, i.e., there exist positive constants $M$ and $\delta$ such that for each $z\in\mathbb{C}$ there exists a compact set $K\subset\Omega^{c}$ such that

[TABLE]

Lemma 3.5**.**

Let $K\subset\mathbb{C}$ be a compact set and $\epsilon>0$.

• (i)

There exists a compact set $K_{\epsilon}\subset\mathbb{C}$ such that $\mathbb{C}\setminus K_{\epsilon}$ has smooth boundary, $K\subset K_{\epsilon}$, and $\operatorname{cap}(K_{\epsilon})\leq\operatorname{cap}(K)+\epsilon$.

• (ii)

Suppose that $K$ has smooth boundary and $K\cap\mathbb{D}(0,1)$ is non-empty. Then there exists a compact set $K_{\epsilon}\subset\overline{\mathbb{D}(0,1)}$ such that $K\cap\overline{\mathbb{D}(0,1)}\subset K_{\epsilon}$, $\operatorname{cap}(K_{\epsilon})\leq\operatorname{cap}(K)+\epsilon$ and $\mathbb{D}(0,1)\setminus K_{\epsilon}$ is a smoothly bounded set.

Proof.

(i) Let $K\subset\mathbb{C}$ be a compact set, $\epsilon>0$ be given. Then, by a result due to Choquet, there exists an open, bounded set $U_{\epsilon}$ such that $K\subset U_{\epsilon}$ and $\operatorname{cap}(U_{\epsilon})\leq\operatorname{cap}(K)+\epsilon$. Moreover, there exists a smoothly bounded, open set $U^{\prime}_{\epsilon}$ such that $K\subset U^{\prime}_{\epsilon}\subset U_{\epsilon}$. Set $K_{\epsilon}=\overline{U^{\prime}}_{\epsilon}$. It then follows from monotonicity of the logarithmic capacity that $\operatorname{cap}(K_{\epsilon})\leq\operatorname{cap}(K)+\epsilon$.

(ii) Let $K\subset\mathbb{C}$ be a compact set with smooth boundary such that $K\cap\mathbb{D}(0,1)$ is non-empty. Let $\epsilon>0$ be given. As in part (i) there exists an open set $U_{\epsilon}$ such that $K\subset U_{\epsilon}$ and $\operatorname{cap}(U_{\epsilon})\leq\operatorname{cap}(K)+\epsilon$. Then there exist an open set $U^{\prime}_{\epsilon}$ such that

[TABLE]

and $\mathbb{D}(0,1)\setminus\overline{U^{\prime}_{\epsilon}}$ has smooth boundary. With $K_{\epsilon}:=\overline{U^{\prime}_{\epsilon}}$, (ii) follows. ∎

Proof of Proposition 3.4.

The proof is done by contraposition, i.e., we assume that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ with constant $C$ while $\rho_{\operatorname{cap}}(\Omega)=\infty$. That is, we assume that for each $M,\delta>0$, there is a point $z_{M,\delta}\in\mathbb{C}$ such that for any compact set $K\subset\Omega^{c}$

[TABLE]

Choose an $M>0$ such that

[TABLE]

Let $\{\delta_{j}\}_{j\in\mathbb{N}}$ be a sequence of positive scalars with $\lim_{j\to\infty}\delta_{j}=0$. For $N>M$ and positive $\epsilon_{j}<\delta_{j}$ we may choose $z_{N,\epsilon_{j}}$ such that any compact set contained in $\Omega^{c}\cap\mathbb{D}(z_{N,\epsilon_{j}},N)$ has logarithmic capacity less than $\epsilon_{j}$. For $j\in\mathbb{N}$, set

[TABLE]

Then $K_{j}^{1}\subset\overline{\mathbb{D}(z_{N,\epsilon_{j}},M)}$ is compact and, by inner regularity of the logarithmic capacity, see [7, Theorem 5.1.2(b)], $\operatorname{cap}(K_{j}^{1})\leq\epsilon_{j}$ for all $j\in\mathbb{N}$. By Lemma 3.5, for any $j\in\mathbb{N}$ there exists a compact set $K_{j}^{2}$ such that $\mathbb{D}(z_{N,\epsilon_{j}},M)\setminus K_{j}^{2}$ has smooth boundary, $K_{j}^{1}\subset K_{j}^{2}$ and $\operatorname{cap}(K_{j}^{2})<\delta_{j}$. Now set

[TABLE]

Then $K_{j}\subset\overline{\mathbb{D}(0,1)}$ is compact such that $\mathbb{D}(0,1)\setminus K_{j}$ has smooth boundary and $\operatorname{cap}(K_{j})<\frac{\delta_{j}}{M}$ for $j\in\mathbb{N}$. Next, set $D_{j}=\mathbb{D}(0,1)\setminus K_{j}$. Then $D_{j}$ is open, has smooth boundary, and $\mathfrak{C}(D_{j})\leq\frac{C}{M}$ for all $j\in\mathbb{N}$, hence

[TABLE]

for some $\epsilon>0$ by (3.6). In the following, we will show that

[TABLE]

This would conclude the proof as (3.8) is a contradiction to (3.7).

Let $\varphi$ be an eigenfunction corresponding to the first eigenvalue for the Dirichlet problem in $\mathbb{D}(0,1)$. Then $\varphi\in\mathcal{C}^{\infty}(\overline{\mathbb{D}(0,1)})$, see e.g., the remark following Theorem 1 in [3, Section 6.5]. Moreover, we may assume that $0\leq\varphi\leq 1$ on $\mathbb{D}(0,1)$. Next, let $h_{j}$ be the harmonic function in $D_{j}$ such that $h_{j}=\varphi$ on $bD_{j}$. Since $bD_{j}$ is smooth and the boundary data $\varphi$ is smooth up to the boundary of $\mathbb{D}(0,1)$, it follows that $h_{j}$ is smooth up to the boundary of $D_{j}$ as well.

Set $\psi_{j}(z)=\varphi(z)-h_{j}(z)$ for $z\in\overline{D}_{j}$.Then $\psi_{j}\in\mathcal{C}^{\infty}(\overline{D_{j}})$, $\psi_{j}=0$ on $bD_{j}$ and $\triangle\psi_{j}=\triangle\varphi$ on $D_{j}$. By (2.7)

[TABLE]

It then follows from monotonicity of the first Dirichlet eigenvalue, that

[TABLE]

As $\operatorname{cap}(\mathbb{D}(0,1)\setminus D_{j})=\operatorname{cap}(K_{j})$ goes to zero as $j\to\infty$, it follows that $\|\varphi\|_{L^{2}(D_{j})}$ approaches $\|\varphi\|_{L^{2}(D)}$. That is,

[TABLE]

To prove that (3.8) holds, it remains to show that $\lim_{j\to\infty}\|h_{j}\|_{L^{2}(D_{j})}=0$. Note first that the maximum principle yields $0<h_{j}\leq 1$ on $D_{j}$, hence $|h_{j}|^{2}\leq h_{j}$. Moreover, if $g_{j}\in\mathcal{C}(\overline{D}_{j})$ is a positive, harmonic function on $D_{j}$ such that $g_{j}=1$ on $bK_{j}\setminus b\mathbb{D}(0,1)$ and $g_{j}\geq 0$ on $b\mathbb{D}(0,1)\cap bD_{j}$, then $0<h_{j}\leq g_{j}$ on $D_{j}$. In particular,

[TABLE]

So it suffices to show that there is such a sequence $\{g_{j}\}_{j\in\mathbb{N}}$ whose $L^{1}(D_{j})$-integral converges to [math].

To construct such $g_{j}$, let $\nu_{j}$ be the equilibrium measure for $K_{j}$, and set

[TABLE]

Note that $J(\nu_{j})=\ln(2)-I(\nu_{j})$. Furthermore, as $\lim_{j\to\infty}\operatorname{cap}(K_{j})=0$, it follows that $\lim_{j\to\infty}I(\nu_{j})=-\infty$, hence $\lim_{j\to\infty}J(\nu_{j})=\infty$. Hence for $j$ sufficiently large we may define

[TABLE]

Then $g_{j}$ is a positive, harmonic function on $D_{j}$, which is non-negative on $b\mathbb{D}(0,1)\cap bD_{j}$. We claim that $g_{j}$ equals $1$ on $bK_{j}\setminus b\mathbb{D}(0,1)$ and is continuous on $\overline{D_{j}}$. To show the former, we first note that any boundary point of $D_{j}$ is a regular boundary point since any smooth defining function of $D_{j}$ serves as a subharmonic barrier function, see [7, Def. 4.1.4]. This implies that the potential function $p_{j}$ associated to the equilibrium measure $\nu_{j}$ of $K_{j}$ is equal to $I(\nu_{j})$ on $bK_{j}\setminus\mathbb{D}(0,1)$, see [7, Theorem 4.2.4]. However, this implies that $g_{j}=1$ on $bK_{j}\setminus b\mathbb{D}(0,1)$. It also implies that $p_{j}\in\mathcal{C}(\overline{D}_{j})$, see [7, Theorem 3.1.3]. Therefore, $g_{j}\in\mathcal{C}(\overline{D}_{j})$. It remains to be shown that $\int_{D_{j}}g_{j}\;dA$ converges to [math] as $j\to\infty$.

We compute

[TABLE]

Note that

[TABLE]

Therefore $\lim_{j\to\infty}\int_{D_{j}}g_{j}(z)\;dA(z)=0$, i.e., $\lim_{j\to\infty}\|h_{j}\|_{L^{2}(D_{j})}=0$, and hence (3.8) holds, which concludes the proof. ∎

4. Proof of “(3)$\Rightarrow$(4)” of Theorem

1.3

Proposition 4.1**.**

Let $\Omega\subset\mathbb{C}$ be an open set. Suppose $\rho_{\operatorname{cap}}(\Omega)<\infty$, i.e., there exist positive constants $M$ and $\delta$ such that for each $z\in\mathbb{C}$ there exists a compact set $K\subset\Omega^{c}$ such that

[TABLE]

Then there exists a bounded function $\varphi\in\mathcal{C}^{\infty}(\Omega)$ and a positive constant $c$ such that $\varphi_{z\bar{z}}(z)\geq c$ for all $z\in\Omega$.

Proof.

We first observe that whenever there exists a compact, non-polar set $K\subset\Omega^{c}$, then $\Omega$ admits a non-constant, bounded, real-analytic, subharmonic function. In fact, let $\nu_{K}$ be the equilibrium measure of $K$ such that $\operatorname{supp}(\nu_{K})\subset K$. The associated potential $p_{K}$ is given by

[TABLE]

By Frostman’s Theorem, $p_{K}(z)\geq\ln\left(\operatorname{cap}(K)\right)$ for any $z\in\Omega$. Thus the values of $e^{-p_{K}(z)}$ are in $(0,1/\operatorname{cap}(K)]$. Moreover, $p_{K}$ is harmonic, hence real-analytic. As $p_{K}$ is also non-constant, it follows that $e^{-p_{K}}$ is a non-constant, bounded, real-analytic, subharmonic function on $\Omega$.

These kinds of functions will be the building blocks for the construction of $\varphi$. In fact, we will show that there exists a sequence $\{K_{j}\}_{j\in\mathbb{N}}$ in $\Omega^{c}$ and constants $c_{1}$, $c_{2}>0$ such that

• (i)

$\varphi(z):=\sum_{j\in\mathbb{N}}e^{-4p_{K_{j}}(z)}$ is a smooth function on $\Omega$,

• (ii)

$0\leq\varphi(z)\leq c_{1}$ for all $z\in\Omega$,

• (iii)

$\triangle\varphi(z)\geq c_{2}$ for all $z\in\Omega$.

Without loss of generality, we may assume that $2\delta<M$. We claim that for each $(j,k)\in\mathbb{Z}\times\mathbb{Z}$, we may choose a compact set $K_{j,k}$ such that $\operatorname{cap}(K_{j,k})\geq\delta$ and

[TABLE]

This can be seen as follows. Suppose that for a given $(j,k)$ there was no such compact set. If there exists a

[TABLE]

then, by hypothesis, the logarithmic capacity of $\overline{\mathbb{D}(z,M)}\cap\Omega^{c}$ is at least $\delta$. This is a contradiction to our assumption since

[TABLE]

See Figure 2. Thus $\Omega^{c}$ contains $\overline{\mathbb{D}\left((2jM,2kM),\delta\right)}$ which is a compact set of logarithmic capacity $\delta$. This proves the claim.

For each $(j,k)\in\mathbb{Z}\times\mathbb{Z}$, choose a compact set $K_{j,k}$ as described above. Let $p_{K_{j,k}}$ be the associated potential; for the sake of brevity, write $p_{j,k}$ in place of $p_{K_{j,k}}$. We shall show that the series

[TABLE]

converges for any $z\in\Omega$. To do this, we will fix a $z\in\Omega$ and show convergence of the series for a particular enumeration of $\mathbb{Z}\times\mathbb{Z}$. As the terms of the series are non-negative, it will then follow that the series converges (to the same value) for any choice of enumeration.

For given $z\in\mathbb{C}$, write $\mathcal{Q}(z,L)$ for the closed square with center $z$ and side length $2L$. For fixed $z\in\Omega$, let $(j_{0},k_{0})\in\mathbb{Z}\times\mathbb{Z}$ such that $z\in\mathcal{Q}\left((2j_{0}M,2k_{0}M),M\right)$. For $\lambda\in\mathbb{N}$, set

[TABLE]

A straightforward computation yields that $\operatorname{card}(A_{\lambda})=8\lambda$ for $\lambda\geq 1$. Next note that

[TABLE]

Furthermore, if $(j,k)\in A_{\lambda}$ for some integer $\lambda\geq 2$ and $w\in K_{j,k}$, then $|z-w|\geq\left(\lambda-1\right)M$. Hence

[TABLE]

Therefore,

[TABLE]

Hence, by the Weierstraß $M$-test, the series is uniformly convergent on $\Omega$. In particular, $\varphi(z):=\sum_{(j,k)\in\mathbb{Z}\times\mathbb{Z}}e^{-4p_{j,k}(z)}$ is a well-defined function which is continuous and bounded on $\Omega$.

To see that $\varphi$ is in fact in $\mathcal{C}^{\infty}(\Omega)$, it suffices to note that all derivatives of the $p_{j,k}$’s are locally, uniformly bounded. For instance, after computing

[TABLE]

we note that for any $z_{0}\in\Omega$, there exists a neighborhood $U_{z_{0}}\subset\Omega$ of $z_{0}$ and a constant $c_{z_{0}}>0$, such that

[TABLE]

Hence for $z\in U_{z_{0}}$, it follows that

[TABLE]

i.e., the series $\sum_{(j,k)\in\mathbb{Z}\times\mathbb{Z}}\frac{\partial}{\partial z}\left(e^{-4p_{j,k}(z)}\right)$ converges locally uniformly. Thus it converges to $\frac{\partial\varphi}{\partial z}$. Similarly, if $D$ represents any differential operator, the series

[TABLE]

is convergent locally uniformly. In particular, the series converges to $D\varphi$, and $\varphi\in\mathcal{C}^{\infty}(\Omega)$.

The last observation yields in particular

[TABLE]

i.e., $\varphi$ is subharmonic on $\Omega$. It remains to be shown that $\varphi$ satisfies condition (iii). For that, it suffices to show that there exists a constant $c>0$ such that for each $z\in\Omega$ there exists a $(j,k)\in\mathbb{Z}\times\mathbb{Z}$ such that

[TABLE]

A straightforward computation yields

[TABLE]

for any given $z\in\Omega$ and $(j,k)\in\mathbb{Z}\times\mathbb{Z}$. Now let $z\in\Omega$ be given. Then, as before, $z\in\mathcal{Q}((2j_{0}M,2k_{0}M),M)$ for some $(j_{0},k_{0})\in\mathbb{Z}\times\mathbb{Z}$. Let $(j_{1},k_{1})=(j_{0}-2,k_{0}-2)$. Note that for any $w\in K_{j_{1},k_{1}}$

[TABLE]

holds. Thus $|\nabla p_{j_{1},k_{1}}(z)|\geq\frac{1}{2\cdot 98M}$. Moreover, $e^{-4p_{j_{1},k_{1}}(z)}\geq\frac{1}{2}M^{-4}$. Hence, $\triangle\varphi(z)\geq\frac{2}{49M^{5}}$ for all $z\in\Omega$. See Figure 3.

∎

5. Proof of “(3)$\Rightarrow$(4)” of Theorem

1.4

Proposition 5.1**.**

Let $\Omega\subset\mathbb{C}$ be an open set such that $\operatorname{cap}(\Omega^{c})>0$. Then there exists a bounded $\varphi\in\mathcal{C}^{\infty}(\Omega)$ such that $\triangle\varphi>0$ on $\Omega$.

Proof.

Note that there exists a compact set $K\subset\Omega^{c}$ such that $\operatorname{cap}(K)>0$. Let $\nu$ be the equilibrium measure of $K$, and $p$ be the associated potential function, i.e.,

[TABLE]

Recall that $p$ is harmonic in $\Omega$ and $p\geq\ln(\operatorname{cap}(K))$. Hence $\varphi:=e^{-p}$ is smooth and bounded on $\Omega$. Moreover, $\triangle\varphi=e^{-p}|\nabla p|^{2}$.

We shall show first that there is an open ball $B$ such that $K\subset B$ and $|\frac{\partial p}{\partial z}(z)|>0$ for all $z\in B^{c}\cap\Omega$. Let $B$, $B^{\prime}$ be concentric open balls, such that $K\subset B^{\prime}$, and the radius of $B$ is twice the radius of $B^{\prime}$. Let $z\in\Omega\cap B^{c}$ be given, $\theta\in[0,2\pi)$ fixed and chosen later. See Figure 4. Then

[TABLE]

where $\alpha(z-w)$ is the branch of the argument of $z-w$ in $[0,2\pi)$. Since the radius of $B$ is twice the radius of $B^{\prime}$, it follows that there is some $c\in(0,\pi/2)$ such that

[TABLE]

This allows us to choose $\theta$ such that $\alpha(z-w)-\theta\in[0,c)\subset[0,\pi)$ for all $w\in K$. Hence $\cos(\alpha(z-w)-\theta)>0$ for all $w\in K$. Thus $|\frac{\partial p}{\partial z}(z)|>0$ for all $z\in\Omega\cap B^{c}$.

Let $B^{\prime\prime}$ be an open set containing $\overline{B}$. Then, since $p$ is smooth in $\Omega$ and $|\nabla p|>0$ on $\Omega\cap B^{c}$ there exists a constant $C>0$ such that $\triangle\varphi>C$ on$\overline{\Omega}\cap\left(\overline{B^{\prime\prime}}\setminus B\right)$. Let $z_{0}$ be the center of $B^{\prime}$, let $\chi\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ such that $\chi(|z-z_{0}|^{2})=|z-z_{0}|^{2}$ for all $z\in B$ and $\text{supp}(\chi(|z-z_{0}|^{2}))\subset B^{\prime\prime}$. Then $\varphi+\epsilon\chi(|z-z_{0}|^{2})$ is a smooth, bounded function on $\Omega$ which is strictly subharmonic everywhere as long as $\epsilon>0$ is sufficiently small. ∎