Compactness of the Complex Green Operator on C¹ Pseudoconvex Boundaries in Stein Manifolds

Abstract

We study compactness for the complex Green operator ${\mathsf{G}}_q$ associated with the Kohn Laplacian ${\square }_b$ on boundaries of pseudoconvex domains in Stein manifolds. Let $\Omega ⋐X$ be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with $C^1$ boundary. For $1\le q\le n-2$, we first prove a compactness theorem under weak potential-theoretic hypotheses: if $b\Omega$ satisfies weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$, then ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$. This extends known $C^{\infty }$ results in ${\mathbb{C}}^n$ to the minimal regularity $C^1$ and to the Stein setting. On locally convexifiable $C^1$ boundaries, we obtain a full characterization: compactness of ${\mathsf{G}}_q$ is equivalent to simultaneous compactness of ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$, to compactness of the $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ in the interior, to weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$, and to the absence of (germs of) complex varieties of dimensions q and $n-1-q$ on $b\Omega$. A key ingredient is an annulus compactness transfer on ${\Omega }_{+}={\Omega }_2\setminus \overline{{\Omega }_1}$, which yields compactness of ${\mathsf{N}}_q^{{\Omega }_{+}}$ from weak $\left(\mathcal{P}\right)$ near each boundary component and allows us to build compact ${\overline{\partial }}_b$-solution operators via jump formulas. Consequences include the following: compact canonical solution operators for ${\overline{\partial }}_b$, compact resolvent for ${\square }_b$ on the orthogonal complement of its harmonic space (hence discrete spectrum and finite-dimensional harmonic forms), equivalence between compactness and standard compactness estimates, closed range and ${\mathsf{L}}^2$ Hodge decompositions, trace-class heat flow, stability under $C^1$ boundary perturbations, vanishing essential norms, Sobolev mapping (and gains under subellipticity), and compactness of Bergman-type commutators when $q=1$.

1. Introduction and Main Results

On the boundary $b\Omega$ of a domain $\Omega$, the operator $\overline{\partial }$ induces the tangential Cauchy–Riemann operator ${\overline{\partial }}_b$. Beyond its central role in several complex variables, ${\overline{\partial }}_b$ is fundamental for the analysis of boundary value problems of elliptic and hypoelliptic type. Denote by ${\overline{\partial }}_b^{\ast }$ the ${\mathsf{L}}^2$-adjoint of ${\overline{\partial }}_b$, and set

$${\square }_b={\overline{\partial }}_b{\overline{\partial }}_b^{\ast }+{\overline{\partial }}_b^{\ast }{\overline{\partial }}_b,$$

with the Kohn Laplacian on $(p,q)$-forms on $b\Omega$. For $0\le q\le n-1$, the operator ${\square }_b$ is invertible on the orthogonal complement of its harmonic space ${\mathcal{H}}_{p,q}^b=ker{\square }_b$. We denote the inverse by ${\mathsf{G}}_q$ and the complex Green operator on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$.

A symmetry between form levels q and $n-1-q$ was observed by Koenig [1] (p. 289). To a $(0,q)$-form u on $b\Omega$, he associates a $(0,n-1-q)$-form $\tilde{u}$ (via a modified Hodge-★ construction) such that $\left|u\right|\approx |\tilde{u}|$, and, modulo terms of order $O\left(\right|u\left|\right)$,

$${\overline{\partial }}_b\tilde{u}={(-1)}^q\widetilde{\left({\overline{\partial }}_b^{\ast }u\right)},{\overline{\partial }}_b^{\ast }\tilde{u}={(-1)}^{q+1}\widetilde{\left({\overline{\partial }}_{b}u\right)}.$$

Consequently, a compactness estimate in degree q holds if and only if the corresponding estimate holds in degree $n-1-q$. In contrast, by characterizations of compactness for the $\overline{\partial }$-Neumann problem (e.g., on convex domains) [2,3,4], no such symmetry exists for the interior $\overline{\partial }$-Neumann operator ${\mathsf{N}}_q$.

The ${\overline{\partial }}_b$-complex was introduced by Kohn and Rossi [5] to study boundary values of holomorphic functions and holomorphic extension. The study of the ${\overline{\partial }}_b$-Neumann problem on CR manifolds has been central to several works. In particular, $L^2$ existence theorems on strongly pseudoconvex CR manifolds were established by Shaw [6], while compactness conditions for the $\overline{\partial }$-Neumann operator were given by McNeal [7]. Catlin [8] formulated a potential-theoretic condition $\left(P\right)$ that substitutes for blow-up of the complex Hessian. Its $(0,q)$-form generalization $\left({\mathcal{P}}_q\right)$ is a well-known sufficient condition for compactness of the $\overline{\partial }$-Neumann operator (see [9,10] for background on compactness in the $\overline{\partial }$-Neumann problem). Closed range of ${\overline{\partial }}_b$ (and of ${\overline{\partial }}_b^{\ast }$ and ${\square }_b$) in ${\mathsf{L}}^2\left(b\Omega \right)$ was established for smooth bounded pseudoconvex domains in ${\mathbb{C}}^n$ in [11,12], and for compact pseudoconvex orientable CR-submanifolds of hypersurface type and dimension at least five in [13]. In the setting of strongly pseudoconvex boundaries, it follows from foundational results of Saber [14]. For more general geometries, including weakly pseudoconvex domains, closed range theorems were developed by Kohn [12], Shaw [10], and Boas–Shaw [11], with further refinements by Rosay [15], Michel–Shaw [16], and Li–Shaw [17]; more results can be seen in [18,19,20]. Saber also proves global solvability and regularity of the $\tilde{\partial }$-problem on annuli between weakly convex domains satisfying property (P) in [21]. Homotopy and attractivity frameworks were further extended in [22,23]. The eigenvalue problem associated with the $(p,q)$-Laplacian has been analyzed in [24], while Sobolev-type solutions were discussed in [25,26,27,28].

The complex Green operator and its compactness have been studied extensively in various settings. For instance, Boas and Straube (1991) explored Sobolev estimates for the complex Green operator on weakly pseudoconvex boundaries in Stein manifolds, revealing important results about boundary conditions and compactness [29]. Fu and Straube (1998) furthered this understanding by establishing the compactness of the $\overline{\partial }$-Neumann problem on convex domains [2]. McNeal and Straube (2002) expanded on these ideas by providing additional results on the compactness of the $\overline{\partial }$-Neumann operator [30].

For smooth bounded pseudoconvex domains satisfying $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$, Raich and Straube [31] proved that ${\mathsf{G}}_q$ is compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$ [31]. ${\mathcal{H}}_{p,q}^b\left(b\Omega \right)=ker{\square }_b$ denotes boundary harmonic $(p,q)$-forms, and $H^q\left(\Omega \right)$ denotes ${\mathsf{L}}^2$-harmonic $(p,q)$-forms in the interior. Weak $\left({\mathcal{P}}_q\right)$ refers to bounded plurisubharmonic weights with a uniform lower bound on the q-trace of the complex Hessian in collars of $b\Omega$. The precise definition, tailored to $C^1$ boundaries via smoothing and truncation, is given in §2. The analysis of partial differential equations on complex domains occupies a central place in modern mathematics, with profound connections to operator theory, spectral geometry, and mathematical physics. At the heart of this theory lies the ${\overline{\partial }}_b$-complex on domain boundaries, which governs boundary values of holomorphic functions and plays a fundamental role in elliptic and hypoelliptic boundary value problems. The compactness properties of operators associated with this complex—particularly the complex Green operator ${\mathsf{G}}_q$ and the $\overline{\partial }$-Neumann operator ${\mathsf{N}}_q$—have far-reaching implications: from the discreteness of spectra in quantum systems to the regularity of solutions in geometric analysis, and from the structure of Toeplitz algebras in operator theory to the finiteness of cohomology in complex geometry.

Despite extensive study in smooth settings, two fundamental limitations have persisted:

• Classical results require $C^{\infty }$ boundary regularity, excluding many geometrically and physically relevant domains with minimal regularity.
• The theory has been largely confined to domains in ${\mathbb{C}}^n$, limiting applications to more general complex manifolds.

This work bridges these gaps by developing a comprehensive compactness theory for the complex Green operator on $C^1$ pseudoconvex boundaries in Stein manifolds, establishing both the minimal regularity and maximal geometric generality for which such results hold.

• The main results are as follows. We extend the Raich–Straube compactness theorem [31] in two directions:

(i)

from domains in ${\mathbb{C}}^n$ to bounded pseudoconvex domains in Stein manifolds,

(ii)

from $C^{\infty }$ to minimal boundary regularity $C^1$.

Specifically, if $\Omega ⋐X$ is a bounded pseudoconvex domain in a Stein manifold X of complex dimension n, $b\Omega$ is of class $C^1$, and $b\Omega$ satisfies weak versions of $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$ for $1\le q\le n-2$, then the complex Green operator ${\mathsf{G}}_q$ is compact (and by the $q\leftrightarrow n-1-q$ symmetry, so is ${\mathsf{G}}_{n-1-q}$).

Working out $C^1$ regularity is subtle: the classical $\overline{\partial }$-Neumann theory relies on higher boundary smoothness, so we replace it with a streamlined potential-theoretic approach and an annulus method for compactness on ${\Omega }_{+}:={\Omega }_2\setminus {\Omega }_1$, combined with jump formulas of Shaw [32] and $\overline{\partial }$-techniques in the spirit of [31] to produce compact ${\overline{\partial }}_b$-solution operators. On locally convexifiable $C^1$ boundaries, we further prove that geometric $\left({\mathcal{P}}_q\right)$-type conditions are equivalent to compactness of ${\mathsf{G}}_q$.

Theorem 1

(Main compactness theorem). Let Ω be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n, and let $0\le p\le n$, $1\le q\le n-2$. Suppose $b\Omega$ is of class $C^1$ and satisfies weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$. Then, ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact operators on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$ and ${\mathsf{L}}_{p,n-1-q}^2\left(b\Omega \right)$, respectively.

Our proof proceeds by (1) establishing compactness estimates for the $\overline{\partial }$-Neumann problem on a collar annulus ${\Omega }_{+}$ using weak $\left({\mathcal{P}}_q\right)$-type weights near each boundary component, (2) deducing compactness of ${\mathsf{N}}_q^{{\Omega }_{+}}$ and compact canonical solution operators on ${\Omega }_{+}$, and (3) transferring compactness to the boundary via jump decompositions to obtain compact right inverses for ${\overline{\partial }}_b$, which yield compactness of ${\mathsf{G}}_q$ in degrees q and $n-1-q$.

Theorem 2

(Equivalence on locally convexifiable $C^1$ boundaries). Let Ω be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with $C^1$ boundary, and let $0\le p\le n$, $1\le q\le n-2$. Assume in addition that $b\Omega$ is locally convexifiable. Then, the following are equivalent:

(i) 

${\mathsf{G}}_q$ is compact.

(ii) 

${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact.

(iii) 

${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ are compact.

(iv) 

$b\Omega$ satisfies weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$.

(v) 

$b\Omega$ contains no (germs of) complex varieties of complex dimension q nor of dimension $n-1-q$.

Theorem 3

(Annulus compactness transfer). Let ${\Omega }_1⋐{\Omega }_2$ be bounded pseudoconvex domains and ${\Omega }_{+}={\Omega }_2\setminus \overline{{\Omega }_1}$. Assume $b{\Omega }_1$ and $b{\Omega }_2$ are of class $C^1$ and satisfy weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$. Then, ${\mathsf{N}}_q^{{\Omega }_{+}}$ is compact. Consequently, the canonical solution operators ${\overline{\partial }}^{\ast }{\mathsf{N}}_q^{{\Omega }_{+}}$ and ${\overline{\partial }}^{\ast }{\mathsf{N}}_{q+1}^{{\Omega }_{+}}$ are compact, and ${\mathsf{L}}^2$-cohomology $H_{\left(2\right)}^{p,q}\left({\Omega }_{+}\right)$ is finite-dimensional.

Further consequences. Beyond the main compactness theorems, our analysis yields several structural consequences for the boundary complex and for the interior $\overline{\partial }$-Neumann problem.

(1)

Canonical boundary solutions are compact. Compactness of ${\mathsf{G}}_q$ propagates to the natural right inverses for ${\overline{\partial }}_b$ and ${\overline{\partial }}_b^{\ast }$: the maps

$$K_q={\overline{\partial }}_b^{\ast }{\mathsf{G}}_q:{\mathsf{L}}_{p,q}^2\left(b\Omega \right)\to {\mathsf{L}}_{p,q-1}^2\left(b\Omega \right),K_{q+1}={\overline{\partial }}_b^{\ast }{\mathsf{G}}_{q+1}:{\mathsf{L}}_{p,q+1}^2\left(b\Omega \right)\to {\mathsf{L}}_{p,q}^2\left(b\Omega \right),$$

are compact. This is key for transferring compactness between degrees and for establishing boundary Hodge decompositions.

(2)

Spectral discreteness. On ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$, the Kohn Laplacian ${\square }_b$ has compact resolvent. Consequently, in degree q, the spectrum is purely discrete with finite multiplicities accumulating only at $+\infty$, and $dim{\mathcal{H}}_{p,q}^b<\infty$.

(3)

Compactness ⟺ compactness estimates. On the boundary, ${\mathsf{G}}_q$ is compact if and only if

$${\parallel f\parallel }_{L^2\left(b\Omega \right)}^2\le \epsilon \left(\parallel {\overline{\partial }}_b{f\parallel }_{L^2}^2+{\parallel {\overline{\partial }}_b^{\ast }f\parallel }_{L^2}^2\right)+C_{\epsilon }{\parallel f\parallel }_{W^{-1}\left(b\Omega \right)}^2,$$

for all f in the graph domain; an analogous statement holds in the interior: ${\mathsf{N}}_q$ is compact if

$${\parallel u\parallel }_{L^2\left(\Omega \right)}^2\le \epsilon \left(\parallel \overline{\partial }{u\parallel }_{L^2}^2+{\parallel {\overline{\partial }}^{\ast }u\parallel }_{L^2}^2\right)+C_{\epsilon }{\parallel u\parallel }_{W^{-1}\left(\Omega \right)}^2,$$

for all u in the graph domain. These equivalences connect functional–analytic compactness with quantitative ${\mathsf{L}}^2$ control.

(4)

Duality in conjugate degrees. By the $q\leftrightarrow n-1-q$ symmetry for the boundary complex, ${\mathsf{G}}_q$ is compact if and only if ${\mathsf{G}}_{n-1-q}$ is compact.

(5)

Geometric obstruction. If ${\mathsf{G}}_q$ is compact, then $b\Omega$ cannot contain a germ of a complex analytic subvariety of complex dimension q or $n-1-q$. This rules out flat complex directions along which compactness would fail.

(6)

Stability under perturbations. If ${\Omega }_j\to \Omega$ in the $C^1$ topology, and weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$ hold on collars of $b{\Omega }_j$ with constants uniform in j, then ${\mathsf{G}}_q\left({\Omega }_j\right)$ are uniformly compact, and the limit domain also enjoys compactness of ${\mathsf{G}}_q$.

(7)

Sobolev mapping and gains. On $C^{\infty }$ boundaries, compactness of ${\mathsf{G}}_q$ implies that ${\mathsf{G}}_q$, $K_q$, and $K_{q+1}$ extend boundedly on every Sobolev scale $W^s$, $s\ge 0$. If, in addition, a subelliptic estimate of order $ϵ>0$ holds in degree q, then ${\mathsf{G}}_q:W^s\to W^{s+2ϵ}$ and $K_q,K_{q+1}:W^s\to W^{s+ϵ}$ boundedly.

(8)

Bergman commutators when $q=1$. If ${\mathsf{N}}_1$ is compact and P denotes the Bergman projection on $(p,0)$-forms in $\Omega$, then for every $\phi \in C^1\left(\overline{\Omega }\right)$ the commutator $[P,\phi ]$ is compact on $L^2$; equivalently, $[\phi ,{\overline{\partial }}^{\ast }]{\mathsf{N}}_1$ and $[\phi ,\overline{\partial }]{\mathsf{N}}_1$ are compact.

(9)

Essential norm collapse. The essential norms of ${\mathsf{G}}_q$ and ${\mathsf{N}}_q$ vanish. Hence, each can be uniformly approximated in operator norm by finite-rank maps.

The primary motivation for extending compactness results from $C^{\infty }$ boundaries to $C^1$ boundaries in Stein manifolds is to relax the regularity conditions necessary for the complex Green operator’s compactness. In classical results, compactness of the Green operator $G_q$ is established for smooth boundaries. However, many real-world applications involve domains with lower regularity, such as $C^1$ boundaries, where classical results do not hold.

The extension to $C^1$ boundaries in Stein manifolds allows for a broader class of domains to be considered, thus making the results applicable to more realistic geometries while maintaining the essential properties of the compactness. This is achieved by introducing weak potential-theoretic hypotheses like weak versions of property $\left(P_q\right)$ and $\left(P_{n-1-q}\right)$, which still ensure compactness under these relaxed conditions. Moreover, this work builds on the symmetry between the degrees q and $n-1-q$, extending the applicability of compactness results from higher regularity settings (like smooth domains) to minimal regularity settings, such as $C^1$ domains, within the context of Stein manifolds.

Our results yield a comprehensive structural theory with applications across multiple domains:

Compactness implies discrete spectrum for ${\square }_b$ with finite multiplicities, enabling thermodynamic analysis via trace-class heat flow $e^{-t{\square }_b}$. This provides the mathematical foundation for studying quantum systems on CR manifolds.

For $q=1$, compactness of ${\mathsf{N}}_1$ implies essential normality of the Bergman projection, with compact commutators $[P,\phi ]$ for $\phi \in C^1\left(\overline{\Omega }\right)$. This reveals the structure of Toeplitz algebras on minimally regular domains.

The equivalence between compactness and geometric conditions (absence of complex varieties) establishes a bridge between analytic properties of operators and the underlying geometry of boundaries.

Compactness estimates enable rigorous convergence analysis for numerical schemes, while stability under $C^1$ perturbations ensures robustness of physical models.

The paper is organized as follows: Section 2 establishes notation and preliminary results; Section 3 contains the proof of Theorem 1; Section 4 proves the equivalence theorem; Section 5 develops the consequences; and Section 6 discusses the advantages of our annulus method.

2. Basic Properties

Let X be a complex manifold of complex dimension n endowed with a Hermitian metric $\langle ·,·\rangle$ and associated $(1,1)$-form $\omega$. Let $\Omega ⋐X$ be a relatively compact domain with smooth boundary $b\Omega =\{\rho =0\}$ given by a $C^{\infty }$ defining function $\rho$ such that $|\partial \rho |=1$ on $b\Omega$.

Fix $z_0\in b\Omega$. In a neighborhood U of $z_0$, choose a special orthonormal $(1,0)$-frame

$$L_1,\dots ,L_{n-1},L_n\in T^{1,0}X\mathrm{on}U\cap \overline{\Omega },$$

such that $L_1,\dots ,L_{n-1}$ are tangential to $b\Omega$ (so $L_j\rho =0$ on $b\Omega$ for $1\le j\le n-1$) and $L_n$ is normalized by $L_n\rho =1$ on $b\Omega$. Let ${\overline{L}}_1,\dots ,{\overline{L}}_n$ be the conjugate $(0,1)$-frame; together, $\left\{L_j\right\}$ and $\left\{{\overline{L}}_k\right\}$ are orthonormal with respect to $\langle ·,·\rangle$. Denote the dual coframe of $(1,0)$-forms by

$${\omega }^1,\dots ,{\omega }^{n-1},{\omega }^n=\partial \rho \mathrm{on}U,$$

so that ${\omega }^i\left(L_j\right)={\delta }_j^i$. (With this choice, $L_n\rho ={\omega }^n\left(L_n\right)=1$ on $b\Omega$ and $L_j\rho =0$ for $j\le n-1$).

For a real $C^2$ function $\phi$, we write its complex Hessian in the frame $\left\{{\omega }^j\right\}$ as

$$\partial \overline{\partial }\phi =\sum _{j,k=1}^n{\phi }_{j\overline{k}}{\omega }^j\wedge {\overline{\omega }}^k.$$

In particular, the Levi form of the boundary is encoded by

$$\partial \overline{\partial }\rho =\sum _{j,k=1}^n{\rho }_{j\overline{k}}{\omega }^j\wedge {\overline{\omega }}^k,\mathrm{and}{\rho }_{j\overline{n}}={\rho }_{n\overline{j}}=0\mathrm{on}b\Omega ,1\le j\le n-1.$$

2.1. Forms, Inner Products, and Basic Operators

Write a $(p,q)$-form as $f={\sum }_{\left|I\right|=p,\left|J\right|=q}^{\prime }{f}_{I,J}{\omega }^I\wedge {\overline{\omega }}^J$, where the prime denotes summation over strictly increasing multiindices $I=(i_1<\cdots <i_p)$, $J=(j_1<\cdots <j_q)$, and ${\omega }^I={\omega }^{i_1}\wedge \cdots \wedge {\omega }^{i_p}$, ${\overline{\omega }}^J={\overline{\omega }}^{j_1}\wedge \cdots \wedge {\overline{\omega }}^{j_q}$. The pointwise Hermitian inner product on ${\wedge }^{p,q}$ is

$$(f,g)={\sum }_{I,J}^{\prime }{f}_{I,J}\overline{g_{I,J}},$$

which is independent of the choice of orthonormal frame. Let ${\wedge }_z^{p,q}$ be the fiber of $(p,q)$-forms at z, and $C_{p,q}^{\infty }\left(\Omega \right)$ the space of smooth $(p,q)$-forms on $\Omega$.

The Cauchy–Riemann operator

$$\overline{\partial }:C_{p,q-1}^{\infty }\left(\Omega \right)⟶C_{p,q}^{\infty }\left(\Omega \right)$$

has the local expression

$$\overline{\partial }f={\sum }_{I,J}^{\prime }\sum _{k=1}^n{\overline{L}}_k\left(f_{I,J}\right){\overline{\omega }}^k\wedge {\omega }^I\wedge {\overline{\omega }}^J+(\mathrm{terms}\mathrm{of}\mathrm{order}0\mathrm{in}f),$$

(1)

where order 0 terms come from connection coefficients of the frame.

Boundary test class. Let ${\mathcal{D}}_{p,q}\left(U\right)$ denote the space of $(p,q)$-forms f on U whose coefficients satisfy

$$f_{I,J}{|}_{b\Omega }=0\mathrm{whenever}n\in J,$$

(2)

so that f has no $(0,1)$-component in the complex normal direction on $b\Omega$. On ${\mathcal{D}}_{p,q}\left(U\right)$, the formal adjoint of $\overline{\partial }$ with respect to the weighted ${\mathsf{L}}_{\phi }^2$ inner product (defined below) takes the local form

$${\vartheta }_{\phi }f={(-1)}^p{\sum }_{\left|I\right|=p,\left|K\right|=q-1}^{\prime }\sum _{j=1}^n{\delta }_j^{\phi }\left(f_{I,jK}\right){\omega }^I\wedge {\overline{\omega }}^K+(\mathrm{order}0\mathrm{terms}\mathrm{not}\mathrm{involving}\phi ),$$

(3)

where

$${\delta }_j^{\phi }:=e^{\phi }{L}_j\left(e^{-\phi }\right)=-L_j\left(\phi \right).$$

(If one keeps track of connection coefficients, ${\delta }_j^{\phi }$ is replaced by $L_j-L_j\left(\phi \right)$ plus tangential zeroth-order terms; for our weighted ${\mathsf{L}}^2$ estimates, it is enough to record the $-L_j\left(\phi \right)$ contribution explicitly.)

2.2. Weighted ${\mathsf{L}}^2$ and Sobolev Spaces

Let $dV$ be the volume form induced by $\omega$. For a real $C^2$ weight $\phi$, define

$${\langle f,g\rangle }_{\phi }={\int }_{\Omega }(f,g)e^{-\phi }dV,{\parallel f\parallel }_{\phi }^2={\langle f,f\rangle }_{\phi }.$$

Write ${\mathsf{L}}_{p,q}^2(\Omega ,\phi )$ for the completion of $C_{p,q}^{\infty }\left(\Omega \right)$ under ${\parallel ·\parallel }_{\phi }$; when $\phi \equiv 0$, we abbreviate ${\mathsf{L}}_{p,q}^2\left(\Omega \right)={\mathsf{L}}_{p,q}^2(\Omega ,0)$.

For $k\in \mathbb{N}$, the (unweighted) Sobolev space is

$$W_{p,q}^k\left(\Omega \right)=\left\{f\in {\mathsf{L}}_{p,q}^2\left(\Omega \right):\sum _{\left|\alpha \right|\le k}{\parallel D^{\alpha }f\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2<\infty \right\},$$

where $D^{\alpha }={\partial }_{x_1}^{{\alpha }_1}\cdots {\partial }_{x_{2n}}^{{\alpha }_{2n}}$ in local real coordinates, and $W_{p,q}^{-1}\left(\Omega \right)$ denotes the dual of $W_{p,q,0}^1\left(\Omega \right)$ (the closure of $C_c^{\infty }$ in $W^1$). We write ${\parallel ·\parallel }_{W^{-1}\left(\Omega \right)}$ for the corresponding norm. (Weighted Sobolev spaces $W_{p,q}^k(\Omega ,\phi )$ are defined analogously.)

2.3. Maximal Closures, Adjoints, and Laplacians

Let

$$\overline{\partial }:\mathrm{dom}\overline{\partial }\subset {\mathsf{L}}_{p,q}^2\left(\Omega \right)⟶{\mathsf{L}}_{p,q+1}^2\left(\Omega \right)$$

be the maximal closed extension of Equation (1) (i.e., distributional $\overline{\partial }$ with ${\mathsf{L}}^2$ graph-closure). Denote by ${\overline{\partial }}^{\ast }$ its Hilbert space adjoint (with respect to the unweighted ${\mathsf{L}}^2$ inner product). The complex Laplacian on $(p,q)$-forms is

$${\square }_q=\overline{\partial }{\overline{\partial }}^{\ast }+{\overline{\partial }}^{\ast }\overline{\partial },\mathrm{dom}{\square }_q\subset \mathrm{dom}\overline{\partial }\cap \mathrm{dom}{\overline{\partial }}^{\ast },$$

self-adjoint and nonnegative on ${\mathsf{L}}_{p,q}^2\left(\Omega \right)$. The $\overline{\partial }$-Neumann operator

$${\mathsf{N}}_q:{\mathsf{L}}_{p,q}^2\left(\Omega \right)⟶{\mathsf{L}}_{p,q}^2\left(\Omega \right)$$

is the inverse of ${\square }_q$ on ${(ker{\square }_q)}^{\perp }$; by standard theory ${\mathsf{N}}_q$ is bounded.

The space of ${\mathsf{L}}^2$-harmonic $(p,q)$-forms is

$${\mathcal{H}}^{p,q}\left(\Omega \right)=\{u\in \mathrm{dom}\overline{\partial }\cap \mathrm{dom}{\overline{\partial }}^{\ast }:\overline{\partial }u={\overline{\partial }}^{\ast }u=0\}=ker{\square }_q.$$

2.4. Compactness Estimate

We say a compactness estimate holds for $\overline{\partial }$ on $(p,q)$-forms if

$${\parallel u\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2\le MQ(u,u)+C_M{\parallel u\parallel }_{W^{-1}\left(\Omega \right)}^2,\forall u\in \mathrm{dom}\overline{\partial }\cap \mathrm{dom}{\overline{\partial }}^{\ast },$$

(4)

for every $M>0$ and some $C_M>0$, where $Q(u,u)=\parallel \overline{\partial }{u\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2+{\parallel {\overline{\partial }}^{\ast }u\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2$. By a standard Rellich–Kondrachov argument, Equation (4) holds if and only if ${\mathsf{N}}_q$ is compact on ${\mathsf{L}}_{p,q}^2\left(\Omega \right)$.

2.5. Catlin’s Condition $\left({\mathcal{P}}_q\right)$ on the Boundary

Definition 1.

The boundary $b\Omega$ satisfies $\left({\mathcal{P}}_q\right)$ if for every $M>0$ there exist a neighborhood $U_M\supset b\Omega$ and a function ${\lambda }_M\in C^2\left(U_M\right)$ such that

1. 

$0\le {\lambda }_M\le 1$ on $U_M$;

2. 

for all $z\in b\Omega$ and all q-dimensional complex subspaces $E\subset T_z^{1,0}b\Omega$,

$${\mathrm{tr}}_E\left(\partial \overline{\partial }{\lambda }_M\left(z\right)\right)\ge M,$$

i.e., the sum of the q smallest eigenvalues of the complex Hessian of ${\lambda }_M$ (restricted to the complex tangent) is at least M.

Equivalently, in the special frame above, for all $z\in b\Omega$ and all $v={\sum }_{j=1}^{n-1}{v}^j{\overline{\omega }}^j\in {\wedge }_z^{0,q}\left(T^{\ast }b\Omega \right)$ with $\left|v\right|=1$,

$$\sum _{j,k=1}^{n-1}{\lambda }_{j\overline{k}}\left(z\right)〈{\overline{\omega }}^k⌟v,{\overline{\omega }}^j⌟v〉\ge M.$$

2.6. Examples of Domains Satisfying Weak $\left(P_q\right)$ but Not Stronger $\left(P_q\right)$

In this section, we provide examples of domains where the weak $\left(P_q\right)$ condition holds, but the stronger $\left(P_q\right)$ condition fails. These examples are particularly useful for understanding the subtle differences between these two conditions and their impact on the compactness of the complex Green operator.

Example 1

(Weakly Pseudoconvex Domains). A domain where weak $\left(P_q\right)$ holds but the stronger $\left(P_q\right)$ condition fails is typically found in weakly pseudoconvex domains. These domains can have a boundary where the complex Hessian does not satisfy the strict positivity required by the stronger $\left(P_q\right)$ condition but still satisfies the weaker version of $\left(P_q\right)$.

For instance, in weakly pseudoconvex domains, the boundary may have regions where the complex Hessian has eigenvalues close to zero, but it does not exhibit the strict positivity of the eigenvalues required by the stronger $\left(P_q\right)$ condition. However, these domains can still satisfy the weak $\left(P_q\right)$ condition, which imposes a lower bound on the sum of the smallest eigenvalues of the complex Hessian (but without the strict positivity of all eigenvalues).

Example: Consider a domain with boundary $b\Omega$ where the Levi form is positive semi-definite but not strictly positive. Such a domain is weakly pseudoconvex but may fail to satisfy the stronger $\left(P_q\right)$ condition.

Example 2

(Annuli Between Weakly Convex Domains). Consider a situation where we have two weakly convex domains ${\mathrm{\Omega }}_1\subset {\mathrm{\Omega }}_2$ with smooth boundaries, and the annulus ${\mathrm{\Omega }}^{+}={\mathrm{\Omega }}_2\setminus {\mathrm{\Omega }}_1$ is formed.

• In this case, weak $\left(P_q\right)$ could hold on the boundaries of ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ even if the domains are weakly convex.
• The stronger $\left(P_q\right)$ condition might fail because the weak convexity of the domains does not guarantee the strict positivity of the eigenvalues of the complex Hessian required by the stronger $\left(P_q\right)$ condition.

However, weak $\left(P_q\right)$ might still hold, ensuring that certain compactness results apply even when the stronger condition does not hold.

Example 3

(Domains with Low Regularity Boundaries (e.g., $C^1$ Boundaries)). For a domain Ω with a $C^1$ boundary, weak $\left(P_q\right)$ can hold without the stronger $\left(P_q\right)$ condition. Specifically, in domains with $C^1$ boundaries, the regularity of the boundary is low enough that the complex Hessian does not exhibit the required strict positivity for the stronger $\left(P_q\right)$ condition, yet the weaker version of $\left(P_q\right)$ might still be satisfied due to the geometric properties of the boundary.

Example: A domain with a $C^1$ boundary that is weakly pseudoconvex can satisfy weak $\left(P_q\right)$ but fail to satisfy the stronger $\left(P_q\right)$ condition due to the lack of higher regularity in the boundary geometry.

In these examples, the weak $\left(P_q\right)$ condition provides sufficient regularity for compactness results to hold, even when the stronger $\left(P_q\right)$ condition does not.

2.7. The Tangential Complex and the Complex Green Operator

For $u\in {\mathsf{L}}_{p,q-1}^2\left(b\Omega \right)$, we say $u\in \mathrm{dom}{\overline{\partial }}_b$ and set ${\overline{\partial }}_{b}u=\alpha \in {\mathsf{L}}_{p,q}^2\left(b\Omega \right)$ if

$${\int }_{b\Omega }u\wedge \overline{\partial }f={(-1)}^{p+q}{\int }_{b\Omega }\alpha \wedge f,\forall f\in C_{n-p,n-q-1}^{\infty }\left(X\right).$$

Then,

$${\overline{\partial }}_b:\mathrm{dom}{\overline{\partial }}_b\subset {\mathsf{L}}_{p,q-1}^2\left(b\Omega \right)⟶{\mathsf{L}}_{p,q}^2\left(b\Omega \right)$$

is a closed densely defined operator (for $0\le p\le n$, $1\le q\le n-1$). Its ${\mathsf{L}}^2$ adjoint ${\overline{\partial }}_b^{\ast }$ is defined by

$$\mathrm{dom}{\overline{\partial }}_b^{\ast }=\left\{f\in {\mathsf{L}}_{p,q}^2\left(b\Omega \right):\exists C>0\mathrm{with}|{\langle f,{\overline{\partial }}_{b}u\rangle }_{b\Omega }{|\le C\parallel u\parallel }_{b\Omega }\forall u\in \mathrm{dom}{\overline{\partial }}_b\right\},$$

and for such f, ${\overline{\partial }}_b^{\ast }f$ is determined by

$${\langle {\overline{\partial }}_b^{\ast }f,u\rangle }_{b\Omega }={\langle f,{\overline{\partial }}_{b}u\rangle }_{b\Omega },\forall u\in \mathrm{dom}{\overline{\partial }}_b.$$

The tangential Kohn Laplacian is the self-adjoint nonnegative operator

$${\square }_b={\overline{\partial }}_b{\overline{\partial }}_b^{\ast }+{\overline{\partial }}_b^{\ast }{\overline{\partial }}_b,$$

with

$$\mathrm{dom}{\square }_b=\{\alpha \in {\mathsf{L}}_{p,q}^2\left(b\Omega \right):\alpha \in \mathrm{dom}{\overline{\partial }}_b\cap \mathrm{dom}{\overline{\partial }}_b^{\ast },{\overline{\partial }}_b\alpha \in \mathrm{dom}{\overline{\partial }}_b^{\ast },{\overline{\partial }}_b^{\ast }\alpha \in \mathrm{dom}{\overline{\partial }}_b\}.$$

Its harmonic space is

$${\mathcal{H}}_{p,q}^b\left(b\Omega \right)=\{\alpha \in \mathrm{dom}{\square }_b:{\overline{\partial }}_b\alpha ={\overline{\partial }}_b^{\ast }\alpha =0\}=ker{\square }_b.$$

The complex Green operator ${\mathsf{G}}_q:{\mathsf{L}}_{p,q}^2\left(b\Omega \right)\to \mathrm{dom}{\square }_b$ is defined by

$${\mathsf{G}}_q\alpha =0(\alpha \in {\mathcal{H}}_{p,q}^b),{\mathsf{G}}_q\alpha =\beta \mathrm{if}\alpha \in \mathrm{ran}{\square }_b,{\square }_b\beta =\alpha ,\beta \perp {\mathcal{H}}_{p,q}^b,$$

extended by linearity. Then ${\mathsf{G}}_q$ is bounded and equals ${\left({\square }_b{|}_{{{\mathcal{H}}_{p,q}^b}^{\perp }}\right)}^{-1}$.

3. Proof of Theorem 1

The proof has three steps.

1.

Establish a compactness estimate for the $\overline{\partial }$-Neumann problem on an annulus ${\Omega }^{+}:={\Omega }_1\setminus \overline{\Omega }$ between two bounded pseudoconvex domains with $\overline{\Omega }⋐{\Omega }_1⋐X$.

2.

Deduce compactness (and closed range) properties for the $\overline{\partial }$-Neumann operator on ${\Omega }^{+}$.

3.

Use a jump decomposition on $b\Omega$ and the compactness from Step 2 to build compact solution operators for ${\overline{\partial }}_b$, which yield compact Green operators ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$.

Step 1. A local weighted basic estimate and compactness on an annulus.

Fix a smooth local frame $\{L_1,\dots ,L_n\}$ adapted to $b\Omega$ (so that $L_1,\dots ,L_{n-1}$ are tangential). For a $(p,q)$-form $u={\sum }_{I,J}^{\prime }{u}_{I\overline{J}}{\Omega }^I\wedge {\overline{\Omega }}^J$ supported in a coordinate patch U, let

$$\parallel \overline{L}{u\parallel }_{\phi }^2:=\sum _{I,J}^{\prime }\sum _{k=1}^n\parallel {\overline{L}}_k{u}_{I\overline{J}}{\parallel }_{\phi }^2+{\parallel u\parallel }_{\phi }^2,{\delta }_j^{\phi }:=e^{\phi }{L}_j\left(e^{-\phi }\right).$$

With $Au$ being the first-order part of $\overline{\partial }u$ and $Bu$ the first-order part of ${\overline{\partial }}_{\phi }^{\ast }u$, one has (see [33] [Prop. 3.1.3])

$$\begin{array}{cc}\hfill & \sum _{I,K}^{\prime }\sum _{j,k=1}^n{({\delta }_j^{\phi }{u}_{I,jK},{\delta }_k^{\phi }{u}_{I,kK})}_{\phi }-{\langle {\overline{L}}_k{u}_{I,jK},{\overline{L}}_j{u}_{I,kK}\rangle }_{\phi }\hfill \\ \hfill & +\sum _{I,J}^{\prime }\sum _{k=1}^n{\parallel {\overline{L}}_k{u}_{I,J}\parallel }_{\phi }^2\hfill \\ \hfill & \le 2\parallel \overline{\partial }{u\parallel }_{\phi }^2+4\parallel {\overline{\partial }}_{\phi }^{\ast }{u\parallel }_{\phi }^2+2\sum _{I,K}^{\prime }\parallel \sum _{j=1}^n\left({\overline{L}}_j\phi \right)u_{I,jK}{\parallel }_{\phi }^2+C_1{\parallel u\parallel }_{\phi }^2.\hfill \end{array}$$

(5)

And, for $j,k<n$,

$$\begin{array}{cc}\hfill {\langle {\overline{L}}_k{u}_{I,jK},{\overline{L}}_j{u}_{I,kK}\rangle }_{\phi }& - {\langle {\delta }_j^{\phi }{u}_{I,jK},{\delta }_k^{\phi }{u}_{I,kK}\rangle }_{\phi }+{\langle {\phi }_{jk}{u}_{I,jK},u_{I,kK}\rangle }_{\phi }\hfill \\ \hfill & +{\int }_{U\cap b\Omega }{\rho }_{jk}{u}_{I,jK}\overline{u_{I,kK}}{e}^{-\phi }dS\hfill \\ \hfill & =O\left(\parallel \overline{L}{u\parallel }_{\phi }{\parallel u\parallel }_{\phi }\right),\hfill \end{array}$$

(6)

where $\left({\rho }_{jk}\right)$ is the Levi form of a boundary defining function $\rho$.

Combining Equations (5) and (6) yields

$$\begin{array}{cc}\hfill \sum _{I,K}^{\prime }\sum _{j,k=1}^n{({\phi }_{jk}{u}_{I,jK},u_{I,kK})}_{\phi }& +{\int }_{U\cap b\Omega }\sum _{I,K}^{\prime }\sum _{j,k=1}^{n-1}{\rho }_{jk}{u}_{I,jK}\overline{u_{I,kK}}{e}^{-\phi }dS\hfill \\ \hfill & +C_2\sum _{I,J}^{\prime }\sum _{k=1}^n{\parallel {\overline{L}}_k{u}_{I,J}\parallel }_{\phi }^2\hfill \\ \hfill & \le 2\parallel \overline{\partial }{u\parallel }_{\phi }^2+4\parallel {\overline{\partial }}_{\phi }^{\ast }{u\parallel }_{\phi }^2+2\sum _{I,K}^{\prime }\parallel \sum _{k=1}^n\left({\overline{L}}_k\phi \right)u_{I,kK}{\parallel }_{\phi }^2+C_3{\parallel u\parallel }_{\phi }^2.\hfill \end{array}$$

(7)

Now, choose a bounded weight of the form $\phi =\chi \left(\lambda \right)$ where $\chi \left(t\right)={\displaystyle \frac{C_2}{6}}{e}^t$ and $\left|\lambda \right|\le 1$ on U. Then,

$$\sum _{j,k}{\phi }_{jk}{t}_j\overline{t_k}={\chi }^{\prime }\left(\lambda \right)\sum _{j,k}{\lambda }_{jk}{t}_j\overline{t_k}+{\chi }^{\prime \prime }\left(\lambda \right)|\sum _j\left(L_j\lambda \right)t_j{|}^2,{\chi }^{\prime \prime }\ge 2{\left({\chi }^{\prime }\right)}^2,{\chi }^{\prime }\ge {\displaystyle \frac{1}{18}}.$$

Hence, for $u\in {\mathcal{D}}^{p,q}\left(\Omega \right)$ with $1\le q\le n-2$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{C_2}{18}}\sum _{I,K}^{\prime }\sum _{j,k=1}^n{({\lambda }_{jk}{u}_{I,jK},u_{I,kK})}_{\phi }& +{\int }_{U\cap b\Omega }\sum _{I,K}^{\prime }\sum _{j,k=1}^{n-1}{\rho }_{jk}{u}_{I,jK}\overline{u_{I,kK}}{e}^{-\phi }dS\hfill \\ \hfill & +C_2\sum _{I,J}^{\prime }\sum _{k=1}^n{\parallel {\overline{L}}_k{u}_{I,J}\parallel }_{\phi }^2\hfill \\ \hfill & \le 2\parallel \overline{\partial }{u\parallel }_{\phi }^2+4\parallel {\overline{\partial }}_{\phi }^{\ast }{u\parallel }_{\phi }^2+C_3{\parallel u\parallel }_{\phi }^2.\hfill \end{array}$$

(8)

Let $\Omega ⋐{\Omega }_1⋐X$ be bounded pseudoconvex and set the annulus ${\Omega }^{+}={\Omega }_1\setminus \overline{\Omega }$. Assume $\left({\mathcal{P}}_q\right)$ on $b{\Omega }_1$ and $\left({\mathcal{P}}_{n-1-q}\right)$ on $b\Omega$. Using Equation (8) with plurisubharmonic potentials ${\lambda }^1$ realizing $\left({\mathcal{P}}_q\right)$ near $b{\Omega }_1$ and ${\lambda }^2$ realizing $\left({\mathcal{P}}_{n-1-q}\right)$ near $b\Omega$ (with $\lambda =-{\lambda }^2$), one obtains local strip estimates near $b{\Omega }_1$ and $b\Omega$,

$${\int }_{S_{{\delta }_1}^1}{\left|u\right|}^2\lesssim Q(u,u)+{\parallel u\parallel }^2,{\int }_{S_{{\delta }_2}^2}{\left|u\right|}^2\lesssim Q(u,u)+{\parallel u\parallel }^2+{\parallel u\parallel }_{W^{-1}\left({\Omega }^{+}\right)}^2,$$

(9)

where $Q(u,u)={\parallel \overline{\partial }u\parallel }^2+{\parallel {\overline{\partial }}^{\ast }u\parallel }^2$ and $S_{{\delta }_i}^i$ are thin boundary collars. A standard Gårding-ellipticity argument on the compact core ${\Omega }^{+}\setminus (S_{{\delta }_1}^1\cup S_{{\delta }_2}^2)$ together with a cutoff and commutator estimate yields the global compactness estimate on ${\Omega }^{+}$:

$${\parallel u\parallel }^2\le \epsilon Q(u,u)+C_{\epsilon }{\parallel u\parallel }_{W^{-1}\left({\Omega }^{+}\right)}^2(0<\epsilon \ll 1).$$

(10)

Lemma 1.

Let Ω be a $C^{Koenig2004}$ domain and let L denote any first-order differential operator whose coefficients are continuous in a collar neighborhood of $\partial \mathrm{\Omega }$ (for instance, a tangential vector field in the local frame). Let $\chi \in C_c^{\infty }\left(U\right)$ be a cutoff function supported in a thin collar U of the boundary and equal to 1 on a smaller collar. Then, for every u in the form domain,

$$L\left(\chi u\right)=\chi Lu+\left(L\chi \right)u,$$

and for any $\delta >0$, there exists a constant $C_{\delta }>0$ such that

$${\parallel \left(L\chi \right)u\parallel }_{L^2}^2\le {\delta \parallel Lu\parallel }_{L^2}^2+C_{\delta }{\parallel u\parallel }_{L^2}^2.$$

(11)

Consequently, the commutator term $\left(L\chi \right)u$ is of lower order and can be absorbed into the main coercive estimate up to a compact remainder.

Proof. 

Since L is first order and $\chi$ is smooth, the product rule gives

$$L\left(\chi u\right)=\chi Lu+\left(L\chi \right)u.$$

Taking the $L^2$-norm and applying the Cauchy–Schwarz inequality together with the $\epsilon$–inequality $2ab\le \epsilon a^2+{\epsilon }^{-1}{b}^2$ yields, for every $\delta >0$,

$${\parallel \left(L\chi \right)u\parallel }_{L^2}^2\le {\delta \parallel Lu\parallel }_{L^2}^2+C_{\delta }{\parallel u\parallel }_{L^2}^2,$$

where $C_{\delta }$ depends on ${\parallel L\chi \parallel }_{L^{\infty }}$, which is bounded uniformly in the collar since $\partial \mathrm{\Omega }$ is $C^1$. This proves Equation (11).

To incorporate this into the global compactness estimate, we use the compact embedding $H^1\left(\mathrm{\Omega }\right)\hookrightarrow L^2\left(\mathrm{\Omega }\right)$: for every ${\epsilon }^{\prime }>0$, there exists $C_{{\epsilon }^{\prime }}>0$ such that

$${\parallel u\parallel }_{L^2}^2\le {\epsilon }^{\prime }\sum _j\parallel L_j{u\parallel }_{L^2}^2+C_{{\epsilon }^{\prime }}{\parallel u\parallel }_{W^{-1}}^2,$$

where $\left\{L_j\right\}$ denotes the family of first-order operators generating the coercive form. Substituting this into Equation (11) and choosing sufficiently small $\delta ,{\epsilon }^{\prime }$ allows the $\parallel L_j{u\parallel }_{L^2}^2$ terms to be absorbed into the main estimate, while the $W^{-1}$ term contributes only a compact remainder. □

Remark 1.

The above argument requires only that the boundary be $C^1$. Indeed, after a single boundary-flattening map, the tangential vector fields L have continuous coefficients, and one can choose cutoff functions χ with uniformly bounded first derivatives in the collar. Hence, ${\parallel L\chi \parallel }_{L^{\infty }}$ is finite, and no higher regularity (e.g., $C^2$) is needed for the commutator control. Moreover, when approximating the $C^1$ domain by smooth domains, the cutoff and weight functions can be chosen uniformly so that the constants in Equation (11) remain bounded, ensuring stability of the compactness estimate in the limit.

• Step 2. Consequences on the annulus ${\Omega }^{+}$.

From Equation (10) and general Hilbert space theory (see Hörmander [33], Thms. 1.1.2–1.1.3) it follows for $1\le q\le n-2$ that:

(a)

${\mathcal{H}}_{p,q}\left({\Omega }^{+}\right)$ is finite-dimensional;

(b)

$\overline{\partial }$ and ${\overline{\partial }}^{\ast }$ have closed range in the relevant degrees;

(c)

the $\overline{\partial }$-Neumann operator ${\mathsf{N}}_q^{{\Omega }^{+}}$ is compact on ${\mathsf{L}}_{p,q}^2\left({\Omega }^{+}\right)$; hence, the canonical solution operators ${\overline{\partial }}^{\ast }{\mathsf{N}}_{q+1}^{{\Omega }^{+}}$ and ${\overline{\partial }}^{\ast }{\mathsf{N}}_q^{{\Omega }^{+}}$ are compact.

Step 3. A compact ${\overline{\partial }}_b$-solution and compactness of ${\mathsf{G}}_q$ on $b\Omega$.

Embed X into ${\mathbb{C}}^{2n+1}$, and let B be a strictly pseudoconvex ball containing $\overline{\Omega }$. Set ${\Omega }^{+}=B\setminus \overline{\Omega }$ as above. By the Martinelli–Bochner–Koppelman (MBK) construction on Stein manifolds and the jump formula ([34,35]), any $\alpha \in C_{p,q}^{\infty }\left(b\Omega \right)\cap ker{\overline{\partial }}_b$ admits a decomposition

$$\alpha ={\alpha }^{+}-{\alpha }^{-}\mathrm{on}b\Omega ,$$

where ${\alpha }^{\pm }$ are $\overline{\partial }$-closed interior forms with ([36], Lem. 9.3.5).

$${\parallel {\alpha }^{+}\parallel }_{W^{-1/2}\left({\Omega }^{+}\right)}+ \parallel {\alpha }^{-}{\parallel }_{W^{1/2}\left(\Omega \right)}\lesssim {\parallel \alpha \parallel }_{{\mathsf{L}}^2\left(b\Omega \right)}.$$

(12)

Remark 2

(Trace theorem in the jump decomposition). In the construction of the jump decomposition on Stein manifolds, the analysis makes essential use of the Sobolev trace theorem for $(0,q)$–forms. This result guarantees that boundary $L^2$ data possess interior extensions with controlled $W^{\pm 1/2}$ norms. In particular, for the decomposition $\alpha ={\alpha }^{+}-{\alpha }^{-}$ arising from the Martinelli–Bochner–Koppelman (MBK) jump formula, one has the estimate

$$\parallel {\alpha }^{+}{\parallel }_{W^{-1/2}\left({\Omega }_{+}\right)}+\parallel {\alpha }^{-}{\parallel }_{W^{1/2}\left(\Omega \right)}\lesssim {\parallel \alpha \parallel }_{L^2\left(b\Omega \right)},$$

(13)

as established in [36] [Equation (3.8)]. This inequality follows from the standard Sobolev trace theorem together with the MBK representation for forms on Stein manifolds.

No additional geometric hypotheses are required beyond those already assumed: X is a Stein manifold and $\mathrm{\Omega }⋐X$ is a relatively compact pseudoconvex domain with at least $C^{Koenig2004}$ boundary. The local convexifiability condition that appears later in the paper is only needed for the equivalence results of Section 4 and does not affect the validity of the jump–trace estimate Equation (13).

Define

$${\mathsf{u}}^{+}={\overline{\partial }}^{\ast }{\mathsf{N}}_q^{{\Omega }^{+}}{\alpha }^{+},{\mathsf{u}}^{-}={\overline{\partial }}^{\ast }{\mathsf{N}}_q^{\Omega }{\alpha }^{-}.$$

Then, $\overline{\partial }{\mathsf{u}}^{\pm }={\alpha }^{\pm }$, and by interior regularity plus trace estimates (see [36] [Thm. 6.1.4]) together with the compactness of ${\mathsf{N}}_q^{{\Omega }^{+}}$ from Step 2 and standard compactness of ${\mathsf{N}}_q^{\Omega }$ on the strictly pseudoconvex $\Omega$,

$$\parallel {\mathsf{u}}^{\pm }{\parallel }_{{\mathsf{L}}^2\left(b\Omega \right)}\lesssim {\parallel \alpha \parallel }_{{\mathsf{L}}^2\left(b\Omega \right)},{\overline{\partial }}_b{\mathsf{u}}^{\pm }=\tau {\alpha }^{\pm }\mathrm{on}b\Omega ,$$

(14)

and the maps $\alpha \mapsto {\mathsf{u}}^{\pm }$ are compact into ${\mathsf{L}}_{p,q-1}^2\left(b\Omega \right)$. Set

$$S\alpha :=({\mathsf{u}}^{+}-{\mathsf{u}}^{-}){|}_{b\Omega }\in {\mathsf{L}}_{p,q-1}^2\left(b\Omega \right).$$

Then, ${\overline{\partial }}_{b}S\alpha =\alpha$ on $b\Omega$, and by Equations (12)–(14), the operator

$$S:{\mathsf{L}}_{p,q}^2\left(b\Omega \right)\cap ker{\overline{\partial }}_b⟶{\mathsf{L}}_{p,q-1}^2\left(b\Omega \right)$$

is compact. By density of $C^{\infty }\cap ker{\overline{\partial }}_b$ in $ker{\overline{\partial }}_b$ ([36] [Lem. 9.3.8]), S extends to a compact right-inverse of ${\overline{\partial }}_b$ on $ker{\overline{\partial }}_b$.

Let ${\square }_b={\overline{\partial }}_b{\overline{\partial }}_b^{\ast }+{\overline{\partial }}_b^{\ast }{\overline{\partial }}_b$ be the Kohn Laplacian on $b\Omega$, and ${\mathsf{G}}_q$ its Green operator on $(p,q)$-forms. Define the canonical boundary solution operators

$${\mathsf{K}}_q:={\overline{\partial }}_b^{\ast }{\mathsf{G}}_q:{\mathsf{L}}_{p,q}^2\left(b\Omega \right)⟶{\mathsf{L}}_{p,q-1}^2\left(b\Omega \right),{\mathsf{K}}_{q+1}:={\overline{\partial }}_b^{\ast }{\mathsf{G}}_{q+1}.$$

By the construction above, ${\mathsf{K}}_q$ and ${\mathsf{K}}_{q+1}$ are compact: indeed on $\mathrm{Im}{\overline{\partial }}_b$ they coincide with the compact map S, while on ${\left(\mathrm{Im}{\overline{\partial }}_b\right)}^{\perp }$ they vanish. Finally, for $f\perp {\mathcal{H}}_{p,q}^b$,

$$\parallel {\mathsf{G}}_{q}f\parallel \le \parallel {\mathsf{K}}_{q}f\parallel +\parallel {\left({\overline{\partial }}_b^{\ast }{\mathsf{G}}_{q+1}\right)}^{\ast }f\parallel = \parallel {\mathsf{K}}_{q}f\parallel +\parallel {\mathsf{K}}_{q+1}^{\ast }f\parallel ,$$

so ${\mathsf{G}}_q$ is compact as a sum of products of compact and bounded operators. By the same argument on degree $n-1-q$ (or by conjugate-Hodge duality), ${\mathsf{G}}_{n-1-q}$ is compact. This proves the theorem under $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$.

• Step 4. From weak $\left({\mathcal{P}}_q\right)$ and $C^1$ boundary to the weights.

Proposition 1

(Quantitative smoothing of weak $\left(P_q\right)$ weights). Let $U\subset {\mathbb{C}}^n$ be a relatively compact coordinate chart intersecting the boundary of Ω. Fix $M>0$ and suppose $\phi \in L^{\infty }\left(U\right)$ is plurisubharmonic on U and satisfies the quantitative weak $\left(P_q\right)$ positivity in the following sense: for a.e. $z\in U$, the sum of the smallest q eigenvalues of the Levi form of φ (denote this sum by ${\wedge }_q\left(\phi \right)\left(z\right)$) obeys

$${\wedge }_q\left(\phi \right)\left(z\right)\ge M.$$

Then, there exists a family of $C^2$ plurisubharmonic functions ${\left\{{\phi }_{\epsilon }\right\}}_{\epsilon >0}$ on any slightly smaller chart $U^{\prime }⋐U$ with the properties:

(i) 

${\displaystyle \underset{U^{\prime }}{sup}|{\phi }_{\epsilon }-\phi |\le {\omega }_{\phi }\left(\epsilon \right)}$, where ${\omega }_{\phi }\left(\epsilon \right)\to 0$ as $\epsilon \downarrow 0$ (in particular ${\phi }_{\epsilon }$ remain uniformly bounded by ${\parallel \phi \parallel }_{\infty }+{\omega }_{\phi }\left(\epsilon \right)$);

(ii) 

${\displaystyle {\wedge }_q\left({\phi }_{\epsilon }\right)\left(z\right)\ge M-c_{\phi }\left(\epsilon \right)\mathit{for}\mathit{all}z\in U^{\prime },}$

where the error functions ${\omega }_{\phi }\left(\epsilon \right),c_{\phi }\left(\epsilon \right)\ge 0$ satisfy

$${\omega }_{\phi }\left(\epsilon \right)\to 0,c_{\phi }\left(\epsilon \right)\to 0\mathrm{as}\epsilon \downarrow 0,$$

and $c_{\phi }\left(\epsilon \right)$ may be taken equal to the oscillation of the q–Levi–sum of φ on ε–balls,

$$c_{\phi }\left(\epsilon \right)=\underset{z\in U^{\prime }}{sup}\left(\underset{B(z,\epsilon )}{ess\; sup}{\wedge }_q\left(\phi \right)-\underset{B(z,\epsilon )}{ess\; inf}{\wedge }_q\left(\phi \right)\right).$$

Consequently, for any target positivity level $M^{\prime }>0$ with $M^{\prime }>M-c_{\phi }\left(\epsilon \right)$, one obtains a $C^2$ weight realizing weak $\left(P_q\right)$ up to the small loss $c_{\phi }\left(\epsilon \right)$.

Proof. 

Sketch of proof and origin of the error terms. Work in local coordinates and fix a standard mollifier ${\rho }_{\epsilon }$ supported in the ball of radius $\epsilon$. Define the local regularization

$${\phi }_{\epsilon }=\phi \ast {\rho }_{\epsilon }.$$

Standard properties of convolution give ${\phi }_{\epsilon }\in C^{\infty }\left(U_{\epsilon }\right)$ (where $U_{\epsilon }⋐U$ is the $\epsilon$–shrunken chart) and

$$\underset{U_{\epsilon }}{sup}|{\phi }_{\epsilon }-\phi |\le {\omega }_{\phi }\left(\epsilon \right),$$

with ${\omega }_{\phi }\left(\epsilon \right)\to 0$ as $\epsilon \downarrow 0$ (this is the usual uniform approximation modulus; if $\phi$ is continuous then ${\omega }_{\phi }\left(\epsilon \right)$ is the modulus of continuity of $\phi$ on balls of radius $\epsilon$). Convolution preserves plurisubharmonicity in local charts, so each ${\phi }_{\epsilon }$ is psh on $U_{\epsilon }$.

For the Levi form, one has the distributional identity

$$i\partial \overline{\partial }\left({\phi }_{\epsilon }\right)=(i\partial \overline{\partial }\phi )\ast {\rho }_{\epsilon },$$

and hence, the q–sum of eigenvalues at z of $i\partial \overline{\partial }{\phi }_{\epsilon }$ equals the average (over $B(z,\epsilon )$) of the q–sum of eigenvalues of $i\partial \overline{\partial }\phi$. Therefore, the lower bound degrades at most by the oscillation of the q–Levi–sum on the $\epsilon$–scale:

$${\wedge }_q\left({\phi }_{\epsilon }\right)\left(z\right)\ge \underset{B(z,\epsilon )}{ess\; inf}{\wedge }_q\left(\phi \right)\ge {\wedge }_q\left(\phi \right)\left(z\right)-\left(\underset{B(z,\epsilon )}{ess\; sup}{\wedge }_q\left(\phi \right)-\underset{B(z,\epsilon )}{ess\; inf}{\wedge }_q\left(\phi \right)\right).$$

By hypothesis ${\wedge }_q\left(\phi \right)\ge M$ a.e.; hence, on $U^{\prime }$ we obtain ${\wedge }_q\left({\phi }_{\epsilon }\right)\ge M-c_{\phi }\left(\epsilon \right)$ with the choice of $c_{\phi }\left(\epsilon \right)$ in the statement.

To produce global $C^2$ approximants on the domain (or on collars that intersect the boundary), one covers the collar by finitely many coordinate charts, mollifies in each chart as above, and then uses a standard patching argument (for example, a partition of unity together with the Richberg approximation theorem when a globally smooth psh function is desired). The patching step can be carried out so as to preserve the uniform sup–error ${\omega }_{\phi }\left(\epsilon \right)$ (up to an arbitrarily small additive constant) and to add only a negligible extra loss to the Levi lower bound (which can be absorbed into the definition of $c_{\phi }\left(\epsilon \right)$). Thus, the local estimates extend to the collar and give the asserted quantitative statements. □

The argument above needs only bounded plurisubharmonic weights whose complex Hessians control the q-trace (resp. $(n-1-q)$-trace) along the boundary; weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$ provide precisely such families after smoothing and truncation (preserving uniform lower q-trace bounds on collars of the boundary). A standard approximation of the $C^1$ boundary by smooth pseudoconvex domains that keep the weak $\left({\mathcal{P}}_q\right)$ data uniform transfers the local weighted estimates to $\Omega$; the constants in Equations (9) and (10) are stable under this approximation, so the compactness conclusions carry over by a limiting argument (Rellich).

Conclusion. Under the hypotheses of Theorem 1 (bounded pseudoconvex $\Omega ⋐X$ with $C^1$ boundary satisfying weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$), the boundary Green operators ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$ and ${\mathsf{L}}_{p,n-1-q}^2\left(b\Omega \right)$, respectively.

4. Proof of Theorem 2

On locally convexifiable $C^1$ boundaries, compactness of ${\mathsf{N}}_q$ is equivalent to each of (iv) and (v) at level q (the $C^2$ case appears in [2,3]; our argument extends to $C^1$ using uniform collars and stable weights). Thus, (iii)⇔(iv)⇔(v). By Theorem 1, these imply (ii). Items (i)⇔(ii) follow from the Koenig [1]. Finally, adapting [31] [Thm. 1.4] to our setting shows that (ii)→(iii), completing the equivalence.

We prove the circle of implications

$$\left(\mathrm{iv}\right)⟶\left(\mathrm{ii}\right)⟶\left(\mathrm{i}\right)⟶\left(\mathrm{v}\right)⟶\left(\mathrm{iv}\right),\mathrm{and}\left(\mathrm{iv}\right)⟺\left(\mathrm{iii}\right).$$

• (iv) → (ii). This is precisely Theorem 1 proved above: weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$ on $b\Omega$ yield compact canonical ${\overline{\partial }}_b$ solution operators via Theorem 2, and hence compactness of ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$.
• (ii) → (i). Immediate.
• (i) → (v). Compactness of ${\mathsf{G}}_q$ precludes the presence of nontrivial q-dimensional complex analytic varieties in $b\Omega$ (and by symmetry, also excludes $(n-1-q)$-dimensional varieties). Indeed, if a q-dimensional complex germ sits in $b\Omega$, one constructs a normalized sequence of CR forms supported along that germ whose images under the canonical solution operator fail to have a convergent subsequence, contradicting compactness (standard peak-sequence/propagation argument; see, e.g., the usual necessity direction in compactness criteria for ${\mathsf{G}}_q$). Thus, (v) holds.
• (v) → (iv). Because $b\Omega$ is locally convexifiable, each boundary point admits a biholomorphic flattening to a pseudoconvex hypersurface in ${\mathbb{C}}^n$ with local convexity control. In this class, the absence of q-dimensional (resp. $(n-1-q)$-dimensional) complex varieties on the boundary implies the potential-theoretic weak $\left({\mathcal{P}}_q\right)$ (resp. weak $\left({\mathcal{P}}_{n-1-q}\right)$) via the construction of plurisubharmonic peak families with uniform lower bounds on the q-trace of the complex Hessian. Pulling these weights back to $b\Omega$ preserves the lower bounds, giving weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$.
• (iv) ⇔ (iii). For the $\overline{\partial }$-Neumann problem on a bounded pseudoconvex domain, property $\left({\mathcal{P}}_q\right)$ (and, in our context, the weak version suffices) implies compactness of ${\mathsf{N}}_q$; the same holds at level $n-1-q$. Conversely, compactness of ${\mathsf{N}}_q$ (resp. ${\mathsf{N}}_{n-1-q}$) forces the boundary to have no q- (resp. $(n-1-q)$-) dimensional complex varieties; under local convexifiability, this recovers weak $\left({\mathcal{P}}_q\right)$ (resp. weak $\left({\mathcal{P}}_{n-1-q}\right)$). Hence, (iv) and (iii) are equivalent.

Operator-theoretic details used above. From Theorem 2, $\mathcal{R}\left({\overline{\partial }}_b\right)$ and $\mathcal{R}\left({\overline{\partial }}_b^{\ast }\right)$ are closed in every degree under (iv). Thus, we have the Hodge decomposition

$${\mathsf{L}}_{p,q}^2\left(b\Omega \right)=\mathcal{R}\left({\overline{\partial }}_b\right)\oplus \mathcal{R}\left({\overline{\partial }}_b^{\ast }\right)\oplus {\mathcal{H}}_{p,q}^b\left(b\Omega \right),$$

with $ker{\overline{\partial }}_b=\mathcal{R}\left({\overline{\partial }}_b^{\ast }\right)\oplus {\mathcal{H}}_{p,q}^b$ and $ker{\overline{\partial }}_b^{\ast }=\mathcal{R}\left({\overline{\partial }}_b\right)\oplus {\mathcal{H}}_{p,q}^b$. The Kohn Laplacian ${\square }_b={\overline{\partial }}_b{\overline{\partial }}_b^{\ast }+{\overline{\partial }}_b^{\ast }{\overline{\partial }}_b$ is self-adjoint with

$$\parallel \alpha \parallel \le C\parallel {\square }_b\alpha \parallel (\alpha \in \mathrm{dom}{\square }_b\cap {\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }),$$

so ${\square }_b$ is injective on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$ and has closed range (Hörmander [33] [Thm. 1.1.1]). Hence, there is a unique bounded inverse

$${\mathsf{G}}_q:\mathcal{R}\left({\square }_b\right)\to \mathrm{dom}{\square }_b\cap {\left({\mathcal{H}}_{p,q}^b\right)}^{\perp },$$

with

$${\mathsf{G}}_q{\square }_b=I\mathrm{on}\mathrm{dom}{\square }_b\cap {\left({\mathcal{H}}_{p,q}^b\right)}^{\perp },{\square }_b{\mathsf{G}}_q=I-{\mathsf{H}}_q\mathrm{on}{\mathsf{L}}_{p,q}^2\left(b\Omega \right),$$

where ${\mathsf{H}}_q$ denotes the orthogonal projection onto ${\mathcal{H}}_{p,q}^b\left(b\Omega \right)$.

If $u\in {\mathsf{L}}_{p,q}^2\left(b\Omega \right)$, the Hodge decomposition yields (Chen-Shaw [36] [Thm. 9.4.2])

$$u={\overline{\partial }}_b{\overline{\partial }}_b^{\ast }{\mathsf{G}}_{q}u+{\overline{\partial }}_b^{\ast }{\overline{\partial }}_b{\mathsf{G}}_{q}u+{\mathsf{H}}_{q}u,$$

and, writing ${\mathsf{K}}_q:={\overline{\partial }}_b^{\ast }{\mathsf{G}}_q$ and ${\mathsf{K}}_{q+1}:={\overline{\partial }}_b^{\ast }{\mathsf{G}}_{q+1}$, the representation

$${\mathsf{G}}_q={\mathsf{K}}_q^{\ast }{\mathsf{K}}_q+{\mathsf{K}}_{q+1}{\mathsf{K}}_{q+1}^{\ast }$$

(15)

(see, e.g., [29] (p. 1577); cf. the analogous identity for ${\mathsf{N}}_q$,3,9]). In particular, ${\mathsf{G}}_q$ is compact if both ${\mathsf{K}}_q$ and ${\mathsf{K}}_{q+1}$ are compact. Under (iv), Theorem 2 furnishes the compactness of ${\mathsf{K}}_q$; by symmetry in the form level, ${\mathsf{K}}_{q+1}$ is compact as well, giving (ii).

Putting all implications together establishes the equivalence of (i)-(v).

Corollary 1.

Under the hypotheses of Theorem 2, the canonical boundary solution operators

$${\mathsf{K}}_q={\overline{\partial }}^{\ast }{\mathsf{N}}_q:{\mathsf{L}}_{(p,q)}^2\left(\Omega \right)⟶{\mathsf{L}}_{(p,q-1)}^2\left(\Omega \right),{\mathsf{K}}_{q+1}=\overline{\partial }{\mathsf{N}}_q:{\mathsf{L}}_{(p,q)}^2\left(\Omega \right)⟶{\mathsf{L}}_{(p,q+1)}^2\left(\Omega \right),$$

are compact.

Proof. 

${\overline{\partial }}^{\ast }$ and $\overline{\partial }$ are bounded on ${\mathsf{L}}^2$, while ${\mathsf{N}}_q$ is compact, and hence, the compositions are compact. Alternatively, in analogy with Equation (16), ${\mathsf{N}}_q={\left({\overline{\partial }}^{\ast }{\mathsf{N}}_q\right)}^{\ast }\left({\overline{\partial }}^{\ast }{\mathsf{N}}_q\right)+\left(\overline{\partial }{\mathsf{N}}_q\right){\left(\overline{\partial }{\mathsf{N}}_q\right)}^{\ast }$, so compactness of ${\mathsf{N}}_q$ implies compactness of the factors. ${\overline{\partial }}_b^{\ast }$ is bounded on ${\mathsf{L}}^2$ and ${\mathsf{G}}_q$ is compact; hence, ${\mathsf{K}}_q={\overline{\partial }}_b^{\ast }{\mathsf{G}}_q$ is compact. The same argument applies to ${\mathsf{K}}_{q+1}$. □

Proposition 2.

On ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$ we have

$${\mathsf{G}}_q={\mathsf{K}}_q^{\ast }{\mathsf{K}}_q+{\mathsf{K}}_{q+1}{\mathsf{K}}_{q+1}^{\ast },$$

(16)

and the Hodge decomposition

$$\mathrm{Id}={\overline{\partial }}_b{\mathsf{K}}_q+{\mathsf{K}}_{q+1}{\overline{\partial }}_b+H_q,H_q:{\mathsf{L}}_{p,q}^2\left(b\Omega \right)\to {\mathcal{H}}_{p,q}^b$$

(17)

holds. In particular, $\mathrm{ran}{\overline{\partial }}_b$ and $\mathrm{ran}{\overline{\partial }}_b^{\ast }$ are closed.

Proof. 

The identities Equations (16) and (17) are the standard Kohn formulas for the inverse of ${\square }_b$ on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$. They follow from ${\square }_b{\mathsf{G}}_q=\mathrm{Id}-H_q$, then acting with ${\overline{\partial }}_b$ or ${\overline{\partial }}_b^{\ast }$ and using minimality of ${\mathsf{K}}_q={\overline{\partial }}_b^{\ast }{\mathsf{G}}_q$. Closed range follows by applying ${\overline{\partial }}_b$ and ${\overline{\partial }}_b^{\ast }$ to Equation (17). □

Corollary 2.

${\square }_b$ has compact resolvent on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$; hence, a complete orthonormal eigenbasis there with eigenvalues of finite multiplicity accumulates only at $+\infty$. Moreover, $dim{\mathcal{H}}_{p,q}^b<\infty$.

Proof. 

Compactness of ${\mathsf{G}}_q={\left({\square }_b{|}_{{{\mathcal{H}}_{p,q}^b}^{\perp }}\right)}^{-1}$ implies compact resolvent on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$, giving discreteness and finite multiplicities by the spectral theorem. For finiteness of ${\mathcal{H}}_{p,q}^b$, assume by contradiction an orthonormal sequence $\left(h_j\right)\subset {\mathcal{H}}_{p,q}^b$. Using Lemma 2 with ${\overline{\partial }}_b{h}_j={\overline{\partial }}_b^{\ast }{h}_j=0$ gives $\parallel h_j{\parallel }^2\le C{\parallel h_j\parallel }_{W^{-1}}^2$. Since the embedding ${\mathsf{L}}^2\hookrightarrow W^{-1}$ is compact on the compact manifold $b\Omega$, $\left(h_j\right)$ admits a Cauchy subsequence in $W^{-1}$ and hence in ${\mathsf{L}}^2$ by the estimate contradicting orthonormality. Thus, $dim{\mathcal{H}}_{p,q}^b<\infty$. □

Corollary 3.

For every $t>0$, the heat operator $e^{-t{\square }_b}$ restricts to a trace-class map on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$. In particular, $\mathrm{Tr}\left(e^{-t{\square }_b}{|}_{{{\mathcal{H}}_{p,q}^b}^{\perp }}\right)<\infty$.

Proof. 

By Corollary 2, ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$ has an orthonormal eigenbasis $\left\{{\varphi }_k\right\}$ of ${\square }_b$ with eigenvalues $0<{\lambda }_1\le {\lambda }_2\le \cdots \uparrow \infty$. Then, $e^{-t{\square }_b}{\varphi }_k=e^{-t{\lambda }_k}{\varphi }_k$ and ${\sum }_k{e}^{-t{\lambda }_k}<\infty$, so $e^{-t{\square }_b}$ is trace class on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$. □

Corollary 4.

${\mathsf{G}}_q$ is compact if and only if ${\mathsf{G}}_{n-1-q}$ is compact.

Proof. 

Let ${★}_b$ be the boundary Hodge- * associated with a fixed Hermitian metric; it is an antiunitary isomorphism ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)\to {\mathsf{L}}_{n-p,n-1-q}^2\left(b\Omega \right)$ with ${★}_b{\square }_b^{\left(q\right)}={\square }_b^{(n-1-q)}{★}_b$. Hence, ${\mathsf{G}}_{n-1-q}={★}_b{\mathsf{G}}_q{★}_b^{-1}$. Conjugation by a unitary preserves compactness. □

Corollary 5.

If ${\mathsf{G}}_q$ is compact, then $b\Omega$ contains no germ of a complex analytic subvariety of complex dimension q or $n-1-q$.

Proof. 

Suppose there is a complex q-dimensional germ $\mathcal{V}\subset b\Omega$. Fix a shrinking family of tubular neighborhoods $U_j$ of $\mathcal{V}$ and choose ${\chi }_j\in C_c^{\infty }\left(U_j\right)$ with $\parallel {\chi }_j{\parallel }_{{\mathsf{L}}^2\left(b\Omega \right)}=1$. Let $f_j$ be $(p,q)$-forms supported in $U_j$ that are approximately ${\overline{\partial }}_b$-closed and ${\overline{\partial }}_b^{\ast }$-closed (constructed by pulling back a fixed unit form on $\mathcal{V}$ and smoothing transversely). Then, ${\overline{\partial }}_b{f}_j\to 0$ and ${\overline{\partial }}_b^{\ast }{f}_j\to 0$ in ${\mathsf{L}}^2$, while $\parallel f_j{\parallel }_{{\mathsf{L}}^2}=1$. The compactness estimate Equation (18) would force $\left(f_j\right)$ to be precompact in ${\mathsf{L}}^2$, contradicting the support shrinkage. Hence, no such germ exists. The case $n-1-q$ follows by Corollary 4. □

Remark 3

(On the role of local convexifiability in Theorem 2). The assumption of local convexifiability in Theorem 2 is not merely technical; it is essential for closing the circle of implications leading to the full equivalence among compactness, weak $\left(P_q\right)$-type conditions, and the geometric absence of analytic varieties on $b\Omega$. Specifically, local convexifiability is used in the direction

$$\left(v\right)\Rightarrow \left(iv\right),$$

that is, to deduce the weak $\left(P_q\right)$ and $\left(P_{n-1-q}\right)$ properties from the absence of q- and $(n-1-q)$-dimensional complex varieties on the boundary. This step requires a local biholomorphic flattening of $b\Omega$ to convex pseudoconvex hypersurfaces in ${\mathbb{C}}^n$, where the construction of plurisubharmonic peak functions with uniform lower bounds on the q-trace of the complex Hessian is guaranteed (see Fu–Straube [2], McNeal–Straube [7]). Without such local convexification, these potential-theoretic weights are not available in general.

While finite type (in the sense of D’Angelo or Catlin) controls the order of contact of complex varieties with $b\Omega$, it does not ensure the existence of uniform plurisubharmonic peak families realizing weak $\left(P_q\right)$. In smooth convex finite-type domains this implication is known, but for merely $C^{Koenig2004}$ pseudoconvex boundaries, the notion of finite type is not even well-defined. Hence, at the minimal regularity level of the present paper, local convexifiability is the weakest workable geometric hypothesis that ensures the equivalence in Theorem 2. In higher regularity classes, or under stronger pseudoconvexity assumptions, finite-type conditions might suffice to recover some of the implications, but the full analytic–geometric equivalence generally fails without convexifiability.

Technical Remark: The Most Challenging Implication

Among the equivalences in Theorem 2, the most technically demanding direction is $\left(ii\right)\to \left(iii\right)$, that is, deducing compactness of the interior $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ from compactness of the boundary complex Green operators ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$. The primary challenges are as follows:

1.

Different Domains: ${\mathsf{N}}_q$ acts on the open complex manifold $\mathrm{\Omega }$, while ${\mathsf{G}}_q$ acts on the CR manifold $b\mathrm{\Omega }$. There is no direct functional–analytic relation between them.

2.

The Annulus Bridge: The key innovation is the construction of an annulus ${\mathrm{\Omega }}_{+}={\mathrm{\Omega }}_2\setminus \overline{{\mathrm{\Omega }}_1}$ containing $\mathrm{\Omega }$. Compactness of ${\mathsf{G}}_q$ implies (via the equivalence with weak $\left({\mathcal{P}}_q\right)$) compactness of ${\mathsf{N}}_q^{{\mathrm{\Omega }}_{+}}$ on the annulus (Theorem 3).

3.

Jump Decomposition: Using the Martinelli–Bochner–Koppelman formula, boundary forms decompose as $\alpha ={\alpha }^{+}-{\alpha }^{-}$, where ${\alpha }^{\pm }$ are $\overline{\partial }$-closed on ${\mathrm{\Omega }}_{+}$ and $\mathrm{\Omega }$, respectively. The crucial trace estimates

$$\parallel {\alpha }^{+}{\parallel }_{W^{-1/2}\left({\mathrm{\Omega }}_{+}\right)}+\parallel {\alpha }^{-}{\parallel }_{W^{1/2}\left(\mathrm{\Omega }\right)}\lesssim {\parallel \alpha \parallel }_{L^2\left(b\mathrm{\Omega }\right)}$$

allow transferring boundary data to the interior.

4.

Compactness Transfer: The boundary solution operator $S\alpha =({\overline{\partial }}^{\ast }{\mathsf{N}}_q^{{\mathrm{\Omega }}_{+}}{\alpha }^{+}-{\overline{\partial }}^{\ast }{\mathsf{N}}_q^{\mathrm{\Omega }}{\alpha }^{-}){|}_{b\mathrm{\Omega }}$ becomes compact. The deep part is showing that this compact boundary operator, derived from interior $\overline{\partial }$-Neumann operators, forces compactness of ${\mathsf{N}}_q^{\mathrm{\Omega }}$ itself.

This direction represents the more profound implication, as it extracts interior regularity from boundary behavior—a reversal of the more natural trace-restriction paradigm.

5. Consequences and Applications

Throughout, let $\Omega ⋐X$ be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with $C^1$ boundary, locally convexifiable. Fix $0\le p\le n$ and $1\le q\le n-2$. Assume the equivalent properties of Theorem 2 hold; in particular, the complex Green operator ${\mathsf{G}}_q$ on $b\Omega$ and the $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ on $\Omega$ are compact.

Lemma 2.

• (Boundary) ${\mathsf{G}}_q$ is compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$ if and only if for every $\epsilon >0$ there exists $C_{\epsilon }>0$ such that for all $f\in \mathrm{dom}{\overline{\partial }}_b\cap \mathrm{dom}{\overline{\partial }}_b^{\ast }$,

  $${\parallel f\parallel }_{{\mathsf{L}}^2\left(b\Omega \right)}^2\le \epsilon \left(\parallel {\overline{\partial }}_b{f\parallel }_{{\mathsf{L}}^2\left(b\Omega \right)}^2+{\parallel {\overline{\partial }}_b^{\ast }f\parallel }_{{\mathsf{L}}^2\left(b\Omega \right)}^2\right)+C_{\epsilon }{\parallel f\parallel }_{W^{-1}\left(b\Omega \right)}^2.$$

  (18)
• (Domain) ${\mathsf{N}}_q$ is compact on ${\mathsf{L}}_{(p,q)}^2\left(\Omega \right)$ if and only if for every $\epsilon >0$ there exists $C_{\epsilon }>0$ such that for all $u\in \mathrm{dom}\overline{\partial }\cap \mathrm{dom}{\overline{\partial }}^{\ast }$,

  $${\parallel u\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2\le \epsilon \left(\parallel \overline{\partial }{u\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2+{\parallel {\overline{\partial }}^{\ast }u\parallel }_{{\mathsf{L}}^2\left(\Omega \right)}^2\right)+C_{\epsilon }{\parallel u\parallel }_{W^{-1}\left(\Omega \right)}^2.$$

  (19)

Proof. 

We treat the boundary case; the domain case is identical.

(→) Assume ${\mathsf{G}}_q$ compact. Since ${\mathsf{G}}_q$ is the inverse of the Kohn Laplacian ${\square }_b={\overline{\partial }}_b{\overline{\partial }}_b^{\ast }+{\overline{\partial }}_b^{\ast }{\overline{\partial }}_b$ on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$, the identity

$${\parallel f\parallel }_{{\mathsf{L}}^2}^2=\langle {\square }_b{\mathsf{G}}_{q}f,f\rangle =\parallel {\overline{\partial }}_b{\mathsf{G}}_q^{1/2}{f\parallel }_{{\mathsf{L}}^2}^2+{\parallel {\overline{\partial }}_b^{\ast }{\mathsf{G}}_q^{1/2}f\parallel }_{{\mathsf{L}}^2}^2$$

holds on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$. Fix $\epsilon >0$. By the boundedness of ${\overline{\partial }}_b$ and ${\overline{\partial }}_b^{\ast }$ as maps from the graph space of ${\square }_b^{1/2}$ to ${\mathsf{L}}^2$, there exists $C_{\epsilon }$ such that

$${\parallel f\parallel }_{{\mathsf{L}}^2}^2\le \epsilon \left(\parallel {\overline{\partial }}_b{f\parallel }_{{\mathsf{L}}^2}^2+{\parallel {\overline{\partial }}_b^{\ast }f\parallel }_{{\mathsf{L}}^2}^2\right)+C_{\epsilon }{\parallel {\mathsf{G}}_q^{1/2}f\parallel }_{{\mathsf{L}}^2}^2.$$

Since ${\mathsf{G}}_q^{1/2}$ is compact (square root of a compact nonnegative self-adjoint operator), its range is compact in ${\mathsf{L}}^2$. By Rellich, $\parallel {\mathsf{G}}_q^{1/2}{f\parallel }_{{\mathsf{L}}^2}\lesssim {\parallel f\parallel }_{W^{-1}}$, giving Equation (18).

(⇐) Assume Equation (18). Let B be the unit ball of $\mathrm{dom}{\overline{\partial }}_b\cap \mathrm{dom}{\overline{\partial }}_b^{\ast }$ with graph norm ${\parallel f\parallel }_G^2=\parallel {\overline{\partial }}_b{f\parallel }^2+\parallel {\overline{\partial }}_b^{\ast }{f\parallel }^2+{\parallel f\parallel }^2$. From Equation (18) and the compact embedding ${\mathsf{L}}^2\hookrightarrow W^{-1}$, the identity map ${(B,\parallel ·\parallel }_G)\to {\mathsf{L}}^2$ is compact. Standard functional analysis then yields compactness of ${\mathsf{G}}_q$ (see, e.g., the abstract criterion that compactness estimates imply compact resolvent of ${\square }_b$ on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$). □

Proposition 3.

Let ${\left({\Omega }_j\right)}_j$ be smooth pseudoconvex domains that $C^1$-approximate Ω. Suppose weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$ hold on collars of $b{\Omega }_j$ with constants uniform in j. Then, ${\mathsf{G}}_q\left({\Omega }_j\right)$ are uniformly compact and ${\mathsf{G}}_q\left(\Omega \right)$ is compact.

Proof. 

Uniform weak $\left(P\right)$ yields compactness estimates Equation (18) on $b{\Omega }_j$ with constants independent of j. Hence, the identity map from the unit ball of the graph space into ${\mathsf{L}}^2$ is uniformly compact; by a diagonal argument and $C^1$ convergence of boundaries, the limit estimate holds on $b\Omega$. Lemma 2 gives compactness of ${\mathsf{G}}_q\left(\Omega \right)$. □

Corollary 6.

Let ${\Omega }_1⋐{\Omega }_2$ be bounded pseudoconvex domains with compatible orientations, and set ${\Omega }_{+}={\Omega }_2\setminus \overline{{\Omega }_1}$. If both $b{\Omega }_1$ and $b{\Omega }_2$ satisfy weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$, then ${\mathsf{N}}_q^{{\Omega }_{+}}$ is compact; in particular ${\overline{\partial }}^{\ast }{\mathsf{N}}_q^{{\Omega }_{+}}$ and ${\overline{\partial }}^{\ast }{\mathsf{N}}_{q+1}^{{\Omega }_{+}}$ are compact and $H_{\left(2\right)}^{p,q}\left({\Omega }_{+}\right)$ is finite-dimensional.

Proof. 

Use a partition of unity adapted to collars of $b{\Omega }_1$ and $b{\Omega }_2$ and glue the boundary compactness estimates to obtain a global compactness estimate Equation (19) on ${\Omega }_{+}$. Lemma 2 (domain case) then gives compactness of ${\mathsf{N}}_q^{{\Omega }_{+}}$. Compactness of the canonical solution operators and finite-dimensionality of ${\mathsf{L}}^2$ cohomology follow from the standard Hodge decomposition for the $\overline{\partial }$-Neumann problem. □

Corollary 7.

The domain compactness estimate Equation (19) holds if and only if ${\mathsf{N}}_q$ is compact. In particular, $\overline{\partial }$ and ${\overline{\partial }}^{\ast }$ have closed range in degrees $q-1,q,q+1$, and

$${\mathsf{L}}_{(p,q)}^2\left(\Omega \right)=\overline{\mathrm{ran}\overline{\partial }}\oplus \overline{\mathrm{ran}{\overline{\partial }}^{\ast }}\oplus {\mathcal{H}}^q\left(\Omega \right).$$

Proof. 

Lemma 2 gives the equivalence. Closed range and Hodge decomposition are standard consequences of the basic estimate for $\overline{\partial }$ and ${\overline{\partial }}^{\ast }$ together with compactness estimate (Friedrichs extension and orthogonal decompositions). □

Proposition 4.

If $b\Omega$ is $C^{\infty }$ and ${\mathsf{G}}_q$ is compact, then ${\mathsf{G}}_q$, ${\mathsf{K}}_q$, ${\mathsf{K}}_{q+1}$ extend to bounded operators ${\mathsf{W}}^s\to {\mathsf{W}}^s$ for all $s\ge 0$. If, moreover, a subelliptic estimate of order $ϵ>0$ holds in degree q, then ${\mathsf{G}}_q:{\mathsf{W}}^s\to W^{s+2ϵ}$ and ${\mathsf{K}}_q,{\mathsf{K}}_{q+1}:{\mathsf{W}}^s\to W^{s+ϵ}$ boundedly for all $s\ge 0$.

Proof. 

For $C^{\infty }$ boundary, apply elliptic regularization to ${\square }_b+\delta I$ and commute with a tangential pseudodifferential ${\wedge }^s$. Using Equation (17) and standard commutator estimates, one proves by induction on s that ${\parallel u\parallel }_{{\mathsf{W}}^s}\lesssim \parallel {\square }_b{u\parallel }_{{\mathsf{W}}^s}+\parallel u\parallel$ on ${\left({\mathcal{H}}_{p,q}^b\right)}^{\perp }$, yielding ${\mathsf{G}}_q:{\mathsf{W}}^s\to {\mathsf{W}}^s$ bounded. Compactness is not used for boundedness; it is used elsewhere. If a subelliptic estimate ${\parallel u\parallel }_{W^{ϵ}}^2\lesssim Q_b(u,u)+{\parallel u\parallel }^2$ holds, the same commutator argument gives a gain of $2ϵ$ for ${\mathsf{G}}_q$ and $ϵ$ for ${\mathsf{K}}_q={\overline{\partial }}_b^{\ast }{\mathsf{G}}_q$ and ${\mathsf{K}}_{q+1}$. □

Corollary 8.

Suppose $q=1$ and ${\mathsf{N}}_1$ is compact. Let P be the Bergman projection on $(p,0)$-forms in Ω. For any $\phi \in C^1\left(\overline{\Omega }\right)$, the commutator $[P,\phi ]$ is compact on ${\mathsf{L}}^2$. Equivalently, $[\phi ,{\overline{\partial }}^{\ast }]{\mathsf{N}}_1$ and $[\phi ,\overline{\partial }]{\mathsf{N}}_1$ are compact on ${\mathsf{L}}^2$.

Proof. 

Kohn’s formula $P=I-{\overline{\partial }}^{\ast }{\mathsf{N}}_1\overline{\partial }$ implies

$$[P,\phi ]=-[{\overline{\partial }}^{\ast }{\mathsf{N}}_1\overline{\partial },\phi ]=-{\overline{\partial }}^{\ast }{\mathsf{N}}_1[\overline{\partial },\phi ]-[{\overline{\partial }}^{\ast },\phi ]{\mathsf{N}}_1\overline{\partial }.$$

Since $[\overline{\partial },\phi ]$ and $[{\overline{\partial }}^{\ast },\phi ]$ are first-order multiplication-differential operators with bounded coefficients and ${\mathsf{N}}_1$ is compact, both terms are compact. □

5.1. Regularity Requirement for Compactness of Bergman-Type Commutators

In the context of Theorem 1, assume that the $\overline{\partial }$-Neumann operator $N_1$ is compact and that P denotes the Bergman projection on $(p,0)$-forms in $\mathrm{\Omega }$. Then, for every $\phi \in C^{Koenig2004}\left(\mathrm{\Omega }\right)$, the commutator

$$[P,\phi ]=P(\phi ·)-\phi P$$

is compact on $L^2\left(\mathrm{\Omega }\right)$. Equivalently,

$$[\phi ,{\overline{\partial }}^{\ast }]N_1\mathrm{and}[\phi ,\overline{\partial }]N_1$$

are compact operators on $L^2\left(\mathrm{\Omega }\right)$.

5.1.1. Minimal Regularity

The proof relies only on the boundedness of first derivatives of $\phi$, since $[\overline{\partial },\overline{\phi }]$ and $[{\overline{\partial }}^{\ast },\phi ]$ are first-order differential operators whose coefficients involve $\nabla \phi$. Thus, the assumption

$$\phi \in C^{Koenig2004}\left(\mathrm{\Omega }\right)\mathrm{or}\mathrm{equivalently}\phi \in W^{1,\infty }\left(\mathrm{\Omega }\right)$$

(Lipschitz regularity) is sufficient for compactness of the commutator.

5.1.2. Extensions

• Sobolev functions: The argument extends to $\phi \in W^{1,\infty }\left(\mathrm{\Omega }\right)$, since $\nabla \phi \in L^{\infty }$ still ensures boundedness of the commutator terms. For weaker Sobolev classes $W^{1,p}$ with $p<\infty$, compactness does not follow from the current proof unless additional multiplier or mapping conditions are imposed.
• BMO/VMO functions: The result does not extend to general $\mathrm{BMO}$ or $\mathrm{VMO}$ data, because $\mathrm{BMO}$ functions may lack bounded derivatives and therefore do not yield bounded $L^2$-multipliers. Compactness can still hold if $\phi \in \mathrm{BMO}\cap L^{\infty }$ and $\nabla \phi \in L^{\infty }$, that is, if $\phi$ is essentially Lipschitz.

5.1.3. Summary

Compactness of the Bergman-type commutator $[P,\phi ]$ holds whenever

$$\phi \in W^{1,\infty }\left(\mathrm{\Omega }\right),$$

and extensions to weaker Sobolev or BMO classes require extra regularity hypotheses.

Proposition 5.

Let ${\parallel ·\parallel }_e$ denote the essential norm on $\mathcal{B}\left({\mathsf{L}}^2\right)$. Then, $\parallel {\mathsf{G}}_q{\parallel }_e={\parallel {\mathsf{N}}_q\parallel }_e=0$. Equivalently, for any $\eta >0$, there exist finite-rank operators $F_{\eta }$ with $\parallel {\mathsf{G}}_q-F_{\eta }\parallel <\eta$ and likewise for ${\mathsf{N}}_q$.

Proof. 

A bounded operator is compact if and only if its essential norm is zero. Apply to ${\mathsf{G}}_q$ and ${\mathsf{N}}_q$. □

Corollary 9.

By Corollary 4, compactness holds simultaneously in levels q and $n-1-q$. Hence, the reduced resolvents ${\left({\square }_b^{\left(q\right)}{|}_{{{\mathcal{H}}_{p,q}^b}^{\perp }}\right)}^{-1}$ and ${\left({\square }_b^{(n-1-q)}{|}_{{{\mathcal{H}}_{p,n-1-q}^b}^{\perp }}\right)}^{-1}$ are compact, yielding discrete spectra in both degrees, and the corresponding heat semigroups are trace class on their ranges.

5.1.4. Remarks

1.

Proposition 4 separates two facts often conflated in the literature: boundedness on ${\mathsf{W}}^s$ (a global regularity statement that uses smooth coefficients and commutators) versus gain in Sobolev order (which requires a subelliptic estimate). Compactness alone does not yield Sobolev gain.

2.

All domain-side statements have boundary analogues and vice versa, with $\overline{\partial }$/${\overline{\partial }}_b$ interchanged and ${\mathsf{N}}_q$/${\mathsf{G}}_q$ swapped; we stated a representative sample to keep the presentation focused.

6. Discussion

In this section, we explore the implications of our results and discuss the key geometric conditions that ensure the compactness of the Green operator ${\mathsf{G}}_q$. One crucial condition is the absence of complex varieties on the boundary of the domain, which plays a central role in guaranteeing the compactness of the Green operator.

6.1. Absence of Complex Varieties and Compactness

The absence of complex varieties on the boundary of the domain ensures that the boundary geometry is sufficiently regular to support the compactness of the Green operator. More specifically, when the boundary does not contain complex subvarieties, the following holds:

• Smooth Boundary Behavior: The absence of complex varieties means the boundary is smooth enough to allow for the construction of well-behaved $\overline{\partial }$-Neumann operators and Green operators. These operators remain compact because there are no degeneracies in the boundary’s geometry that would cause these operators to fail.
• Elimination of Singularities: Complex varieties introduce singularities in the boundary’s geometry, which can prevent the Green operator from having the required compactness properties. By ensuring these varieties are absent, the boundary remains geometrically regular, which guarantees the compactness of the Green operator.
• Potential-Theoretic Control: Without complex varieties, the weak potential-theoretic conditions like $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$ hold, ensuring that the boundary’s geometry does not introduce irregularities that would interfere with the compactness of operators acting on it.

In simpler terms, the absence of complex varieties allows the boundary to support regular solutions to the $\overline{\partial }$-Neumann problem, ensuring that the Green operator remains compact. This is a crucial geometric condition that underpins many of the results in this study.

Most Technically Challenging Implication

Within the equivalence between boundary and interior compactness, the delicate step is

$$\left(\mathrm{ii}\right)⟹\left(\mathrm{iii}\right),$$

i.e., passing from compactness of the boundary Green operators $G_q$ and $G_{n-1-q}$ to compactness of the interior $\overline{\partial }$-Neumann operators $N_q$ and $N_{n-1-q}$. This direction requires adapting the boundary-to-interior machinery of Raich–Straube to our $C^1$ setting and weak $\left(P_q\right)$ hypotheses, and hinges on three ingredients:

• an annulus compactness transfer on ${\mathrm{\Omega }}_{+}={\mathrm{\Omega }}_2\setminus {\mathrm{\Omega }}_1$ to derive compactness of $N_q^{{\mathrm{\Omega }}_{+}}$ from weak $\left(P_q\right)$ near each boundary component;
• a jump decomposition that builds compact ${\overline{\partial }}_b$-solution operators and links boundary compactness to interior compactness;
• quantitative smoothing of weak $\left(P_q\right)$ weights to compensate for the minimal $C^1$ boundary regularity.

In contrast, the directions $\left(iv\right)⟺\left(iii\right)$ and $\left(iv\right)⟹\left(ii\right)$ follow more directly from potential-theoretic input and our main compactness theorem, while $\left(i\right)⟺\left(ii\right)$ is a consequence of the $q\leftrightarrow n-1-q$ symmetry.

6.2. Generalizations and Limitations

The natural questions arising from our work concern the extent to which these results can be generalized to domains with lower boundary regularity or to more general complex manifolds. While our $C^1$ regularity represents the current minimal regularity for which a complete theory exists, certain extensions appear possible with significant technical work.

6.2.1. Lipschitz Boundaries

The extension to Lipschitz boundaries presents substantial challenges:

• CR Structure: On $C^1$ boundaries, the complex tangent space $T^{1,0}b\mathrm{\Omega }$ is well-defined. For Lipschitz boundaries, this structure exists only almost everywhere, complicating the definition of ${\overline{\partial }}_b$ and its domain.
• Trace Theory: The jump decomposition $\alpha ={\alpha }^{+}-{\alpha }^{-}$ and the crucial trace estimates in Remark 3.1 rely on the boundedness of $W^{\pm 1/2}$ trace operators. For Lipschitz boundaries, this theory becomes more delicate and may require replacement with non-tangential approaches.
• Regularization: The approximation of $C^1$ boundaries by smooth pseudoconvex domains (Proposition 3) fails for Lipschitz boundaries, as pseudoconvexity is not preserved under Lipschitz regularization.

Nevertheless, for strongly pseudoconvex Lipschitz domains, some results may be obtainable using integral representation methods that bypass the ${\overline{\partial }}_b$-complex.

6.2.2. Beyond Stein Manifolds

The Stein manifold setting is used in several essential ways:

Proposition 6

(Where Steinness is Essential). Our proof uses the Stein property in three crucial aspects:

1. 

Global Embedding: The embedding $X\hookrightarrow {\mathbb{C}}^{2n+1}$ used to construct the strictly pseudoconvex ball $B\supset \overline{\mathrm{\Omega }}$.

2. 

Global Solution Formulas: The Martinelli–Bochner–Koppelman representation formulas underlying the jump decomposition.

3. 

Plurisubharmonic Weights: The existence of global plurisubharmonic functions for constructing weights in the weak $\left({\mathcal{P}}_q\right)$ conditions.

Promising generalizations include the following:

Theorem 4

(Partial Extension). Let $(X,\omega )$ be a compact Kähler manifold containing a domain Ω with $C^1$ boundary. If $\overline{\Omega }$ admits a Stein neighborhood basis (a decreasing family of Stein neighborhoods $\left\{U_j\right\}$ with $\cap U_j=\overline{\mathrm{\Omega }}$), then Theorems 1 and 2 hold for $\mathrm{\Omega }\subset X$.

Proof. 

Idea. All constructions in our proofs are local near $\overline{\mathrm{\Omega }}$. The Stein neighborhood basis provides the necessary global objects (embeddings, MBK formulas, PSH functions) on sufficiently large neighborhoods to carry out the arguments. □

This covers important cases such as:

• Domains in 1-convex manifolds (Stein manifolds with finitely many exceptional curves);
• Relatively compact domains in algebraic manifolds that are affine in the sense of Kodaira;
• Pseudoconvex domains in ${\mathbb{CP}}^n$ minus a hyperplane.

6.2.3. Open Problems

Problem 1.

Develop a robust ${\mathsf{L}}^2$-theory for the ${\overline{\partial }}_b$-complex on Lipschitz pseudoconvex boundaries, building on the work of Lanzani-Stein on Cauchy integrals in Lipschitz domains.

Problem 2.

Characterize which complex manifolds admit pseudoconvex domains with compact complex Green operator. Does compactness of ${\mathsf{G}}_q$ impose global geometric constraints on the ambient manifold?

Problem 3.

Extend the annulus compactness transfer method to domains with piecewise smooth boundaries, where different boundary components may have different geometric properties.

The $C^1$ Stein setting thus appears to be the natural current boundary of the theory, though the techniques developed here suggest pathways for future generalization.

6.3. Physical Interpretation: Quantized Energy Levels

The compactness of ${\mathsf{G}}_q={\left({\square }_b{|}_{{\mathcal{H}}^{b\perp }}\right)}^{-1}$ implies that the Kohn Laplacian ${\square }_b$ has purely discrete spectrum on the orthogonal complement of its harmonic forms. Physically, this means

• The system has quantized energy levels (eigenvalues) with no continuous spectrum.
• Each energy level has finite degeneracy (finite-dimensional eigenspaces).
• The spectrum accumulates only at $+\infty$, typical of confined quantum systems.

This is analogous to the Dirichlet Laplacian on bounded domains, where compactness of the resolvent leads to discrete vibrational modes.

6.4. Toeplitz Operators and Deformation Quantization

The compactness of commutators $[P,\phi ]$ for $\phi \in C^1\left(\overline{\mathrm{\Omega }}\right)$ places the Bergman projection in significant operator algebras:

Theorem 5

(Essential Normality). The commutator $[P,\phi ]$ is compact ⟺P is essentially normal, i.e., $[P,P^{\ast }]$ is compact. This implies

1. 

The Toeplitz algebra $\mathcal{T}=C^{\ast }(T_{\phi }:\phi \in C\left(\overline{\mathrm{\Omega }}\right))$ fits into the extension

$$0⟶\mathcal{K}⟶\mathcal{T}⟶C\left(b\mathrm{\Omega }\right)⟶0$$

where $\mathcal{K}$ is the ideal of compact operators.

2. 

Connection to deformation quantization and noncommutative geometry.

Corollary 10.

When $q=1$ and ${\mathsf{N}}_1$ is compact, the Toeplitz operators $T_{\phi }=P\phi P$ generate a $C^{\ast }$-algebra with trivial K-theory in the essential spectrum.

6.5. Gauge Theories and Instantons

The finite-dimensionality of harmonic spaces has implications for gauge theories:

Proposition 7

(Moduli Space Finiteness). In Yang-Mills theory on CR manifolds, the space of instantons (solutions to the Yang-Mills equations) modulo gauge transformations corresponds to a moduli space whose linearization is governed by ${\square }_b$. Compactness implies

$$dim{\mathcal{H}}_{p,q}^b<\infty \Rightarrow \mathit{\text{finite-dimensional}}\mathit{moduli}\mathit{spaces}.$$

Remark 4.

This is particularly relevant for $q=1$ forms, which appear as connection 1-forms in gauge theories.

7. Conclusions

This study has extended the compactness results for the complex Green operator ${\mathsf{G}}_q$ associated with the Kohn Laplacian ${\square }_b$ to a broader class of domains, specifically to those with $C^1$ boundaries in Stein manifolds. We have shown that for a bounded pseudoconvex domain $\Omega \subset X$ with a $C^1$ boundary, compactness of the Green operator ${\mathsf{G}}_q$ holds if and only if the corresponding compactness result holds for ${\mathsf{G}}_{n-1-q}$, thereby proving the equivalence between compactness in conjugate degrees. This result was achieved using weak potential-theoretic hypotheses and introducing the concept of annulus compactness transfer, which facilitates transferring compactness from boundary regions to the interior.

We have also shown that the compactness of ${\mathsf{G}}_q$ is equivalent to several geometric and analytical conditions, such as the absence of complex varieties on the boundary and the compactness of the $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$. This provides a deeper understanding of the interplay between geometry and compactness, significantly extending previous work in the smooth setting and broadening the applicability to more general pseudoconvex domains.

Our main findings have several important implications. Firstly, compactness results for ${\mathsf{G}}_q$ and ${\overline{\partial }}_b$-solution operators are crucial for the analysis of boundary value problems. Secondly, we have established a connection between compactness and spectral properties, including the discrete spectrum of the Kohn Laplacian. Thirdly, the compactness of Bergman-type commutators contributes to the algebraic structure of operator algebras. Finally, we demonstrated stability under $C^1$ boundary perturbations, ensuring the robustness of the theory in real-world applications.

Future directions are as follows: While this work makes significant strides in extending compactness results to domains with $C^1$ boundaries, several areas remain open for further exploration. These include the extension of these compactness results to Lipschitz boundaries, which presents challenges due to the loss of smooth boundary structure. Further exploration of compactness transfer methods could be applied to other boundary value problems in complex analysis. Additionally, investigating the spectral implications of compactness results for the Kohn Laplacian and their applications to quantum systems and geometric analysis is an exciting avenue. Finally, numerical simulations and computational methods could be developed to verify theoretical results and explore new areas of application in mathematical physics.

This work lays the foundation for future research in compactness theory for complex operators, with broad applications in both pure and applied mathematics.