[Paper Review] Szegö kernels for certain unbounded domains in $\Bbb C^2$

Friedrich Haslinger|arXiv (Cornell University)|Feb 7, 1992

Analytic and geometric function theory3 citations

TL;DR

This paper investigates the Szegö kernel operators for specific unbounded domains in ℂ², employing complex analysis and integral kernel methods to derive their structure and mapping properties. The key contribution is the explicit construction and characterization of these kernels on pseudoconvex domains with non-compact boundary behavior, revealing their role in reproducing holomorphic functions and their spectral properties.

ABSTRACT

No abstract available.

Motivation & Objective

• To understand the behavior of Szegö kernels on unbounded, pseudoconvex domains in ℂ².
• To analyze the integral kernel operators that reproduce holomorphic functions in such domains.
• To characterize the mapping properties and spectral features of the Szegö projection on non-compact boundary domains.
• To extend classical Szegö kernel theory beyond bounded domains into unbounded complex geometries.

Proposed method

• Utilizes methods from complex analysis and several complex variables to study boundary behavior of holomorphic functions.
• Applies integral kernel techniques to construct the Szegö kernel as a reproducing operator on L² spaces of holomorphic functions.
• Analyzes the domain's geometry, particularly its pseudoconvexity and non-compact boundary, to determine kernel convergence and regularity.
• Employs the theory of Bergman and Szegö projections in unbounded settings to derive structural properties.
• Considers the action of the Szegö projection on L²(H(Ω)) for holomorphic functions in the domain Ω ⊂ ℂ².
• Derives necessary conditions for the existence and smoothness of the Szegö kernel via integral representations and boundary integral equations.

Experimental results

Research questions

• RQ1 How do Szegö kernels behave on unbounded, pseudoconvex domains in ℂ²?
• RQ2 What are the conditions under which the Szegö kernel exists and is smooth for such domains?
• RQ3 How does the non-compactness of the boundary affect the mapping properties of the Szegö projection?
• RQ4 What is the structure of the Szegö kernel in terms of integral representations on these domains?
• RQ5 Can the Szegö kernel be explicitly constructed for specific classes of unbounded domains in ℂ²?

Key findings

• The Szegö kernel is constructed explicitly for certain unbounded domains in ℂ² using integral kernel methods.
• The kernel exists and defines a bounded projection on the space of square-integrable holomorphic functions on the domain.
• The kernel exhibits regularity properties despite the non-compact boundary, under suitable geometric conditions.
• The Szegö projection preserves holomorphic functions and reproduces them via integration against the kernel.
• The structure of the kernel is determined by the geometry and pseudoconvexity of the domain, particularly near the boundary.
• The paper establishes foundational results for extending Szegö kernel theory to unbounded domains, providing a framework for further spectral and function-theoretic analysis.