Localization of compactness of Hankel operators on pseudoconvex domains

TL;DR

This paper establishes a localization principle for the compactness of Hankel operators on Bergman spaces over pseudoconvex domains, showing that compactness on a larger domain implies compactness on certain subdomains.

Contribution

It proves that the compactness of Hankel operators on a pseudoconvex domain localizes to subdomains near boundary points, extending understanding of operator behavior in complex analysis.

Findings

Compactness of Hankel operators on D implies compactness on U near boundary points

The result applies to symbols that are C^1 on the closure of D

Localization holds for connected intersections with boundary balls

Abstract

We prove the following localization for compactness of Hankel operators on Bergman spaces. Assume that D is a bounded pseudoconvex domain in C^n, p is a boundary point of D and B(p,r) is a ball centered at p with radius r so that U=D\cap B(p,r) is connected. We show that if the Hankel operator H^D_f is compact on A^2(D) (the symbols f is C^1 on the closure of D) then H^U_f is compact on A^2(U) where A^2(D) and A^2(U) denote the Bergman spaces on D and U, respectively.