Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

TL;DR

This paper explores how the boundary geometry of pseudoconvex domains influences the compactness of Hankel operators, establishing conditions under which boundary analytic discs determine holomorphicity of functions.

Contribution

It demonstrates that for certain pseudoconvex domains, compactness of Hankel operators implies boundary holomorphicity of functions along analytic discs, with a complete characterization in convex C^2 domains.

Findings

Compactness of H_f implies f is holomorphic along boundary analytic discs in certain domains.

In convex C^2 domains, the boundary holomorphicity condition is both necessary and sufficient for H_f compactness.

The boundary geometry, such as convexity or Levi form rank, critically affects Hankel operator properties.

Abstract

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in C^n. We show that, if D is convex or the Levi form of the boundary of D is of rank at least n-2, then compactness of the Hankel operator H_f implies that f is holomorphic "along" analytic discs in the boundary. Furthermore, when D is convex in C^2 we show that the condition on f is necessary and sufficient for compactness of H_f