Hardy Spaces for a Class of Singular Domains

TL;DR

This paper develops a framework for Hardy spaces on singular domains, generalizes key lemmas, observes stabilization phenomena, and constructs Hardy space filtrations on complex domains using holomorphic maps.

Contribution

It introduces a new framework for Hardy spaces on complements of hypersurfaces, generalizes a classical rigidity lemma, and constructs Hardy space filtrations on complex domains.

Findings

Generalization of Kerzman and Stein's rigidity lemma

Observation of stabilization in egg domains

Construction of Hardy space filtrations on Hartogs triangles

Abstract

We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific settings. For punctured planar domains, we prove a generalization of a famous rigidity lemma of Kerzman and Stein. A stabilization phenomenon is observed for egg domains. Finally, using proper holomorphic maps, we derive a filtration of Hardy spaces for certain power-generalized Hartogs triangles, although these domains fall outside the scope of the original framework.