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Hardy Spaces for a Class of Singular Domains

Anne-Katrin Gallagher, Purvi Gupta, Loredana Lanzani, Liz Vivas

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.

Hardy Spaces for a Class of Singular Domains

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.

Paper Structure

This paper contains 19 sections, 14 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Function-theoretic context
  3. Boundary-based approach to Hardy spaces
  4. Description of results
  5. Structure of this paper.
  6. Motivating example
  7. Hardy Space for the unit disk
  8. Hardy spaces on the punctured disk
  9. Hardy spaces on hypersurface-deleted domains
  10. Requirements on the parent space
  11. Requirements on the hypersurface
  12. The inheritance scheme
  13. Planar domains
  14. Hypersurface-deleted egg domains as examples of finite stabilization
  15. Hartogs triangles: an application
  16. ...and 4 more sections

Key Result

Proposition 3.3

Suppose that $\mathcal{A}(\Omega,\nu)$ is weakly admissible. Then for any $z\in\Omega$, there exists a unique bounded linear functional such that $\operatorname{Ev}_z(F|_T)=F(z)$ for any $F \in \mathcal{A}(\Omega,\nu)$. Furthermore, there exists a unique function $s:\Omega\times T\rightarrow \mathbb{C}$ such that

Theorems & Definitions (31)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Lemma 3.9
  • ...and 21 more