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A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains

Steven R. Bell, Loredana Lanzani, Nathan A. Wagner

TL;DR

The paper addresses representing boundary data on planar Lipschitz domains as a sum of interior and exterior holomorphic boundary values by proving an $h+H$ decomposition for $u\in L^p(b\Omega)$. It develops Hardy-space tools on Lipschitz domains, establishes the boundedness of the Cauchy transform, and constructs a robust regularity theory that extends from smooth to Lipschitz boundaries, yielding a Banach direct-sum decomposition $L^p(b\Omega) \simeq h^p(b\Omega) \oplus h^p_-(b\Omega)$. Key contributions include a constructive existence proof, a new higher-dimensional regularity result for holomorphic Hardy spaces, and applications to the Dirichlet/Poisson problems as well as a pseudo-local Szegő projection property. These results provide a fresh perspective on classical boundary-value problems and offer a framework for higher-dimensional generalizations of the Cauchy-transform kernel on non-smooth domains.

Abstract

We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $Ω\subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in $Ω$ while the other is holomorphic in $\mathbb C \setminus \overlineΩ$ and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the $L^p$ setting relies upon a classical albeit little-known regularity theorem for the holomorphic Hardy space $h^p(bΩ)$ of planar domains for which we provide a new proof that is valid also in higher dimensions. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on $L^p(bΩ)$. These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.

A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains

TL;DR

The paper addresses representing boundary data on planar Lipschitz domains as a sum of interior and exterior holomorphic boundary values by proving an decomposition for . It develops Hardy-space tools on Lipschitz domains, establishes the boundedness of the Cauchy transform, and constructs a robust regularity theory that extends from smooth to Lipschitz boundaries, yielding a Banach direct-sum decomposition . Key contributions include a constructive existence proof, a new higher-dimensional regularity result for holomorphic Hardy spaces, and applications to the Dirichlet/Poisson problems as well as a pseudo-local Szegő projection property. These results provide a fresh perspective on classical boundary-value problems and offer a framework for higher-dimensional generalizations of the Cauchy-transform kernel on non-smooth domains.

Abstract

We decompose - integrable functions on the boundary of a simply connected Lipschitz domain into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in while the other is holomorphic in and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the setting relies upon a classical albeit little-known regularity theorem for the holomorphic Hardy space of planar domains for which we provide a new proof that is valid also in higher dimensions. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on . These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.
Paper Structure (13 sections, 6 theorems, 62 equations)

This paper contains 13 sections, 6 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Smooth domains: antiderivatives of Hardy space functions and uniqueness of the $h+H$ decomposition in $L^p(b\Omega)$
  3. Proof of Theorem \ref{['T:3']} for smooth $\Omega$
  4. Uniqueness of the $h + H$ decomposition in $L^p(b\Omega)$ for smooth $\Omega$
  5. Proof of Theorem \ref{['T:1']}
  6. Background on Lipschitz Domains
  7. Hardy spaces for $\Omega$ and for the interior of its complement; the Cauchy Transform for $\Omega$
  8. Proof of Theorem \ref{['T:3']} and of Theorem \ref{['T:1']}: Part i.
  9. Proof of Theorem \ref{['T:1']}: Part ii.
  10. Proof of Theorem \ref{['T:1']}: Part iii.
  11. Further Results
  12. Something about Dirichlet and Poisson.
  13. A pseudo-local property of the Szegő projection.

Key Result

Theorem 1.1

Suppose $\Omega$ is a bounded Jordan domain with Lipschitz boundary. In any of the cases i. to iv., we have where $\mathcal{C}$ is the Cauchy transform associated to $\Omega$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 4.1
  • Theorem 4.2