Hardy Spaces ($0<p<\infty$) over Lipschitz Domains

TL;DR

This paper establishes an isomorphism between Hardy spaces over Lipschitz domains and classical Hardy spaces on the upper half-plane, demonstrating boundedness of the conformal mapping and boundary limit properties of functions.

Contribution

It proves the isomorphism between Hardy spaces on Lipschitz domains and the classical Hardy spaces, including boundedness of the conformal mapping and boundary behavior of functions.

Findings

$H^p( ext{Lipschitz domain})$ is isomorphic to $H^p( ext{upper half-plane})$

Conformal map induces bounded isomorphism between Hardy spaces

Functions have non-tangential boundary limits and are Cauchy integrals for $p \\geq 1$

Abstract

Let $0<p<\infty$, $\Gamma$ be a Lipschitz curve on the complex plane~$\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of analytic functions $F$ satisfying $\sup_{\tau>0}(\int_{\Gamma} |F(\zeta+\mathrm{i}\tau)|^p |\,\mathrm{d}\zeta|)^{\frac1p}< \infty$. We denote the conformal mapping from $\mathbb{C}_+$ onto $\Omega_+$ as $\Phi$, and prove that, $H^p(\Omega_+)$ is isomorphic to $H^p(\mathbb{C}_+)$, the classical Hardy space on the upper half plane~$\mathbb{C}_+$, under the mapping $T\colon F\to F(\Phi)\cdot (\Phi')^{\frac1p}$. Besides, $T$ and $T^{-1}$ are both bounded. We also prove that if $F(w)\in H^p(\Omega_+)$, then $F(w)$ has non-tangential boundary limit $F(\zeta)$ a.e. on $\Gamma$, and, if $1\leqslant p< \infty$, $F(w)$ is the Cauchy integral on $\Gamma$ of $F(\zeta)$.