Hardy Spaces ($1<p<\infty$) over Lipschitz Domains

TL;DR

This paper extends the theory of Hardy spaces to Lipschitz domains in the complex plane, establishing boundary limits and isomorphisms with classical Hardy spaces via conformal mappings for the range 1<p<∞.

Contribution

It proves boundary limit properties and an isomorphism between Hardy spaces on Lipschitz domains and the upper half-plane, generalizing classical results.

Findings

Functions in $H^p( ext{domain})$ have non-tangential boundary limits almost everywhere.

$H^p( ext{domain})$ is isomorphic to $H^p( ext{upper half-plane})$ via a conformal map.

The isomorphism involves a weighted composition with the derivative of the conformal map.

Abstract

Let $\Gamma$ 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 holomorphic functions $F$ satisfying $\sup_{\tau>0}(\int_{\Gamma} |F(\zeta+\mathrm{i}\tau)|^p |\,\mathrm{d}\zeta|)^{\frac1p}< \infty$. We mainly focus on the case of $1<p<\infty$ in this paper, and prove that if $F(w)\in H^p(\Omega_+)$, then $F(w)$ has non-tangential boundary limit $F(\zeta)$ a.e. on $\Gamma$, and $F(w)$ is the Cauchy integral of $F(\zeta)$. We denote the conformal mapping from $\mathbb{C}_+$ onto $\Omega_+$ as $\Phi$, and then prove that, $ H^p(\Omega_+)$ is isomorphic to $H^p(\mathbb{C}_+)$, the classical Hardy space on upper half plane, under the mapping $T\colon F\to F(\Phi(z))\cdot (\Phi'(z))^\frac{1}{p}$, where $F\in H^p(\Omega_+)$.