Hardy Spaces over Half-strip Domains

TL;DR

This paper extends the theory of Hardy spaces to half-strip domains, establishing boundary limits, boundedness of Cauchy transforms, and isomorphisms with classical Hardy spaces, thus broadening the understanding of complex analysis in these regions.

Contribution

It introduces Hardy spaces on half-strip domains, proves boundary limit properties, boundedness of Cauchy transforms, and establishes isomorphisms with classical Hardy spaces.

Findings

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

Cauchy transforms of $L^p$ functions on the boundary are in $H^p$ spaces, showing boundedness.

There is an isomorphism between Hardy spaces on half-strip domains and classical Hardy spaces over half-planes.

Abstract

We define Hardy spaces $H^p(\Omega_\pm)$ on half-strip domain~$\Omega_+$ and $\Omega_-= \mathbb{C}\setminus\overline{\Omega_+}$, where $0<p<\infty$, and prove that functions in $H^p(\Omega_\pm)$ has non-tangential boundary limit a.e. on $\Gamma$, the common boundary of $\Omega_\pm$. We then prove that Cauchy integral of functions in $L^p(\Gamma)$ are in $H^p(\Omega_\pm)$, where $1<p<\infty$, that is, Cauchy transform is bounded. Besides, if $1\leqslant p<\infty$, then $H^p(\Omega_\pm)$ functions are the Cauchy integral of their non-tangential boundary limits. We also establish an isomorphism between $H^p(\Omega_\pm)$ and $H^p(\mathbb{C}_\pm)$, the classical Hardy spaces over upper and lower half complex planes.