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

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 $F(\zeta)$.