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

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