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 ${\sup }_{\tau >0}({\int }_{\Gamma }|F(\zeta +\mathrm{i}\tau ){|}^p|\mathrm{d}\zeta |{)}^{\frac{1}{p}}<\infty$. 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 $T:F\to F(\Phi )\cdot ({\Phi }^{\prime }{)}^{\frac{1}{p}}$. 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 $1⩽p<\infty$, $F(w)$ is the Cauchy integral on $\Gamma$ of $F(\zeta )$.

18 pages. arXiv admin note: text overlap with arXiv:1708.01188