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

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 ${\sup }_{\tau >0}({\int }_{\Gamma }|F(\zeta +\mathrm{i}\tau ){|}^p|\mathrm{d}\zeta |{)}^{\frac{1}{p}}<\infty$. 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 $T:F\to F(\Phi (z))\cdot ({\Phi }^{\prime }(z){)}^{\frac{1}{p}}$, where $F\in H^p({\Omega }_{+})$.

25 pages

Cited by in corpus (1)

• Notes of Boundedness on Cauchy Integrals on Lipschitz Curves ($p=2$)