Hardy Spaces over Half-strip Domains

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 $1⩽p<\infty$, 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.

38 pages