Hardy Spaces over Half-strip Domains

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 $1\leqslant p<\infty$, 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.