Composition operators on generalized Hardy spaces

Abstract

Let ${\Omega }_1,{\Omega }_2\subset \mathbb{C}$ be bounded domains. Let $\varphi :{\Omega }_1\to {\Omega }_2$ holomorphic in ${\Omega }_1$ and belonging to $W_{{\Omega }_2}^{1,\infty }({\Omega }_1)$. We study the composition operators $f\mapsto f\circ \varphi$ on generalized Hardy spaces on ${\Omega }_2$, recently considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or sufficient conditions on $\varphi$, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact. Some of our results are new even for Hardy spaces of analytic functions.