Composition operators on generalized Hardy spaces

Let $Ω_1,Ω_2\subset {\mathbb C}$ be bounded domains. Let $ϕ:Ω_1\rightarrow Ω_2$ holomorphic in $Ω_1$ and belonging to $W^{1,\infty}_{Ω_2}(Ω_1)$. We study the composition operators $f\mapsto f\circϕ$ on generalized Hardy spaces on $Ω_2$, recently considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or sufficient conditions on $ϕ$, 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.

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