Composition operators between Hardy and weighted Bergman spaces on convex domains in $\textbf {C}^ N$

Abstract:

Suppose $\Omega$ is a bounded, strongly convex domain in ${{\mathbf {C}}^N}$ with smooth boundary and $\phi :\Omega \to \Omega$ is an arbitrary holomorphic map. While in general the composition operator ${C_\phi }$ need not map the Hardy space ${H^p}(\Omega )$ into itself when $N > 1$, our main theorem shows that ${C_\phi }$ does map ${H^p}(\Omega )$ boundedly into a certain weighted Bergman space on $\Omega$, where the weight function depends on the dimension N. We also consider properties of ${C_\phi }$ on ${H^p}(\Omega )$ when $\phi (\Omega )$ is contained in an approach region in $\Omega$.