Compact composition operators on Bergman-Orlicz spaces

Abstract

We construct an analytic self-map $\varphi$ of the unit disk and an Orlicz function $\Psi$ for which the composition operator of symbol $\varphi$ is compact on the Hardy-Orlicz space $H^\Psi$, but not on the Bergman-Orlicz space ${\mathfrak B}^\Psi$. For that, we first prove a Carleson embedding theorem and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order $2$). We show that this Carleson function is equivalent to the Nevanlinna counting function of order $2$.