Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk

Abstract

We prove that, for every $\hat{I}\pm >-1$, the pull-back measure $\ddot{I}•({\mathcal{A}}_{\hat{I}}\pm )$ of the measure $d{\mathcal{A}}_{\hat{I}}\pm (z)=(\hat{I}\pm +1)(1-|z{|}^2{)}^{\hat{I}}\pm d\mathcal{A}(z)$, where $\mathcal{A}$ is the normalized area measure on the unit disk $\D$, by every analytic self-map $ϕ\colon \D \to \D$ is not only an $(\hat{I}\pm +2)$-Carleson measure, but that the measure of the Carleson windows of size $\eps h$ is controlled by $\eps^{α+ 2}$ times the measure of the corresponding window of size $h$. This means that the property of being an $(\hat{I}\pm +2)$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces.