Infinitesimal Carleson Property for Weighted Measures Induced by Analytic Self-Maps of the Unit Disk

Abstract

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