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

TL;DR

This paper demonstrates that weighted measures induced by analytic self-maps of the unit disk exhibit an infinitesimal Carleson property, with implications for the compactness of composition operators on weighted Bergman-Orlicz spaces.

Contribution

It establishes that these measures are not only Carleson measures but also satisfy an infinitesimal scale control, extending understanding of their behavior at small scales.

Findings

Measures are $( ext{alpha}+2)$-Carleson measures at all scales

Carleson window measures are controlled by $ ext{epsilon}^{ ext{alpha}+2}$

Application to characterizing compactness of composition operators

Abstract

We prove that, for every $\alpha > -1$, the pull-back measure $\phi ({\cal A}_\alpha)$ of the measure $d{\cal A}_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\cal A} (z)$, where ${\cal A}$ is the normalized area measure on the unit disk $\D$, by every analytic self-map $\phi \colon \D \to \D$ is not only an $(\alpha + 2)$-Carleson measure, but that the measure of the Carleson windows of size $\eps h$ is controlled by $\eps^{\alpha + 2}$ 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.