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

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.