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

Abstract

We prove that, for every α>−1, the pull-back measure φ(Aα) of the measure dAα(z)=(α+1)(1−|z|2)αdA(z), where A is the normalized area measure on the unit disk D, by every analytic self-map φ:D→D is not only an (α+2)-Carleson measure, but that the measure of the Carleson windows of size εhεh is controlled by εα+2 times the measure of the corresponding window of size h. This means that the property of being an (α+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.