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

Daniel Li (1) , Hervé Queffélec (2) , Luis Rodriguez-Piazza (3)

1 LML - Laboratoire de Mathématiques de Lens
2 LPP - Laboratoire Paul Painlevé
3 Departamento de Analisis Matematico

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.

Keywords: Calderón-Zygmund decomposition; Carleson measure; weighted Bergman space

Domains: Functional Analysis [math.FA]; Classical Analysis and ODEs [math.CA]

Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk. 2012. ⟨hal-00704738⟩