Infinitesimal Carleson Property for Weighted Measures Induced by Analytic Self-Maps of the Unit Disk |
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Authors: | Daniel Li Hervé Queffélec Luis Rodríguez-Piazza |
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Institution: | 1. Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Univ Lille-Nord-de-France, UArtois, Faculté des Sciences Jean Perrin, rue Jean Souvraz SP 18, 62300, Lens, France 2. Laboratoire Paul Painlevé, Univ Lille-Nord-de-France USTL, U.M.R. CNRS 8524, 59 655, Villeneuve d’Ascq Cedex, France 3. Departamento de Análisis Matemático & IMUS, Facultad de Matemáticas, Universidad de Sevilla, Apartado de Correos 1160, 41 080, Seville, Spain
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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. |
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