Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk
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چکیده
We prove that, for every α > −1, the pull-back measure φ(Aα) of the measure dAα(z) = (α+1)(1− |z| ) dA(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 εh is controlled by ε 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. Mathematics Subject Classification 2010. Primary: 30J99 – Secondary: 30H05 – 30C99 – 30H20 – 46E15 – 47B33 Key-words. Calderón-Zygmund decomposition ; Carleson measure ; weighted Bergman space Daniel Li, Univ Lille-Nord-de-France UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Faculté des Sciences Jean Perrin, Rue Jean Souvraz, S.P. 18, F-62 300 LENS, FRANCE – [email protected] Hervé Queffélec, Univ Lille-Nord-de-France USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524, F-59 655 VILLENEUVE D’ASCQ Cedex, FRANCE – [email protected] Luis Rodríguez-Piazza, Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático, Apartado de Correos 1160, 41 080 SEVILLA, SPAIN – [email protected]
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