Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar¹

¹ UFR Mathématique et Informatique Université de Bordeaux I 351, Cours de la Libération 33405 Talence, France

Annales Polonici Mathematici, Tome 94 (2008) no. 1, pp. 79-87

Résumé

Let $S$ be a sequence of points in the unit ball ${\mathbb B}$ of ${\mathbb{C}}^{n}$ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure $ \mu _{S}:=\sum_{a\in S}^{}{(1-\vert{ a}\vert ^{2})^{n}\delta _{a}}$ is bounded, by use of the Wirtinger inequality. Conversely, if $X$ is an analytic subset of ${\mathbb{B}}$ such that any $\delta $-separated sequence $S$ has its associated measure $\mu _{S}$ bounded by $C/\delta ^{n},$ then $X$ is the zero set of a function in the Nevanlinna class of ${\mathbb{B}}.$As an easy consequence, we prove that if $S$ is a dual bounded sequence in $H^{p}({\mathbb{B}}),$ then $\mu _{S}$ is a Carleson measure, which gives a short proof in one variable of a theorem of L. Carleson and in several variables of a theorem of P. Thomas.

Détail

DOI : 10.4064/ap94-1-6

Keywords: sequence points unit ball mathbb mathbb which separated hyperbolic distance contained zero set nevanlinna function prove associated measure sum vert vert delta bounded wirtinger inequality conversely analytic subset mathbb delta separated sequence has its associated measure bounded delta zero set function nevanlinna class mathbb easy consequence prove dual bounded sequence mathbb carleson measure which gives short proof variable theorem carleson several variables theorem thomas