Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

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.