Contractive inequalitie for Bergman spaces and multiplicative Hankel forms.

Abstract

Abstract. We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimet- ric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield correspond- ing inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multi- plicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two types of forms which does not exist in lower dimensions. Finally, we produce some counterexamples concerning Carleson measures on the infinite polydisc.