Contractive inequalities for Bergman spaces and multiplicative Hankel forms

Abstract

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding 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 multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc.

This paper has been accepted for publication in Transactions of the AMS

References in corpus (4)

• The multiplicative Hilbert matrix
• An inequality of Hardy--Littlewood type for Dirichlet polynomials
• On Helson matrices: moment problems, non-negativity, boundedness, and finite rank
• Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes

Cited by in corpus (8)

• Norms of composition operators on the $H^2$ space of Dirichlet series
• Hypercontractive inequalities for weighted Bergman spaces
• A monotonicity theorem for subharmonic functions on manifolds
• Contractive inequalities for mixed norm spaces and the Beta function
• About a conjecture of Lieb-Solovej
• Volterra operators and Hankel forms on Bergman spaces of Dirichlet series
• Norms of inclusions between some spaces of analytic functions
• A Sharp Inequality of Hardy-Littlewood Type Via Derivatives