Contractive inequalities for Bergman spaces and multiplicative Hankel forms

Bayart, Frédéric; Brevig, Ole Fredrik; Haimi, Antti; Ortega-Cerdà, Joaquim; Perfekt, Karl-Mikael

doi:10.1090/tran/7290

Contractive inequalities for Bergman spaces and multiplicative Hankel forms
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by Frédéric Bayart, Ole Fredrik Brevig, Antti Haimi, Joaquim Ortega-Cerdà and Karl-Mikael Perfekt PDF

Trans. Amer. Math. Soc. 371 (2019), 681-707 Request permission

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 types of forms which does not exist in lower dimensions. Finally, we produce some counterexamples concerning Carleson measures on the infinite polydisc.

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