Functionals with extrema at reproducing kernels

Abstract

We show that certain monotone functionals on the Hardy spaces and convex functionals on the Bergman spaces are maximized at the normalized reproducing kernels among the functions of norm 1, thus proving the contractivity conjecture of Pavlović and of Brevig, Ortega-Cerdà, Seip and Zhao and the Wehrl-type entropy conjecture for the SU(1, 1) group of Lieb and Solovej, respectively.

Hypercontractive inequalities for weighted Bergman spaces

“…The main tools in our approach are recent inequalities due to Kulikov ([10]) and the lemma on convexity of certain integral means of analytic functions ( [15]). Let us say that, in [10], Kulikov has proved a conjecture of Pavlović from [14] and [15] and Bayart, Brevig, Haimi, Ortega-Cerda and Perfekt from [1], where several results that support this and some other conjectures are provided. Moreover, he proved a more general conjecture of Lieb and Solovej from [11].…”

Hypercontractive inequalities for weighted Bergman spaces

“…The main tools in our approach are recent inequalities due to Kulikov ([10]) and the lemma on convexity of certain integral means of analytic functions ([15]). Let us say that, in [10], Kulikov has proved a conjecture of Pavlović from [14] and [15] and Bayart, Brevig, Haimi, Ortega‐Cerda, and Perfekt from [1], where several results that support this and some other conjectures are provided. Moreover, he proved a more general conjecture of Lieb and Solovej from [11].…”

Contractive inequalities between Dirichlet and Hardy spaces

“…Contractive inclusions between spaces of analytic functions have attracted the attention of the experts because of their multiple applications. For this work is especially relevant the following inequality, which was conjectured by Brevig, Ortega-Cerdà, Seip and Zhao [4] and Lieb and Solovej [11] and recently proved by Kulikov [10].…”

“…Although it was not explicitly stated in [10], from Theorem A we can derive a complete characterization of contractive inclusions between weighted Bergman spaces.…”