Functionals with extrema at reproducing kernels

TL;DR

This paper proves that specific monotone and convex functionals on Hardy and Bergman spaces reach their maximum at normalized reproducing kernels, confirming several longstanding conjectures in functional analysis and quantum information theory.

Contribution

It establishes that certain functionals are maximized at normalized reproducing kernels, resolving the contractivity and Wehrl-type entropy conjectures.

Findings

Maxima of monotone functionals on Hardy spaces occur at reproducing kernels.

Convex functionals on Bergman spaces are maximized at these kernels.

The results confirm conjectures by Pavlović, Brevig et al., and Lieb and Solovej.

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 Pavlovi\'c and of Brevig, Ortega-Cerd\`a, Seip and Zhao and the Wehrl-type entropy conjecture for the $SU(1,1)$ group of Lieb and Solovej, respectively.