Hypercontractive inequalities for weighted Bergman spaces

TL;DR

This paper establishes precise hypercontractive inequalities for weighted Bergman spaces on the unit disk, resolving a specific case of a known problem and providing additional estimates for certain parameter ranges.

Contribution

It provides sharp $L^p\rightarrow L^q$ hypercontractive inequalities for weighted Bergman spaces, advancing understanding of their functional analysis properties.

Findings

Sharp inequalities for $q\geq 2$ in weighted Bergman spaces.

Partial estimates for $0<q<2$.

Resolution of a problem from Janson's work.

Abstract

\begin{abstract} We obtain sharp $L^p\rightarrow L^q$ hypercontractive inequalities for the weighted Bergman spaces on the unit disk $\mathbb{D}$ with the usual weights \\ $\frac{\alpha-1}{\pi}(1-|z|^2)^{\alpha-2},\alpha>1$ for $q\geq 2,$ thus solving an interesting case of a problem from \cite{JANSON}. We also give some estimates for $0<q<2.$ \end{abstract}