Hypercontractivity for log-subharmonic functions

TL;DR

This paper establishes strong hypercontractivity inequalities for log-subharmonic functions across various measures on real space, revealing new log-Sobolev inequalities with improved constants and extending results to higher dimensions.

Contribution

It proves strong hypercontractivity for log-subharmonic functions under multiple measures and introduces sharper log-Sobolev inequalities in this context.

Findings

Strong hypercontractivity holds for Gaussian, Bernoulli, and uniform measures.

A weaker hypercontractivity property applies to all symmetric measures on \\RR.

Log-Sobolev inequalities with improved constants are established for these measures.

Abstract

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on $\RR^n$ and different classes of measures: Gaussian measures on $\RR^n$, symmetric Bernoulli and symmetric uniform probability measures on $\RR$, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for {\em any} symmetric measure on $\RR$. For all measures on $\R$ for which we know the (SHC) holds, we prove that a log--Sobolev inequality holds in the log-subharmonic category with a constant {\em smaller} than the one for Gaussian measure in the classical context. This result is extended to all dimensions for compactly-supported measures.