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arXiv · 0802.4260

Hypercontractivity for log-subharmonic functions

Piotr Graczyk·Todd Kemp·Jean-Jacques Loeb·Tomasz Zak
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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.

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Piotr Graczyk, Todd Kemp, Jean-Jacques Loeb, Tomasz Zak. 2008-10-20. Hypercontractivity for log-subharmonic functions. https://arxiv.org/abs/0802.4260

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