Hypercontractivity for log-subharmonic functions

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on ${\text{RR}}^n$ and different classes of measures: Gaussian measures on ${\text{RR}}^n$, symmetric Bernoulli and symmetric uniform probability measures on $\text{RR}$, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for {\em any} symmetric measure on $\text{RR}$. For all measures on $\mathbb{R}$ 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.