Contraction property of Fock type space of log-subharmonic functions in ${\mathbb{R}}^m$

Abstract

We prove a contraction property of Fock type spaces ${\mathcal{L}}_{\alpha }^p$ of log-subharmonic functions in ${\mathbb{R}}^n$. To prove the result, we demonstrate a certain monotonic property of measures of the superlevel set of the function $u(x)=|f(x){|}^p{e}^{-\frac{\alpha }{2}p|x{|}^2}$, provided that $f$ is a certain log-subharmonic function in ${\mathbb{R}}^m$. The result recover a contraction property of holomorphic functions in the Fock space ${\mathcal{F}}_{\alpha }^p$ proved by Carlen in \cite{carlen}.