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}.