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

TL;DR

This paper establishes a contraction property for Fock type spaces of log-subharmonic functions in Euclidean space, extending known results from holomorphic functions to a broader class of functions.

Contribution

It proves a new contraction property for Fock type spaces of log-subharmonic functions, generalizing previous results for holomorphic functions.

Findings

Demonstrates a monotonic property of measures of superlevel sets of certain functions.

Establishes a contraction property for Fock type spaces of log-subharmonic functions.

Recovers a known contraction property for holomorphic functions in Fock spaces.

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}.