Characterizations of BMO Associated with Gauss Measures via Commutators of Local Fractional Integrals

TL;DR

This paper characterizes the BMO space associated with the Gauss measure using commutators of local fractional integral and maximal operators, establishing boundedness properties and new characterizations.

Contribution

It provides new characterizations of BMO$( ext{Gauss})$ via commutators of local fractional integrals and maximal operators, extending previous understanding in Gaussian harmonic analysis.

Findings

Boundedness of local fractional integral operators from $L^p( ext{Gauss})$ to $L^{p/(1-p\beta)}( ext{Gauss})$

Characterizations of BMO$(\text{Gauss})$ via commutators of these operators

Extension of classical harmonic analysis results to Gaussian measure setting

Abstract

Let $d\gamma(x)\equiv\pi^{-n/2}e^{-|x|^2}dx$ for all $x\in{\mathbb R}^n$ be the Gauss measure on ${\mathbb R}^n$. In this paper, the authors establish the characterizations of the space BMO$(\gamma)$ of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order $\beta$ is bounded from $L^p(\gamma)$ to $L^{p/(1-p\beta)}(\gamma)$, or from the Hardy space $H^1(\gamma)$ of Mauceri and Meda to $L^{1/(1-\beta)}(\gamma)$ or from $L^{1/\beta}(\gamma)$ to BMO$(\gamma)$, where $\beta\in(0, 1)$ and $p\in(1, 1/\beta)$.