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

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 $p\in(1, 1/\beta)$.