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

Let $d\gamma (x)\equiv {\pi }^{-n\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2}dx$ 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 $L^{p\mathrm{/}(1-p\beta )}(\gamma )$, or from the Hardy space $H^1(\gamma )$ of Mauceri and Meda to $L^{1\mathrm{/}(1-\beta )}(\gamma )$ or from $L^{1\mathrm{/}\beta }(\gamma )$ to BMO$(\gamma )$, where $\beta \in (0,1)$ and $p\in (1,1\mathrm{/}\beta )$.