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

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

2009-03-23

arxiv.org/abs/0903.3780v2

Mathematics

Classical Analysis and ODEs

25 pages; Israel J. Math. (to appear)

Scientific paper

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

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