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On commutators of fractional integrals

Authors: Xuan Thinh Duong and Li Xin Yan
Journal: Proc. Amer. Math. Soc. 132 (2004), 3549-3557
MSC (2000): Primary 42B20, 47B38
Published electronically: July 14, 2004
MathSciNet review: 2084076
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Abstract: Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb R}^n)$ with Gaussian kernel bounds, and let $L^{-\alpha/2}$ be the fractional integrals of $L$ for $0<\alpha< n$. For a BMO function $b(x)$ on ${\mathbb R}^n$, we show boundedness of the commutators $ [b, L^{-\alpha/2}](f)(x)= b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x) $ from $L^p({\mathbb R}^n)$ to $L^q({\mathbb R}^n)$, where $1< p < \tfrac{n}{\alpha}, \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Our result of this boundedness still holds when ${\mathbb R}^n$ is replaced by a Lipschitz domain of ${\mathbb R}^n$ with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form.

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Additional Information

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, New South Wales 2109, Australia

Li Xin Yan
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Address at time of publication: Department of Mathematics, Macquarie University, New South Wales 2109, Australia

Keywords: Gaussian bound, fractional integrals, {\rm BMO}, commutator
Received by editor(s): January 3, 2003
Published electronically: July 14, 2004
Additional Notes: Both authors were supported by a grant from Australia Research Council, and the second author was also partially supported by the NNSF of China (Grant No. 10371134).
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society