On commutators of fractional integrals
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- by Xuan Thinh Duong and Li Xin Yan PDF
- Proc. Amer. Math. Soc. 132 (2004), 3549-3557 Request permission
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.References
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Additional Information
- Xuan Thinh Duong
- Affiliation: Department of Mathematics, Macquarie University, New South Wales 2109, Australia
- MR Author ID: 271083
- Email: duong@ics.mq.edu.au
- 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
- MR Author ID: 618148
- Email: mcsylx@zsu.edu.cn
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3549-3557
- MSC (2000): Primary 42B20, 47B38
- DOI: https://doi.org/10.1090/S0002-9939-04-07437-4
- MathSciNet review: 2084076