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On commutators of fractional integrals

Authors:	Xuan Thinh Duong Li Xin Yan

Institution:	Department of Mathematics, Macquarie University, New South Wales 2109, Australia ; Department of Mathematics, Zhongshan University, Guangzhou, 510275, People's Republic of China

Abstract:	Let 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 for $0<\alpha< n$ . For a BMO function 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.

Keywords:	Gaussian bound fractional integrals {\rm BMO} commutator

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