On commutators of fractional integrals

Duong, Xuan; Yan, Li

doi:10.1090/S0002-9939-04-07437-4

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.

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