On commutators of fractional integrals

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.