On commutators of fractional integrals

Let L be the infinitesimal generator of an analytic semigroup on L ²(ℝⁿ) with Gaussian kernel bounds, and let L ^-α/2 be the fractional integrals of L for 0 < α < n. For a BMO function b(x) on ℝⁿ, we show boundedness of the commutators [b, L^-α/2](f)(x) = b(x)L^-α/2(f)(x) -L^-α/2(bf)(x) from L^p(ℝⁿ) to L ^q(ℝⁿ), where 1 < p < n/α, 1/q = 1/p - α/n. Our result of this boundedness still holds when ℝⁿ is replaced by a Lipschitz domain of ℝⁿ 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.