On commutators of fractional integrals

Abstract

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) 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 R n , 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 (R n ) to L q (R n ), where 1 < p < n α, 1 q = 1 p - α n. Our result of this boundedness still holds when R n is replaced by a Lipschitz domain of R n with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrodinger operators and second-order elliptic operators of divergence form.