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 Lp(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.
|Number of pages||9|
|Journal||Proceedings of the American Mathematical Society|
|Publication status||Published - Dec 2004|