Sharp bounds for general commutators on weighted Lebesgue spaces

Abstract

We show that if a linear operator T is bounded on weighted Lebesgue space L2(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a bound that is a power (k + 1) of the A2 constant of the weight. Sharp extrapolation provides corresponding Lp(w) estimates. In particular these estimates hold for T any Calder´on-Zygmund singular integral operator. The results are sharp in terms of the growth of the operator norm with respect to the Ap constant of the weight for all 1 < p < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.