SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES

We show that if a linear operator T is bounded on a weighted Lebesgue space L-2(w) and obeys a linear bound with respect to the A(2) constant of the weight, then its commutator [b, T] with a function b in BMO will obey a quadratic bound with respect to the A(2) constant of the weight. We also prove that the kth-order commutator T-b(k) = [b, T-b(k-1)] will obey a bound that is a power (k + 1) of the A(2) constant of the weight. Sharp extrapolation provides corresponding L-p(w) estimates. In particular these estimates hold for T any Calderon-Zygmund singular integral operator. The results are sharp in terms of the growth of the operator norm with respect to the A(p) constant of the weight for all 1 < p < infinity, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.