Sharp bounds for general commutators on weighted Lebesgue spaces

Abstract

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 $k$th-order commutator $T^k_b=[b,T^{k-1}_b]$ 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 Calderón-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 $1<p<\infty$, all $k$, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.