Sharp bounds for general commutators on weighted Lebesgue spaces

TL;DR

This paper establishes sharp bounds for commutators of operators on weighted Lebesgue spaces, showing their operator norms grow quadratically or as a power of the A_2 constant, with results applicable across all p, k, and dimensions.

Contribution

It provides the first sharp bounds for general commutators on weighted Lebesgue spaces, extending known linear bounds to quadratic and higher powers with respect to the A_2 constant.

Findings

Commutators obey quadratic bounds in A_2 for bounded operators.

Higher-order commutators grow as powers of the A_2 constant.

Results are sharp and applicable across all p, k, and dimensions.

Abstract

We show that if an operator T is bounded on 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^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. 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.