Weighted Inequalities and Dyadic Harmonic Analysis

Abstract

We survey the recent solution of the so-called A 2 conjecture, that states: all Calderón–Zygmund singular integral operators are bounded on L 2(w) with a bound that depends linearly on the A 2 characteristic of the weight w. We also survey corresponding results for commutators. We highlight the interplay of dyadic harmonic analysis in the solution of the A 2 conjecture, especially Hytönen’s representation theorem for Calderón–Zygmund singular integral operators in terms of Haar shift operators. We describe Chung’s dyadic proof of the corresponding quadratic bound on L 2(w) for the commutator of the Hilbert transform with a BMO function, and we deduce sharpness of the bounds for the dyadic paraproduct on L p (w) that were obtained extrapolating Beznosova’s linear bound on L 2(w). We show that if an operator T is bounded on the weighted Lebesgue space L r (w) and its operator norm is bounded by a power α of the A r characteristic of the weight, then its commutator [T,b] with a function b in BMO will be bounded on L r (w) with an operator norm bounded by the increased power \(\alpha +\max \{ 1, \frac{1} {r-1}\}\) of the A r characteristic of the weight. The results are sharp in terms of the growth of the operator norm with respect to the A r characteristic of the weight for all 1<r<∞.