Weighted Inequalities and Dyadic Harmonic Analysis

Abstract

We survey the recent solution of the so-called A 2 conjecture, 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, as well as 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 α + max{1, 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 < ∞.

Dyadic analysis meets number theory

“…Indeed, it suffices if the statement of [5,Claim 1.13] holds for a subsequence {n i } i≥1 . This is clear from the proof of [5, Theorem 1.12] (see, [5, Page 272, Line [19][20][21][22]). In Proposition 2.1, we have shown that the ratio Om(p,q) p m−1 stabilizes for m sufficiently large (which is n in [5], which is stronger than necessary.…”

“…The study of these weights has been extensive, more information and some recent applications can be found in [9], [21], [22], [19], [14], [15], [4] among many others. There is also interesting complimentary work done in [7].…”

“…In the authors previous works together ( [2], [3]), as well as in other places ( [23], [19], [1]), a class of numbers called the far numbers (specifically "far from the dyadic rationals") play a large role in understanding distinct dyadic systems, which are a set of grids with the property that every cube is contained in a cube from one of the grids of roughly the same size. Distinct dyadic systems are highly useful ( [6], [8], [10], [17], [18], [20], [21] to name just a few), and in our recent works we were able to completely characterize them in both R and R n ( [2], [3]). Essentially, far numbers are bounded away from the dyadic numbers on every scale (see (2.7) for a precise definition), and by shifting a dyadic grid by a far number, one can create a distinct dyadic systems for small scale cubes.…”

The structure of weight and function classes with coprime bases

“…Measures and weight and function classes are frequent tools used in harmonic analysis and its applications (such as in [3], [8], [9], to name a few). The study of intersection properties of different classes of these objects is delicate, but results greatly increase our understanding of how these classes interact, uncovering their underlying structure (see [6], [7], [10]).…”

A structure theorem on doubling measures with different bases

“…and reverse Hölder weights are all automatically doubling, and the doubling property is the key point of the definition of spaces of homogeneous type. For more background and applications of doubling measures (particularly from a more modern perspective), see, for example [4], [5], [7], [9], [10], [11], [12], [13]. In particular, dyadic doubling measures have been central to a rich area of study, see, for example [6].…”