Two-weight inequality for the Hilbert transform: A real variable characterization, II

Abstract

Let and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A 2 condition, and satisfy the testing conditions below, for the Hilbert transform H , Z I H. 1 I / 2 dw .I /; Z I H.w1 I / 2 d w.I /; with constants independent of the choice of interval I . Then H. / maps L 2 . / to L 2 .w/, verifying a conjecture of Nazarov, Treil, and Volberg. The proof uses basic tools of nonhomogeneous analysis with two components particular to the Hilbert transform. The first component is a global-to-local reduction which is a consequence of prior work by Lacey, Sawyer, Shen, and Uriarte-Tuero. The second component, an analysis of the local part, is the particular contribution of this article.

Entropy conditions in two weight inequalities for singular integral operators

“…We should mention here a breakthrough paper [13] where it was proved that the estimate (1.2) holds for p = 2 if and only if (1.2) and the "dual" estimate with u and v interchanged hold uniformly for indicators of intervals (and the so-called Poisson A 2 condition is satisfied); this result solves a long standing conjecture by Nazarov-Treil-Voberg.…”

“…Paper [13] was a culmination of the long line of research started by Nazarov and us [24] (see also the last two chapters of a book [35]), and continued in joint works of Lacey, Sawyer, Uriarte-Tuero, and Shen: [15], [16], [17]. In particular, a very interesting achievment of the latter group was the introduction of a certain "energy", a very interesting positive quantity capturing "the specifics of the tail" of the Hilbert transform.…”

Two weight L estimates for paraproducts in non-homogeneous settings

“…The recent achievements in the theory of L 2 two weight estimates include sufficiency of Sawyer type estimates for Hilbert Transform [7], [5], for the Cauchy Transform [8], for the Riesz transforms [9].…”

The two-weight inequality for the Hilbert transform with general measures

“…A lot of deep work in the last 10 years or more was devoted to obtaining conditional characterizations under increasingly general side conditions on and , including the doubling property [, Theorem 15.1] and so‐called ‘pivotal’ and ‘energy’ hypotheses. In a culmination of these efforts, a complete solution was finally obtained by Lacey, Sawyer, Shen, and Uriarte‐Tuero . Indeed, their recent result goes beyond the weighted problem as stated, in that their characterization is equally valid for any pair of positive Radon measures and , subject only to the restriction that they do not share a common point mass; this manifestly covers the case of weighted measures.…”