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

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 has two components, a global-to-local reduction, carried out in this article, and an analysis of the local problem, to be elaborated in a future Part II version of this article.

A Characterization of Two-Weight Trace Inequalities for Positive Dyadic Operators in the Upper Triangle Case

“…In his excellent survey of the A 2 theorem [3] Tuomas P. Hytönen introduces another proof of Proposition 1.2, which uses the "parallel corona" decomposition from the recent work of Lacey, Sawyer, Shen and Uriarte-Tuero [5] on the two weight boundedness of the Hilbert transform. Following Hytönen's arguments and applying a basic lemma due to [1], we shall establish the following two weight T 1 theorem for positive dyadic operators in the upper triangle case.…”

$$L^{p}(\mu ) \rightarrow L^{q}(\nu )$$ L p ( μ ) → L q ( ν ) Characterization for Well Localized Operators

“…We note here that this trick of introducing the constants c Q is from [5]. From the testing condition (4.1) and the Carleson embedding theorem 2.2 it follows that…”

“…Our proof will follow the technique of "parallel stopping cubes" as in [2]. This method was first introduced in [5] (only in the older arXiv versions) and was used in the investigations of the two weight inequality for the Hilbert transform.…”

Two weight norm inequalities for the $g$ function

“…This is then combined with the averaging over good Whitney regions of Martikainen-Mourgoglou [8], followed by a stopping time construction from [5]. Also see [4] for the application of these two ideas in the local T b setting.…”