Let and be locally finite positive Borel measures on which do not share a common point mass. Assume that the pair of weights satisfy a Poisson condition, and satisfy the testing conditions below, for the Hilbert transform ,
with constants independent of the choice of interval . Then maps to , 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.
Michael T. Lacey. Eric T. Sawyer. Chun-Yen Shen. Ignacio Uriarte-Tuero. "Two-weight inequality for the Hilbert transform: A real variable characterization, I." Duke Math. J. 163 (15) 2795 - 2820, 1 December 2014. https://doi.org/10.1215/00127094-2826690