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

Abstract

Let $\sigma$ and $w$ be locally finite positive Borel measures on $\mathbb{R}$ 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 $H$,

$${\int }_{I}H\left(\sigma {\mathbf{1}}_I{)}^{2}dw\lesssim \sigma \right(I),{\int }_{I}H(w{\mathbf{1}}_I{)}^{2}d\sigma \lesssim w\left(I\right),$$

with constants independent of the choice of interval $I$. Then $H(\sigma \cdot )$ maps $L^2\left(\sigma \right)$ to $L^2\left(w\right)$, 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.