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

Abstract

The two‐weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures σ and w on ℝ. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two‐weight problem by Lacey, Sawyer, Shen, and Uriarte‐Tuero. Our characterization is in terms of Sawyer‐type testing conditions and a variant of the two‐weight A2 condition, where σ and w are integrated over complementary intervals only. A key novelty of the proof is a two‐weight inequality for the Poisson integral with ‘holes’.