The two weight inequality for the hilbert transform: a primer

Abstract

Given a pair of weights w, σ, the two weight inequality for the Hilbert transform is of the form ∥H(σf)∥L2(w)≲∥f∥L2(σ). Recent work of Lacey-Sawyer-Shen-Uriarte-Tuero and Lacey have established a conjecture of Nazarov-Treil-Volberg, giving a real-variable characterization of which pairs of weights this inequality holds, provided the pair of weights do not share a common point mass. In this paper, the characterization is proved, collecting details from across several papers; counterexamples are detailed; and areas of application are indicated.