The Two Weight Inequality for Hilbert Transform, Coronas, and Energy Conditions

Michael T. Lacey, Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero

Published 2011-08-11Version 1

We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on $ L ^2 $ of the weights. Such a characterization is known subject to certain side conditions. We give a new proof, simpler in many details, of the best such result. In addition, we analyze underlying assumptions in the proof, especially in terms of two alternate side conditions. A new characterization in the case of one doubling weight is given.