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

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