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

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.

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