  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.
