  The two weight inequality for the Hilbert transform arises in the settings of
analytic function spaces, operator theory, and spectral theory, and what would
be most useful is a characterization in the simplest real-variable terms. We
show that the $L^2$ to $L^2$ inequality holds if and only if two L^2 to
weak-L^2 inequalities hold. This is a corollary to a characterization in terms
of a two-weight Poisson inequality, and a pair of testing inequalities on
bounded functions.
