  We characterize two-weight inequalities for certain maximal truncations of
the Hilbert transform in terms of testing conditions on simpler functions. For
1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling,
and we consider maximal truncations T_# of the Hilbert transform. The norm
estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms
of an A_p condition on the weights and two testing conditions. The first is the
norm condition above, but the function f varies over bounded functions
supported on a cube. The second is a dual weak-type condition, for arbitrary
functions. This result should be compared to the result of Nazarov, Treil and
Volberg, arXiv:math/0702758. Additional results are obtained for 2<p<\infty,
and for the weak type inequality.
