  We begin an investigation into extending the T1 theorem of David and
Journ\'e, and the corresponding cancellation conditions of Stein, to more
general pairs of distinct doubling weights. For example, assuming the measures
satisfy a fractional A infinity condition and are comparable in the sense of
Coifman and Fefferman, we characterize the two weight norm inequality for a
strongly elliptic fractional Calder\'on-Zygmund singular integral, in terms of
the one-tailed fractional Muckenhoupt conditions, and the usual cube testing
conditions. We then apply this result to give a version, in the setting of two
comparable fractional A infinity weights, of Stein's characterization of
cancellation conditions on a kernel K in order that there exists a bounded
operator T that is associated with K. More generally we prove a T1 theorem
involving a bilinear indicator/cube testing inequality in place of the weak
boundedness property of David and Journe\'e - where we must test over all
bounded functions instead of just Holder continuous functions. We use a proof
strategy based on an adaptation of the `pivotal' argument of Nazarov, Treil and
Volberg to the weighted Alpert wavelets of Rahm, Sawyer and Wick using a
Parallel Corona decomposition of Lacey, Sawyer, Shen and Uriarte-Tuero.
