  This paper is the third in an investigation begun in arXiv:1906.05602 and
arXiv:1907.07571 of extending the T1 theorem of David and Journ\'e, and optimal
cancellation conditions, to more general weight pairs. The main result here is
that the familiar T1 testing conditions over indicators of cubes, together with
the one-tailed A2 conditions, imply polynomial testing. Analogous results for
fractional singular integrals hold as well. Applications include a T1 theorem
for fractional CZO's T in the case of doubling measures when one of the weights
is A infinity, and then to optimal cancellation conditions for such CZO's in
similar situations.
