  We show that for two doubling measures on n dimensional Euclidean space, any
gradient-elliptic smooth fractional Calderon-Zygmund operator T, and any fixed
dyadic grid D, the operator norm of T from one weighted L2 space to the other
is comparable to the supremum of the norms of Th and T*k as h and k vary over
all weighted Haar wavelets associated with the grid D and the respective
measures. We then extend this theorem to weighted Lp spaces for p between 1 and
infinity, using frames in place of orthonormal bases, and Haar testing
characteristics modelled on the vector-valued characteristics arising in the
work of Hytonen and Vuorinen in Lp spaces. Finally, we briefly pose these
questions in the context of orthonormal bases in arbitrary Hilbert spaces, and
in the context of frames in arbitrary Banach spaces.
