  If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and
omega are doubling measures, then the two weight L^{p} norm inequality holds if
and only if the quadratic triple testing conditions of Hyt\"onen and Vuorinen
hold. We also show that these quadratic triple testing conditions can be
relaxed to quadratic local testing conditions, quadratic offset Muckenhoupt
conditions, and a quadratic weak boundedness property.
