Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures

Abstract: 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.