  We prove a T1 theorem for fractional vector Riesz transforms mapping one
weighted Sobolev space to another, where the weights are doubling measures on
Euclidean space. Boundedness is characterized by the classical A_2 condition
and two dual testing conditions on indicators of cubes. We also show the
equivalence of various weighted Sobolev norms when the measure is doubling,
something that fails in general.
