The two weight T1 theorem for fractional Riesz transforms when one measure is supported on a curve

Let σ and ω be locally finite positive Borel measures on ℝn. We assume that at least one of the two measures σ and ω is supported on a regular C1,δ curve in ℝn. Let Rα,n be the α-fractional Riesz transform vector on ℝn. We prove the T1 theorem for Rα,n: namely that Rα,n is bounded from L2(σ) to L2(σ) if and only if the A2α$${\cal A}_2^\alpha $$ conditions with holes hold, the punctured A2α$$A_2^\alpha $$ conditions hold, and the cube testing …