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

TL;DR

This paper extends a T1 theorem for fractional Riesz transforms to measures supported on curves, including cases with common point masses, broadening the scope from previous work on lines and circles.

Contribution

It generalizes the T1 theorem to regular C(1,delta) curves and allows for common point masses, expanding the applicability of the previous results.

Findings

Extended T1 theorem to measures on curves

Included cases with common point masses

Applied results to Cauchy transform on circles

Abstract

Using our T1 theorem with an energy side condition allowing common point masses, we extend our previous work in arXiv:1310.4484v3 on one measure supported on a line, to include regular C(1,delta) curves and to permit common point masses. In the special case of the Cauchy transform with one measure supported on the circle, this gives a slightly different conclusion than that in arXiv:1310.4820v4.