  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.
