Compactness and essential norms for two-weight fractional Riesz transforms on curves

We prove a compact two-weight $T1$ theorem for the $\alpha$-fractional Riesz vector, $0\le \alpha <n$, when one measure is supported on a compact regular $C^{1,\mathrm{D}\mathrm{i}\mathrm{n}\mathrm{i}}$ curve $\mathrm{\Gamma }$. The measures may have common point masses. Under the global boundedness assumptions, we give a two-sided estimate for the essential norm in terms of Muckenhoupt and testing constants computed after restricting both measures to small cubes. Compactness is equivalent to the uniform vanishing of these local bounds as the cube side length tends to zero. The smooth truncations have a unique weak operator limit and converge in operator norm in the compact case. For arclength measure and distance power weights, we determine the compactness ranges for Green gradients on bounded tubes and for Newtonian gradients and Biot–Savart fields on the whole space. The whole-space operators in these ranges are compact but are not Hilbert–Schmidt. When $\alpha =n-1$ and both measures have positive continuous arclength densities $a,b$ on the same closed curve, we obtain the exact essential norm $\pi c_{n-1,n}{\max }_{\mathrm{\Gamma }}\sqrt{ab}$.