Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a $C^{1,\mathrm{Dini}}$ curve

Abstract

Let $0\leq\alpha<n$, and let $\sigma$ and $\omega$ be locally finite positive Borel measures on $\R^n$, possibly with common point masses. Suppose that one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve. We prove that the $\alpha$-fractional Riesz vector maps $L^2(\sigma)$ to $L^2(\omega)$ if and only if the two $\cA_2^\alpha$ conditions with holes, the two punctured $A_2^\alpha$ conditions, and the forward and dual testing conditions over rotated cubes hold. The theorem includes curves whose tangent is not H\"older continuous of any positive order. Straightening such a curve produces a kernel whose first derivatives need not satisfy the H\"older estimates required by the energy-side-condition $T1$ theorem. We prove both strong energy conditions in the straightened coordinates using only a sufficiently small graph slope, and show that these conditions are preserved under biLipschitz changes of variables. The Dini condition makes the errors in the kernel and testing comparisons summable. We then transfer the energy conditions back and apply the $T1$ theorem to the original smooth Riesz kernel. We also obtain a two-weight criterion for Green gradients with curve-supported sources. For arclength measure on the curve and observation measure $\dist(x,\Gamma)^\beta\,dx$, the range is $\beta>n-3$ in a bounded tube and $n-3<\beta<n-2$ in $\R^n$. The endpoints fail logarithmically whenever the measure is locally finite. In dimension three, the corresponding whole-space range for the Biot--Savart field with arbitrary vector densities is $0<\beta<1$.