Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a C1,Dini curve
Abstract
Let 0≤α<n, and let σ and ω be locally finite positive Borel measures on Rn, possibly with common point masses. Suppose that one measure is supported on a compact regular C1,Dini curve. We prove that the α-fractional Riesz vector maps L2(σ) to L2(ω) if and only if the two cA2α conditions with holes, the two punctured A2α conditions, and the forward and dual testing conditions over rotated cubes hold. The theorem includes curves whose tangent is not Hölder continuous of any positive order. Straightening such a curve produces a kernel whose first derivatives need not satisfy the Hölder 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,Γ)β dx, the range is β>n−3 in a bounded tube and n−3<β<n−2 in Rn. 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<β<1.
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BibTeX
@misc{li2026twoweight,
title = {{Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a $C^1,\mathrmDini$ curve}},
author = {Ji Li and Chun-Yen Shen and Eric Sawyer and Brett Wick},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16123},
eprint = {2610.16123},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16123},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a C^1,Dini curve
AU - Ji Li
AU - Chun-Yen Shen
AU - Eric Sawyer
AU - Brett Wick
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16123
UR - https://arxiv.org/abs/2610.16123
AB - Let 0≤α<n, and let σ and ω 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,Dini curve. We prove that the α-fractional Riesz vector maps L^2(σ) to L^2(ω) if and only if the two cA_2^α conditions with holes, the two punctured A_2^α conditions, and the forward and dual testing conditions over rotated cubes hold. The theorem includes curves whose tangent is not Hölder continuous of any positive order. Straightening such a curve produces a kernel whose first derivatives need not satisfy the Hölder 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,Γ)^β dx, the range is β>n−3 in a bounded tube and n−3<β<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<β<1.
ER -