Compactness and essential norms for two-weight fractional Riesz transforms on curves
Abstract
We prove a compact two-weight T1 theorem for the α-fractional Riesz vector, 0≤α<n, when one measure is supported on a compact regular C1,Dini curve Γ. 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 α=n−1 and both measures have positive continuous arclength densities a,b on the same closed curve, we obtain the exact essential norm πcn−1,nmaxΓ√(ab).
Related
- Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a C1,Dini curve
- Boundedness of Multilinear Hilbert Transforms Along Moment Curves
- Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators
- Compactness and Essential Distances of Weighted Bilinear Hardy-Steklov Operators
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BibTeX
@misc{li2026compactness,
title = {{Compactness and essential norms for two-weight fractional Riesz transforms on curves}},
author = {Ji Li and Chun-Yen Shen and Chaojie Wen and Brett D. Wick and Liangchuan Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15002},
eprint = {2610.15002},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15002},
}
RIS
TY - UNPB
TI - Compactness and essential norms for two-weight fractional Riesz transforms on curves
AU - Ji Li
AU - Chun-Yen Shen
AU - Chaojie Wen
AU - Brett D. Wick
AU - Liangchuan Wu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15002
UR - https://arxiv.org/abs/2610.15002
AB - We prove a compact two-weight T1 theorem for the α-fractional Riesz vector, 0≤α<n, when one measure is supported on a compact regular C^1,Dini curve Γ. 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 α=n−1 and both measures have positive continuous arclength densities a,b on the same closed curve, we obtain the exact essential norm πc_n−1,nmax_Γ√(ab).
ER -