Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs
Abstract
In this paper, we focus on the positive reflectionless measures rigidity problem proposed by Xavier Tolsa in [16]. Let 1≤ n<d, we consider positive reflectionless measures for the n-dimensional Riesz transform in ℝd. Under polynomial upper growth and a lower mass bound at large scales about one point, we obtain two flatness results for reflectionless measures. The first concerns globally defined Lipschitz graph measures with finite Ḣ1/2 energy and bounded, uniformly positive parameter densities whose deviations from a constant belong to L2. The second case concerns a reflectionless measure that coincides with a flat measure outside a compact set. In both cases, the measure is a constant multiple of the n-dimensional Hausdorff measure on an affine n-plane. For integers s>n/2, we further obtain Hs estimates controlling the graph gradient and density perturbation by the normalized Riesz field under smallness of the slope and density perturbation in L∞. The constants are independent of the supports.
Related
- Reflectionless measures with singular continuous components on Widom sets
- Rigidity of Lipschitz map using harmonic map heat flow
- Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains
- Shorter proof of dimension-free Lp estimates for maximal Riesz transforms
- Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms
BibTeX
@misc{chen2026rigidity,
title = {{Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs}},
author = {Lu Chen and Haolin Liu and Hongyu Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32179},
eprint = {2609.32179},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32179},
}
RIS
TY - UNPB
TI - Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs
AU - Lu Chen
AU - Haolin Liu
AU - Hongyu Liu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32179
UR - https://arxiv.org/abs/2609.32179
AB - In this paper, we focus on the positive reflectionless measures rigidity problem proposed by Xavier Tolsa in [16]. Let 1≤ n<d, we consider positive reflectionless measures for the n-dimensional Riesz transform in ℝ^d. Under polynomial upper growth and a lower mass bound at large scales about one point, we obtain two flatness results for reflectionless measures. The first concerns globally defined Lipschitz graph measures with finite Ḣ^1/2 energy and bounded, uniformly positive parameter densities whose deviations from a constant belong to L^2. The second case concerns a reflectionless measure that coincides with a flat measure outside a compact set. In both cases, the measure is a constant multiple of the n-dimensional Hausdorff measure on an affine n-plane. For integers s>n/2, we further obtain H^s estimates controlling the graph gradient and density perturbation by the normalized Riesz field under smallness of the slope and density perturbation in L^∞. The constants are independent of the supports.
ER -