Reflectionless measures with singular continuous components on Widom sets
Abstract
We give a detailed implementation of the mass-splitting construction of Nazarov, Volberg and Yuditskii for reflectionless measures with singular continuous components. The finite approximants are allowed to have isolated atoms. An exact local operation replaces each active atom by a short permanent interval and two child atoms, thereby eliminating the dependence of a future splitting scale on an already prescribed band width. Summable estimates for completed generations and fixed cylinders retain positive mass on a null Cantor set and exclude atoms in the limit. Uniform analytic estimates near the permanent intervals establish reflectionlessness and exclude endpoint singularities. A separate Green function estimate for thin pairs of intervals gives the Widom condition and regularity of the limiting compact set. The resulting probability measure has both a nonzero absolutely continuous component and a nonzero singular continuous component.
Related
- Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs
- Boundary-layer asymptotics for Gaussian-smoothed singular measures
- Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations
- Gaussian measures under the Navier-Stokes and heat flows are mutually singular
- Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
BibTeX
@misc{li2026reflectionless,
title = {{Reflectionless measures with singular continuous components on Widom sets}},
author = {Long Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27133},
eprint = {2609.27133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27133},
}
RIS
TY - UNPB
TI - Reflectionless measures with singular continuous components on Widom sets
AU - Long Li
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.27133
UR - https://arxiv.org/abs/2609.27133
AB - We give a detailed implementation of the mass-splitting construction of Nazarov, Volberg and Yuditskii for reflectionless measures with singular continuous components. The finite approximants are allowed to have isolated atoms. An exact local operation replaces each active atom by a short permanent interval and two child atoms, thereby eliminating the dependence of a future splitting scale on an already prescribed band width. Summable estimates for completed generations and fixed cylinders retain positive mass on a null Cantor set and exclude atoms in the limit. Uniform analytic estimates near the permanent intervals establish reflectionlessness and exclude endpoint singularities. A separate Green function estimate for thin pairs of intervals gives the Widom condition and regularity of the limiting compact set. The resulting probability measure has both a nonzero absolutely continuous component and a nonzero singular continuous component.
ER -