Weighted Fourier capacities and Frostman restriction estimates

@misc{leon2026weighted,
  title = {{Weighted Fourier capacities and Frostman restriction estimates}},
  author = {Rodrigo Leon and Vicente Vergara},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20800},
  eprint = {2610.20800},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20800},
}

TY  - UNPB
TI  - Weighted Fourier capacities and Frostman restriction estimates
AU  - Rodrigo Leon
AU  - Vicente Vergara
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20800
UR  - https://arxiv.org/abs/2610.20800
AB  - We study a weighted Fourier capacity attached to compact sets, $ c_{p,k}(K)=\inf\left\{\|k\widehat u\|_{L^p(\mathbb{R}^d)}:u\in C_0^\infty(\mathbb{R}^d),\ u\geq 1 \text{ on } K\right\},$ where k is a positive tempered weight. The paper develops complementary upper and lower estimates for this quantity. The upper bounds are obtained from Frostman data, Fourier restriction estimates, and Bessel-potential majorants, which together provide admissible smooth majorants of the compact set. The lower bounds follow from a dual testing argument against positive measures supported on K, yielding estimates in terms of the weighted L^p' size of their Fourier transforms. For sets of positive Lebesgue measure, testing with χ_K dx connects the capacity with the weighted integrability of χ_K̂. A Lebedev-type Sobolev criterion then gives non-triviality ranges governed by the upper Minkowski dimension of ∂ K and the growth of the weight. We also record consequences for Ahlfors-David regular sets and positive-measure Cantor-type examples.
ER  -