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PreprintarXiv licencearXiv

Sharp volume bounds for BRK-type sets of surfaces of revolution

  • Xianghong Chen
  • Tongou Yang
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Abstract

We prove sharp volume bounds (up to dimensional constants) for the δ-neighbourhoods of a general family of Besicovitch-Rado-Kinney (BRK) type sets associated with surfaces of revolution in ℝn, n≥ 3, which, in particular, include spheres, elliptic paraboloids, and cones. As a result, we obtain sharp Lp-Lq bounds for their associated Wolff-type maximal operators for all 1≤ p,q≤ ∞. In ℝ3, it is worth noting that our result necessitates a variant of Sogge's cinematic curvature condition.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp volume bounds for BRK-type sets of surfaces of revolution}},
  author = {Xianghong Chen and Tongou Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39059},
  eprint = {2609.39059},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39059},
}
RIS
TY  - UNPB
TI  - Sharp volume bounds for BRK-type sets of surfaces of revolution
AU  - Xianghong Chen
AU  - Tongou Yang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39059
UR  - https://arxiv.org/abs/2609.39059
AB  - We prove sharp volume bounds (up to dimensional constants) for the δ-neighbourhoods of a general family of Besicovitch-Rado-Kinney (BRK) type sets associated with surfaces of revolution in ℝ^n, n≥ 3, which, in particular, include spheres, elliptic paraboloids, and cones. As a result, we obtain sharp L^p-L^q bounds for their associated Wolff-type maximal operators for all 1≤ p,q≤ ∞. In ℝ^3, it is worth noting that our result necessitates a variant of Sogge's cinematic curvature condition.
ER  -