Sharp L^p--L^q bounds for the strong circular maximal operator

Abstract

We study L^p--L^q boundedness of the strong circular maximal operator M, the maximal function over axis-parallel ellipses centered at the origin whose two semiaxes range independently over [1,2]. We prove that M is bounded from L^p to L^q whenever $(1/p,1/q)$ lies in the interior of the triangle with vertices (0,0), $(2/7,1/7)$, and $(1/3,1/3)$, the two-parameter analogue of the triangle with vertices (0,0), $(2/5,1/5)$, and $(1/2,1/2)$ obtained by Schlag and Schlag--Sogge for the circular maximal function. The range is sharp up to the boundary; in particular, weak type (p,q) bounds fail for every p≤3. The proof is based on a bilinear estimate for frequencies in separated angular sectors, uniform as the sectors approach the coordinate axes. We also prove the analogous sharp result for axis dilations of a translated circle c+𝕊¹ with fixed c≠0, where the vertex $(2/7,1/7)$ is replaced by $(1/4,1/8)$.

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BibTeX RIS

BibTeX

@misc{lee2026sharp,
  title = {{Sharp $L^p$--$L^q$ bounds for the strong circular maximal operator}},
  author = {Jin Bong Lee and Sanghyuk Lee and Seunghwan Lee and Jeongtae Oh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34691},
  eprint = {2610.34691},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34691},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Sharp L^p--L^q bounds for the strong circular maximal operator
AU  - Jin Bong Lee
AU  - Sanghyuk Lee
AU  - Seunghwan Lee
AU  - Jeongtae Oh
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34691
UR  - https://arxiv.org/abs/2610.34691
AB  - We study L^p--L^q boundedness of the strong circular maximal operator M, the maximal function over axis-parallel ellipses centered at the origin whose two semiaxes range independently over [1,2]. We prove that M is bounded from L^p to L^q whenever $(1/p,1/q)$ lies in the interior of the triangle with vertices (0,0), $(2/7,1/7)$, and $(1/3,1/3)$, the two-parameter analogue of the triangle with vertices (0,0), $(2/5,1/5)$, and $(1/2,1/2)$ obtained by Schlag and Schlag--Sogge for the circular maximal function. The range is sharp up to the boundary; in particular, weak type (p,q) bounds fail for every p≤3. The proof is based on a bilinear estimate for frequencies in separated angular sectors, uniform as the sectors approach the coordinate axes. We also prove the analogous sharp result for axis dilations of a translated circle c+𝕊^1 with fixed c≠0, where the vertex $(2/7,1/7)$ is replaced by $(1/4,1/8)$.
ER  -

Abstract

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