The strong ℓr spherical maximal function
Abstract
Let d≥3 and 2≤ r<∞. We consider the strong maximal operator associated with Euclidean surface averages over the ℓr unit sphere and independent coordinate dilations. We prove that this operator is bounded on Lp(ℝd) if and only if p>max{(d+1)/(d−1),r/(d−1)}. We also obtain Lp→ Lq bounds for the corresponding local maximal operator. For each fixed p in certain ranges, the resulting range of q is optimal up to endpoints. The proof uses a dyadic surface decomposition and frequency-localized L2 estimates. We establish these estimates by combining exact multiplier factorization via iterated stationary phase with scalar Fourier inversion and TT* arguments. In dimension three, these estimates are combined with quantitative multiparameter local smoothing.
Related
- The strong type endpoint estimate for the local spherical maximal function
- Sharp Lp estimates for the strong spherical maximal operator
- Strong-type endpoint obstructions for bilinear spherical maximal operators
- Bilinear Korányi spherical maximal function on the Heisenberg group
- Improved weighted bounds for the strong maximal function
BibTeX
@misc{lee2026strong,
title = {{The strong $\ell^r$ spherical maximal function}},
author = {Jin Bong Lee and Sanghyuk Lee and Jeongtae Oh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04968},
eprint = {2610.04968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04968},
}
RIS
TY - UNPB
TI - The strong ℓ^r spherical maximal function
AU - Jin Bong Lee
AU - Sanghyuk Lee
AU - Jeongtae Oh
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04968
UR - https://arxiv.org/abs/2610.04968
AB - Let d≥3 and 2≤ r<∞. We consider the strong maximal operator associated with Euclidean surface averages over the ℓ^r unit sphere and independent coordinate dilations. We prove that this operator is bounded on L^p(ℝ^d) if and only if p>max{(d+1)/(d−1),r/(d−1)}. We also obtain L^p→ L^q bounds for the corresponding local maximal operator. For each fixed p in certain ranges, the resulting range of q is optimal up to endpoints. The proof uses a dyadic surface decomposition and frequency-localized L^2 estimates. We establish these estimates by combining exact multiplier factorization via iterated stationary phase with scalar Fourier inversion and TT^* arguments. In dimension three, these estimates are combined with quantitative multiparameter local smoothing.
ER -