PreprintCC0arXiv
Strong-type endpoint obstructions for bilinear spherical maximal operators
Abstract
For the bilinear spherical averages in dimension d≥2, we show that the lacunary maximal operator is not bounded from L1(ℝd)× Lq(ℝd) to Lr(ℝd) whenever 1≤ q≤∞ and 1/r=1+1/q. The obstruction is realized by fixed nonnegative inputs; a second construction proves quantitative failure on smooth compactly supported functions and finitely many dyadic scales. The same conclusions hold for the full maximal operator. We give a sharp Lorentz-index criterion in the range q>d/(d−1) and extend the negative result to all multilinear spherical maximal operators with an L1 input. We indicate the precise strong-type boundary statements to which these conclusions apply, without drawing conclusions about separate weak-type or interior estimates.
Related
- Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces
- Bilinear Korányi spherical maximal function on the Heisenberg group
- Endpoint power weight inequalities for spherical maximal functions
- Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators
- The strong type endpoint estimate for the local spherical maximal function
BibTeX
@misc{vergara2026strongtype,
title = {{Strong-type endpoint obstructions for bilinear spherical maximal operators}},
author = {Vicente Vergara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14201},
eprint = {2610.14201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14201},
note = {CC0},
}
RIS
TY - UNPB
TI - Strong-type endpoint obstructions for bilinear spherical maximal operators
AU - Vicente Vergara
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14201
UR - https://arxiv.org/abs/2610.14201
AB - For the bilinear spherical averages in dimension d≥2, we show that the lacunary maximal operator is not bounded from L^1(ℝ^d)× L^q(ℝ^d) to L^r(ℝ^d) whenever 1≤ q≤∞ and 1/r=1+1/q. The obstruction is realized by fixed nonnegative inputs; a second construction proves quantitative failure on smooth compactly supported functions and finitely many dyadic scales. The same conclusions hold for the full maximal operator. We give a sharp Lorentz-index criterion in the range q>d/(d−1) and extend the negative result to all multilinear spherical maximal operators with an L^1 input. We indicate the precise strong-type boundary statements to which these conclusions apply, without drawing conclusions about separate weak-type or interior estimates.
ER -