Stein's generalized spherical maximal operator
Abstract
This survey concerns Stein's analytic family of generalized spherical means and the associated maximal operator. We review Stein's original theorem, the sharp necessary conditions for boundedness, and the complete theory for radial functions. We then explain the connection between positive results and local smoothing estimates for the half-wave propagator and, more generally, for Fourier integral operators. This point of view yields the sharp two-dimensional result through the work of Guth, H. Wang, and R. Zhang and, when combined with the recent local smoothing theorems, an improved higher-dimensional range. As a byproduct, the maximal operator associated with the first radial derivative of spherical means is bounded on Lp(R4) for 2<p<3, a case left open in recent work of Kwaśnicki. We keep track of the relation between the local smoothing exponents and the resulting maximal estimates, and identify the region that remains open.
Related
- Sharp Lp estimates for the strong spherical maximal operator
- Sharp power-weight estimates for the fractional spherical maximal operator
- Characterization of the Boundedness of a Generalized Fractional Maximal Operator in Weighted Lorentz Spaces for Certain Parameter Values
- Fefferman--Stein type inequalities via area and maximal functions for Schrödinger operators with applications
- A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform
BibTeX
@misc{roncal2026steins,
title = {{Stein's generalized spherical maximal operator}},
author = {Luz Roncal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05154},
eprint = {2611.05154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05154},
}
RIS
TY - UNPB
TI - Stein's generalized spherical maximal operator
AU - Luz Roncal
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05154
UR - https://arxiv.org/abs/2611.05154
AB - This survey concerns Stein's analytic family of generalized spherical means and the associated maximal operator. We review Stein's original theorem, the sharp necessary conditions for boundedness, and the complete theory for radial functions. We then explain the connection between positive results and local smoothing estimates for the half-wave propagator and, more generally, for Fourier integral operators. This point of view yields the sharp two-dimensional result through the work of Guth, H. Wang, and R. Zhang and, when combined with the recent local smoothing theorems, an improved higher-dimensional range. As a byproduct, the maximal operator associated with the first radial derivative of spherical means is bounded on L^p(R^4) for 2<p<3, a case left open in recent work of Kwaśnicki. We keep track of the relation between the local smoothing exponents and the resulting maximal estimates, and identify the region that remains open.
ER -