Endpoint gradient bounds for centered convex maximal operators and probability kernels
Abstract
We establish the endpoint gradient estimate for centered homothetic maximal averages in every dimension d≥2 over each fixed convex body with C1,θ boundary, 0<θ≤1. In particular, the centered ball maximal operator satisfies the W1,1 gradient bound in every dimension. Neither strict convexity nor a lower bound on the support gap is required. The geometric argument combines concentration in gradient values with nonhomogeneous calibrated rows and a summable radial correction. It extends the scalar refinement, direction-history construction and global assembly of the planar centered-disk proof of OpenAI. A separate transfer theorem gives sharp singular-derivative identities for bounded compact probability kernels once local BV output is known, and gradient continuity under an additional spatial-tail condition. Quantitative contact and translation estimates yield further profile and kernel examples. The estimates retain the fixed body's regularity data; the endpoint for arbitrary C1 and nonsmooth convex bodies remains open here.
Related
- Fourier decay and endpoint estimates for lacunary maximal operators
- Strong-type endpoint obstructions for bilinear spherical maximal operators
- Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces
- Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators
- Non-centered maximal operators over ℓq balls for exponential and Laguerre-type measures
BibTeX
@misc{long2026endpoint,
title = {{Endpoint gradient bounds for centered convex maximal operators and probability kernels}},
author = {Zhouyu Long},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35760},
eprint = {2610.35760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35760},
}
RIS
TY - UNPB
TI - Endpoint gradient bounds for centered convex maximal operators and probability kernels
AU - Zhouyu Long
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35760
UR - https://arxiv.org/abs/2610.35760
AB - We establish the endpoint gradient estimate for centered homothetic maximal averages in every dimension d≥2 over each fixed convex body with C^1,θ boundary, 0<θ≤1. In particular, the centered ball maximal operator satisfies the W^1,1 gradient bound in every dimension. Neither strict convexity nor a lower bound on the support gap is required. The geometric argument combines concentration in gradient values with nonhomogeneous calibrated rows and a summable radial correction. It extends the scalar refinement, direction-history construction and global assembly of the planar centered-disk proof of OpenAI. A separate transfer theorem gives sharp singular-derivative identities for bounded compact probability kernels once local BV output is known, and gradient continuity under an additional spatial-tail condition. Quantitative contact and translation estimates yield further profile and kernel examples. The estimates retain the fixed body's regularity data; the endpoint for arbitrary C^1 and nonsmooth convex bodies remains open here.
ER -