Hausdorff-Choquet Angular Spaces and Weak-Type (1,1) Bounds for Rough Maximal Operators
Abstract
In the present paper, we consider the maximal operator ℳΩf(x) := supr>01/rn ∫|y|<r |f(x−y)| | Ω (y/|y|) | dy. A longstanding open conjecture raised by E.M.Stein asks whether the maximal operator ℳΩ is of weak type (1,1) when Ω is merely in L1(𝕊n−1). We partially settle this problem by proving weak type (1,1) bounds of ℳΩ with kernel Ω∈𝒳(𝕊n−1), yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here𝒳(𝕊n−1) is the space related to the Hausdorff--Choquet angular space and Llog+ L(𝕊n−1) subsetneq 𝒳(𝕊n−1)⊂ L1(𝕊n−1). Prior to our work, the condition $Ω\in L\log^+\!L(\mathbb S^{n-1}) $ remained the best known size restriction guaranteeing the weak-type$ (1,1)$ bounds for all dimensions n > 2. To our knowledge, the present paper is the first to relax this threshold for n>2. Finally, for the general Hausdorff--Choquet scale ℋ𝒞α, we show that α=(n−1)/2 is the sharp exponent for uniform weak type (1,1) estimates.
Related
- Spherical maximal operators on rank one Riemannian symmetric spaces of noncompact type
- Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators
- One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity
- Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces
- Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces
BibTeX
@misc{chen2026hausdorffchoquet,
title = {{Hausdorff-Choquet Angular Spaces and Weak-Type $(1,1)$ Bounds for Rough Maximal Operators}},
author = {Yanping Chen and Zhengyang Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03785},
eprint = {2609.03785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03785},
}
RIS
TY - UNPB
TI - Hausdorff-Choquet Angular Spaces and Weak-Type (1,1) Bounds for Rough Maximal Operators
AU - Yanping Chen
AU - Zhengyang Ji
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.03785
UR - https://arxiv.org/abs/2609.03785
AB - In the present paper, we consider the maximal operator ℳ_Ωf(x) := sup_r>01/r^n ∫_|y|<r |f(x−y)| | Ω (y/|y|) | dy. A longstanding open conjecture raised by E.M.Stein asks whether the maximal operator ℳ_Ω is of weak type (1,1) when Ω is merely in L^1(𝕊^n−1). We partially settle this problem by proving weak type (1,1) bounds of ℳ_Ω with kernel Ω∈𝒳(𝕊^n−1), yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here𝒳(𝕊^n−1) is the space related to the Hausdorff--Choquet angular space and Llog^+ L(𝕊^n−1) subsetneq 𝒳(𝕊^n−1)⊂ L^1(𝕊^n−1). Prior to our work, the condition $Ω\in L\log^+\!L(\mathbb S^{n-1}) $ remained the best known size restriction guaranteeing the weak-type$ (1,1)$ bounds for all dimensions n > 2. To our knowledge, the present paper is the first to relax this threshold for n>2. Finally, for the general Hausdorff--Choquet scale ℋ𝒞_α, we show that α=(n−1)/2 is the sharp exponent for uniform weak type (1,1) estimates.
ER -