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Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators
Abstract
Let d≥ 2 and T*Ω,b be the commutator of the rough maximal singular integral on ℝd defined by T*Ω,bf(x)=supε>0|∫|x−y|>ε(b(x)−b(y))(Ω(x−y))/(|x−y|d)f(y) dy|. Assume that Ω∈ L(log L)2(𝕊d−1) has mean zero and that b∈BMO(ℝd). We prove that, for every λ>0, |{x∈ℝd:T*Ω,bf(x)>λ}|≲Ω,b∫ℝd|f(x)|/λlog (e+|f(x)|/λ) dx. The argument starts from a Calderón-Zygmund decomposition of f and a dyadic linearization of the maximal truncation. A decomposition of Ω by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized L1 and L3 bounds and using an analytic-family argument for the commutators.
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BibTeX
@misc{lin2026weak,
title = {{Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators}},
author = {Shenglong Lin and Qingying Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34311},
eprint = {2609.34311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34311},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators
AU - Shenglong Lin
AU - Qingying Xue
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34311
UR - https://arxiv.org/abs/2609.34311
AB - Let d≥ 2 and T^*_Ω,b be the commutator of the rough maximal singular integral on ℝ^d defined by T^*_Ω,bf(x)=sup_ε>0|∫_|x−y|>ε(b(x)−b(y))(Ω(x−y))/(|x−y|^d)f(y) dy|. Assume that Ω∈ L(log L)^2(𝕊^d−1) has mean zero and that b∈BMO(ℝ^d). We prove that, for every λ>0, |{x∈ℝ^d:T^*_Ω,bf(x)>λ}|≲_Ω,b∫_ℝ^d|f(x)|/λlog (e+|f(x)|/λ) dx. The argument starts from a Calderón-Zygmund decomposition of f and a dyadic linearization of the maximal truncation. A decomposition of Ω by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized L^1 and L^3 bounds and using an analytic-family argument for the commutators.
ER -