Sharp weighted weak-type bounds for commutators of singular integrals
Abstract
We establish sharp weak-type bounds for commutators of Calderón--Zygmund operators and BMO(ℝn) symbols with respect to Ap weights. We prove that the bound \looseness=-1 |[b,T]|Lp(w)→ Lp,∞(w)≲ [w]Apmax(2,p') for p ∈ (1,∞) and w ∈ Ap is sharp for general T ∈ CZO(ℝn) and b ∈ BMO(ℝn). In the case of the standard logarithmic symbol, we show that the estimate improves to |[log|·|,T]|Lp(w)→ Lp,∞(w)≲ [w]Apmax(2,1/p+p'/p) and that this dependence is optimal for this symbol. Our upper bound for the logarithmic symbol follows from the pointwise control of the commutator by a sum of the Hardy--Littlewood maximal operator and the squares of the radial Hardy operator and its adjoint, while our lower bounds are consequences of a general result for singular integral operators.
Related
- Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities
- Rough spectral asymptotics for commutators of general singular integrals
- Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals
- Weak type (1,1) bounds for rough oscillatory singular integrals
- Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals
BibTeX
@misc{mair2026sharp,
title = {{Sharp weighted weak-type bounds for commutators of singular integrals}},
author = {Adam Mair and Óscar Andrés Ramírez and Cody B. Stockdale},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31493},
eprint = {2609.31493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31493},
}
RIS
TY - UNPB
TI - Sharp weighted weak-type bounds for commutators of singular integrals
AU - Adam Mair
AU - Óscar Andrés Ramírez
AU - Cody B. Stockdale
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31493
UR - https://arxiv.org/abs/2609.31493
AB - We establish sharp weak-type bounds for commutators of Calderón--Zygmund operators and BMO(ℝ^n) symbols with respect to A_p weights. We prove that the bound \looseness=-1 |[b,T]|_L^p(w)→ L^p,∞(w)≲ [w]_A_p^max(2,p') for p ∈ (1,∞) and w ∈ A_p is sharp for general T ∈ CZO(ℝ^n) and b ∈ BMO(ℝ^n). In the case of the standard logarithmic symbol, we show that the estimate improves to |[log|·|,T]|_L^p(w)→ L^p,∞(w)≲ [w]_A_p^max(2,1/p+p'/p) and that this dependence is optimal for this symbol. Our upper bound for the logarithmic symbol follows from the pointwise control of the commutator by a sum of the Hardy--Littlewood maximal operator and the squares of the radial Hardy operator and its adjoint, while our lower bounds are consequences of a general result for singular integral operators.
ER -