Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications
Abstract
In this article, we establish sharp quantitative weighted estimates for the lifted Hardy--Littlewood (resp Sobolev sharp) maximal operators {ℳθ}θ∈(0,∞) (resp {ℳθ♯}θ∈(0,∞)). Specifically, for p∈(1,∞), γ∈ℝ∖{0}, υ∈ Ap, and f∈ Lpυ, we prove that supθ,λ∈(0,∞)λp undersetℳθ(f)(x,t)>λtγ ∫ℝn∫0∞ tγp−1 dt υ(x) dx ≲ [υ]Apαp,γ |f|Lpυp, where αp,γ:=p/(p−1) for γ∈[−n,0) and αp,γ:=1 otherwise and where the implicit positive constant depends only on n, p, and γ. For ℳθ♯, the corresponding estimate holds with |f|Lpυp replaced by |∇ f|Lpυp for any f∈Ẇ1,pυ, and with αp,γ:=p/(p−1) for γ∈[−1,0) and αp,γ:=1 otherwise. We show that, for an A1 weight υ, the weight constant in the above estimates becomes [υ]A1 for the sharp ranges p∈(1,∞) and γ∈ℝ∖{0}, or p=1 and γ∈ℝ∖[−n,0] (resp. γ∈ℝ∖[−1,0]). We also prove that αp,γ is sharp, and we establish exact limiting formulae and characterize Muckenhoupt weights via weak-type bounds. As applications, we characterize the weighted Hardy(--Sobolev) space H1υ with υ∈ A1 in terms of the truncated Riesz transforms with optimal range γ∈ ℝ∖{0}. More applications include the sharp weighted Caffarelli--Silvestre extension and the generalization to ball Banach function spaces.
Related
- Maximal velocity estimates for Hardy operators
- Sharp weighted estimates for the multilinear maximal operator
- Orthonormal maximal estimates for fractional Schrödinger operators along Hölder curves
- Fourier decay and endpoint estimates for lacunary maximal operators
- Maximal estimates of the Schrödinger operators with quadratic potential for orthonormal systems
BibTeX
@misc{geng2026sharp,
title = {{Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications}},
author = {Meng Geng and Dachun Yang and Wen Yuan and Yirui Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33998},
eprint = {2610.33998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33998},
}
RIS
TY - UNPB
TI - Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications
AU - Meng Geng
AU - Dachun Yang
AU - Wen Yuan
AU - Yirui Zhao
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33998
UR - https://arxiv.org/abs/2610.33998
AB - In this article, we establish sharp quantitative weighted estimates for the lifted Hardy--Littlewood (resp Sobolev sharp) maximal operators {ℳ_θ}_θ∈(0,∞) (resp {ℳ_θ^♯}_θ∈(0,∞)). Specifically, for p∈(1,∞), γ∈ℝ∖{0}, υ∈ A_p, and f∈ L^p_υ, we prove that sup_θ,λ∈(0,∞)λ^p undersetℳ_θ(f)(x,t)>λt^γ ∫_ℝ^n∫_0^∞ t^γp−1 dt υ(x) dx ≲ [υ]_A_p^α_p,γ |f|_L^p_υ^p, where α_p,γ:=p/(p−1) for γ∈[−n,0) and α_p,γ:=1 otherwise and where the implicit positive constant depends only on n, p, and γ. For ℳ_θ^♯, the corresponding estimate holds with |f|_L^p_υ^p replaced by |∇ f|_L^p_υ^p for any f∈Ẇ^1,p_υ, and with α_p,γ:=p/(p−1) for γ∈[−1,0) and α_p,γ:=1 otherwise. We show that, for an A_1 weight υ, the weight constant in the above estimates becomes [υ]_A_1 for the sharp ranges p∈(1,∞) and γ∈ℝ∖{0}, or p=1 and γ∈ℝ∖[−n,0] (resp. γ∈ℝ∖[−1,0]). We also prove that α_p,γ is sharp, and we establish exact limiting formulae and characterize Muckenhoupt weights via weak-type bounds. As applications, we characterize the weighted Hardy(--Sobolev) space H^1_υ with υ∈ A_1 in terms of the truncated Riesz transforms with optimal range γ∈ ℝ∖{0}. More applications include the sharp weighted Caffarelli--Silvestre extension and the generalization to ball Banach function spaces.
ER -