On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions

Abstract

For the one-sided Hardy--Littlewood maximal operator M_d⁺ on ℝ^d, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak (p,p) inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint p=1 in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman--Stein-type question and the related weighted weak-type (1,1) question \cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every d≥ 3 and p>1. More precisely, for every 1<p<∞ and every d≥ 3, there exist weights w and v such that A_p,d⁺(w,v)<∞, |M_d⁺|_{L^p(v)→ L^p,∞(w)}=∞. The proof uses a finite two-dimensional Hardy operator whose weak operator norm is bounded below by c_p(log N)^1/p'. This operator is embedded into the three-dimensional lattice maximal operator and then transferred to the continuous setting. A tensor extension yields the same failure in every dimension d>3.

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BibTeX RIS

BibTeX

@misc{wang2026twoweight,
  title = {{On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions}},
  author = {Dinghuai Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08975},
  eprint = {2609.08975},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08975},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions
AU  - Dinghuai Wang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08975
UR  - https://arxiv.org/abs/2609.08975
AB  - For the one-sided Hardy--Littlewood maximal operator M_d^+ on ℝ^d, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak (p,p) inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint p=1 in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman--Stein-type question and the related weighted weak-type (1,1) question \cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every d≥ 3 and p>1. More precisely, for every 1<p<∞ and every d≥ 3, there exist weights w and v such that A_p,d^+(w,v)<∞, |M_d^+|_L^p(v)→ L^p,∞(w)=∞. The proof uses a finite two-dimensional Hardy operator whose weak operator norm is bounded below by c_p(log N)^1/p'. This operator is embedded into the three-dimensional lattice maximal operator and then transferred to the continuous setting. A tensor extension yields the same failure in every dimension d>3.
ER  -

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