发表机构
Anhui Normal University（安徽师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明高维（$d\geq 3$）单侧 Hardy--Littlewood 极大算子的双权 Muckenhoupt 条件不再刻画弱型不等式，通过构造反例并利用有限二维 Hardy 算子嵌入与张量扩展实现。
AI 中文摘要
对于 $\mathbb R^d$ 上的单侧 Hardy--Littlewood 极大算子 $M_d^+$，Sawyer 于 1986 年证明自然双权 Muckenhoupt 条件在一维情形刻画了弱 $(p,p)$ 不等式，Forzani、Martín-Reyes 和 Ombrosi 于 2011 年在二维情形证明了该结果。在三维情形的端点 $p=1$ 处，Ombrosi 和 Nazarov 最近对 Fefferman--Stein 型问题以及相关的加权弱型 $(1,1)$ 问题均给出了否定答案（个人通信）。本文证明对于所有 $d\geq 3$ 和 $p>1$，双权刻画均不成立。更精确地，对于每个 $1<p<\infty$ 和每个 $d\geq 3$，存在权函数 $w$ 和 $v$ 使得 $A_{p,d}^+(w,v)<\infty$ 且 $\\|M_d^+\\|_{L^p(v)\to L^{p,\infty}(w)}=\infty$。证明使用了一个有限二维 Hardy 算子，其弱算子范数以 $c_p(\log N)^{1/p'}$ 为下界。该算子被嵌入三维格点极大算子，然后转移到连续情形。张量扩展得到在所有维度 $d>3$ 中同样的失败结果。
英文摘要
For the one-sided Hardy--Littlewood maximal operator $M_d^+$ on $\mathbb R^d$, 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 $d\geq 3$ and $p>1$. More precisely, for every $1<p<\infty$ and every $d\geq 3$, there exist weights $w$ and $v$ such that $$ A_{p,d}^+(w,v)<\infty, \qquad \|M_d^+\|_{L^p(v)\to L^{p,\infty}(w)}=\infty. $$ 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 $d>3$.
Comments20 pages
