AI 中文总结
本文刻画了无限有根$k$-叉树上Hardy--Littlewood极大算子在加权$L^p$空间有界性的权条件，引入双层Muckenhoupt权类并给出等价刻画、最优常数估计及加权向量值不等式应用。
AI 中文摘要
设$k\geq 2$为整数，$T$为无限有根$k$-叉树，$M$为$T$上的Hardy--Littlewood极大算子。对于任意$p\in(0,\infty)$，我们刻画了使得$M$在$L^p(w)$上有界的权$w$。为此，我们引入了双层Muckenhoupt权类$\mathscr A_p$，并证明对于任意$p\in(\frac{1}{2},\infty)$，$M$在$L^p(w)$上的有界性、$w\in\mathscr A_p$以及球面平均算子在$L^p(w)$上的指数衰减有界性三者相互等价；而当$p\in(0,\frac{1}{2}]$时，不存在使得$M$在$L^p(w)$上有界的权$w$。此外，对于任意$p\in(\frac{1}{2},\infty)$，我们建立了$M$在$L^p(w)$上有界性的定量估计，其中权常数的指数$\frac{1}{p}$是最优的。对于任意$p\in(1,\infty)$，我们还分别通过全局Sawyer型检验条件和距离关联集的加权乘积测度估计，得到了$M$在$L^p(w)$上有界性的另外两个等价刻画。作为应用，对于任意$p\in(\frac{1}{2},\infty)$，在$M$在$L^p(w)$上有界的假设下，我们建立了指数衰减核算子在$L^p(w)$上的有界性以及加权Fefferman--Stein向量值不等式。
英文摘要
Let $k\geq 2$ be an integer, $T$ an infinite rooted $k$-ary tree, and $M$ the Hardy--Littlewood maximal operator on $T$. For any $p\in(0,\infty)$, we characterize the weight $w$ such that $M$ is bounded on $L^p(w)$. To this end, we introduce a two-layer Muckenhoupt weight class $\mathscr A_p$ and prove that, for any $p\in(\frac{1}{2},\infty)$, the boundedness of $M$ on $L^p(w)$, $w\in\mathscr A_p$, and the exponential decay boundedness of spherical averaging operators on $L^p(w)$ are mutually equivalent, and that, when $p\in(0,\frac{1}{2}]$, there exists no weight $w$ such that $M$ is bounded on $L^p(w)$. Moreover, for any $p\in(\frac{1}{2},\infty)$, we establish the quantitative estimate, with the optimal exponent $\frac{1}{p}$ of the weight constant, for the boundedness of $M$ on $L^p(w)$. For any $p\in(1,\infty)$, we also obtain two further equivalent characterizations of the boundedness of $M$ on $L^p(w)$, respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any $p\in(\frac{1}{2},\infty)$, under the assumption that $M$ is bounded on $L^p(w)$, we establish the boundedness of exponentially decaying kernel operators on $L^p(w)$ and weighted Fefferman--Stein vector-valued inequalities.
Comments34 pages
