标题： 蜘蛛网和Gromov双曲空间上Hardy--Littlewood极大算子的尖锐$L^p$估计 显示英文标题

标题： Spider's webs and sharp $L^p$ bounds for the Hardy--Littlewood maximal operator on Gromov hyperbolic spaces

摘要： 在本文中我们证明，如果$1<a\leq b<a^2$和$X$是一个局部加倍的$\delta$-双曲的完备连通长度度量测度空间，并且在无穷远处具有$(a,b)$-夹紧的指数增长，则中心 Hardy--Littlewood 极大算子$\mathcal M$在$L^p(X)$上对于所有$p>\tau$是有界的，并且它是弱类型$(\tau,\tau)$，其中$\tau := \log_ab$。 证明中的关键步骤是针对具有在无穷远处指数增长的$(a,b)$-紧致的Gromov双曲空间的一个新的结构定理，该定理通过本文引入并称为蜘蛛网的某些图对$X$进行离散化，这些图具有“良好的连通性性质”。我们的结果适用于有界几何的树以及具有紧致负曲率的Cartan--Hadamard流形，在这些情况下提供了新的有界性结果。指标$\tau$是最优的，因为在如果$p<\tau$时，存在满足上述假设的$X$，使得$\mathcal M$不是弱类型的$(p,p)$。 此外，如果 $b>a^2$，则存在满足上述假设的空间 $X$，使得 $\mathcal M$在 $L^p(X)$上有界当且仅当 $p=\infty$。 显示英文摘要

摘要： In this paper we prove that if $1<a\leq b<a^2$ and $X$ is a locally doubling $\delta$-hyperbolic complete connected length metric measure space with $(a,b)$-pinched exponential growth at infinity, then the centred Hardy--Littlewood maximal operator $\mathcal M$ is bounded on $L^p(X)$ for all $p>\tau$, and it is of weak type $(\tau,\tau)$, where $\tau := \log_ab$. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with $(a,b)$-pinched exponential growth at infinity, consisting in a discretisation of $X$ by means of certain graphs, introduced in this paper and called spider's webs, with ``good connectivity properties". Our result applies to trees with bounded geometry, and Cartan--Hadamard manifolds of pinched negative curvature, providing new boundedness results in these settings. The index $\tau$ is optimal in the sense that if $p<\tau$, then there exists $X$ satisfying the assumptions above such that $\mathcal M$ is not of weak type $(p,p)$. Furthermore, if $b>a^2$, then there are examples of spaces $X$ satisfying the assumptions above such that $\mathcal M$ bounded on $L^p(X)$ if and only if $p=\infty$.