Let X be the direct product of X i where X i is smooth manifold for 1 ≤ i ≤ k . As is known, if every X i satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one X i does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties of M when X 1 has exponential volume growth and X 2 satisfies the doubling condition. Next, we consider the product space of two weighted hyperbolic spaces X 1 = (ℍ n +1 , d , y α − n −1 dydx ) and X 2 = (ℍ n +1 , d , y β − n −1 dydx ) which both have exponential volume growth. The mapping properties of M are discussed for every α,β≠n2 \alpha,\beta \ne {n \over 2} . Furthermore, let X = X 1 × X 2 × … X k where X i = (ℍ n i +1 , y α i − n i −1 dydx ) for 1 ≤ i ≤ k . Under the condition αi>ni2 {\alpha_i} > {{{n_i}} \over 2} , we also obtained the mapping properties of M .

中文翻译：

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产品流形上的居中 Hardy-Littlewood 极大函数

令 X 是 X i 的直接乘积，其中 X i 是 1 ≤ i ≤ k 的光滑流形。众所周知，如果每个 X i 都满足加倍体积条件，则 X 上的居中 Hardy-Littlewood 极大函数 M 是弱 (1,1) 有界的。在本文中，我们考虑乘积流形 X，其中至少有一个 X i 不满足加倍体积条件。准确地说，我们首先研究了当 X 1 具有指数体积增长且 X 2 满足倍增条件时 M 的映射性质。接下来，我们考虑两个加权双曲空间的乘积空间 X 1 = (ℍ n +1 , d , y α - n -1 dydx ) 和 X 2 = (ℍ n +1 , d , y β - n -1 dydx ) 两者都有指数级的体积增长。对每个 α,β≠n2 \alpha,\beta \ne {n \over 2} 讨论 M 的映射性质。此外，令 X = X 1 × X 2 × … X k 其中 X i = (ℍ ni +1 , y α i - ni -1 dydx ) 对于 1 ≤ i ≤ k 。在条件 αi>ni2 {\alpha_i} > {{{n_i}} \over 2} 下，我们也得到了 M 的映射性质。