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Estimations of the maximal Hardy-Littlewood function. (English) Zbl 1154.22013
Let \((M,\delta)\) be a metric space equipped with a Borel measure \(\mu\) which is finite on all balls related to the metric \(\delta\). Let \(f\mapsto f^*\) denote the (centered) Hardy-Littlewood maximal operator in this context. The paper treats the problem of proving inequalities of the form \(\| f^*\| _p\leq C(p)\| f\| _p\), for all \(p>p_0\) with some \(p_0>1\), under certain assumptions on the geometry of \((M,\delta,\mu)\). The two specific situations considered include: i) the case of a connected, unimodular, noncompact, nonamenable Lie group equipped with control distance and a bi-invariant Haar measure; ii) the case of a Cartan-Hadamard manifold with some restrictions imposed on the curvature, which is equipped with a Riemannian metric and a Riemannian measure.
MSC:
22E30 Analysis on real and complex Lie groups
43A90 Harmonic analysis and spherical functions
60B99 Probability theory on algebraic and topological structures
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