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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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