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Maximal characterisation of local Hardy spaces on locally doubling manifolds. (English) Zbl 1491.46020

Summary: We prove a radial maximal function characterisation of the local atomic Hardy space \({{\mathfrak{h}}}^1(M)\) on a Riemannian manifold \(M\) with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to \({{\mathfrak{h}}}^1(M)\) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama [Trans. Am. Math. Soc. 262, 579–592 (1980; Zbl 0503.46020)].

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0503.46020
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References:

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