Maximal characterisation of local Hardy spaces on locally doubling manifolds

Abstract

We prove a radial maximal function characterisation of the local atomic Hardy space $${{\\mathfrak {h}}}^1(M)$$\n \n \n \n h\n \n 1\n \n \n (\n M\n )\n \n \n 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 $${{\\mathfrak {h}}}^1(M)$$\n \n \n \n h\n \n 1\n \n \n (\n M\n )\n \n \n 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.\xa0Uchiyama.