Maximal characterisation of local Hardy spaces on locally doubling manifolds

TL;DR

This paper characterizes local Hardy spaces on certain Riemannian manifolds using maximal functions, providing new equivalences and decompositions that extend previous results to more general spaces.

Contribution

It establishes a radial maximal function characterization of local Hardy spaces on manifolds with positive injectivity radius and Ricci curvature bounds, generalizing Uchiyama's decomposition.

Findings

Characterization of h^1(M) via local heat and Poisson maximal functions

Decomposition of Hölder cut-offs on Ahlfors-regular spaces

Extension of Uchiyama's approximation results to broader settings

Abstract

We prove a radial maximal function characterisation of the local atomic Hardy space 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 h^1(M) if and only if either its local heat maximal function or its local Poisson maximal function are integrable. A key ingredient is a decomposition of H\"older 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.