Maximal characterisation of local Hardy spaces on locally doubling manifolds

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