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We prove nontangential and radial maximal function characterizations forHardy spaces associated to a non-negative self-adjoint operator satisfyingGaussian estimates on a space of homogeneous type with finite measure. This notonly addresses an open point in the literature, but also gives a completeanswer to the question posed by Coifman and Weiss in the case of finitemeasure. We then apply our results to give maximal function characterizationsfor Hardy spaces associated to second order elliptic operators with Neumann andDirichlet boundary conditions, Schr\"odinger operators with Dirichlet boundaryconditions, and Fourier--Bessel operators.

Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications