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In this paper, we study Bessel operators and Bessel Laplace equations studiedby Weinstein, Huber, and related the harmonic function theory introduced byMuckenhoupt--Stein. We establish the Moser type inequality for these harmonicfunctions, which is missing in this setting before. We then apply it to give adirect proof for the equivalence of characterizations of the Hardy spacesassociated to Bessel operator via non-tangential maximal function and radialmaximal function defined in terms of the Poisson semigroup.

A Moser type inequality for Bessel Laplace equations and applications