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In dimension one we give a maximal function characterisation of the Hardyspace H^1(g) for the Gauss measure g, introduced by G. Mauceri and S. Meda. Inarbitrary dimension, we give a description of the nonnegative functions inH^1(g) and use it to prove that L^p(g) is a contained in H^1(g) for1<p\le\infty.

A maximal function characterisation of the Hardy space for the Gauss measure