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

Abstract. An atomic Hardy space H1(γ) associated to the Gauss measure γ in Rn has been introduced by the first two authors. We first prove that it is equivalent to use (1,r)- or (1,∞)-atoms to define this H1(γ). For n = 1, a maximal function characterization of H1(γ) is found. In arbitrary dimension, we give a description of the nonnegative functions in H1(γ) and use it to prove that Lp(γ) ⊂ H1(γ) for 1 < p ≤ ∞.