Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

Abstract

In this paper we introduce the atomic Hardy space H1 ((0, ∞), γα) associated with the non-doubling probability measure dγα(x) = 2x 2α+1 Γ(α+1) e −x 2 dx on (0, ∞), for α > − 1 2 . We obtain characterizations of H1 ((0, ∞), γα) by using two local maximal functions. We also prove that the truncated maximal function defined through the heat semigroup generated by the Laguerre differential operator is bounded from H1 ((0, ∞), γα) into L1 ((0, ∞), γα).