Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

In this paper we introduce the atomic Hardy space ${\mathcal{H}}^1((0,\mathrm{\infty }),{\gamma }_{\alpha })$ associated with the non-doubling probability measure $d{\gamma }_{\alpha }(x)=\frac{2x^{2\alpha +1}}{\mathrm{\Gamma }(\alpha +1)}{e}^{-x^2}dx$ on $(0,\mathrm{\infty })$, for $\alpha >-\frac{1}{2}$. We obtain characterizations of ${\mathcal{H}}^1((0,\mathrm{\infty }),{\gamma }_{\alpha })$ 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 ${\mathcal{H}}^1((0,\mathrm{\infty }),{\gamma }_{\alpha })$ into $L^1((0,\mathrm{\infty }),{\gamma }_{\alpha })$.