Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

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