Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

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