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Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

In this paper we introduce the atomic Hardy space $\mathcal{H}^1((0,\infty),γ_α)$ associated with the non-doubling probability measure $dγ_α(x)=\frac{2x^{2α+1}}{Γ(α+1)}e^{-x^2}dx$ on $(0,\infty)$, for ${α>-\frac12}$. We obtain characterizations of $\mathcal{H}^1((0,\infty),γ_α)$ 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 $\mathcal{H}^1((0,\infty),γ_α)$ into $L^1((0,\infty),γ_α)$.