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 \(L^1((0,\infty ),\gamma _\alpha )\).

在本文中，我们介绍与非倍增概率测度\(d\gamma _ \alpha) 相关的原子 Hardy 空间 \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) (x)=\frac{2x^{2\alpha +1}}{\Gamma (\alpha +1)}e^{-x^2}dx\)上\((0,\infty )\)，对于\({\alpha >-\frac{1}{2}}\)。我们通过使用两个局部极大函数获得\(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\)的表征。我们还证明，通过拉盖尔微分算子生成的热半群定义的截断极大函数有界于\(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\)到\( L^1((0,\infty ),\gamma _\alpha )\)。