Let X be a metric measure space with a doubling measure and L be a nonnegative self-adjoint operator acting on L2(X)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(X)$$\end{document}. Assume that L generates an analytic semigroup e-tL\documentclass[12pt]{minimal}
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\begin{document}$$e^{-tL}$$\end{document} whose kernels pt(x,y)\documentclass[12pt]{minimal}
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\begin{document}$$p_t(x,y)$$\end{document} satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables x and y. In this article, we continue a study in Song and Yan (Adv Math 287:463–484, 2016) to give an atomic decomposition for the Hardy spaces HL,maxp(X)\documentclass[12pt]{minimal}
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\begin{document}$$ H^p_{L,\mathrm{max}}(X)$$\end{document} in terms of the nontangential maximal function associated with the heat semigroup of L, and hence, we establish characterizations of Hardy spaces associated with an operator L, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of HL,maxp(X)\documentclass[12pt]{minimal}
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\begin{document}$$ H^p_{L, \mathrm{max}}(X)$$\end{document} in terms of the radial maximal function.
