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\begin{document}$${\mathcal {X}}$$\end{document} be a metric space with doubling measure and L be a non-negative self-adjoint operator on L2(X)\documentclass[12pt]{minimal}
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\begin{document}$$L^2({\mathcal {X}})$$\end{document} whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function φ:X×[0,∞)→[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi :\ {\mathcal {X}}\times [0,\infty ) \rightarrow [0,\infty )$$\end{document} satisfies that φ(x,·)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi (x,\cdot )$$\end{document} is an Orlicz function and φ(·,t)∈A∞(X)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi (\cdot ,t)\in {{\mathbb {A}}}_{\infty }({\mathcal {X}})$$\end{document} (the class of uniformly Muckenhoupt weights). Let Hφ,L(X)\documentclass[12pt]{minimal}
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\begin{document}$$H_{\varphi ,\,L}({\mathcal {X}})$$\end{document} be the Musielak–Orlicz–Hardy space defined via the Lusin area function associated with the heat semigroup of L. In this article, the authors characterize the space Hφ,L(X)\documentclass[12pt]{minimal}
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\begin{document}$$H_{\varphi ,\,L}({\mathcal {X}})$$\end{document} by means of atoms, non-tangential and radial maximal functions associated with L. In particular, when μ(X)<∞\documentclass[12pt]{minimal}
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\begin{document}$$\mu ({\mathcal {X}})<\infty $$\end{document}, the local non-tangential and radial maximal function characterizations of Hφ,L(X)\documentclass[12pt]{minimal}
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\begin{document}$$H_{\varphi ,\,L}({\mathcal {X}})$$\end{document} are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the “geometric” Musielak–Orlicz–Hardy spaces Hφ,r(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$H_{\varphi ,\,r}(\Omega )$$\end{document} and Hφ,z(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$H_{\varphi ,\,z}(\Omega )$$\end{document} on the strongly Lipschitz domain Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} in Rn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^n$$\end{document} associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when φ(x,t):=t\documentclass[12pt]{minimal}
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\begin{document}$$\varphi (x,t):=t$$\end{document} for any x∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$x\in {\mathbb {R}}^n$$\end{document} and t∈[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$t\in [0,\infty )$$\end{document}, the equivalent characterizations of Hφ,z(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$H_{\varphi ,\,z}(\Omega )$$\end{document} given in this article improve the known results via removing the assumption that Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is unbounded.
