Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{ R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{ R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in { \mathbb A}_{ \infty}(\mathbb{ R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{ \varphi,\,L}(\mathbb{ R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{ \varphi,\,L}(\mathbb{ R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schr\"odinger operators on $\mathbb{ R}^n$ with non-negative potentials belonging to the reverse H\"older class, and second-order divergence form elliptic operators on $\mathbb{ R}^n$ with bounded measurable real coefficients.
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