  Let $X$ be a space of homogeneous type. Assume that $L$ is an non-negative
second-order self-adjoint operator on $L^2\left(X\right)$ with (heart) kernel
associated to the semigroup $e^{ - tL}$ that satisfies the Gaussian upper
bound. In this paper, the authors introduce a new characterization of the
Musielak-Orlicz-Hardy Space $H_{\varphi, L}\left(X\right)$ associated with $L$
in terms of the Lusin area function where $\varphi$ is a growth function.
Further, the authors prove that the Musielak-Orlicz-Hardy Space
$H_{L,G,\varphi}\left(X\right)$ associated with $L$ in terms of the
Littlewood-Paley function is coincide with $H_{\varphi, L}\left(X\right)$ and
their norms are equivalent.

Classical Analysis and ODEs;Analysis of PDEs