Littlewood–Paley $g_{\lambda}^*$-function characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Let $({\mathcal X}, d, \mu)$ be a space of homogeneous type, in the sense of Coifman and Weiss, and $\varphi\colon\ \mathcal{X}\times[0, \infty)\rightarrow[0, \infty)$ satisfy that, for almost every $x\in\mathcal{X}$, $\varphi(x, \cdot)$ is an Orlicz function and that $\varphi(\cdot, t)$ is a Muckenhoupt weight uniformly in $t\in[0, \infty)$. In this article, by using the aperture estimate of Littlewood–Paley auxiliary functions on the Musielak–Orlicz space $L^{\varphi}(\mathcal{X})$, we obtain the Littlewood–Paley $g_{\lambda}^*$-function characterization of Musielak–Orlicz Hardy space $H^{\varphi}(\mathcal{X})$. Particularly, the range of $\lambda$ coincides with the best-known one.