Musielak-Orlicz BMO-Type Spaces Associated with Generalized Approximations to the Identity

Let $\mathcal{X}$ be a space of homogenous type and $\varphi:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ a growth function such that $\varphi(\cdot,t)$ is a Muckenhoupt weight uniformly in $t$ and $\varphi(x,\cdot)$ an Orlicz function of uniformly upper type 1 and lower type $p\in(0,1]$. In this article, the authors introduce a new Musielak-Orlicz BMO-type space $\mathrm{BMO}^{\varphi}_A(\mathcal{X})$ associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces $\mathrm{BMO}^{\varphi}_{A,\,\mathrm{max}}(\mathcal{X})$ and $\widetilde{\mathrm{BMO}}^{\varphi}_A(\mathcal{X})$. Moreover, two variants of the John-Nirenberg inequality on $\mathrm{BMO}^{\varphi}_A(\mathcal{X})$ are obtained. As an application, the authors further prove that the space $\mathrm{BMO}^{\varphi}_{\sqrt{\Delta}}(\mathbb{R}^n)$, associated with the Poisson semigroup of the Laplace operator $\Delta$ on $\mathbb{R}^n$, coincides with the space $\mathrm{BMO}^{\varphi}(\mathbb{R}^n)$ introduced by L. D. Ky.