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

Let $\mathcal{X}$ be a space of homogenous type and $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ a growth function such that $φ(\cdot,t)$ is a Muckenhoupt weight uniformly in $t$ and $φ(x,\cdot)$ 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 $\mathrm{BMO}^φ_A(\mathcal{X})$ 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 $\mathrm{BMO}^φ_{A,\,\mathrm{max}}(\mathcal{X})$ and $\widetilde{\mathrm{BMO}}^φ_A(\mathcal{X})$. Moreover, two variants of the John-Nirenberg inequality on $\mathrm{BMO}^φ_A(\mathcal{X})$ are obtained. As an application, the authors further prove that the space $\mathrm{BMO}^φ_{\sqrtΔ}(\mathbb{R}^n)$, associated with the Poisson semigroup of the Laplace operator $Δ$ on $\mathbb{R}^n$, coincides with the space $\mathrm{BMO}^φ(\mathbb{R}^n)$ introduced by L. D. Ky.

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