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

Abstract

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

Acta Math. Sin. (Engl. Ser.) (to appear)