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

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Let X be a space of homogenous type and φ:X×[0,∞) →[0,∞) a growth function such that φ(·,t) is a Muckenhoupt weight uniformly in t and φ(x,·) an Orlicz function of uniformly upper type 1 and lower type p∈(0,1]. In this article, the authors introduce a new Musielak-Orlicz BMO-type space BMOφₐ(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 BMOφA,max(X) and BMOφₐ(X). Moreover, two variants of the John-Nirenberg inequality on BMOφₐ(X) are obtained. As an application, the authors further prove that the space BMOφ√Δ(Rⁿ), associated with the Poisson semigroup of the Laplace operator Δ on Rⁿ, coincides with the space BMOφ(Rⁿ) introduced by L. D. Ky.