BMO-scale of distribution on ℝ n

Abstract

Let S′ be the class of tempered distributions. For ƒ ∈ S′ we denote by J−α ƒ the Bessel potential of ƒ of order α. We prove that if J−α ƒ ∈ BMO, then for any λ ∈ (0, 1), J−α(f)λ ∈ BMO, where (f)λ = λ−nf(φ(λ−1)), φ ∈ S. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the VMO space.