Let $S^{\text{'}}$ be the class of tempered distributions. For $f\in S^{\text{'}}$ we denote by $J^{-\alpha }f$ the Bessel potential of $f$ of order $\alpha$. We prove that if $J^{-\alpha }f\in \mathrm{B}MO$, then for any $\lambda \in (0,1)$, $J^{-\alpha }{\left(f\right)}_{\lambda }\in \mathrm{B}MO$, where ${\left(f\right)}_{\lambda }={\lambda }^{-n}f\left(\phi ({\lambda }^{-1}·)\right)$, $\phi \in S$. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order $\alpha >0$ belongs to the $\mathrm{V}MO$ space.

Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.