Remarks on Chemin's space of homogeneous distributions

This article focuses on Chemin’s space ${\mathcal{S}}_h^{\mathrm{\prime }}$ of homogeneous distributions, which was introduced to serve as a basis for realizations of subcritical homogeneous Besov spaces. We will discuss how this construction fails in multiple ways for supercritical spaces. In particular, we study its intersection $X_h:={\mathcal{S}}_h^{\mathrm{\prime }}\cap X$ with various Banach spaces $X$, namely supercritical homogeneous Besov spaces and the Lebesgue space $L^{\mathrm{\infty }}$. For each $X$, we find out if the intersection $X_h$ is dense in $X$. If it is not, then we study its closure $C=\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}(X_h)$ and prove that the quotient $X\mathrm{/}C$ is not separable and that $C$ is not complemented in $X$.