Remarks on Chemin's space of homogeneous distributions

This paper focuses on Chemin's space Sh′$\mathcal {S}^{\prime }_h$ of homogeneous distributions, which was introduced to serve as a basis for the 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 Xh:=Sh′∩X$X_h := \mathcal {S}^{\prime }_h \cap X$ with various Banach spaces X$X$, namely supercritical homogeneous Besov spaces and the Lebesgue space L∞$L^\infty$. For each X$X$, we investigate whether the intersection Xh$X_h$ is dense in X$X$. If it is not, then we study its closure C=clos(Xh)$C = {\rm clos}(X_h)$ and prove that the quotient X/C$X/C$ is not separable and that C$C$ is not complemented in X$X$.