Remarks on Chemin's space of homogeneous distributions

This article focuses on Chemin's space \(\mathcal{S}'_h\) 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 \(X_h := \mathcal{S}'_h \cap X\) with various Banach spaces \(X\), namely supercritical homogeneous Besov spaces and the Lebesgue space \(L^\infty\). For each \(X\), we find out if the intersection \(X_h\) is dense in \(X\). If it is not, then we study its closure \(C = {\rm clos}(X_h)\) and prove that the quotient \(X/C\) is not separable and that \(C\) is not complemented in \(X\).