Remarks on Chemin's space of homogeneous distributions

TL;DR

This paper examines Chemin's space of homogeneous distributions, highlighting its limitations in supercritical spaces and analyzing the density and structural properties of its intersections with various Banach spaces.

Contribution

It reveals the failure of Chemin's space in supercritical contexts and characterizes the density and complementability of its intersections with key Banach spaces.

Findings

Chemin's space fails in supercritical cases

Intersections with certain Banach spaces are not dense

Closures of intersections lead to non-separable quotients

Abstract

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 $X$.