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arXiv · 2207.07415

Remarks on Chemin's space of homogeneous distributions

Dimitri Cobb
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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 $X$.

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Dimitri Cobb. 2022-07-15. Remarks on Chemin's space of homogeneous distributions. https://arxiv.org/abs/2207.07415

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