A generalization of a localization property of Besov spaces

Abstract

The notion of a localization property of Besov spaces is introduced by G. Bourdaud, where he has provided that the Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$, with $s\in\mathbb{R}$ and $p,q\in[1,+\infty]$ such that $p\neq q$, are not localizable in the $\ell^{p}$ norm. Further, he has provided that the Besov spaces $B^{s}_{p,q}$ are embedded into localized Besov spaces $(B^{s}_{p,q})_{\ell^{p}}$ (i.e., $B^{s}_{p,q}\hookrightarrow(B^{s}_{p,q})_{\ell^{p}},$ for $p\geq q$). Also, he has provided that the localized Besov spaces $(B^{s}_{p,q})_{\ell^{p}}$ are embedded into the Besov spaces $B^{s}_{p,q}$ (i.e., $(B^{s}_{p,q})_{\ell^{p}}\hookrightarrow B^{s}_{p,q},$ for $p\leq q$). In particular, $B_{p,p}^{s}$ is localizable in the $\ell^{p}$ norm, where $\ell^{p}$ is the space of sequences $(a_{k})_{k}$ such that $\|(a_{k})\|_{\ell^{p}}<\infty$. In this paper, we generalize the Bourdaud theorem of a localization property of Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$ on the $\ell^{r}$ space, where $r\in[1,+\infty]$. More precisely, we show that any Besov space $B^{s}_{p,q}$ is embedded into the localized Besov space $(B^{s}_{p,q})_{\ell^{r}}$ (i.e., $B^{s}_{p,q}\hookrightarrow(B^{s}_{p,q})_{\ell^{r}},$ for $r\geq\max(p,q)$). Also we show that any localized Besov space $(B^{s}_{p,q})_{\ell^{r}}$ is embedded into the Besov space $B^{s}_{p,q}$ (i.e., $(B^{s}_{p,q})_{\ell^{r}}\hookrightarrow B^{s}_{p,q},$ for $r\leq\min(p,q)$). Finally, we show that the Lizorkin-Triebel spaces $F^{s}_{p,q}(\mathbb{R}^{n})$, where $s\in\mathbb{R}$ and $p\in[1,+\infty)$ and $q\in[1,+\infty]$ are localizable in the $\ell^{p}$ norm (i.e., $F^{s}_{p,q}=(F^{s}_{p,q})_{\ell^{p}}$).