A generalization of a localization property of Besov spaces
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
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., $F^{s}_{p,q}=(F^{s}_{p,q})_{\ell^{p}}$).
Authors and Affiliations
N. Ferahtia, S. E. Allaoui
Keywords
Related Articles
- EP ID EP532703
- DOI 10.15330/cmp.10.1.71-78
- Views 68
- Downloads 0
Coupled fixed point results on metric spaces defined by binary operations
In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, unifo...
On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis
Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful appr...
On the growth of a klasss of Dirichlet series absolutely convergent in half-plane
In terms of generalized orders it is investigated a relation between the growth of a Dirichlet series $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with the abscissa of asolute convergence $A\in (-\infty,+\infty...
Hypercyclic operators on algebra of symmetric analytic functions on $\ell_p$
In the paper, it is proposed a method of construction of hypercyclic composition operators on $H(\mathbb{C}^n)$ using polynomial automorphisms of $\mathbb{C}^n$ and symmetric analytic functions on $\ell_p.$ In particular...
Homomorphisms and functional calculus in algebras of entire functions on Banach spaces
In the paper the homomorphisms of algebras of entire functions on Banach spaces to a commutative Banach algebra are studied. In particular, it is proposed a method of constructing of homomorphisms vanishing on homogeneou...