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Recent Publications
Localization of some functional spaces
and we denote by Eul. i.e., we have the following relation
with a designate the translation operator, and ' is a function of D(Rn), positive and not identically null.
Citation
M. FERAHTIA Nassim, (2024-11-27), "Localization of some functional spaces", [national] Second National Conference on Mathematics and Applications , University of M'sila-Algeria
Uniform Localization of some finite dimensional vector spaces
$p\in[1,+\infty]$, $\tau\in\mathbb{R}$ and $\tau\geq 0,$ where we concretely characterize these spaces i.e., we show that the uniform localized Lebesgue-type spaces $L_{p}^{\tau}(\mathbb{R}^{n})_{ul}$ are described without using an auxiliary function $\varphi$.
Citation
M. FERAHTIA Nassim, (2024-09-17), "Uniform Localization of some finite dimensional vector spaces", [international] The 4th International Conference On Applied Algebra ICAA '2024, 17-18th September 2024, Barika, Algeria , University Center of Barika, Algeria
A composition property on certain uniform localized Lizorkin-Triebel spaces
and we denote by $E_{lu}$. i.e., we have the following relation
$$\sup_{a\in \mathbb{R}^{n}}\|(\tau_{a}\varphi)f\|_{E}<\infty,$$
with $\tau_{a}$ designate the translation operator, and $\varphi$ is a function of $\mathcal{D}(\mathbb{R}^{n})$, positive and not identically null.
\medskip
\noindent In this work, we will study the composition in the uniform localized Lizorkin-Triebel spaces $F^{s}_{p,q}(\mathbb{R}^{n})_{lu}$, with
$0<s<1$, $p\in[1,+\infty[$ and $q\in[1,+\infty]$. We search to characterize the functions that operate by composition on the left, on space
$F^{s}_{p,q}(\mathbb{R}^{n})_{lu}$. Where we show that the Lipschitz conditions are necessary for $0<s<1$.
Citation
M. FERAHTIA Nassim, (2023-06-19), "A composition property on certain uniform localized Lizorkin-Triebel spaces", [national] The first National Conference On Applied Algebra , University center of Barika, Algeria
Transformations intégrales dans les espaces Lp
Citation
M. FERAHTIA Nassim, (2023-03-15), "Transformations intégrales dans les espaces Lp", [national] University of M'sila
A composition property on certain uniform localized Besov spaces
and we denote by $E_{lu}$. i.e., we have the following relation
$$\sup_{a\in \mathbb{R}^{n}}\|(\tau_{a}\varphi)f\|_{E}<\infty,$$
with $\tau_{a}$ designate the translation operator, and $\varphi$ is a function of $\mathcal{D}(\mathbb{R}^{n})$, positive and not identically null.
\medskip
\noindent In this work, we will study the composition in the uniform localized Besov space $B^{s}_{p,q}(\mathbb{R}^{n})_{lu}$, with
$0<s<1$ and $p,q\in[1,+\infty]$. We search to characterize the functions that operate by composition on the left, on space
$B^{s}_{p,q}(\mathbb{R}^{n})_{lu}$. Where we show that the Lipschitz conditions are necessary for $0<s<1$.
Citation
M. FERAHTIA Nassim, (2022-07-20), "A composition property on certain uniform localized Besov spaces", [international] The 6th International conference of Mathematical Sciences (ICMS2022) , Maltepe university, istanbul Turkey
Localisations sur les espaces de Lizorkin-Triebel et composition dans certains espaces de Besov localisés uniformes
$p,~q,~r\in\left[1,+\infty\right]$. Also, we have proved that the Lizorkin-Triebel spaces are localizable in the $\ell^{p}$ norm. Furthermore, we were interested in composition operators $T_{f}(g)=f\circ g$
on some Besov spaces with vector valued, where we gave sufficient conditions so that the operator $T_{f}$ operates on $B^{s}_{p,q}(\mathbb{R}^{n},\mathbb{R}^{m})$. Finally, we studied the uniform
localization on Lebesgue type space $L^{\tau}_{p}(\mathbb{R}^{n})$ and the Besov type space $B^{s,\tau}_{p,q}(\mathbb{R}^{n})$, where we have characterized
concretely these spaces , i.e., we have shown that $L^{\tau}_{p}(\mathbb{R}^{n})_{lu}$ and $B^{s,\tau}_{p,q}(\mathbb{R}^{n})_{lu}$ are described without using an auxiliary function $\varphi$,
for $0<s<1$, $p,~q\in\left[1,+\infty\right]$, and $\tau\in\mathbb{R}$.
Citation
M. FERAHTIA Nassim, (2021-04-07), "Localisations sur les espaces de Lizorkin-Triebel et composition dans certains espaces de Besov localisés uniformes", [national] University of msila
A GENERALIZATION OF A LOCALIZATION PROPERTY OF 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 $p,q\in[1,+\infty]$ are localizable in the $\ell^{p}$ norm (i.e., $F^{s}_{p,q}=(F^{s}_{p,q})_{\ell^{p}}$).
Citation
M. FERAHTIA Nassim, Allaoui Salah Eddine, , (2018-05-28), "A GENERALIZATION OF A LOCALIZATION PROPERTY OF BESOV SPACES", [national] Carpathian mathematical publications , Vasyl Stefanyk Precarpathian National University