arXiv subjects
Publications and source records attributed to Douadi Drihem.
In this paper, we study the Hardy-H\'{e}non parabolic equation \begin{equation*} \partial_t u=\Delta u+a|x|^{-\gamma}|u|^{\beta}u, \qquad t>0,\; x\in\mathbb{R}^{n}\setminus\{0\}, \quad \beta>0,\; a\in\mathbb{R}, \end{equation*} under suitable assumptions on the parameter $\gamma$. We establish the local and global well-posedness of mild solutions in homogeneous Herz spaces.
In this paper, we prove that Herz-type Sobolev spaces form a Banach algebra and establish a Rellich--Kondrachov compactness theorem for these spaces. These results extend the corresponding classical theory and further demonstrate that Herz-type Sobolev spaces provide a natural generalization of the classical Sobolev spaces.
Based on mixed-norm Herz spaces, we introduce the classes of mixed-norm Herz-type Besov and Triebel-Lizorkin spaces. We establish their $\varphi$-transform characterization in the sense of Frazier and Jawerth and prove Sobolev, Franke and Jawerth embedding theorems for these spaces. The obtained Sobolev embeddings extend and improve several known embedding results for mixed-norm Besov and Triebel-Lizorkin spaces. As a consequence of our results, the Franke and Jawerth embeddings for mixed-norm Besov and Triebel-Lizorkin spaces are established.
In this paper, we introduce a new family of function spaces of Besov and Triebel-Lizorkin type. We present the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev and Franke-Jewarth embeddings. Also, we establish the smooth atomic, molecular and wavelet decomposition of these function spaces. Characterizations by ball means of differences are given. Finally, we investigate a series of examples which play an important role in the study of function spaces of Besov-Triebel-Lizorkin type.
In this paper, we identify the duals of Triebel-Lizorkin spaces of generalized smoothness. In some particular cases these function spaces are just weighted Triebel-Lizorkin spaces. To do these, we will be working at the level of sequence spaces. The $\varphi $-transform characterization of these function spaces in the sense of Frazier and Jawerth, and new weighted version of vector-valued maximal inequality of Fefferman and Stein are the main tools.
The aim of this paper is to present some necessary conditions for the boundedness of the mapping $f\mapsto |f|^{\mu },\mu >0$ on Besov spaces equipped with power weights.
This paper is concerned with proving some embeddings of the form \begin{equation*} F_{p_{1},q}^{s_{1}}\cdot B_{p_{2},\infty }^{s_{2}}\cdot ...\cdot B_{p_{m},\infty }^{s_{m}}\hookrightarrow F_{p,q}^{s_{1}},\quad m\geq 2. \end{equation*} The different embeddings obtained here are under certain restrictions on the parameters. In particular, we improve some results of pointwise multiplication on Triebel-Lizorkin spaces. Franke-Jawerth embeddings, the $\pm $ - method of Gustaffson-Peetre and the relation between Hardy spaces and Triebel-Lizorkin spaces are the main tools.
The aim of this paper is twofold. Firstly, we chatacterize the Besov spaces $\dot{B}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and the Triebel-Lizorkin spaces $\dot{F}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ for $q=\infty $. Secondly, under some suitable assumptions on the $p$-admissible weight sequence $\{t_{k}\}$, we prove that \begin{equation*} \dot{A}_{p,q}(\mathbb{R}^{n},\{t_{k}\})=\dot{A}_{p,q}(\mathbb{R} ^{n},t_{j}),\quad j\in \mathbb{Z}, \end{equation*} in the sense of equivalent quasi-norms, with $\dot{A}$ $\in \{\dot{B},\dot{F}\}$. Moreover, we find a necessary and sufficient conditions for the coincidence of the spaces $\dot{A}_{p,q}(\mathbb{R}^{n},t_{i}),i\in \{1,2\}$.
Let $G:\mathbb{R\rightarrow R}$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on $G$ such that \begin{equation*} \{G(f):f\in \dot{K}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset \dot{K}_{p,q}^{\alpha }F_{\beta }^{s} \end{equation*} holds. Here $\dot{K}_{p,q}^{\alpha }F_{\beta }^{s}$ are Herz-type Triebel-Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel-Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabolic equations \begin{equation*} \partial _{t}u-\Delta u=G(u) \end{equation*} with initial data in Herz-type Triebel-Lizorkin spaces. Our results cover the results obtained with initial data in some know function spaces such us fractional Sobolev spaces. Some limit cases are given.
Let $G:\mathbb{R\rightarrow R}$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove that \begin{equation*} \{G(f):f\in A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha }) \end{equation*} implies $G$ is a linear function. Here $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot|^{\alpha })$ stands for either the Besov space $B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ or the Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$. These spaces unify and generalize many classical function spaces such as Sobolev spaces of power weights. One of the main difficulties to study this problem is that the norm of the $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ spaces with $\alpha \neq 0$ is not translation invariant, so some new techniques must be developed.
In this paper, the author introduce Triebel-Lizorkin spaces with general smoothness. We present the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. Also, we establish the smooth atomic and molecular decomposition of these function spaces. To do these we need a generalization of some maximal inequality to the case of general weights.
In this paper we introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the $\varphi $-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained.
In this paper, we present the complex interpolation of Besov and Triebel-Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel-Lizorkin spaces. An application, we obtain the complex interpolation between the weighted Triebel-Lizorkin spaces $\dot{F}_{p_{0},q_{0}}^{s_{0}}(\omega_{0})$ and $\dot{F}_{\infty ,q_{1}}^{s_{1}}(\omega_{1})\ $with suitable assumptions on the parameters $\ s_{0},s_{1},p_{0},$ $q_{0}$ and$\ q_{1}$, and the pair of weights $(\omega_{0},\omega_{1})$.
In this paper, we present more regularity conditions which ensure the boundedness of dilation operators on Besov and Triebel-Lizorkin spaces equiped with general weights.
We introduce Herz-Sobolev spaces, which unify and generalize the classical Sobolev spaces. We will give a proof of the Sobolev-type embedding for these function spaces. All these results generalize the classical results on Sobolev spaces. Some remarks on Caffarelli--Kohn--Nirenberg inequality are given.
In this paper the authors obtain a new equivalent norms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of Calderon reproducing formula, maximal inequalities and variable exponent technique, but allowing the parameters to vary from point to point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on these exponents.
We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. More Precisely, we investigate the inequalities \begin{equation*} \big\|f\big\|_{\dot{k}_{v,\sigma }^{\alpha_{1},r}}\leq c\big\|f\big\|_{\dot{K}_{u}^{\alpha_{2},\delta }}^{1-\theta }\big\|f\big\|_{\dot{K}_{p}^{\alpha_{3},\delta_{1}}A_{\beta }^{s}}^{\theta }, \end{equation*} and \begin{equation*} \big\|f\big\|_{\mathcal{E}_{p,2,u}^{\sigma }}\leq c\big\|f\big\|_{\mathcal{M}_{\mu }^{\delta }}^{1-\theta }\big\|f\big\|_{\mathcal{N}_{q,\beta ,v}^{s}}^{\theta }, \end{equation*} with some appropriate assumptions on the parameters, where $\dot{k}_{v,\sigma }^{\alpha_{1},r}$ is the Herz-type Bessel potential spaces, which are just the Sobolev spaces if $\alpha_{1}=0,1<r=v<\infty $ and $% \sigma \in \mathbb{N}_{0}$, and $\dot{K}_{p}^{\alpha_{3},\delta_{1}}A_{\beta }^{s}$ are Besov or Triebel-Lizorkin spaces if $\alpha_{3}=0$ and$\ \delta_{1}=p$. To do these, we study when distributions belonging to these spaces can be interpreted as functions in $L_{\mathrm{loc}}^{1}$. The usual Littlewood-Paley technique, Sobolev and Franke embeddings are the main tools of this paper. Some remarks on Hardy-Sobolev inequalities are given.
We study complex interpolation of variable Triebel-Lizorkin spaces, especially we present the complex interpolation of $F_{p(\cdot),q}^{α}$ and $F_{p(\cdot ),p(\cdot )}^{α(\cdot )}$ spaces. Also, some limiting cases are given.