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# [FORTHCOMING] Caractérisation de sous-espaces vectoriels fermés des espaces de Morrey et approximation

## [FORTHCOMING] Characterization of some closed linear subspaces of Morrey spaces and approximation

Nouffou Diarra Université Félix Houphouët Boigny
Côte d’Ivoire
Ibrahim Fofana Université Félix Houphouët Boigny
Côte d’Ivoire

Publié le 17 février 2022   DOI :

### Keywords

Let $1\leq q\leq\alpha < \infty. \left\{(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d):\alpha\leq p\leq\infty \right\}$ is a nondecreasing family of Banach spaces such that the Lebesgue space is $L^{\alpha}(\mathbb{R}^d)$ its minimal element and the classical Morrey space $\mathcal{M}_{q}^{\alpha}(\mathbb{R}^d)$ is its maximal element. In this note we investigate some closed linear subspaces of $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$. We give a characterization of the closure in $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$ of the set of all its compactly supported elements and study the action of some classical operators on it. We also describe the closure in $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$ of the set $\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$ of all infinitely differentiable and compactly supported functions on $\mathbb{R}^{d}$ as an intersection of other linear subspaces of $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$ and obtain the weak density of $\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$ in some of these subspaces. We establish a necessary condition on a function $f$ in order that its Riesz potential $I_{\gamma}(|f|) \;(0<\gamma<1)$ be in a given Lebesgue space.

Let $1\leq q\leq\alpha < \infty. \left\{(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d):\alpha\leq p\leq\infty \right\}$ is a nondecreasing family of Banach spaces such that the Lebesgue space is $L^{\alpha}(\mathbb{R}^d)$ its minimal element and the classical Morrey space $\mathcal{M}_{q}^{\alpha}(\mathbb{R}^d)$ is its maximal element. In this note we investigate some closed linear subspaces of $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$. We give a characterization of the closure in $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$ of the set of all its compactly supported elements and study the action of some classical operators on it. We also describe the closure in $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$ of the set $\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$ of all infinitely differentiable and compactly supported functions on $\mathbb{R}^{d}$ as an intersection of other linear subspaces of $(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$ and obtain the weak density of $\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$ in some of these subspaces. We establish a necessary condition on a function $f$ in order that its Riesz potential $I_{\gamma}(|f|) \;(0<\gamma<1)$ be in a given Lebesgue space.