@ARTICLE{10.21494/ISTE.OP.2023.0980, 

TITLE={Characterization of some closed linear subspaces of Morrey spaces and approximation}, 
  
AUTHOR={Nouffou Diarra, Ibrahim Fofana, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={14}, 

NUMBER={Issue 3 (June 2023)}, 
	
YEAR={2023}, 

URL={http://openscience.fr/Characterization-of-some-closed-linear-subspaces-of-Morrey-spaces-and}, 
	
DOI={10.21494/ISTE.OP.2023.0980}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={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.}}