Characterization of some closed linear subspaces of Morrey spaces and approximation

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.

Closed linear subspaces Approximation Adams-Spanne type theorem Riesz potential Fractional maximal operator

Closed linear subspaces Approximation Adams-Spanne type theorem Riesz potential Fractional maximal operator