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