The distance in Morrey spaces to $C_{\mathrm{comp}}^{\infty }$

Abstract

In this paper we characterize the distance between the function $f$ and the set $C_{\mathrm{comp}}^{\infty }({\mathbb{R}}^d)$ in generalized Morrey spaces $L_{p,\ddot{I}•}({\mathbb{R}}^d)$ with variable growth condition. We also prove that the bi-dual of ${\overline{C_{\mathrm{comp}}^{\infty }({\mathbb{R}}^d)}}^{L_{p,\ddot{I}•}({\mathbb{R}}^d)}$ is $L_{p,\ddot{I}•}({\mathbb{R}}^d)$. As an application of the characterization of the distance we show the boundedness of Calderón-Zygmund operators on ${\overline{C_{\mathrm{comp}}^{\infty }({\mathbb{R}}^d)}}^{L_{p,\ddot{I}•}({\mathbb{R}}^d)}$. By the duality we also see that these operators are bounded on its dual and bi-dual spaces.

25 pages