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

Abstract

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