The distance in Morrey spaces to $C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}$

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