Some approximation properties in fractional Musielak-Sobolev spaces

In this article we show some density properties of smooth and compactly supported functions in fractional Musielak-Sobolev spaces essentially extending the results of Fiscella et al. (Ann Acad Sci Fenn Math 40(1):235-253, 2015) obtained in the fractional Sobolev setting. The proofs of these properties are mainly based on a basic technique of convolution (which makes functions C infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{\infty }$$\end{document}), joined with a cut-off (which makes their support compact), with some care needed in order not to exceed the original support.