Some properties of anisotropic Sobolev spaces

In the theory of mixed problems for symmetric hyperbolic systems under characteristic boundary conditions, and in particular the equations of ideal Magneto-hydrodynamics with perfectly conducting wall boundary condition, the natural functional setting is provided by the anisotropic Sobolev spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $H^m_\ast (\Omega ),\; H^m_{\ast \ast }(\Omega )$\end{document}. In the present paper we show some properties of such function spaces which provide basic tools for calculations.