Some inequalities related to Sobolev norms

In this paper, we study some properties related to the new characterizations of Sobolev spaces introduced in Bourgain and Nguyen (C R Acad Sci, 343:75–80, [2006]), Nguyen (J Funct Anal 237: 689–720, [2006]; J Eur Math Soc 10:191–229, [2008]). More precisely, we establish variants of the Poincaré inequality, the Sobolev inequality, and the Rellich–Kondrachov compactness theorem, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_{\mathbb{R}^N} |\nabla g|^p \;dx}$$\end{document} is replaced by some quantity of the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\delta} (g) =\mathop{\int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N}}_{|g(x) - g(y)| > \delta}\frac{\delta^p}{|x-y|^{N+p}}\, dx \, dy.$$\end{document}