Brezis-Gallouet's inequality in higher-dimensional spaces (Q674682)

Brezis-Gallouet's inequality (it is called the B-G's inequality) is the following \[ \| u\|_{L^\infty(\Omega)}\leq c(1+ \sqrt{\log(1+\| u\|_{H^2(\Omega)})}),\quad \Omega\subset \mathbb{R}^2, \] where \(u\in H^2(\Omega)\), and \(\| u\|_{H^1(\Omega)}\leq 1\). In this paper, the author gives extensions of this inequality up to higher dimensions by different methods from \textit{H. Brezis} and \textit{S. Wainger's} method [Commun. Partial Differ. Equa. 5, 773-789 (1980; Zbl 0437.35071)]. Theorem 1. Suppose \(\Omega\subset\mathbb{R}^n\) is a bounded domain, the boundary \(\partial\Omega\) is smooth enough, \(u\in H^{1+n/2}(\Omega)\), and \(\| u\|_{H^{n/2}(\Omega)\leq 1}\). Then there exists a constant \(c\) such that \[ \| u\|_{L^\infty(\Omega)}\leq c(1+ n\sqrt{\log(1+\| u\|_{H^{1+n/2}(\Omega)})}). \] Theorem 2. Suppose \(\Omega\) is the same as Theorem 1, \(u\in H^{\sigma- 1+n/2}(\Omega)\), where \(\sigma\geq 2\), and \(\| u\|_{H^{{n\over 2}}(\Omega)\leq 1}\), then there exists a constant \(c\) such that \[ \| u\|_{L^\infty(\Omega)}\leq c(1+ n\sqrt{\log(1+\| u\|_{H^{\sigma- 1+n/2}(\Omega)})}). \]