Sobolev and isoperimetric inequalities for degenerate metrics
Let \(X_1, \dots, X_m\) be continuous vector fields in an open subset \(\Omega \subseteq \mathbb{R}^N\). A Sobolev inequality holds for \(p\geq 1\) if there exist \(q>p\) and \(C= C(\Omega, p,q)\) such that \[ \Biggl( \int_\Omega |u|^q dx\Biggr)^{1/q} \leq C \Biggl( \int_\Omega \biggl(\sum_j \langle X_j, \nabla u\rangle^2 \biggr)^{p/2} dx\Biggr)^{1/p}, \] for any \(u\in C_0^\infty (\Omega)\). A Sobolev inequality with \(p=1\) and with the corresponding optimal choice of \(q\) will be called a geometric inequality, it is equivalent to a suitable form of the isoperimetric inequality. In the present paper Sobolev's inequality is proved, also the geometric inequality for some classes of vector fields.
scientific article; zbMATH DE number 734586
scientific article