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Sobolev and isoperimetric inequalities for degenerate metrics. (English) Zbl 0830.46027
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.
Reviewer: J.Wloka (Kiel)

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D15 Inequalities for sums, series and integrals
52A38 Length, area, volume and convex sets (aspects of convex geometry)
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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