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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