Sobolev and isoperimetric inequalities for degenerate metrics
type: Item
instance of: scholarly article
title: Sobolev and isoperimetric inequalities for degenerate metrics
zbMATH Open document ID: 0830.46027
DOI: 10.1007/BF01450501
author: Bruno Franchi
author: R. L. Wheeden
author: Sylvestre (Sylvain) Gallot
published in: Mathematische Annalen
publication date: 1996-02-04T00:00:00Z
full work available at URL: https://eudml.org/doc/165267
review text: 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.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 734586
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: geometric inequality
zbMATH Keywords: isoperimetric inequality
reviewed by: Josef Wloka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 734586