Sobolev inequalities on Lie groups and symmetric spaces
type: Item
instance of: scholarly article
title: Sobolev inequalities on Lie groups and symmetric spaces
zbMATH Open document ID: 0697.22013
DOI: 10.1016/0022-1236(89)90063-3
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let G be a connected real Lie group and m(g) the modular function relating left and right Haar measure. Put \(\nabla f=(X_ 1f,...,X_ kf)\). The problem studied here is to find necessary and sufficient conditions on various indices in order that an inequality of Sobolev type (or some related inequality), \[ \| m^{-1/n}f\|_{np/(n-p)}\leq C\| \nabla f\|_ p\quad (n>p\geq 1), \] should hold for all \(f\in C_ 0^{\infty}(G)\). Amongst these indices, the so-called spectral gap associated to the vector-fields \(X_ j\) plays an essential rôle.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 4142346
zbMATH Keywords: connected real Lie group
zbMATH Keywords: inequality of Sobolev type
zbMATH Keywords: spectral gap
author: N. Th. Varopoulos
reviewed by: Edwin J. Akutowicz
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90063-3
OpenAlex ID: W2151868502
cites work: Balls and metrics defined by vector fields. I: Basic properties
cites work: Analysis on Lie groups
cites work: Q4146925
cites work: The spaces \(L^ p\), with mixed norm
cites work: Q3743593
cites work: Q3726372
cites work: Q3726373
cites work: On L(p,q) spaces
cites work: Q5550080
cites work: Q5607484
cites work: Q5734920
cites work: Finitely generated fuchsian groups.
cites work: Q5682519
cites work: Q3331943
cites work: Q3810417
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description: scientific article; zbMATH DE number 4142346