Bourgain–Brezis inequalities on symmetric spaces of non-compact type

Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L1 vector fields, with vector fields in a critical Sobolev space on M:|∫M〈f,ϕ〉dV|≤C‖f‖L1(dV)‖∇ϕ‖Lm(dV). This estimate provides a remedy for the failure of a critical Sobolev...

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Bibliographic Details
Published inJournal of functional analysis Vol. 273; no. 4; pp. 1504 - 1547
Main Authors Chanillo, Sagun, Van Schaftingen, Jean, Yung, Po-Lam
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.08.2017
Subjects
Critical Sobolev space
Iwasawa decomposition
Lie group
Symmetric space
secondary
Lie group
Symmetric space
Critical Sobolev space
primary
Iwasawa decomposition
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Summary:Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L1 vector fields, with vector fields in a critical Sobolev space on M:|∫M〈f,ϕ〉dV|≤C‖f‖L1(dV)‖∇ϕ‖Lm(dV). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2017.05.005