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The Euclidean Onofri inequality in higher dimensions

Dolbeault, Jean; Del Pino, Manuel (2013), The Euclidean Onofri inequality in higher dimensions, International Mathematics Research Notices, 2013, 15, p. 3600-3611. http://dx.doi.org/10.1093/imrn/rns119

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00658665/fr/
Date
2013
Journal name
International Mathematics Research Notices
Volume
2013
Number
15
Publisher
Oxford University Press
Pages
3600-3611
Publication identifier
Metadata
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Author(s)
Dolbeault, Jean cc
Del Pino, Manuel
Abstract (EN)
The classical Onofri inequality in the two-dimensional sphere assumes a natural form in the plane when transformed via stereographic projection. We establish an optimal version of a generalization of this inequality in the d-dimensional Euclidean space for any d≥2, by considering the endpoint of a family of optimal Gagliardo-Nirenberg interpolation inequalities. Unlike the two-dimensional case, this extension involves a rather unexpected Sobolev-Orlicz norm, as well as a probability measure no longer related to stereographic projection.
Subjects / Keywords
stereographic projection; optimal constants; extremal functions; interpolation; Gagliardo-Nirenberg inequalities; Onofri inequalities; logarithmic Sobolev inequality; Sobolev inequality