A dual form of the sharp Nash inequality and its weighted generalization

Eric Carlen, Elliott H. Lieb

Research output: Contribution to journalArticle

Abstract

The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L 2 norm to the infimal convolution of the L∞ and H -1 norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.

Original languageEnglish (US)
Pages (from-to)129-144
Number of pages16
JournalBulletin of the London Mathematical Society
Volume51
Issue number1
DOIs
StatePublished - Feb 1 2019

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Convolution
Duality
Sharp Constants
Norm
Sobolev Inequality
Minimization Problem
Generalization
Form

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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A dual form of the sharp Nash inequality and its weighted generalization. / Carlen, Eric; Lieb, Elliott H.

In: Bulletin of the London Mathematical Society, Vol. 51, No. 1, 01.02.2019, p. 129-144.

Research output: Contribution to journalArticle

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