A DUAL FORM OF THE SHARP NASH INEQUALITY AND ITS WEIGHTED GENERALIZATION (Tosio Kato Centennial Conference)

Abstract

The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality should also have an interesting dual form. We provide one here. This dual inequality relates the L^{2} norm to the infimal convolution of the L^{infty} 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 weighted generalizations of the Nash inequality and the dual of these weighted variants.