The best-constant problem for a family of Gagliardo-Nirenberg inequalities on a compact Riemannian manifold

Abstract

The best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo-Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.