Conformal invariants measuring the best constants for Gagliardo-Nirenberg-Sobolev inequalities

TL;DR

This paper introduces conformal invariants linked to Gagliardo-Nirenberg-Sobolev inequalities, generalizing classical constants, and characterizes their minimizers and critical points in various geometric settings.

Contribution

It develops a new family of conformal invariants for Gagliardo-Nirenberg-Sobolev inequalities using a minimization approach of the weighted scalar curvature functional.

Findings

Minimizers are critical points only for specific p and q values.

On Euclidean space, the invariants characterize sharp GNS constants with p=2(q-1) or q=2(p-1).

Provides a geometric framework linking invariants to classical inequalities.

Abstract

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for Gagliardo-Nirenberg-Sobolev inequalities $\|w\r|_q \leq C\|\nabla w\|_2^\theta \|w\|_p^{1-\theta}$. These invariants are constructed via a minimization procedure for the weighted scalar curvature functional in the conformal class of a smooth metric measure space. We then describe critical points which are also critical points for variations in the metric or the measure. When the measure is assumed to take a special form --- for example, as the volume element of an Einstein metric --- we use this description to show that minimizers of our invariants are only critical for certain values of $p$ and $q$. In particular, on Euclidean space our result states that either $p=2(q-1)$ or $q=2(p-1)$, giving a new characterization of the GNS inequalities whose sharp constants were computed by Del Pino and Dolbeault.