Conformal invariants measuring the best constants for Gagliardo-Nirenberg-Sobolev 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 $\Vert w{\mathring{|}}_q\le C\Vert \nabla w{\Vert }_2^{\theta }\Vert w{\Vert }_p^{1-\theta }$. 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.

20 pages

References in corpus (2)

• Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation
• Smooth metric measure spaces, quasi-Einstein metrics, and tractors