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

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 {double pipe}w{double pipe}_q≤ C{double pipe}∇w{double pipe}₂^θ{double pipe}w{double pipe}_p^1-θ. 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.