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 β₯wβ£Λβqββ€Cβ₯βwβ₯2ΞΈββ₯wβ₯p1βΞΈβ. 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(qβ1)
or q=2(pβ1), giving a new characterization of the GNS inequalities whose
sharp constants were computed by Del Pino and Dolbeault.Comment: 20 page