## 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 {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.

Original language | English (US) |
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Pages (from-to) | 507-526 |

Number of pages | 20 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 48 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 2013 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics