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

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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}2θ{double pipe}w{double pipe}p1-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 languageEnglish (US)
Pages (from-to)507-526
Number of pages20
JournalCalculus of Variations and Partial Differential Equations
Volume48
Issue number3-4
DOIs
StatePublished - Nov 2013

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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