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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 \({\| w \|_q \leq C \| \nabla w \|_2^{\theta} \| w \|_p^{1-\theta}}\) . 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.

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  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Jeffrey S. Case

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  1. Jeffrey S. Case
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Communicated by A.Chang.

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Case, J.S. Conformal invariants measuring the best constants for Gagliardo–Nirenberg–Sobolev inequalities. Calc. Var. 48, 507–526 (2013). https://doi.org/10.1007/s00526-012-0559-3

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