The Caffarelli–Kohn–Nirenberg inequality on metric measure spaces

Tokura, Willian Isáo; Adriano, Levi; Xia, Changyu

doi:10.1007/s00229-020-01206-1

manuscripta math.

2020

DOI: 10.1007/s00229-020-01206-1

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The Caffarelli–Kohn–Nirenberg inequality on metric measure spaces

Willian Isáo Tokura

Levi Adriano

Changyu Xia

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2024

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Sharp Sobolev inequalities on noncompact Riemannian manifolds with $$\textsf{Ric}\ge 0$$ via optimal transport theory

Kristály

2024

Calc. Var.

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In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using $$L^1$$ L 1 -optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp $$L^p$$ L p -Sobolev and $$L^p$$ L p -logarithmic Sobolev inequalities (both for $$p>1$$ p > 1 and $$p=1$$ p = 1 ) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

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Sharp Sobolev inequalities on noncompact Riemannian manifolds with $$\textsf{Ric}\ge 0$$ via optimal transport theory

Kristály

2024

Calc. Var.

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show abstract

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The Caffarelli–Kohn–Nirenberg inequality on metric measure spaces

Cited by 1 publication

References 34 publications

Sharp Sobolev inequalities on noncompact Riemannian manifolds with $$\textsf{Ric}\ge 0$$ via optimal transport theory

Sharp Sobolev inequalities on noncompact Riemannian manifolds with $$\textsf{Ric}\ge 0$$ via optimal transport theory

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