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Riemannian metrics on \(\mathbb{R}^n\) and Sobolev-type inequalities. (English. Russian original) Zbl 1360.53037

Dokl. Math. 94, No. 2, 510-513 (2016); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 470, No. 2, 1137-140 (2016).
Authors’ abstract: Poincaré-type estimates for a logarithmically concave measure \(\mu\) on a convex set \(\Omega\) are obtained. For this purpose, \(\Omega\) is endowed with a Riemannian metric \(g\) in which the Riemannian manifold with measure \((\Omega, g, \mu )\) has nonnegative Bakry-Emery tensor and, as a corollary, satisfies the Brascamp-Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincaré, Hardy, or log-Sobolev type inequalities and other results.

MSC:

53C20 Global Riemannian geometry, including pinching
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

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