A Sharp Sobolev Interpolation Inequality on Finsler Manifolds

Advances in Mathematics

2018

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By using optimal mass transportation and a quantitative Hölder inequality, we provide estimates for the Borell-Brascamp-Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb inequalities (including Brunn-Minkowski and Prékopa-Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge-Ampère equation, we give a new proof of Dubuc's characterization of the equality in Borell-Brascamp-Lieb inequalities in the Euclidean setting. When the n-dimensional Riemannian manifold has Ricci curvature Ric(M ) ≥ (n − 1)k for some k ∈ R, it turns out that equality in the Borell-Brascamp-Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature k. A precise characterization is provided for the equality in the Lott-Sturm-Villani-type distorted Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented.

Section: Equality In Borell-brascamp-lieb Inequality: Finsler Casementioning

confidence: 99%

Equality in Borell–Brascamp–Lieb inequalities on curved spaces

Balogh

Advances in Mathematics

2018

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“…On one hand, inspired by [10], a systematic study of the Hardy inequality is carried out by Berchio, Ganguly and Grillo [7], D'Ambrosio and Dipierro [15], Kombe and Özaydin [31,32], Yang, Su and Kong [47] in the Riemannian setting, as well as by Kristály and Repovš [29] and Yuan, Zhao and Shen [48] in the Finsler setting. On the other hand, Caffarelli-Kohn-Nirenberg-type inequalities are studied by do Carmo and Xia [14], Erb [16] and Xia [46] on Riemannian manifolds, and by Kristály [26] and Kristály and Ohta [28] on Finsler manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…p for s > 0 instead of u s (x) = e −sρ 2 x 0 (x) for s > 0. The case when λ F (M ) = 1 and m = m BH has been considered by Kristály[26, Theorem 1.2].…”

mentioning

confidence: 99%

Sharp uncertainty principles on general Finsler manifolds

Huang

Zhao

2018

Preprint

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The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non-Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and S-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. Our results complement in several aspects those obtained recently in the Riemannian setting by Kristály [J. Math. Pures Appl. (9) 119 ( 2018), 326-346], their optimality being supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

“…On the other hand, due to Ledoux [17], if (M, g) has nonnegative Ricci curvature, inequality (FSI) c 0 holds if and only if (M, g) is isometric to the Euclidean space R n . Further first-order Sobolevtype inequalities on Riemannian/Finsler manifolds can be found in Bakry, Concordet and Ledoux [2], Druet, Hebey and Vaugon [8], do Carmo and Xia [6], Xia [24]- [26], Kristály [15]; moreover, similar Sobolev inequalities are also considered on 'nonnegatively' curved metric measure spaces formulated in terms of the Lott-Sturm-Villani-type curvature-dimension condition or the Bishop-Gromov-type doubling measure condition, see Kristály [14] and Kristály and Ohta [16].…”

Section: Introductionmentioning

confidence: 99%

Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature

Barbosa

Bull. London Math. Soc.

2017

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Abstract. Let (M, g) be an n−dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying ρ∆gρ ≥ n − 5 ≥ 0, where ∆g is the Laplace-Beltrami operator on (M, g) and ρ is the distance function from a given point. If (M, g) supports a second-order Sobolev inequality with a constant C > 0 close to the optimal constant K0 in the second-order Sobolev inequality in R n , we show that a global volume non-collapsing property holds on (M, g). The latter property together with a Perelman-type construction established by Munn (J. Geom. Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of (M, g). Furthermore, it turns out that (M, g) supports the second-order Sobolev inequality with the constant C = K0 if and only if (M, g) is isometric to the Euclidean space R n .