Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications

Abstract

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent n ≥ 3, then it has exactly the n-dimensional volume growth. As an application, if an n-dimensional Finsler manifold of non-negative n-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.

A Sharp Sobolev Interpolation Inequality on Finsler Manifolds

“…Contrary to [5], we are dealing with not necessarily reversible Finsler manifolds, which is one of the most relevant features of such non-Riemannian structures.…”

“…(d) Theorems 1.1-1.3 can be compared with the results in Kristály and Ohta [5], where a Caffarelli-Kohn-Nirenberg inequality is studied in a generic metric measure setting with applications on reversible Finsler manifolds. Contrary to [5], we are dealing with not necessarily reversible Finsler manifolds, which is one of the most relevant features of such non-Riemannian structures.…”

New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case

“…Since µ > n, the sequence {u j } is bounded in W 1,n G (M); in particular, by relation (33) and the latter estimate one can guarantee the existence of c 4 > 0 (depending only on n, µ and c) such that for every j ∈ N,…”

“…where c ∈ R and c 4 > 0 are from (33) and (35), respectively. Note that there exists σ l ∈ G such thatx l = σ l (x) for every l ∈ {1, ..., N}.…”

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

“…When (R n , F ) is a Minkowski space, then W 1,2 0 (Ω, F ) is the usual Sobolev space W 1,2 0 (Ω) for every open set Ω ⊂ R n , see Kristály and Ohta [19]; indeed, in this case there exits C 0 ≥ 1 such that…”