Volume growths versus Sobolev inequalities

Abstract: Our study deals with fine volume growth estimates on metric measures spaces supporting various Sobolev-type inequalities. We first establish the sharp Nash inequality on Riemannian manifolds with non-negative Ricci curvature and positive asymptotic volume ratio. The proof of the sharpness of this result provides a generic approach to establish a unified volume growth estimate whenever a wide range of Gagliardo-Nirenberg-Sobolev interpolation inequality holds on general, not necessarily smooth metric measure spaces without any curvature restriction. This result provides a partial answer to the question of Ledoux [Ann. Fac. Sci. Toulouse Math., 2000] in a broader setting, as it includes the usual Gagliardo-Nirenberg, Sobolev and Nash inequalities, as well as their borderlines, i.e., the logarithmic-Sobolev, Faber-Krahn, Morrey and Moser-Trudinger inequalities, respectively. As a first application, we prove sharp Gagliardo-Nirenberg-Sobolev interpolation inequalities -- with their borderlines -- in the setting of metric measure spaces verifying the curvature-dimension condition ${\sf CD}(0,N)$ in the sense of Lott-Sturm-Villani. In addition, the equality cases are also characterized in terms of the $N$-volume cone structure of the ${\sf CD}(0,N)$ space together with the precise profile of extremizers. In addition, we also provide the heat kernel estimate with sharp constant on Riemannian manifolds with non-negative Ricci curvature, answering another question of Bakry, Concordet and Ledoux [ESAIM Probab. Statist., 1997].