Volume growths versus Sobolev inequalities

The paper deals with fine volume growth estimates on metric measures spaces supporting various Sobolev-type inequalities. Given a generic metric measure space, we first prove a quantitative volume growth of metric balls under the validity of a Sobolev-type inequality (including Gagliardo-Nirenberg, Sobolev and Nash inequalities, as well as their borderlines, i.e., the logarithmic-Sobolev, Faber-Krahn, Morrey and Moser-Trudinger inequalities, respectively), answering partially a question of Ledoux [Ann. Fac. Sci. Toulouse Math., 2000] in a broader setting. We then 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.