2025/01/27 by Alexandru Kristály, Kristály, Alexandru
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2501.16199
openalex publication_date 2025/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.