Volume growths versus Sobolev inequalities
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915478227648512 |
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| author | Kristály, Alexandru |
| author_facet | Kristály, Alexandru |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Volume growths versus Sobolev inequalities Kristály, Alexandru Analysis of PDEs 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. |
| title | Volume growths versus Sobolev inequalities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.16199 |