Hölder regularity of harmonic functions on metric measure spaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912844699664384 |
|---|---|
| author | Gao, Jin Yang, Meng |
| author_facet | Gao, Jin Yang, Meng |
| contents | We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński carpet cable system, resolving an open problem left by Devyver, Russ, Yang (Int. Math. Res. Not. IMRN (2023), no. 18, 15537-15583). Second, we prove that two-sided heat kernel estimates alone imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on strongly recurrent metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hölder regularity of harmonic functions on metric measure spaces Gao, Jin Yang, Meng Analysis of PDEs Functional Analysis Metric Geometry 28A80, 35K08 We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński carpet cable system, resolving an open problem left by Devyver, Russ, Yang (Int. Math. Res. Not. IMRN (2023), no. 18, 15537-15583). Second, we prove that two-sided heat kernel estimates alone imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on strongly recurrent metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds. |
| title | Hölder regularity of harmonic functions on metric measure spaces |
| topic | Analysis of PDEs Functional Analysis Metric Geometry 28A80, 35K08 |
| url | https://arxiv.org/abs/2407.20789 |