Elliptic Harnack inequality and its applications on Finsler metric measure spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866909470928404480 |
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| author | Cheng, Xinyue Liu, Liulin Zhang, Yu |
| author_facet | Cheng, Xinyue Liu, Liulin Zhang, Yu |
| contents | In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature ${\rm Ric}_{\infty}$ has non-positive lower bound and the distortion $τ$ is of linear growth, $|τ|\leq ar+b$, where $a,b$ are some non-negative constants, $r=d(x_0,x)$ is the distance function for some point $x_{0} \in M$. We obtain an elliptic $p$-Harnack inequality for positive harmonic functions from a local uniform Poincaré inequality and a mean value inequality. As applications of the Harnack inequality, we derive the Hölder continuity estimate and a Liouville theorem for positive harmonic functions. Furthermore, we establish a gradient estimate for positive harmonic functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic Harnack inequality and its applications on Finsler metric measure spaces Cheng, Xinyue Liu, Liulin Zhang, Yu Differential Geometry 53C60, 53B40, 53C21, 58C35 In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature ${\rm Ric}_{\infty}$ has non-positive lower bound and the distortion $τ$ is of linear growth, $|τ|\leq ar+b$, where $a,b$ are some non-negative constants, $r=d(x_0,x)$ is the distance function for some point $x_{0} \in M$. We obtain an elliptic $p$-Harnack inequality for positive harmonic functions from a local uniform Poincaré inequality and a mean value inequality. As applications of the Harnack inequality, we derive the Hölder continuity estimate and a Liouville theorem for positive harmonic functions. Furthermore, we establish a gradient estimate for positive harmonic functions. |
| title | Elliptic Harnack inequality and its applications on Finsler metric measure spaces |
| topic | Differential Geometry 53C60, 53B40, 53C21, 58C35 |
| url | https://arxiv.org/abs/2501.18814 |