Abstract
We study functional and geometric inequalities on complete Finsler measure spaces with the weighted Ricci curvature \(\textrm{Ric}_\infty \) bounded below. We first obtain some local uniform Poincaré inequalities and Sobolev inequalities. Then, we prove a mean value inequality for nonnegative subsolutions of elliptic equations. Further, we derive local and global Harnack inequalities for positive harmonic functions. Finally, we establish a global gradient estimate for positive harmonic functions on forward complete non-compact Finsler measure spaces. Besides, as an application of the mean value inequality, we prove a Liouville type theorem for harmonic functions.
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The authors would like to express their sincere thanks to the anonymous referee for his/her careful reading and valuable suggestions.
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X. Cheng is supported by the National Natural Science Foundation of China (12371051, 12141101, 11871126). Y. Feng is supported by the Chongqing Postgraduate Research and Innovation Project (CYB23231).
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Cheng, X., Feng, Y. Some Functional Inequalities and Their Applications on Finsler Measure Spaces. J Geom Anal 34, 127 (2024). https://doi.org/10.1007/s12220-024-01583-z
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DOI: https://doi.org/10.1007/s12220-024-01583-z
Keywords
- Finsler measure space
- Volume comparison
- Sobolev inequality
- Mean value inequality
- Harnack inequality
- Gradient estimate