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Ultracontractivity and Functional Inequalities on Infinite Graphs

Yong Lin Department of Mathematics, Renmin University of China, Beijing 100872, China Shuang Liu Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Hongye Song Department of Mathematics, Renmin University of China, Beijing 100872, China; Beijing International Studies University, Beijing 100024, China

Differential Geometry Functional Analysis mathscidoc:2207.10006

Discrete & Computational Geometry, 61, 198–211, 2018.6
We prove the equivalence between some functional inequalities and the ultracontractivity property of the heat semigroup on infinite graphs. These functional inequalities include Sobolev inequalities, Nash inequalities, Faber–Krahn inequalities, and log-Sobolev inequalities. We also show that, under the assumptions of volume growth and CDE(n, 0), which is regarded as the natural notion of curvature on graphs, these four functional inequalities and the ultracontractivity property of the heat semigroup are all true on graphs.
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@inproceedings{yong2018ultracontractivity,
  title={Ultracontractivity and Functional Inequalities on Infinite Graphs},
  author={Yong Lin, Shuang Liu, and Hongye Song},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707152902810984564},
  booktitle={Discrete & Computational Geometry},
  volume={61},
  pages={198–211},
  year={2018},
}
Yong Lin, Shuang Liu, and Hongye Song. Ultracontractivity and Functional Inequalities on Infinite Graphs. 2018. Vol. 61. In Discrete & Computational Geometry. pp.198–211. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707152902810984564.
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