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Ultracontractivity and Functional Inequalities on Infinite Graphs

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Abstract

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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Acknowledgements

Y.L. is supported by the National Natural Science Foundation of China (grant no. 11671401). S.L. is supported by the Certificate of China Postdoctoral Science foundation Grant (grant no. 2018M631435). The authors would like to thank the anonymous referees for their extraordinarily careful reading of the manuscript, leading to various improvements.

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Authors and Affiliations

  1. Department of Mathematics, Renmin University of China, Beijing, 100872, China

    Yong Lin & Hongye Song

  2. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Shuang Liu

  3. Beijing International Studies University, Beijing, 100024, China

    Hongye Song

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  1. Yong Lin
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  2. Shuang Liu
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  3. Hongye Song
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Correspondence to Shuang Liu.

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Lin, Y., Liu, S. & Song, H. Ultracontractivity and Functional Inequalities on Infinite Graphs. Discrete Comput Geom 61, 198–211 (2019). https://doi.org/10.1007/s00454-018-0014-0

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  • DOI: https://doi.org/10.1007/s00454-018-0014-0

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