Ultracontractivity and Functional Inequalities on Infinite Graphs

Yong Lin; Shuang Liu; Hongye Song

doi:10.1007/s00454-018-0014-0

Ultracontractivity and Functional Inequalities on Infinite Graphs

Yong Lin, Shuang Liu^*, Hongye Song

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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.

Original language	English
Pages (from-to)	198-211
Number of pages	14
Journal	Discrete and Computational Geometry
Volume	61
Issue number	1
DOIs	https://doi.org/10.1007/s00454-018-0014-0
Publication status	Published - 15 Jan 2019
Externally published	Yes

Keywords

CDE(n, K)
Laplacian
Sobolev-type inequalities
Ultracontractivity

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10.1007/s00454-018-0014-0

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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.",

keywords = "CDE(n, K), Laplacian, Sobolev-type inequalities, Ultracontractivity",

author = "Yong Lin and Shuang Liu and Hongye Song",

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T1 - Ultracontractivity and Functional Inequalities on Infinite Graphs

AU - Lin, Yong

AU - Liu, Shuang

AU - Song, Hongye

PY - 2019/1/15

Y1 - 2019/1/15

N2 - 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.

AB - 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.

KW - CDE(n, K)

KW - Laplacian

KW - Sobolev-type inequalities

KW - Ultracontractivity

UR - http://www.scopus.com/inward/record.url?scp=85048799471&partnerID=8YFLogxK

U2 - 10.1007/s00454-018-0014-0

DO - 10.1007/s00454-018-0014-0

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VL - 61

SP - 198

EP - 211

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -

Ultracontractivity and Functional Inequalities on Infinite Graphs

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Keywords

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