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Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs
[Bornes supérieures gaussiennes, doublement du volume et inégalités de Sobolev sur les graphes]
Matthias Keller1 ; Christian Rose1
1Universität Potsdam, Institut für Mathematik, 14476 Potsdam, Germany
Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 965-1005

We investigate the equivalence of Sobolev inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain the equivalence up to constants. If arbitrary measures are considered, we incorporate a new local regularity condition. Furthermore, new correction functions for the Gaussian, doubling, and Sobolev dimension are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Sobolev dimension can be related to the doubling dimension and the vertex degree growth.

Nous étudions l’équivalence entre les inégalités de Sobolev et la conjonction des bornes supérieures gaussiennes du noyau de chaleur et du doublement de volume à grande échelle sur les graphes. Pour la mesure normalisante, nous obtenons l’équivalence à constante près. Pour des mesures arbitraires, nous incorporons une nouvelle condition de régularité locale. De plus, nous introduisons de nouvelles fonctions de correction pour les dimensions gaussienne, de doublement et de Sobolev. Pour les dimensions gaussienne et de doublement, les fonctions de correction variables tendent toujours vers $1$ à l’infini. Par ailleurs, la dimension de Sobolev variable peut être mise en relation avec la dimension de doublement et la croissance du degré des sommets.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.340
Classification : 39A12, 35K08, 60J74
Keywords: Graph, heat kernel, Sobolev, Gaussian bound, unbounded geometry
Mots-clés : Graphe, noyau de la chaleur, Sobolev, borne gaussienne, géométrie non bornée

Matthias Keller  1   ; Christian Rose  1

1 Universität Potsdam, Institut für Mathematik, 14476 Potsdam, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Matthias Keller; Christian Rose. Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 965-1005. doi: 10.5802/jep.340
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