Riesz transforms for bounded Laplacians on graphs

  • Li Chen
  • Thierry Coulhon
  • Bobo HuaEmail author


We study several problems related to the \(\ell ^p\) boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of the gradient of a function, we prove for \(p\in (1,2]\) an \(\ell ^p\) estimate for the gradient of the continuous time heat semigroup, an \(\ell ^p\) interpolation inequality as well as the \(\ell ^p\) boundedness of the modified Littlewood–Paley–Stein function for a graph with bounded Laplacian. This yields an analogue to Dungey’s results in [21] while removing some additional assumptions. Coming back to the classical notion of the gradient, we give a counterexample to the interpolation inequality and hence to the boundedness of Riesz transforms for bounded Laplacians for \(1<p<2\). Finally, we prove the boundedness of the Riesz transform for \(1< p<\infty \) under the assumption of positive spectral gap.



  1. 1.
    Auscher, P., Coulhon, T.: Riesz transform on manifolds and Poincaré inequalities. Ann. ScI. Norm. Super. Pisa Cl. Sci. (5) 4(3), 531–555 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. (4) 37(6), 911–957 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bakry, D.: Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée (French) [A study of Riesz transforms in Riemannian manifolds with minorized Ricci curvature]. In: Séminaire de Probabilités, XXI, vol. 1247 of Lecture Notes in Math., pp. 137–172. Springer, Berlin (1987)Google Scholar
  4. 4.
    Bañuelos, R., Bogdan, K., Luks, T.: Hardy-Stein identities and square functions for semigroups. J. Lond. Math. Soc. 94(2), 462–478 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators, vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham (2014)Google Scholar
  6. 6.
    Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., Yau, S.-T.: Li-Yau inequality on graphs. J. Differ. Geom. 99(3), 359–405 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Badr, N., Russ, E.: Interpolation of Sobolev spaces, Littlewood–Paley inequalities and Riesz transforms on graphs. Publ. Mat. 53(2), 273–328 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Carron, G.: Riesz transforms on connected sums. Ann. Inst. Fourier (Grenoble) 57(7), 2329–2343 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, L.: Sub-Gaussian heat kernel estimates and quasi Riesz transforms for \(1\le p\le 2\). Publ. Mat. 59(2), 313–338 (2015)MathSciNetGoogle Scholar
  10. 10.
    Chen, L., Coulhon, T., Feneuil, J., Russ, E.: Riesz transform for \(1 \le p \le 2\) without gaussian heat kernel bound. J. Geom. Anal. 27(2), 1489–1514 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Coulhon, T., Duong, X.T.: Riesz transforms for \(1\le p\le 2\). Trans. Am. Math. Soc. 351(3), 1151–1169 (1999)zbMATHGoogle Scholar
  12. 12.
    Coulhon, T., Duong, X.T.: Riesz transform and related inequalities on noncompact Riemannian manifolds. Commun. Pure Appl. Math. 56(12), 1728–1751 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Coulhon, T., Duong, X.T., Li, X.D.: Littlewood–Paley–Stein functions on complete Riemannian manifolds for \(1\le p\le 2\). Studia Math. 154(1), 37–57 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Coulhon, T., Grigor’yan, A., Zucca, F.: The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 57(4), 559–587 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Coulhon, T., Sikora, A.: Riesz meets Sobolev. Colloq. Math. 118(2), 685–704 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Davies, E.B.: Heat Kernels and Spectral Theory, Vol. 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989)Google Scholar
  17. 17.
    Davies, E.B.: Large deviations for heat kernels on graphs. J. Lond. Math. Soc. s2–47(1), 65–72 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Davies, E.B.: Linear Operators and Their Spectra, Vol. 106 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2007)Google Scholar
  19. 19.
    Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion, Contemp. Math., vol. 73, pp. 25–40. American Mathematical Society, Ithaca, NY, Providence, RI (1988)Google Scholar
  20. 20.
    Dungey, N.: Riesz transforms on a discrete group of polynomial growth. Bull. Lond. Math. Soc. 36(6), 833–840 (2004)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dungey, N.: A Littlewood-Paley-Stein estimate on graphs and groups. Studia Math. 189(2), 113–129 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Grigor’yan, A.: Heat kernel and analysis on manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, R.I. (2009)Google Scholar
  23. 23.
    Hua, B., Jost, J.: \(L^q\) harmonic functions on graphs. Isr. J. Math. 202(1), 475–490 (2014)zbMATHGoogle Scholar
  24. 24.
    Hua, B., Keller, M.: Harmonic functions of general graph Laplacians. Calc. Var. Part. Differ. Equations 51(1–2), 343–362 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ji, L., Kunstmann, P., Weber, A.: Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap. Bull. Sci. Math. 134(1), 37–43 (2010)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Komatsu, H.: Fractional powers of operators. Pac. J. Math. 19, 285–346 (1966)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Lohoué, N.: Estimation des fonctions de Littlewood–Paley–Stein sur les variétés riemanniennes à courbure non positive. Ann. Sci. École Norm. Sup. (4) 20(4), 505–544 (1987)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lin, Y., Yau, S.-T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett. 17(2), 343–356 (2010)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Morgan, F.: Manifolds with density. Not. Am. Math. Soc. 52(8), 853–858 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis, 2nd edn. Academic Press, New York (1980)zbMATHGoogle Scholar
  32. 32.
    Russ, E.: Riesz transforms on graphs for \(1\le p\le 2\). Math. Scand. 87(1), 133–160 (2000)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Russ, E.: \(H^1\)-\(L^1\) boundedness of Riesz transforms on Riemannian manifolds and on graphs. Potential Anal. 14(3), 301–330 (2001)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood–Paley Theory. Annals of Mathematics Studies, no. 63. Princeton University Press, Princeton, University of Tokyo Press, Tokyo (1970)Google Scholar
  35. 35.
    Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Wei, G., Wylie, W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Differ. Geom. 83(2), 277–405 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Woess, W.: Random Walks on Infinite Graphs and Groups, Vol. 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Université de Cergy-PontoiseCergyFrance
  3. 3.School of Mathematical SciencesLMNS, Fudan UniversityShanghaiChina
  4. 4.Shanghai Center for Mathematical SciencesFudan UniversityShanghaiChina

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