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Weighted Norm Inequalities on Graphs

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Abstract

Let (Γ,μ) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ∇. We assume that μ is doubling, a uniform lower bound for p(x,y) when p(x,y)>0, and gaussian upper estimates for the iterates of p. Under these conditions (and in some cases assuming further some Poincaré inequality) we study the comparability of (I−P)1/2 f and ∇f in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood–Paley–Stein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.

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References

  1. Auscher, P., Coulhon, T.: Riesz transforms on manifolds and Poincaré inequalities. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4(5), 1–25 (2005)

    MathSciNet  Google Scholar 

  2. Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators. J. Funct. Anal. 241(2), 703–746 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights. Adv. Math. 212(1), 225–276 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights. Math. Z. 260(3), 527–539 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transforms on manifolds and heat kernel regularity. Ann. Sci. Ecole Norm. Super. 37(6), 911–957 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Badr, N., Russ, E.: Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs. Publ. Mat. 53(2), 273–328 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Bernicot, F., Zhao, J.: Abstract Hardy spaces. J. Funct. Anal. 255(7), 1761–1796 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)

    MATH  Google Scholar 

  9. Coulhon, T., Grigor’yan, A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8, 656–701 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Delmotte, T.: Parabolic Harnack inequality. Rev. Mat. Iberoam. 15(1), 181–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Franchi, B., Pérez, C., Wheeden, R.L.: Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type. J. Funct. Anal. 153, 108–146 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145, 688 (2000)

    MathSciNet  Google Scholar 

  13. Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math. 167(2), 575–599 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Martell, J.M.: Desigualdades con pesos en el Análisis de Fourier: de los espacios de tipo homogéneo a las medidas no doblantes. Ph.D. Thesis, Universidad Autónoma de Madrid (2001)

  15. Russ, E.: Riesz transforms on graphs for 1≤p≤2. Math. Scand. 87, 133–160 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Russ, E.: H 1−L 1 boundedness of Riesz transforms on Riemannian manifolds and on graphs. Potential Anal. 14, 301–330 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Camille Jordan, Université de Lyon et CNRS, 43, boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Nadine Badr

  2. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, 28049, Madrid, Spain

    José María Martell

Authors
  1. Nadine Badr
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  2. José María Martell
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Corresponding author

Correspondence to José María Martell.

Additional information

Communicated by Loukas Grafakos.

The second author was supported by MICINN Grant MTM2010-16518, and by CSIC PIE 200850I015.

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Badr, N., Martell, J.M. Weighted Norm Inequalities on Graphs. J Geom Anal 22, 1173–1210 (2012). https://doi.org/10.1007/s12220-011-9233-9

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  • DOI: https://doi.org/10.1007/s12220-011-9233-9

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