Skip to main content
SpringerLink
Log in

Estimates of the Green function for X-elliptic operators

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this note we establish two-sided pointwise estimates near the pole of the Green function for \(X\)-elliptic operators in divergence form enjoying the doubling condition and the Poincaré inequality. As a step towards this result, we also prove a nonhomogeneous Harnack inequality, some representation formulas and a result on approximation by regular domains.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bramanti, M., Brandolini, L., Lanconelli, E., Uguzzoni, F.: Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities. Mem. Amer. Math. Soc. 204(961), vi+123 (2010)

    MathSciNet  Google Scholar 

  2. Cancelier, C., Xu, C.J.: Remarques sur les fonctions de Green associées aux opérateurs de Hörmander. C. R. Acad. Sci. Paris Sér. I Math. 330, 433–436 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Di Fazio, G., Gutiérrez, C.E., Lanconelli, E.: Covering theorems, inequalities on metric spaces and applications to PDE’s. Math. Ann. 341, 255–291 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fabes, E., Jerison, D., Kenig, C.: The Wiener test for degenerate elliptic equations. Ann. Inst. Fourier (Grenoble) 32, 151–182 (1982)

    Article  MathSciNet  Google Scholar 

  5. Franchi, B., Lanconelli, E.: Une métrique associée à une classe d’opérateurs elliptiques dégénérés. Conference on linear partial and pseudodifferential operators (Torino, 1982). Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 105–114 (1984)

  6. Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(10), 523–541 (1983)

    MathSciNet  Google Scholar 

  7. Franchi, B., Lanconelli, E.: An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality. Comm. Partial Differ. Equ. 9, 1237–1264 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Franchi, B., Serapioni, R.: Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. Un. Mat. Ital. B 7(11), 83–117 (1997)

    MathSciNet  Google Scholar 

  9. Franchi, B., Lu, G., Wheeden, R.L.: A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type. Internat. Math. Res. Notices 1, 1–14 (1996)

    Article  MathSciNet  Google Scholar 

  10. Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49, 1081–1144 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin (1983)

  12. Gutiérrez, C.E., Lanconelli, E.: Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for X-elliptic operators. Comm. Partial Differ. Equ. 28, 1833–1862 (2003)

    Article  MATH  Google Scholar 

  13. Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(688), x+101 (2000)

    MathSciNet  Google Scholar 

  14. Jin, Y.Y.: Hölder continuity for a class of X-elliptic equations with singular lower order term. Appl. Math. J. Chin. Univ. Ser. B 24, 56–64 (2009)

    Article  MATH  Google Scholar 

  15. Kogoj, A.E., Lanconelli, E.: Liouville theorem for X-elliptic operators. Nonlinear Anal. 70, 2974–2985 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lanconelli, E., Kogoj, A.E.: \(X\)-elliptic operators and \(X\)-control distances. Contributions in honor of the memory of Ennio De Giorgi. Ricerche Mat. 49, 223–243 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Lanconelli, E., Uguzzoni, F.: Potential analysis for a class of diffusion equations: a Gaussian bounds approach. J. Differ. Equ. 248, 2329–2367 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 3(17), 43–77 (1963)

    MathSciNet  Google Scholar 

  19. Mazzoni, G.: Green function for X-elliptic operators. Manuscr. Math. 115, 207–238 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Phillips, R.S., Sarason, L.: Elliptic-parabolic equations of the second order. J. Math. Mech. 17, 891–917 (1967)

    MathSciNet  Google Scholar 

  21. Zheng, S., Feng, Z.: Green functions for a class of nonlinear degenerate operators with X-ellipticity. Trans. Amer. Math. Soc. 364, 3627–3655 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has been originated from an idea of Ermanno Lanconelli. It is a pleasure to thank him and dedicate this paper to him on the occasion of his 70th birthday.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 , Bologna, Italy

    Francesco Uguzzoni

Authors
  1. Francesco Uguzzoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Uguzzoni.

Additional information

Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Uguzzoni, F. Estimates of the Green function for X-elliptic operators. Math. Ann. 361, 169–190 (2015). https://doi.org/10.1007/s00208-014-1072-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-014-1072-0

Mathematics Subject Classification (2010)

Search

Navigation