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Poincaré-Type Inequalities for Green’s Operator on Harmonic Forms

Chapter

Abstract

In this paper, we prove the local Poincaré-type inequalities with singular factors for Green’s operator applied to harmonic forms and extend the local results into the global cases in John domains. We also establish the local Poincaré-type inequalities with Orlicz norms for Green’s operator on harmonic forms and global inequalities in \(L^{\varphi }(m)\)-averaging or averaging domains.

References

  1. 1.
    Agarwal, R.P., Ding, S., Nolder, C.A.: Inequalities for Differential Forms. Springer, New York (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Buckley, S.M., Koskela, P.: Orlicz-Hardy inequalities. Ill. J. Math. 48, 787–802 (2004)MATHMathSciNetGoogle Scholar
  3. 3.
    Cartan, H.: Differential Forms. Houghton Mifflin, Boston (1970)MATHGoogle Scholar
  4. 4.
    Chanillo, S., Wheeden, R.L.: Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Am. J. Math. 107, 1191–1226 (1985)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Ding, S.: Integral estimates for the Laplace-Beltrami and Green’s operators applied to differential forms. Zeitschrift fur Analysis und ihre Anwendungen (J. Anal. Appl.) 22, 939–957 (2003)Google Scholar
  6. 6.
    Ding, S.: Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds. Proc. Am. Math. Soc. 132, 2367–2375 (2004)CrossRefMATHGoogle Scholar
  7. 7.
    Ding, S.: \(L^{\varphi }(\mu )\)-averaging domains and the quasihyperbolic metric. Comput. Math. Appl. 47, 1611–1618 (2004)Google Scholar
  8. 8.
    Ding, S., Nolder, C.A.: Weighted Poincaré-type inequalities for solutions to the A-harmonic equation. Ill. J. Math. 2, 199–205 (2002)MathSciNetGoogle Scholar
  9. 9.
    Franchi, B., Wheeden, R.L.: Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type. J. Inequal. Appl. 3(1), 65–89 (1999)MATHMathSciNetGoogle Scholar
  10. 10.
    Franchi, B., Gutiérrez, C.E., Wheeden, R.L.: Weighted Sobolev-Poincaré inequalities for Grushin type operators. Commun. Partial Differ. Equ. 19, 523–604 (1994)CrossRefMATHGoogle Scholar
  11. 11.
    Franchi, B., Lu, G., Wheeden, R.L.: Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations. Potential theory and degenerate partial differential operators (Parma). Potential Anal. 4, 361–375 (1995)Google Scholar
  12. 12.
    Franchi, B., Pérez, C., Wheeden, R.L.: Sharp geometric Poincaré inequalities for vector fields and non-doubling measures. Proc. Lond. Math. Soc. 80, 665–689 (2000)CrossRefMATHGoogle Scholar
  13. 13.
    Iwaniec, T.: p-harmonic tensors and quasiregular mappings. Ann. Math. (2) 136(3), 589–624 (1992)Google Scholar
  14. 14.
    Iwaniec, T., Lutoborski, A.: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, 25–79 (1993)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Liu, B.: A r λ(Ω)-weighted imbedding inequalities for A-harmonic tensors. J. Math. Anal. Appl. 273(2), 667–676 (2002)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Liu, B.: L p-estimates for the solutions of A-harmonic equations and the related operators. In: The Proceedings of the 6th International Conference on Differential Equations and Dynamical Systems (2008)Google Scholar
  17. 17.
    Nolder, C.A.: Hardy-Littlewood theorems for A-harmonic tensors. Ill. J. Math. 43, 613–631 (1999)MATHMathSciNetGoogle Scholar
  18. 18.
    Scott, C.: L p-theory of differential forms on manifolds. Trans. Am. Math. Soc. 347, 2075–2096 (1995)MATHGoogle Scholar
  19. 19.
    Staples, S.G.: L p-averaging domains and the Poincaré inequality. Ann. Acad. Sci. Fenn. Ser. AI Math. 14, 103–127 (1989)MATHMathSciNetGoogle Scholar
  20. 20.
    Stroffolini, B.: On weakly A-harmonic tensors. Stud. Math. 114(3), 289–301 (1995)MATHMathSciNetGoogle Scholar
  21. 21.
    Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Springer, New York (1983)CrossRefMATHGoogle Scholar
  22. 22.
    Xing, Y.: Weighted integral inequalities for solutions of the A-harmonic equation. J. Math. Anal. Appl. 279, 350–363 (2003)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Xing, Y.: Two-weight imbedding inequalities for solutions to the A-harmonic equation. J. Math. Anal. Appl. 307, 555–564 (2005)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsSeattle UniversityWAUSA
  2. 2.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China

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