Two-Weighted Intergal Inequalities for Differential Forms

  • Xiuyin, Shang (College of Science, Agricultural University of Hebei) ;
  • Zhihua, Gu (College of Science, Agricultural University of Hebei) ;
  • Zengbo, Zhang (College of Science, Agricultural University of Hebei)
  • Received : 2006.01.06
  • Accepted : 2008.05.14
  • Published : 2009.09.30


In this paper, we make use of the weight to obtain some two-weight integral inequalities which are generalizations of the Poincar$\'{e}$ inequality. These inequalities are extensions of classical results and can be used to study the integrability of differential forms and to estimate the integrals of differential forms. Finally, we give some applications of this results to quasiregular mappings.


differential form;Poincar$\'{e}$ inequality;A-harmonic equation;H$\"{o}$lder inequality


  1. J. M. Ball, Convexity Conditions and Existence Theorems in Nonlinear Elasticity, Arch. Rational Mech. Anal., 63(1977), 337-403.
  2. J. M. Ball, F. Murat, $W^{1,p}$-quasi-Convexity and Variational Problems for Multiple Integrals, J. Funct. Anal., 58(1984), 225-253.
  3. T. Iwaniec, p-Harmonic Tensors and Quasiregular Mappings, Ann. Math., 136(1992), 589-624.
  4. T. Iwaniec, G.Martin, Quasiregular Mappings in Even Dimensions, Acta Math., 170(1993), 29-81.
  5. C. A. Nolder, Hardy-Littlewood Theorems for A-Harmonic Tensors, Illinois J. Math. Anal. Appl., 43(1999), 613-631.
  6. B. Stroffolini, On Weakly A-Harmonic Tensors, Studia Math., 114(3)(1995), 289-301.
  7. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1970.
  8. J. Heinonen, T. Kilpelainen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993.
  9. S. Ding, L. Li, Weighted Norm Inequalities for Conjugate A-Harmonic Tensors, J. Math. Anal. Appl., 203(1996), 279-289.
  10. S. Ding, Estimates of Weighted Integrals for Differential Forms, J. Math. Anal. Appl., 256(2001), 312-323.
  11. S. Ding, P. Shi, Weighted Poincare-Type Inequalities for Differential Forms in $L^{s}(\mu)$-Averaging Domains, J. Math. Anal. Appl., 227(1998), 200-215.
  12. S. Ding, D. Sylvester, Weak Reverse Holder Inequalities and Imbedding Inequalities for Solutions to the A-Harmonic Equation, Non. Anal., 51(2002), 783-800.
  13. S. Ding, Weighted Integral Inequalities for Solutions to the A-Harmonic Equations, to appear.
  14. B. Bojarski, T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in Rn, Ann. Acad. Sci. Fenn. Ser. AI Math, 8(1983), 257-324.