Mathematical Notes

, Volume 68, Issue 2, pp 201–216 | Cite as

Tangential boundary behavior of functions of several variables

  • V. G. Krotov


In this paper we study the asymptotic behavior of functions defined on domains of a multidimensional real or complex space when the point tends to the boundary in the approach region with different orders of tangency. The main results are related to the boundary behavior of functions from Hardy-Sobolev spaces in a multidimensional complex ball and of solutions to elliptic boundary-value problems in a Lipschitz domain of a real Euclidean space. The methods used are based on two-weighted estimates for tangential maximal functions in an abstract ball. The boundary of this ball is a space equipped with measure and quasimetric.

Key words

tangential boundary behavior Hardy-Sobolev space elliptic boundary-value problem Dirichlet problem two-weighted estimate 


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  1. 1.
    A. Nagel, W. Rudin, and J. Shapiro, “Tangential boundary behavior of functions in Dirichlet-type spaces,”Ann. of Math.,116, No. 2, 331–360 (1982).CrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Nagel and E. M. Stein, “On certain maximal functions and approach regions,”Adv. Math.,54, No. 1, 83–106 (1984).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    P. Ahern and A. Nagel, “StrongL p-estimates for maximal functions with respect to singular measure with applications to exceptional sets,”Duke Math. J.,53, No. 2, 359–393 (1986).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    V. G. Krotov, “Estimates for maximal operators associated with boundary behavior, and their applications,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],190, 117–138 (1989).MathSciNetGoogle Scholar
  5. 5.
    V. G. Krotov, “Boundary behavior of functions from Hardy-type spaces,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv] 54, No. 1, 957–974 (1990).MATHGoogle Scholar
  6. 6.
    J. Sueiro, “Tangential boundary limits and exceptional sets for harmonic functions in Dirichlet-type spaces,”J. Math. Ann.,286, No. 4, 661–678 (1990).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Cifuentes, J. Dorronsoro, and J. Sueiro, “Boundary tangential convergence in spaces of homogeneous type,”Trans. Amer. Math. Soc.,332, No. 1, 331–350 (1992).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C. Cascante and J. M. Ortega, “Tangential-exceptional sets for Hardy-Sobolev spaces,”Illinois J. Math.,39, No. 1, 68–85 (1995).MATHMathSciNetGoogle Scholar
  9. 9.
    G. Verhota, “The Dirichlet problem for the polyharmonic equation in Liphschitz domains,”Indiana Univ. Math. J.,39, No. 3, 671–702 (1990).CrossRefMathSciNetGoogle Scholar
  10. 10.
    V. G. Krotov, “Weighted inequalities related to the boundary behavior of functions from multidimensional Hardy-Sobolev classes and of the solutions of elliptic boundary-value problems” in:Boundary-Value Problems, Special Functions and Fractional Calculus. Proc. of the Internat. Conference Dedicated to the 90th Anniversary of Academician F. D. Gakhov [in Russian], 16–20 February, 1996, Minsk (1996), pp. 172–177.Google Scholar
  11. 11.
    R. R. Coifman and G. Weiss, “Extensions of Hardy spaces and their use in analysis,”Bull. Amer. Math. Soc.,83, No. 4, 569–645 (1977).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    W. Rudin, Function Theory in the Unit Ball of ℂn, Springer-Verlag, Heidelberg (1981).Google Scholar
  13. 13.
    V. G. Krotov, “The differential properties on the boundary of functions holomorphic in the unit ball of ℂn,”Mat. Zametki [Math. Notes],45, No. 2, 51–59 (1989).MATHMathSciNetGoogle Scholar
  14. 14.
    J. Pipher and G. C. Verhota, “Dilatation invariant estimates and the boundary Gårding inequality for the higher order elliptic operators,”Ann. of Math.,142, No. 1, 1–36 (1995).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    P. Ahern and W. Cohn, “Exceptional sets for Hardy-Sobolev functions,”Indiana Univ. Math. J.,38, No. 2, 417–452 (1989).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • V. G. Krotov
    • 1
  1. 1.Belarus State UniversityUSSR

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