Weighted Sobolev Space Estimates for a Class of Singular Integral Operators

Part of the International Mathematical Series book series (IMAT, volume 13)


The aim of this paper is to prove the boundedness of a category of integral operators mapping functions from Besov spaces on the boundary of a Lipschitz domain \(\Omega \subseteq \mathbb{R}^n \) into functions belonging to weighted Sobolev spaces in Ω. The model we have in mind is the Poisson integral operator
$$(PIf)(x): = - \int_{\partial \Omega } {\partial _{v(y)} G(x,y)f(y)d\sigma (y),} {\rm }x \in \Omega ,$$
where G(·; ·) is the Green function for the Dirichlet Laplacian in Ω, \(\partial _v \) is the normal derivative, and σ is the surface area on \(\partial \Omega \), in the case where \(\Omega \subseteq \mathbb{R}^n \) is a bounded Lipschitz domain satisfying a uniform exterior ball condition.


Integral Operator Green Function Besov Space Lipschitz Domain Singular Integral Operator 
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  1. 1.
    Adolfsson, V., Pipher, J.: The inhomogeneous Dirichlet problem for Δ 2 in Lipschitz domains. J. Funct. Anal. 159, no. 1, 137–190 (1998)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Calderón, A., Torchinsky, A.: Parabolic maximal functions associated with a distribution II. Adv. Math. 24, 101–171 (1977)MATHCrossRefGoogle Scholar
  3. 3.
    Dahlberg, B.: Estimates of harmonic measure. Arch. Rat. Mech. Anal. 65, 275–288 (1977)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dahlberg, B.E.J. : L q-estimates for Green potentials in Lipschitz domains. Math. Scand. 44, no. 1, 149–170 (1979)MATHMathSciNetGoogle Scholar
  5. 5.
    Fabes, E., Mendez, O., Mitrea, M.: Boundary layers on Sobolev–Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, no. 2, 323–368 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1985)MATHGoogle Scholar
  7. 7.
    Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37, no. 3, 303–342 (1982)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jerison, D., Kenig, C.E.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46, no. 1, 80–147 (1982)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, no. 1, 161–219 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kalton, N., Mayboroda, S., Mitrea, M.: Interpolation of Hardy–Sobolev– Besov–Triebel–Lizorkin spaces and applications to problems in partial differential equations. Contemp. Math. 445, 121–177 (2007)MathSciNetGoogle Scholar
  11. 11.
    Kenig, C.E.: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. Am. Math. Soc., Providence, RI (1994)MATHGoogle Scholar
  12. 12.
    Kozlov, V.A., Maz'ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Am. Math. Soc., Providence, RI (1997)MATHGoogle Scholar
  13. 13.
    Mayboroda, S., Mitrea, M.: Sharp estimates for Green potentials on nonsmooth domains. Math. Res. Lett. 11, 481–492 (2004)MATHMathSciNetGoogle Scholar
  14. 14.
    Mayboroda, S., Mitrea, M.: The solution of the Chang–Krein–Stein conjecture. In: Proc. Conf. Harmonic Analysis and its Applications (March 24–26, 2007), pp. 61–154. Tokyo Woman's Cristian University, Tokyo (2007)Google Scholar
  15. 15.
    Maz'ya, V.G.: Solvability in W2 2 of the Dirichlet problem in a region with a smooth irregular boundary (Russian). Vestn. Leningr. Univ. 22, no. 7, 87–95 (1967)Google Scholar
  16. 16.
    Maz'ya, V.G.: The coercivity of the Dirichlet problem in a domain with irregular boundary (Russian). Izv. VUZ, Ser. Mat. no. 4, 64–76 (1973)Google Scholar
  17. 17.
    Maz'ya, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions. Pitman, Boston etc. (1985) Russian edition: Leningrad. Univ. Press, Leningrad (1986)Google Scholar
  18. 18.
    Maz'ya, V., Mitrea, M., Shaposhnikova, T.: The Dirichlet Problem in Lipschitz Domains with Boundary Data in Besov Spaces for Higher Order Elliptic Systems with Rough Coefficients. Preprint (2008)Google Scholar
  19. 19.
    Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem. J. Funct. Anal. 176, no. 1, 1–79 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators. de Gruyter, Berlin–New York (1996)MATHGoogle Scholar
  21. 21.
    Rychkov, V.: On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. (2) 60, no. 1, 237–257 (1999)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, N.J. (1970)MATHGoogle Scholar
  23. 23.
    Triebel, H.: Theory of Function Spaces. Birkhäuser, Berlin (1983)Google Scholar
  24. 24.
    Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of Missouri at ColumbiaColumbiaUSA
  2. 2.LATP - UMR 6632 Faculté des Sciences, de Saint-Jérôme - Case Cour A Université Aix-Marseille 3Marseille Cédex 20France

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