Weighted Sobolev Space Estimates for a Class of Singular Integral Operators

  • Dorina Mitrea
  • Marius Mitrea
  • Sylvie Monniaux
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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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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