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Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group

Abstract

We study the boundary values of the functions of the Sobolev function spaces W l and the Nikol’skiĭ function spaces H l which are defined on an arbitrary domain of a Carnot group. We obtain some reversible characteristics of the traces of the spaces under consideration on the boundary of the domain of definition and sufficient conditions for extension of the functions of these spaces outside the domain of definition. In some cases these sufficient conditions are necessary.

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Dedicated to Yu. G. Reshetnyak on the occasion of his 7th birthday

Original Russian Text © S. K. Vodop’yanov and I. M. Pupyshev, 2006, published in Matematicheskie Trudy, 2006, Vol. 9, No. 2, pp. 23–46.

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Vodop’yanov, S.K., Pupyshev, I.M. Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group. Sib. Adv. Math. 17, 62–78 (2007). https://doi.org/10.3103/S105513440701004X

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Key words

  • Sobolev space
  • Nikol’skiĭ space
  • Carnot group
  • trace
  • boundary values
  • Whitney’s Theorem
  • extension of functions