Doklady Mathematics

, Volume 95, Issue 1, pp 79–83 | Cite as

On the Whitney problem for weighted Sobolev spaces

Mathematics

Abstract

Given a closed weakly regular d-thick subset S of ℝn, we prove the existence of a bounded linear extension operator Ext: Tr|SWp1 (ℝn, γ) → Wp1(ℝn, γ) for p ∈ (1, ∞), 0 ≤ dn, r ∈ (max{1, nd}, p), l ∈ ℕ, and \(\gamma \in {A_{\frac{p}{r}}}\)(ℝn). In particular, we prove that a linear bounded trace space exists in the case where S is the closure of an arbitrary domain in ℝn, γ ≡ 1, and p > n − 1. The obtained results supplement those of previous studies, in which a similar problem was considered either in the case of p ∈ (n, ∞) without constraints on the set S or in the case of p ∈ (1, ∞) under stronger constraints on the set S.

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia

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