Lipschitz spaces generated by the Sobolev-Poincaré inequality and extensions of Sobolev functions

2013/10/02 by Pavel Shvartsman, Shvartsman, Pavel
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.FA

paper · pdf · doi:10.48550/arxiv.1310.0795

49 pages

arxiv created 2013/10/02 · openalex publication_date 2013/10/02 · arxiv updated 2013/10/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let d be a metric on Rⁿ and let C^m,(d)(Rⁿ) be the space of C^m-function on Rⁿ whose partial derivatives of order m belong to the space Lip(Rⁿ;d). We show that the homogeneous Sobolev space L^m+1_p(Rⁿ),p>n, can be represented as a union of C^m,(d)(Rⁿ)-spaces where d belongs to a family of metrics on Rⁿ with certain "nice" properties. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of Rⁿ. In particular, we generalize the classical Whitney extension theorem for the space C^m(Rⁿ) to the case of the Sobolev space L^m_p(Rⁿ) whenever m≥ 1 and p>n.

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