Extension criteria for homogeneous Sobolev spaces of functions of one variable

• Pavel Shvartsman ^[1]
  1. [1] Technion – Israel Institute of Technology

     Technion – Israel Institute of Technology

     Israel

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 1, 2021, págs. 361-414
• Idioma: inglés
• DOI: 10.4171/rmi/1210
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• Resumen

  • For each p>1 and each positive integer m, we give intrinsic characterizations of the restriction of the homogeneous Sobolev space Lmp(R) to an arbitrary closed subset E of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of Lmp(R) spaces in the following sense: For every p∈(1,∞], it provides almost optimal Lmp-extensions of functions defined on E. The operator norm of this extension operator is bounded by a constant depending only on m. This enables us to prove several constructive Lmp-extension criteria expressed in terms of m-th order divided differences of functions.