JournalsrmiVol. 37, No. 1pp. 361–414

Extension criteria for homogeneous Sobolev spaces of functions of one variable

  • Pavel Shvartsman

    Technion - Israel Institute of Technology, Haifa, Israel
Extension criteria for homogeneous Sobolev spaces of functions of one variable cover

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Abstract

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

Cite this article

Pavel Shvartsman, Extension criteria for homogeneous Sobolev spaces of functions of one variable. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 361–414

DOI 10.4171/RMI/1210