JournalsrmiVol. 36, No. 7pp. 1917–1956

A coordinate-free proof of the finiteness principle for Whitney’s extension problem

  • Jacob Carruth

    University of Texas at Austin, USA
  • Abraham Frei-Pearson

    University of Texas at Austin, USA
  • Arie Israel

    University of Texas at Austin, USA
  • Bo'az Klartag

    The Weizmann Institute of Science, Rehovot, Israel
A coordinate-free proof of the finiteness principle for Whitney’s extension problem cover
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Abstract

We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space Cm1,1(Rn)C^{m−1,1}(\mathbb R^n). While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry. These techniques may be relevant to the study of Whitney-type extension problems on sub-Riemannian manifolds where global coordinates are generally unavailable.

Cite this article

Jacob Carruth, Abraham Frei-Pearson, Arie Israel, Bo'az Klartag, A coordinate-free proof of the finiteness principle for Whitney’s extension problem. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 1917–1956

DOI 10.4171/RMI/1186