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

Jacob Carruth ^[2] ; Abraham Frei-Pearson ^[2] ; Arie Israel ^[2] ; Bo'az Klartag ^[1]
1. [1] Weizmann Institute of Science
  
  Weizmann Institute of Science
  
  Israel
2. [2] University of Texas at Austi
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 7, 2020, págs. 1917-1956
Idioma: inglés
DOI: 10.4171/rmi/1186
Enlaces
- Texto completo
Resumen
- We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space Cm−1,1(Rn). 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.