Mityagin's Extension Problem. Progress Report

TL;DR

This paper investigates the extension property of Whitney jets on compact sets in Euclidean space, demonstrating that it cannot be fully characterized by geometric densities or growth conditions.

Contribution

It proves the non-existence of a complete geometric characterization of the extension property in terms of Hausdorff contents or Markov's factors.

Findings

No complete geometric description of the extension property exists.

Extension property cannot be characterized by densities of Hausdorff contents.

Growth of Markov's factors does not determine the extension property.

Abstract

Given a compact set $K\subset {\Bbb R}^d,$ let ${\mathcal E}(K)$ denote the space of Whitney jets on $K$. The compact set $K$ is said to have the extension property if there exists a continuous linear extension operator $W:{\mathcal E}(K) \longrightarrow C^{\infty}({\Bbb R}^d)$. In 1961 B. S. Mityagin posed a problem to give a characterization of the extension property in geometric terms. We show that there is no such complete description in terms of densities of Hausdorff contents or related characteristics. Also the extension property cannot be characterized in terms of growth of Markov's factors for the set.