Whitney extension operators without loss of derivatives

TL;DR

This paper characterizes when a linear extension operator for Whitney jets exists without loss of derivatives, using a simple geometric condition related to the set being sufficiently large in all directions.

Contribution

It provides a new geometric criterion for the existence of optimal Whitney extension operators without derivative loss.

Findings

Characterizes existence of extension operators via geometric condition

Provides a simple criterion based on set size in all directions

Establishes optimal continuity estimates for extensions

Abstract

For a compact set, we characterize the existence of a linear extension operator E for the space of Whitney jets without loss of derivatives, that is, E satisfies the best possible continuity estimates: The supremum of all partial derivatives up to order n of E(f) is less or equal than a constant times the n-th Whitney norm of f. The characterization is a surprisingly simple purely geometric condition telling in a way that at all its points, the set is big enough in all directions.