The norm of linear extension operators for $C^{m-1,1}({\mathbb{R}}^n)$

Abstract

Fix integers $m\ge 2$, $n\ge 1$. We prove the existence of a bounded linear extension operator for $C^{m-1,1}({\mathbb{R}}^n)$ with operator norm at most $\exp (\gamma D^k)$, where $D:=\left(\genfrac{}{}{0ex}{}{m+n-1}{n}\right)$ is the number of multiindices of length $n$ and order at most $m-1$, and $\gamma ,k>0$ are absolute constants (independent of $m,n,E$). Upper bounds on the norm of this operator are relevant to basic questions about fitting a smooth function to data. Our results improve on a previous construction of extension operators of norm at most $\exp (\gamma D^k{2}^D)$. Along the way, we establish a finiteness theorem for $C^{m-1,1}({\mathbb{R}}^n)$ with improved bounds on the involved constants.

arXiv comment88 pages. Updated version with new introduction