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

Fix integers $m\ge 2$, $n\ge 1$. We prove the existence of a bounded linear extension operator for $C^{m-1,1}(\R^n)$ with operator norm at most $\exp(\gamma D^k)$, where $D := \binom{m+n-1}{n}$ 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 $C^{m-1,1}(\R^n)$ with improved bounds on the involved constants.