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

TL;DR

This paper establishes the existence of a bounded linear extension operator for $C^{m-1,1}( ^n)$ with an explicit exponential bound on its norm, improving previous results and aiding smooth data fitting.

Contribution

It provides a new construction of extension operators with better norm bounds for $C^{m-1,1}( ^n)$, advancing understanding of smooth function extension.

Findings

Existence of extension operator with norm at most $ ext{exp}( ext{constant} imes D^k)$

Improved bounds on constants in finiteness theorems

Enhanced methods for fitting smooth functions to data

Abstract

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.