Higher order Lipschitz Sandwich theorems

Abstract

We investigate the consequence of two $\mathrm{Lip}\left(\gamma \right)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0>\epsilon >0$ and $\gamma >\eta >0$, there is a constant $\delta =\delta (\gamma ,\eta ,\epsilon ,K_0)>0$ for which the following is true. Let $\Sigma \subset R^d$ be closed and $f,h:\Sigma \to R$ be $\mathrm{Lip}\left(\gamma \right)$ functions whose $\mathrm{Lip}\left(\gamma \right)$ norms are both bounded above by $K_0$. Suppose $B\subset \Sigma$ is closed and that $f$ and $h$ coincide throughout $B$. Then, over the set of points in $\Sigma$ whose distance to $B$ is at most $\delta$, we have that the $\mathrm{Lip}\left(\eta \right)$ norm of the difference $f-h$ is bounded above by $\epsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two $\mathrm{Lip}\left(\gamma \right)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $\Sigma$ be closed; in particular, the case that $\Sigma$ is finite is covered by our results. The restriction that is sharp in the sense that our result is false for $\eta :=\gamma$.