Higher Order Lipschitz Sandwich Theorems
Terry Lyons, Andrew D. McLeod
TL;DR
This work develops higher-order Lipschitz Sandwich theorems for Lip$(\gamma)$-maps on closed subsets of Banach spaces, showing that proximity on a $\delta_0$-cover subset propagates to Lip$(\eta)$-closeness on the ambient set for any $0<\eta<\gamma$. The approach combines Stein's extension framework with precise remainder estimates, nested embedding properties, and local-to-global patching arguments to yield both pointwise and full Lip$(\eta)$ control. A key consequence is cost-effective approximation: Lip$(\gamma)$-functions are determined up to $\varepsilon$ in Lip$(\eta)$ by finitely many samples, with explicit covering-number bounds. Collectively, the results extend Whitney/Stein-type extension theory to higher-order Lip spaces in Banach settings and provide quantitative tools for approximating rough-path-relevant regularities in practical contexts.
Abstract
We investigate the consequence of two Lip$(γ)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $γ> η> 0$ there is a constant $δ= δ(γ,η,\varepsilon,K_0) > 0$ for which the following is true. Let $Σ\subset \mathbb{R}^d$ be closed and $f , h : Σ\to \mathbb{R}$ be Lip$(γ)$ functions whose Lip$(γ)$ norms are both bounded above by $K_0$. Suppose $B \subset Σ$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $Σ$ whose distance to $B$ is at most $δ$ we have that the Lip$(η)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(γ)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $Σ$ be closed; in particular, the case that $Σ$ is finite is covered by our results. The restriction that $η< γ$ is sharp in the sense that our result is false for $η:= γ$.