Higher order Lipschitz Sandwich theorems - Lyons, T; McLeod, AD (2024)
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 K0 > ε > 0 and γ > η > 0 there is a constant δ = δ(γ, η, ε, K0) > 0 for which the following is true. Let Σ ⊂ R d be closed and f, h : Σ → R be Lip(γ) functions whose Lip(γ) norms are both bounded above by K0. Suppose B ⊂ 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 ε. 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.