Higher order Lipschitz Sandwich theorems

We investigate the consequence of two Lip(gamma) 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 of Rd be closed and f,h:Sigma -> R be Lip(gamma) functions whose Lip(gamma) norms are both bounded above by K-0. Suppose B subset of 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 Lip(eta) 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 Lip(gamma) 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 eta<gamma is sharp in the sense that our result is false for eta:=gamma.