Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces

Abstract

Abstract. We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have C p -smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩ U → R and every ε > 0, there exists a C p -smooth Lipschitz function F :If we are given a separable subspace Y of a Banach space X and a continuous (resp. Lipschitz) function f : Y → R, under what conditions can we ensure the existence of a C p -smooth (Lipschitz) perturbed extension of f ? That is, for a givenSuch an F will be called a smooth perturbed extension of f .Of course there are several conditions under which the answer is "yes" in a trivial way (for the non-Lipschitz case). For instance, when X has C p -smooth partitions of unity, and also when Y has a C p -smooth bump function and the subspace Y is complemented in X, such perturbed extensions F are easily proved to exist. However, in the Lipschitz case, or when the space X does not have smooth partitions of unity and Y is not complemented in X, it is not quite clear what the answer is.These questions are interesting in the theory of Banach spaces because, for instance, while trying to prove a theorem, one might be able to construct a continuous (resp. Lipschitz) function f with some nice properties, but defined only on a certain separable subspace Y of X, and then one might want to obtain a smooth (resp. and Lipschitz) function F defined on the whole of X that behaves on Y almost the same way as f does.In this paper we will try to give a solution to the above question in the Lipschitz case. This problem is clearly related to the question concerning uniform approximation of Lipschitz functions by smooth Lipschitz functions on infinite-dimensional spaces, which has remained unasked and open until recent times: in [7] R. Fry has shown that, on any separable Banach space with a C 1 -smooth bump function, uniformly continuous functions can be approximated by C 1 -smooth Lipschitz functions, uniformly on bounded sets.

On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach–Finsler manifolds

“…The notion of smooth sup-partitions of unity on Banach spaces was introduced by R. Fry [9] to solve the problem of approximation of real-valued, bounded and Lipschitz functions defined on a Banach space with separable dual by C 1 -smooth and Lipschitz functions. Subsequent generalizations of this result and related results were given in [3] and [13], by means of the existence of smooth sup-partitions of unity on the Banach space. This concept can be considered in the context of Finsler manifolds as well.…”

“…(2) Every Hilbert space H admits property ( * 1 ) (see [16]). Also, from the construction of the functions K(•) with inf-sup-convolution formulas, it can be easily checked that the constant C 0 can be taken as 1 for every Hilbertian norm || • || considered in H. (3) Every separable Banach space with a C k -smooth and Lipschitz bump function satisfies property ( * k ) (see [3], [5], [9] and [13]). Moreover, the constant C 0 can be obtained to be independent of the equivalent norm considered in X.…”

“…The first collorary is related to the concept of uniformly bumpable Banach manifold. Let us recall that the existence of smooth and Lipschitz bump functions on a Banach space is an essential tool to obtain approximation of Lipschitz functions by Lipschitz and smooth functions defined on Banach spaces (see [3,9,13]). A generalization of this concept to manifolds is the notion of uniformly bumpable manifold, which was introduced by Azagra, Ferrera and López-Mesas [1] for Riemannian manifolds.…”

“…R. Fry introduced "sup-partitions of unity", a key tool to answer positively to this problem for the case of bounded and Lipschitz functions defined on Banach spaces with separable dual [9]. Later on, D. Azagra, R. Fry and V. Montesinos extended this result in [3]. In particular, they obtained C k smoothness of the Lipschitz approximating functions whenever X is separable and admits a Lipschitz C k -smooth bump function.…”

“…In Section 5, we follow the ideas of [9], [3] and [13] to establish a characterization of the class of separable C Finsler manifolds M in the sense of Neeb-Upmeier which are C k -smooth uniformly bumpable (see Definition 4.1) as those having the property that every Lipschitz function f defined on M can be uniformly approximated by a C k -smooth and Lipschitz function g such that Lip(g) ≤ C Lip(f ) and C only depends on M .…”