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

Abstract

Let us consider a Riemannian manifold M (either separable or non-separable). We prove that, for every ε > 0 , every Lipschitz function f : M → R can be uniformly approximated by a Lipschitz, C 1 -smooth function g with Lip ( g ) ≤ Lip ( f ) + ε . As a consequence, every Riemannian manifold is uniformly bumpable. These results extend to the non-separable setting those given in [1] for separable Riemannian manifolds. The results are presented in the context of C ℓ Finsler manifolds modeled on Banach spaces. Sufficient conditions are given on the Finsler manifold M (and the Banach space X where M is modeled), so that every Lipschitz function f : M → R can be uniformly approximated by a Lipschitz, C k -smooth function g with Lip ( g ) ≤ C Lip ( f ) (for some C depending only on X ). Some applications of these results are also given as well as a characterization, on the separable case, of the class of C ℓ Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the C 1 Finsler manifold M and X , to ensure the existence of Lipschitz and C 1 -smooth extensions of every real-valued function f defined on a submanifold N of M provided f is C 1 -smooth on N and Lipschitz with the metric induced by M .