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

摘要

Let us consider a Riemannian manifold M (either separable or non-separable). We prove that, for every ε>0, every Lipschitz function f:M→? 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? 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→? can be uniformly approximated by a Lipschitz, C ~κ-smooth function g with Lip(g)≤CLip(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? Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the C1 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.