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

Let us consider a Riemannian manifold MM (either separable or non-separable). We prove that, for every ε>0ε>0, every Lipschitz function f:M→Rf:M→R can be uniformly approximated by a Lipschitz, C1C1-smooth function gg 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 CℓCℓ Finsler manifolds modeled on Banach spaces. Sufficient conditions are given on the Finsler manifold MM (and the Banach space XX where MM is modeled), so that every Lipschitz function f:M→Rf:M→R can be uniformly approximated by a Lipschitz, CkCk-smooth function gg with Lip(g)≤CLip(f) (for some CC depending only on XX). Some applications of these results are also given as well as a characterization, on the separable case, of the class of CℓCℓ Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the C1C1 Finsler manifold MM and XX, to ensure the existence of Lipschitz and C1C1-smooth extensions of every real-valued function ff defined on a submanifold NN of MM provided ff is C1C1-smooth on NN and Lipschitz with the metric induced by MM.