Jimenez Sevilla, Maria del Mar and Sánchez González, Luis (2011) On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler manifolds. Nonlinear Analysis: Theory, Methods and Applications, 74 (11). pp. 3487-3500. ISSN 0362-546X
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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, C1-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 Cℓ 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, Ck-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 Cℓ 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 C1-smooth extensions of every real-valued function fdefined on a submanifold N of M provided f is C1-smooth on N and Lipschitz with the metric induced.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Riemannian manifolds; Finsler manifolds; Geometry of Banach spaces; Smooth approximation of Lipschitz functions; Smooth extension of Lipschitz functions. |
| Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |
| ID Code: | 13814 |
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| Deposited On: | 10 Nov 2011 13:01 |
| Last Modified: | 10 Nov 2011 13:01 |
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