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

Jiménez Sevilla, María del Mar and Sánchez González , Luis Francisco (2011) On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach–Finsler manifolds. Nonlinear Analysis: Theory, Methods & 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 . 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 (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 f defined on a submanifold N of M provided f is C1-smooth on N and Lipschitz with the metric induced by M.

Item Type:Article
Additional Information:The authors wish to thank Jesús Jaramillo for many helpful discussions. Supported in part by DGES (Spain) Project MTM2009-07848. L. Sánchez-González has also been supported by grant MEC AP2007-00868.
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:16311
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