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Smooth approximation of Lipschitz functions on Finsler manifolds

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Garrido, M. Isabel and Jaramillo Aguado, Jesús Ángel and Rangel, Yenny C. (2013) Smooth approximation of Lipschitz functions on Finsler manifolds. Journal of function spaces and applications . ISSN 0972-6802

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Official URL: http://www.hindawi.com/journals/jfsa/2013/164571/abs/


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Abstract

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz function f : M -> R defined on a connected, second countable Finsler manifold M, for each positive continuous function epsilon : M -> (0, infinity) and each r > 0, there exists a C-1-smooth Lipschitz function g : M -> R such that vertical bar f(x) - g(x)vertical bar <= epsilon(x), for every x is an element of M, and Lip(g) <= Lip(f) + r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebra C-b(1)(M) of all C-1 functions with bounded derivative on a complete quasi-reversible Finsler manifold M, we obtain a characterization of algebra isomorphisms T : C-b(1)(N) -> C-b(1)(M) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.


Item Type:Article
Uncontrolled Keywords:Riemannian-manifolds; isometries
Subjects:Sciences > Mathematics > Differential geometry
ID Code:23180
Deposited On:15 Oct 2013 10:18
Last Modified:12 Dec 2018 15:12

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