### Impacto

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

PDF
Restringido a Repository staff only hasta 31 December 2020. 333kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0362546X11001106

## 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 |

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 |

References: | D. Azagra, J. Ferrera, F. López-Mesas, Y. Rangel. Smooth approximation of Lipschitz functions on Riemannian manifolds. J. Math. Anal. Appl., 326 (2007), pp. 1370–1378 D. Azagra, J. Ferrera, F. López-Mesas. Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal., 220 (2005), pp. 304–361 I. Garrido, J.A. Jaramillo, Y.C. Rangel. Algebras of differentiable functions on Riemannian manifolds. Bull. Lond. Math. Soc., 41 (2009), pp. 993–1001 J.M. Lasry, P.L. Lions. A remark on regularization in Hilbert spaces. Israel J. Math., 55 (3) (1986), pp. 257–266 R. Fry. Approximation by functions with bounded derivative on Banach spaces. Bull. Aust. Math. Soc., 69 (2004), pp. 125–131 D. Azagra, R. Fry, A. Montesinos. Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces. Proc. Amer. Math. Soc., 133 (2005), pp. 727–734 P. Hájek, M. Johanis. Smooth approximations. J. Funct. Anal., 259 (2010), pp. 561–582 D. Azagra, R. Fry, L. Keener. Smooth extension of functions on separable Banach spaces. Math. Ann., 347 (2) (2010), pp. 285–297 P. Hájek, M. Johanis.Uniformly Gâteaux smooth approximation on c0(Γ). J. Math. Anal. Appl., 350 (2009), pp. 623–629 R.S. Palais. Lusternik–Schnirelman theory on Banach manifolds. Topology, 5 (1966), pp. 115–132 K.H. Neeb. A Cartan–Hadamard theorem for Banach–Finsler manifolds. Geom. Dedicata, 95 (2002), pp. 115–156 H. Upmeier. Symmetric Banach Manifolds and Jordan C∗-Algebras, North-Holland Math. Stud., vol. 104 (1985) M.E.Rudin.A new proof that metric spaces are paracompact. Proc. Amer. Math. Soc., 20 (2) (1969), p. 603 M. Jiménez-Sevilla, L. Sánchez-González. Smooth extension of functions on a certai. class of non-separable Banach spaces. J. Math. Anal. Appl., 378 (2011), pp. 173–183 Y.C. Rangel, Algebras de funciones diferenciables en variedades, Ph.D. Dissertation (Departmento de Analisis Matematico, Facultad de Matematicas, Universidad Complutense de Madrid), 2008. R. Deville, G. Godefroy, V. Zizler. Smoothness and Renormings in Banach Spaces, Pitman Monographies and Surveys in Pure and Applied Mathematics, vol. 64 (1993) M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, V. Zizler. Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Math., vol. 8, Springer-Verlag, New York (2001) S. Lang.Fundamentals of Differential Geometry,GTM,vol. 191, Springer-Verlag, New York (1999) K. Deimling. Nonlinear Functional Analysis. Springer-Verlang, New York (1985) P.J. Rabier. Ehresmann fibrations and Palais–Smale conditions for morphisms of Finsler manifolds. Ann. of Math., 146 (1997), pp. 647–691 I. Garrido, O. Gutú, J.A. Jaramillo. Global inversion and covering maps on length spaces. Nonlinear Anal., 73 (5) (2010), pp. 1364–1374 D.Azagra,R.Fry,J.Gómez Gil,J.A.Jaramillo,M.Lovo C1-fine approximation of functions on Banach spaces with unconditional bases. Quart. J. Math. Oxford ser.,56 (2005), pp. 13–20 N. Moulis. Approximation de fonctions différentiables sur certains espaces de Banach. Ann. Inst. Fourier (Grenoble), 21 (1971), pp. 293–345 S.B. Myers. Algebras of differentiable functions. Proc. Amer. Math. Soc., 5 (1954), pp. 917–922 M. Nakai. Algebras of some differentiable functions on Riemannian manifolds. Jpn. J. Math., 29 (1959), pp. 60–67 |

Deposited On: | 12 Sep 2012 11:13 |

Last Modified: | 24 Jun 2013 16:43 |

Repository Staff Only: item control page