# Some remarks about metric spaces, spherical mappings, functions and their derivatives.

Publicacions Matemàtiques (1996)

- Volume: 40, Issue: 2, page 411-430
- ISSN: 0214-1493

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topSemmes, Stephen. "Some remarks about metric spaces, spherical mappings, functions and their derivatives.." Publicacions Matemàtiques 40.2 (1996): 411-430. <http://eudml.org/doc/41266>.

@article{Semmes1996,

abstract = {If p ∈ Rn, then we have the radial projection map from Rn \{p\} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In this paper we shall discuss these topics mostly in the context of metric doubling measures, which provides a nontrivial setting in which these mappings exist and can be used. This provides an alternative approach (or understanding) of the results in [DS], and a variation on the themes of [Se4].},

author = {Semmes, Stephen},

journal = {Publicacions Matemàtiques},

keywords = {Espacios métricos; Análisis funcional; Funciones de variable compleja; Función derivada; Geometría euclídea; Desigualdades; Formas diferenciales; Medidas de Borel; spherical mapping; metric doubling measure; partition of unity; Sobolev inequality},

language = {eng},

number = {2},

pages = {411-430},

title = {Some remarks about metric spaces, spherical mappings, functions and their derivatives.},

url = {http://eudml.org/doc/41266},

volume = {40},

year = {1996},

}

TY - JOUR

AU - Semmes, Stephen

TI - Some remarks about metric spaces, spherical mappings, functions and their derivatives.

JO - Publicacions Matemàtiques

PY - 1996

VL - 40

IS - 2

SP - 411

EP - 430

AB - If p ∈ Rn, then we have the radial projection map from Rn {p} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In this paper we shall discuss these topics mostly in the context of metric doubling measures, which provides a nontrivial setting in which these mappings exist and can be used. This provides an alternative approach (or understanding) of the results in [DS], and a variation on the themes of [Se4].

LA - eng

KW - Espacios métricos; Análisis funcional; Funciones de variable compleja; Función derivada; Geometría euclídea; Desigualdades; Formas diferenciales; Medidas de Borel; spherical mapping; metric doubling measure; partition of unity; Sobolev inequality

UR - http://eudml.org/doc/41266

ER -

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