# Regular mappings between dimensions

Publicacions Matemàtiques (2000)

- Volume: 44, Issue: 2, page 369-417
- ISSN: 0214-1493

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topDavid, Guy, and Semmes, Stephen. "Regular mappings between dimensions." Publicacions Matemàtiques 44.2 (2000): 369-417. <http://eudml.org/doc/41404>.

@article{David2000,

abstract = {The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat like projections. In particular, they can map sets with Hausdorff dimension s to sets of Hausdorff dimension t. We mostly consider the case of mappings between Euclidean spaces, and show in particular that if f : Rs → Rn is an (s, t)-regular mapping, then for each ball B in Rs there is a linear mapping λ: Rs → Rs−t and a subset E of B of substantial measure such that the pair (f, λ) is bilipschitz on E. We also compare these mappings in comparison with “nonlinear quotient mappings” from [6].},

author = {David, Guy, Semmes, Stephen},

journal = {Publicacions Matemàtiques},

keywords = {Aplicación lipschitziana; Espacio topológico regular; Integrales singulares; regular mappings; Lipschitz functions; Lipschitz projections; Hausdorff dimension},

language = {eng},

number = {2},

pages = {369-417},

title = {Regular mappings between dimensions},

url = {http://eudml.org/doc/41404},

volume = {44},

year = {2000},

}

TY - JOUR

AU - David, Guy

AU - Semmes, Stephen

TI - Regular mappings between dimensions

JO - Publicacions Matemàtiques

PY - 2000

VL - 44

IS - 2

SP - 369

EP - 417

AB - The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat like projections. In particular, they can map sets with Hausdorff dimension s to sets of Hausdorff dimension t. We mostly consider the case of mappings between Euclidean spaces, and show in particular that if f : Rs → Rn is an (s, t)-regular mapping, then for each ball B in Rs there is a linear mapping λ: Rs → Rs−t and a subset E of B of substantial measure such that the pair (f, λ) is bilipschitz on E. We also compare these mappings in comparison with “nonlinear quotient mappings” from [6].

LA - eng

KW - Aplicación lipschitziana; Espacio topológico regular; Integrales singulares; regular mappings; Lipschitz functions; Lipschitz projections; Hausdorff dimension

UR - http://eudml.org/doc/41404

ER -

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