Regular mappings between dimensions

David, 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 -