Regular mappings between dimensions

Guy David; Stephen Semmes

Publicacions Matemàtiques (2000)

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

Abstract

top
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].

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.