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].
@article{urn:eudml:doc:41404,
title = {Regular mappings between dimensions},
journal = {Publicacions Matem\`atiques},
volume = {44},
year = {2000},
pages = {369-417},
mrnumber = {MR1800814},
zbl = {1041.42010},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41404}
}
David, Guy; Semmes, Stephen. Regular mappings between dimensions. Publicacions Matemàtiques, Tome 44 (2000) pp. 369-417. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41404/