Regular mappings between dimensions

Abstract

The notions of Lipschitz and bilipschitz provide classes of connected to the geometry of metric spaces in certain ways. A notion between these two is given by 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 called $(s,t)$-regular mappings. These are the same as ordinary regular 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 between Euclidean spaces, and show in particular that if $f\colon {\mathbf R}^s\to {\mathbf R}^n$ is an $(s,t)$-regular mapping, then for each ball $B$ in ${\mathbf R}^s$ there is a linear mapping $\lambda \colon {\mathbf R}^s\to {\mathbf R}^{s-t}$ and a subset $E$ of $B$ of substantial measure such that the pair $(f,\lambda )$ is bilipschitz on $E$. We also compare these in comparison with nonlinear quotient mappings from [6].