Revistes Catalanes amb Accés Obert (RACO)

Regular mappings between dimensions

G. David, S. Semmes


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 $f\colon {\mathbf R}^s\to {\mathbf R}^n$ 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 mappings in comparison with "nonlinear quotient mappings" from [6].

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