Dimension distortion by Sobolev mappings in foliated metric spaces

TL;DR

This paper investigates how supercritical Sobolev mappings can increase the dimension of subsets within metric measure spaces, especially in the context of foliations, providing quantitative estimates and demonstrating the sharpness of these results.

Contribution

It introduces quantitative estimates for dimension distortion by Sobolev mappings in foliated metric spaces, including sharpness and prevalence results, extending understanding beyond Euclidean spaces.

Findings

Quantitative bounds on dimension increase under Sobolev mappings.

Examples demonstrating sharpness of the bounds.

Prevalence of maximal dimension distortion mappings.

Abstract

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincar\'e inequality. For foliations of a metric space X defined by a David--Semmes regular mapping we quantitatively estimate, in terms of Hausdorff dimension, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups. Examples are given demonstrating the extent to which our results are sharp. In fact, we show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space; the latter result is new even in Euclidean space.