## 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 Poincaré inequality. 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. For foliations of a metric space X defined by a David-Semmes regular mapping π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, 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.

Original language | English (US) |
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Pages (from-to) | 232-254 |

Number of pages | 23 |

Journal | Analysis and Geometry in Metric Spaces |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

## Keywords

- Ahlfors regularity
- David-semmes regular mapping
- Foliation
- Poincaré inequality
- Sobolev mapping

## ASJC Scopus subject areas

- Analysis
- Geometry and Topology
- Applied Mathematics