Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

TL;DR

This paper studies how the integrability properties of derivatives in Orlicz-Sobolev mappings influence the Hausdorff dimension of image sets in Euclidean spaces, using generalized Hausdorff measures.

Contribution

It extends the understanding of dimension distortion under Sobolev mappings by analyzing the effects of Orlicz space integrability conditions.

Findings

Derivatives' integrability controls image set size

Generalized Hausdorff measures quantify distortion

Results apply to mappings with Orlicz-Sobolev regularity

Abstract

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of $\mathbb{R}^n$ affect the sizes of the images of sets of Hausdorff dimension less than $n$. We measure the sizes of the image sets in terms of generalized Hausdorff measures.