Distortion of Hausdorff measures under Orlicz--Sobolev maps

TL;DR

This paper develops a comprehensive theory on how Orlicz-Sobolev maps distort Hausdorff measures, generalizing classical results and revealing new phenomena related to measure and integrability flexibility.

Contribution

It introduces an explicit formula for gauge function distortion under Orlicz-Sobolev maps, extending classical Sobolev-Hausdorff measure results to more general settings.

Findings

Derived explicit distortion formula for gauge functions

Reproduced and improved classical Sobolev-Hausdorff results

Discovered new phenomena related to measure and integrability flexibility

Abstract

A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved.