Distortion of Hausdorff measures under Orlicz–Sobolev maps

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 functions 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 provided in terms of the Young function defining the Orlicz–Sobolev space. 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 nonpower type, are also recovered and, when not sharp, improved. 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. For instance, perhaps surprisingly, Hausdorff measures, other than the Lebesgue measure, which are not distorted by classical Sobolev maps are exhibited. The relevant measures are associated with gauge functions with a logarithmic decay near zero, which cannot be attained by the standard theory.