Luzin's Condition (N) and Modulus of Continuity

TL;DR

This paper investigates when mappings in specific Sobolev-Orlicz spaces satisfy Luzin's condition (N), providing bounds on exceptional sets and analyzing the implications of violations, including mappings of measure-zero sets to positive measure.

Contribution

It establishes Luzin's condition (N) for Sobolev-Orlicz space mappings with certain moduli of continuity and bounds the size of exceptional sets where the condition fails.

Findings

Luzin's condition (N) holds under specific Sobolev-Orlicz space conditions.

Exceptional sets where (N) fails are quantitatively bounded.

Mappings violating (N) can map measure-zero sets to positive measure.

Abstract

In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.