Luzin's condition (N) and modulus of continuity

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.

Keywords

• Luzin's condition (N)
• Sobolev mapping
• Sobolev-Orlicz space
• modulus of continuity