# On Luzin N-property and uncertainty principle for the Sobolev mappings

## TL;DR

This paper investigates the Luzin N-property for Sobolev mappings, identifying critical dimensions where it fails, providing counterexamples, and extending results to fractional Sobolev spaces with implications for the Morse--Sard theorem.

## Contribution

It characterizes the conditions under which the Luzin N-property holds for Sobolev spaces, including critical cases and extensions to fractional spaces, with new counterexamples and applications.

## Key findings

- N-property holds except at critical dimension t_*
- Counterexample constructed at the critical dimension
- Extensions to fractional Sobolev spaces and applications to Morse--Sard theorem

## Abstract

We study Luzin N-property with respect to the Hausdorff measures for Sobolev spaces W^k_p(R^n,R^d). We prove that such N-property holds except for one critical dimensional value t_*=n-(k-1)p; for this critical value the N-property fails in general, and we constructed the corresponding nontrivial counterexample (based on the theory of lacunary Fourier series). Nevertheless, this N-property holds if we assume in addition that the highest k-derivatives belongs to the Lorentz space L_{p,1} instead of L_p. We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini type theorems for $N$-properties and discuss their applications to the Morse--Sard theorem and its recent extensions.