#### Vol. 12, No. 5, 2019

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On the Luzin $N\mskip-2mu$-property and the uncertainty principle for Sobolev mappings

### Adele Ferone, Mikhail V. Korobkov and Alba Roviello

Vol. 12 (2019), No. 5, 1149–1175
##### Abstract

We say that a mapping $v:{ℝ}^{n}\to {ℝ}^{d}$ satisfies the $\left(\tau ,\sigma \right)$-$N$-property if ${\mathsc{ℋ}}^{\sigma }\left(v\left(E\right)\right)=0$ whenever ${\mathsc{ℋ}}^{\tau }\left(E\right)=0$, where ${\mathsc{ℋ}}^{\tau }$ means the Hausdorff measure. We prove that every mapping $v$ of Sobolev class ${W}_{p}^{k}\left({ℝ}^{n},{ℝ}^{d}\right)$ with $kp>n$ satisfies the $\left(\tau ,\sigma \right)$-$N$-property for every $0<\tau \ne {\tau }_{\ast }:=n-\left(k-1\right)p$ with

We prove also that for $k>1$ and for the critical value $\tau ={\tau }_{\ast }$ the corresponding $\left(\tau ,\sigma \right)$-$N$-property fails in general. Nevertheless, this $\left(\tau ,\sigma \right)$-$N$-property holds for $\tau ={\tau }_{\ast }$ if we assume in addition that the highest derivatives ${\nabla }^{k}v$ belong to the Lorentz space ${L}_{p,1}\left({ℝ}^{n}\right)$ 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 $N$-Nproperties and discuss their applications to the Morse–Sard theorem and its recent extensions.

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##### Keywords
Sobolev–Lorentz mappings, fractional Sobolev classes, Luzin $N\mskip-2mu$-property, Morse–Sard theorem, Hausdorff measure
##### Mathematical Subject Classification 2010
Primary: 46E35, 58C25
Secondary: 26B35, 46E30
##### Milestones
Received: 26 June 2017
Revised: 12 July 2018
Accepted: 12 August 2018
Published: 15 December 2018
##### Authors
 Adele Ferone Dipartimento di Matematica e Fisica Università degli studi della Campania “Luigi Vanvitelli” Caserta Italy Mikhail V. Korobkov School of Mathematical Sciences Fudan University Shanghai China Novosibirsk State University Novosibirsk Russia Alba Roviello Dipartimento di Matematica e Fisica Università degli studi della Campania “Luigi Vanvitelli” Caserta Italy