#### 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.

##### 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