## Analysis & PDE

• Anal. PDE
• Volume 12, Number 5 (2019), 1149-1175.

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

#### Abstract

We say that a mapping $v:ℝn→ℝd$ satisfies the $(τ,σ)$-$N$-property if $ℋσ(v(E))=0$ whenever $ℋτ(E)=0$, where $ℋτ$ means the Hausdorff measure. We prove that every mapping $v$ of Sobolev class $Wpk(ℝn,ℝd)$ with $kp>n$ satisfies the $(τ,σ)$-$N$-property for every $0<τ≠τ∗:=n−(k−1)p$ with

We prove also that for $k>1$ and for the critical value $τ=τ∗$ the corresponding $(τ,σ)$-$N$-property fails in general. Nevertheless, this $(τ,σ)$-$N$-property holds for $τ=τ∗$ if we assume in addition that the highest derivatives $∇kv$ belong to the Lorentz space $Lp,1(ℝn)$ instead of $Lp$.

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.

#### Article information

Source
Anal. PDE, Volume 12, Number 5 (2019), 1149-1175.

Dates
Revised: 12 July 2018
Accepted: 12 August 2018
First available in Project Euclid: 5 January 2019

https://projecteuclid.org/euclid.apde/1546657228

Digital Object Identifier
doi:10.2140/apde.2019.12.1149

Mathematical Reviews number (MathSciNet)
MR3892399

Zentralblatt MATH identifier
07006757

#### Citation

Ferone, Adele; Korobkov, Mikhail V.; Roviello, Alba. On the Luzin $N$-property and the uncertainty principle for Sobolev mappings. Anal. PDE 12 (2019), no. 5, 1149--1175. doi:10.2140/apde.2019.12.1149. https://projecteuclid.org/euclid.apde/1546657228

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