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

Ferone, Adele; Korobkov, Mikhail V.; Roviello, Alba

doi:10.2140/apde.2019.12.1149

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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

DOI: 10.2140/apde.2019.12.1149

Abstract

We say that a mapping $v : ℝ^{n} \to ℝ^{d}$ satisfies the $(τ, σ)$ - $N$ -property if $ℋ^{σ} (v (E)) = 0$ whenever $ℋ^{τ} (E) = 0$ , where $ℋ^{τ}$ means the Hausdorff measure. We prove that every mapping $v$ of Sobolev class $W_{p}^{k} (ℝ^{n}, ℝ^{d})$ with $k p > n$ satisfies the $(τ, σ)$ - $N$ -property for every $0 < τ \neq τ_{*} : = n - (k - 1) p$ with

σ = σ (τ) : = \{\begin{matrix} τ & if τ > τ_{*}, \\ p τ ∕ (k p - n + τ) & if 0 < τ < τ_{*} . \end{matrix}

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 $\nabla^{k} v$ belong to the Lorentz space $L_{p, 1} (ℝ^{n})$ 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

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

Vol. 12, No. 5, 2019