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2019 On the Luzin $N$-property and the uncertainty principle for Sobolev mappings
Adele Ferone, Mikhail V. Korobkov, Alba Roviello
Anal. PDE 12(5): 1149-1175 (2019). DOI: 10.2140/apde.2019.12.1149

Abstract

We say that a mapping v:nd satisfies the (τ,σ)-N-property if σ(v(E))=0 whenever τ(E)=0, where τ means the Hausdorff measure. We prove that every mapping v of Sobolev class Wpk(n,d) with kp>n satisfies the (τ,σ)-N-property for every 0<ττ:=n(k1)p with

σ = σ ( τ ) : = τ  if  τ > τ , p τ ( k p n + τ )  if  0 < τ < τ .

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(n) 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.

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Adele Ferone. Mikhail V. Korobkov. Alba Roviello. "On the Luzin $N$-property and the uncertainty principle for Sobolev mappings." Anal. PDE 12 (5) 1149 - 1175, 2019. https://doi.org/10.2140/apde.2019.12.1149

Information

Received: 26 June 2017; Revised: 12 July 2018; Accepted: 12 August 2018; Published: 2019
First available in Project Euclid: 5 January 2019

zbMATH: 07006757
MathSciNet: MR3892399
Digital Object Identifier: 10.2140/apde.2019.12.1149

Subjects:
Primary: 46E35, 58C25
Secondary: 26B35, 46E30

Rights: Copyright © 2019 Mathematical Sciences Publishers

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