Analysis & PDE

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

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

Adele Ferone, Mikhail V. Korobkov, and Alba Roviello

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 58C25: Differentiable maps
Secondary: 26B35: Special properties of functions of several variables, Hölder conditions, etc. 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Sobolev–Lorentz mappings fractional Sobolev classes Luzin $N\mskip-2mu$-property Morse–Sard theorem Hausdorff measure


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.

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