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

We say that a mapping v : n d 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 (k 1)p with

σ = σ(τ) := τ  if τ > τ, pτ(kp 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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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
Received: 26 June 2017
Revised: 12 July 2018
Accepted: 12 August 2018
Published: 15 December 2018
Adele Ferone
Dipartimento di Matematica e Fisica
Università degli studi della Campania “Luigi Vanvitelli”
Mikhail V. Korobkov
School of Mathematical Sciences
Fudan University
Novosibirsk State University
Alba Roviello
Dipartimento di Matematica e Fisica
Università degli studi della Campania “Luigi Vanvitelli”