Analysis & PDE
- Anal. PDE
- Volume 12, Number 5 (2019), 1149-1175.
On the Luzin $N$-property and the uncertainty principle for Sobolev mappings
We say that a mapping satisfies the --property if whenever , where means the Hausdorff measure. We prove that every mapping of Sobolev class with satisfies the --property for every with
We prove also that for and for the critical value the corresponding --property fails in general. Nevertheless, this --property holds for if we assume in addition that the highest derivatives belong to the Lorentz space instead of .
We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for -Nproperties and discuss their applications to the Morse–Sard theorem and its recent extensions.
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
Permanent link to this document
Digital Object Identifier
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.)
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