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On the luzin N-property and the uncertainty principle for Sobolev mappings. / Ferone, Adele; Korobkov, Mikhail V.; Roviello, Alba.

In: Analysis and PDE, Vol. 12, No. 5, 01.01.2019, p. 1149-1175.

Research output: Contribution to journal › Article › peer-review

Harvard

Ferone, A, Korobkov, MV & Roviello, A 2019, 'On the luzin N-property and the uncertainty principle for Sobolev mappings', Analysis and PDE, vol. 12, no. 5, pp. 1149-1175. https://doi.org/10.2140/apde.2019.12.1149

APA

Ferone, A., Korobkov, M. V., & Roviello, A. (2019). On the luzin N-property and the uncertainty principle for Sobolev mappings. Analysis and PDE, 12(5), 1149-1175. https://doi.org/10.2140/apde.2019.12.1149

Vancouver

Ferone A, Korobkov MV, Roviello A. On the luzin N-property and the uncertainty principle for Sobolev mappings. Analysis and PDE. 2019 Jan 1;12(5):1149-1175. doi: 10.2140/apde.2019.12.1149

Author

Ferone, Adele ; Korobkov, Mikhail V. ; Roviello, Alba. / On the luzin N-property and the uncertainty principle for Sobolev mappings. In: Analysis and PDE. 2019 ; Vol. 12, No. 5. pp. 1149-1175.

BibTeX

@article{8f758c9abb964a518816bdb850f2191c,
title = "On the luzin N-property and the uncertainty principle for Sobolev mappings",
abstract = "We say that a mapping v: ℝn → ℝd satisfies the (τ, σ ) -N-property if Hσ( v (E)) = 0 whenever Hτ (E) = 0, where Hτ means the Hausdorff measure. We prove that every mapping v of Sobolev class W p k (ℝn,ℝd ) with kp > n satisfies the (τ, σ )-N-property for every 0 < τ ≠ τ*: = n - (k -1)p with 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(ℝn ) 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.",
keywords = "Fractional Sobolev classes, Hausdorff measure, Luzin N-property, Morse-Sard theorem, Sobolev-Lorentz mappings, fractional Sobolev classes, SARD THEOREM, DIFFERENTIABILITY, INEQUALITY, DISTORTION, MAPS, HAUSDORFF MEASURES, HOMEOMORPHISM",
author = "Adele Ferone and Korobkov, {Mikhail V.} and Alba Roviello",
note = "Publisher Copyright: {\textcopyright} 2019 Mathematical Sciences Publishers.",
year = "2019",
month = jan,
day = "1",
doi = "10.2140/apde.2019.12.1149",
language = "English",
volume = "12",
pages = "1149--1175",
journal = "Analysis and PDE",
issn = "2157-5045",
publisher = "Mathematical Sciences Publishers",
number = "5",

}

RIS

TY - JOUR

T1 - On the luzin N-property and the uncertainty principle for Sobolev mappings

AU - Ferone, Adele

AU - Korobkov, Mikhail V.

AU - Roviello, Alba

N1 - Publisher Copyright: © 2019 Mathematical Sciences Publishers.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We say that a mapping v: ℝn → ℝd satisfies the (τ, σ ) -N-property if Hσ( v (E)) = 0 whenever Hτ (E) = 0, where Hτ means the Hausdorff measure. We prove that every mapping v of Sobolev class W p k (ℝn,ℝd ) with kp > n satisfies the (τ, σ )-N-property for every 0 < τ ≠ τ*: = n - (k -1)p with 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(ℝn ) 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.

AB - We say that a mapping v: ℝn → ℝd satisfies the (τ, σ ) -N-property if Hσ( v (E)) = 0 whenever Hτ (E) = 0, where Hτ means the Hausdorff measure. We prove that every mapping v of Sobolev class W p k (ℝn,ℝd ) with kp > n satisfies the (τ, σ )-N-property for every 0 < τ ≠ τ*: = n - (k -1)p with 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(ℝn ) 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.

KW - Fractional Sobolev classes

KW - Hausdorff measure

KW - Luzin N-property

KW - Morse-Sard theorem

KW - Sobolev-Lorentz mappings

KW - fractional Sobolev classes

KW - SARD THEOREM

KW - DIFFERENTIABILITY

KW - INEQUALITY

KW - DISTORTION

KW - MAPS

KW - HAUSDORFF MEASURES

KW - HOMEOMORPHISM

UR - http://www.scopus.com/inward/record.url?scp=85058893158&partnerID=8YFLogxK

U2 - 10.2140/apde.2019.12.1149

DO - 10.2140/apde.2019.12.1149

M3 - Article

AN - SCOPUS:85058893158

VL - 12

SP - 1149

EP - 1175

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 5

ER -

ID: 17928039