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Sibirskii Matematicheskii Zhurnal, 2019, Volume 60, Number 5, Pages 1171–1185
DOI: https://doi.org/10.33048/smzh.2019.60.514 (Mi smj3141) 		 
This article is cited in 3 scientific papers (total in 3 papers)

The Morse–Sard theorem and Luzin $N$-property: a new synthesis for smooth and Sobolev mappings

A. Feronea, M. V. Korobkovbc, A. Rovielloa

a Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, Caserta, Italy
b Fudan University, Shanghai, China
c Novosibirsk State University, Novosibirsk, Russia
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DOI: https://doi.org/10.33048/smzh.2019.60.514
Abstract: Considering regular mappings of Euclidean spaces, we study the distortion of the Hausdorff dimension of a given set under restrictions on the rank of the gradient on the set. This problem was solved for the classical cases of $k$-smooth and Hölder mappings by Dubovitskii, Bates, and Moreira. We solve the problem for Sobolev and fractional Sobolev classes as well. Here we study the Sobolev case under minimal integrability assumptions that guarantee in general only the continuity of a mapping (rather than differentiability everywhere). Some new facts are found out in the classical smooth case. The proofs are mostly based on our previous joint papers with Bourgain and Kristensen (2013, 2015).
Keywords: Morse–Sard theorem, Luzin $N$-property, Hausdorff measure, Hölder mappings, Sobolev–Lorentz mappings, Bessel potential spaces.
Funding agency Grant number
Ministry of Education and Science of the Russian Federation 1.8126.2017/8.9
Russian Foundation for Basic Research 18-01-00649_à
M. V. Korobkov was partially supported by the Ministry of Education and Science of the Russian Federation (Grant 1.8126.2017/8.9) and the Russian Federation for Basic Research (Grant 18-01-00649). The last part of the paper was written during Korobkov’s visit to the University of Campania “Luigi Vanvitelli” in 2019 which has been supported by the GNAMPA foundation.
Received: 21.02.2019
Revised: 21.02.2019
Accepted: 12.03.2019
English version:
Siberian Mathematical Journal, 2019, Volume 60, Issue 5, Pages 916–926
DOI: https://doi.org/10.1134/S0037446619050148
Bibliographic databases:
Document Type: Article
UDC: 517.2
Language: Russian
Citation: A. Ferone, M. V. Korobkov, A. Roviello, “The Morse–Sard theorem and Luzin $N$-property: a new synthesis for smooth and Sobolev mappings”, Sibirsk. Mat. Zh., 60:5 (2019), 1171–1185; Siberian Math. J., 60:5 (2019), 916–926
Citation in format AMSBIB
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\paper The Morse--Sard theorem and Luzin $N$-property: a new synthesis for smooth and Sobolev mappings
\jour Sibirsk. Mat. Zh.
\yr 2019
\vol 60
\issue 5
\pages 1171--1185
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\crossref{https://doi.org/10.33048/smzh.2019.60.514}
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\transl
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