S. K. Vodop'yanov, A. D.-O. Ukhlov, “Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I”, Mat. Tr., 6:2 (2003), 14–65; Siberian Adv. Math., 14:4 (2004), 78

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Matematicheskie Trudy, 2003, Volume 6, Number 2, Pages 14–65 (Mi mt91)

This article is cited in 64 scientific papers (total in 64 papers)

Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I

S. K. Vodop'yanov^a, A. D.-O. Ukhlov^b

^a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
^b Khabarovsk State University of Technology

Full-text PDF (4333 kB) Citations (64)

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Abstract: We study the properties of mappings inducing a bounded operator of Lebesgue or Sobolev spaces by change of variable and the properties of the operator of extension of functions in Sobolev classes beyond the domain of definition. Throughout, we deduce and apply the properties of quasiadditive functions on open subsets of homogeneous spaces. We estimate the integral of the upper derivative of a set function which implies an easy proof of the Lebesgue Integral Differentiation Theorem and existence of a density almost everywhere.
The article consists of two sections. In Section 1, apart from studying the properties of quasiadditive functions, we find necessary and sufficient conditions on a mapping inducing a bounded extension operator in Lebesgue spaces (in Sobolev spaces with weak first-order derivatives).

Key words: quasiadditive set function, Lebesgue space, Sobolev space, embedding theorems.

Received: 02.04.2002

Bibliographic databases:

UDC: 517.518.1+517.54

Language: Russian

Citation: S. K. Vodop'yanov, A. D.-O. Ukhlov, “Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I”, Mat. Tr., 6:2 (2003), 14–65; Siberian Adv. Math., 14:4 (2004), 78–125

Citation in format AMSBIB

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\jour Siberian Adv. Math.

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Linking options:

https://www.mathnet.ru/eng/mt91

https://www.mathnet.ru/eng/mt/v6/i2/p14

Cycle of papers

Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I
S. K. Vodop'yanov, A. D.-O. Ukhlov
Mat. Tr., 2003, 6:2, 14–65
Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. II
S. K. Vodop'yanov, A. D.-O. Ukhlov
Mat. Tr., 2004, 7:1, 13–49

This publication is cited in the following 64 articles:

S. K. Vodopyanov, S. V. Pavlov, “Funktsionalnye svoistva predelov sobolevskikh gomeomorfizmov s integriruemym iskazheniem”, Funktsionalnye prostranstva. Differentsialnye operatory. Problemy matematicheskogo obrazovaniya, SMFN, 70, no. 2, Rossiiskii universitet druzhby narodov, M., 2024, 215–236
S. K. Vodopyanov, “Funktsionalno-geometricheskie svoistva predelov ACL-otobrazhenii s integriruemym iskazheniem”, Sib. matem. zhurn., 65:5 (2024), 820–840
M. B. Karmanova, “Ploschad obrazov klassov izmerimykh mnozhestv na gruppakh Karno s sublorentsevoi strukturoi”, Sib. matem. zhurn., 65:5 (2024), 926–952
M. B. Karmanova, “Metric characteristics of classes of compact sets on Carnot groups with sub-Lorentzian structure”, Vladikavk. matem. zhurn., 26:3 (2024), 56–64
S. K. Vodopyanov, “Composition operators in Sobolev spaces on Riemannian manifolds”, Siberian Math. J., 65:6 (2024), 1305–1326
M. B. Karmanova, “Area of images of measurable sets on depth 2 Carnot manifolds with sub-Lorentzian structure”, Vladikavk. matem. zhurn., 26:4 (2024), 78–86
Izv. Math., 87:4 (2023), 683–725
M. B. Karmanova, “Sub-Riemannian Co-Area Formula for Classes of Noncontact Mappings of Carnot Groups”, Math. Notes, 115:3 (2024), 439–443
M. B. Karmanova, “Klassy nekontaktnykh otobrazhenii grupp Karno i metricheskie svoistva”, Sib. matem. zhurn., 64:6 (2023), 1199–1223
D. A. Sboev, “Prostranstva $BV$ i ogranichennye operatory kompozitsii $BV$ -funktsii na gruppakh Karno”, Sib. matem. zhurn., 64:6 (2023), 1304–1326
M. B. Karmanova, “Lipschitz images of open sets on sub-Lorentzian structures”, Siberian Adv. Math., 34:1 (2024), 67–79
Hashash P., Ukhlov A., “On Lipschitz Approximations in Second Order Sobolev Spaces and the Change of Variables Formula”, J. Math. Anal. Appl., 506:2 (2022), 125659
Gol'dshtein V., Pchelintsev V., Ukhlov A., “Quasiconformal Mappings and Neumann Eigenvalues of Divergent Elliptic Operators”, Complex Var. Elliptic Equ., 67:9 (2022), 2281–2302
S. K. Vodopyanov, “Coincidence of set functions in quasiconformal analysis”, Sb. Math., 213:9 (2022), 1157–1186
S. K. Vodopyanov, N. A. Evseev, “Functional and analytical properties of a class of mappings of quasiconformal analysis on Carnot groups”, Siberian Math. J., 63:2 (2022), 233–261
M. B. Karmanova, “Sub-Lorentzian coarea formula for mappings of Carnot groups”, Siberian Math. J., 63:3 (2022), 485–508
M. B. Karmanova, “Sub-riemannian properties of the level sets of noncontact mappings of Heisenberg groups”, Siberian Adv. Math., 33:1 (2023), 28–38
M. B. Karmanova, “The coarea formula for vector functions on Carnot groups with sub-Lorentzian structure”, Siberian Math. J., 62:2 (2021), 239–261
S. K. Vodopyanov, “On the equivalence of two approaches to problems of quasiconformal analysis”, Siberian Math. J., 62:6 (2021), 1010–1025
Gol'dshtein V., Pchelintsev V., Ukhlov A., “Spectral Stability Estimates of Neumann Divergence Form Elliptic Operators”, Math. Rep., 23:1-2 (2021), 131–147

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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