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Mat. Tr., 2003, Volume 6, Number 2, Pages 14–65 (Mi mt91)  

This article is cited in 43 scientific papers (total in 43 papers)

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

S. K. Vodop'yanova, A. D.-O. Ukhlovb

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

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.

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English version:
Siberian Advances in Mathematics, 2004, 14:4, 78–125

Bibliographic databases:

UDC: 517.518.1+517.54
Received: 02.04.2002

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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\by S.~K.~Vodop'yanov, A.~D.-O.~Ukhlov
\paper Set Functions and Their Applications in the~Theory of Lebesgue and Sobolev Spaces.~I
\jour Mat. Tr.
\yr 2003
\vol 6
\issue 2
\pages 14--65
\mathnet{http://mi.mathnet.ru/mt91}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2033646}
\zmath{https://zbmath.org/?q=an:1089.47027|1050.47030}
\elib{https://elibrary.ru/item.asp?id=9530086}
\transl
\jour Siberian Adv. Math.
\yr 2004
\vol 14
\issue 4
\pages 78--125


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    This publication is cited in the following articles:
    1. S. K. Vodop'yanov, A. D.-O. Ukhlov, “Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. II”, Siberian Adv. Math., 15:1 (2005), 91–125  mathnet  mathscinet  zmath  elib
    2. S. K. Vodop'yanov, D. V. Isangulova, “Differentiability of the mappings of Carnot–Caratheodory spaces in the Sobolev and $BV$-topologies”, Siberian Math. J., 48:1 (2007), 37–55  mathnet  crossref  mathscinet  zmath  isi  elib
    3. Vodop'yanov S.K., Karmanova M.B., “A coarea formula for smooth contact mappings of carnot manifolds”, Dokl. Math., 76:3 (2007), 908–912  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    4. Vodopyanov S.K., “Geometry of Carnot-Caratheodory spaces and differentiability of mappings”, Interaction of Analysis and Geometry, Contemporary Mathematics Series, 424, 2007, 247–301  crossref  mathscinet  zmath  isi
    5. Karmanova M., “Geometric measure theory formulas on rectifiable metric spaces”, Interaction of Analysis and Geometry, Contemporary Mathematics Series, 424, 2007, 103–136  crossref  mathscinet  zmath  isi
    6. Reshetnyak Yu.G., “Sobolev-type classes of mappings with values in metric spaces”, Interaction of Analysis and Geometry, Contemporary Mathematics Series, 424, 2007, 209–226  crossref  mathscinet  zmath  isi
    7. Vodopyanov S.K., “Foundations of the theory of mappings with bounded distortion on carnot groups”, Interaction of Analysis and Geometry, Contemporary Mathematics Series, 424, 2007, 303–344  crossref  mathscinet  zmath  isi
    8. S. K. Vodop'yanov, “Differentiability of mappings in the geometry of Carnot manifolds”, Siberian Math. J., 48:2 (2007), 197–213  mathnet  crossref  mathscinet  zmath  isi
    9. A. S. Romanov, “Absolute continuity of the Sobolev type functions on metric spaces”, Siberian Math. J., 49:5 (2008), 911–918  mathnet  crossref  mathscinet  isi
    10. Karmanova M., “Rectifiable sets and coarea formula for metric-valued mappings”, J. Funct. Anal., 254:5 (2008), 1410–1447  crossref  mathscinet  zmath  isi  elib  scopus
    11. Ukhlov A., Vodopyanov S.K., “Mappings associated with weighted Sobolev spaces”, Complex Analysis and Dynamical Systems III, Contemporary Mathematics Series, 455, 2008, 369–382  crossref  mathscinet  zmath  isi
    12. Vodop'yanov S.K., “Spaces of differential forms and mappings with controlled distortion”, Dokl. Math., 79:1 (2009), 105–109  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    13. Gol'dshtein V., Ukhlov A., “Weighted Sobolev spaces and embedding theorems”, Trans. Amer. Math. Soc., 361:7 (2009), 3829–3850  crossref  mathscinet  zmath  isi  scopus
    14. Karmanova M., Vodop'yanov S., “Geometry of Carnot-Caratheodory Spaces, Differentiability, Coarea and Area Formulas”, Analysis and Mathematical Physics, Trends in Mathematics, 2009, 233–335  mathscinet  zmath  isi
    15. S. K. Vodop'yanov, “Spaces of differential forms and maps with controlled distortion”, Izv. Math., 74:4 (2010), 663–689  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. Ukhlov A., Vodop'yanov S.K., “Mappings with bounded (P, Q)-distortion on Carnot groups”, Bull Sci Math, 134:6 (2010), 605–634  crossref  mathscinet  zmath  isi  elib  scopus
    17. Ukhlov A., “Composition operators in weighted Sobolev spaces on Carnot groups”, Acta Math Hungar, 133:1–2 (2011), 103–127  crossref  mathscinet  zmath  isi  elib  scopus
    18. S. K. Vodopyanov, “Regularity of mappings inverse to Sobolev mappings”, Sb. Math., 203:10 (2012), 1383–1410  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    19. Karmanova M.B., “Graphs of Lipschitz Functions and Minimal Surfaces on Carnot Groups”, Dokl. Math., 86:1 (2012), 500–505  crossref  mathscinet  zmath  isi  elib  scopus
    20. M. B. Karmanova, “The graphs of Lipschitz functions and minimal surfaces on Carnot groups”, Siberian Math. J., 53:4 (2012), 672–690  mathnet  crossref  mathscinet  isi
    21. Karmanova M. Vodopyanov S., “A Coarea Formula for Smooth Contact Mappings of Carnot-Carath,Odory Spaces”, Acta Appl. Math., 128:1 (2013), 67–111  crossref  mathscinet  zmath  isi  elib  scopus
    22. S. K. Vodop'yanov, N. A. Evseev, “Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings”, Siberian Math. J., 55:5 (2014), 817–848  mathnet  crossref  mathscinet  isi
    23. A. N. Baykin, S. K. Vodop'yanov, “Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion”, Siberian Math. J., 56:2 (2015), 237–261  mathnet  crossref  mathscinet  isi  elib  elib
    24. M. B. Karmanova, “The area formula for graphs on $4$-dimensional $2$-step sub-Lorentzian structures”, Siberian Math. J., 56:5 (2015), 852–871  mathnet  crossref  crossref  isi  elib  elib
    25. M. V. Tryamkin, “The morphism property of subelliptic equations on the roto-translation group”, Siberian Math. J., 56:5 (2015), 936–954  mathnet  crossref  crossref  isi  elib  elib
    26. N. A. Evseev, “Change of variables operators in weighted Sobolev spaces on Carnot groups”, J. Math. Sci., 221:6 (2017), 826–832  mathnet  crossref  crossref
    27. N. A. Evseev, “Composition operators in weighted Sobolev spaces on the Carnot group”, Siberian Math. J., 56:6 (2015), 1042–1059  mathnet  crossref  crossref  mathscinet  isi  elib
    28. M. B. Karmanova, “Graph surfaces over three-dimensional Lie groups with sub-Riemannian structure”, Siberian Math. J., 56:6 (2015), 1080–1092  mathnet  crossref  crossref  mathscinet  isi  elib
    29. M. V. Tryamkin, “Modulus inequalities for mappings with weighted bounded $(p,q)$-distortion”, Siberian Math. J., 56:6 (2015), 1114–1132  mathnet  crossref  crossref  mathscinet  isi  elib
    30. Vodopyanov S.K., “on Admissible Changes of Variables For Sobolev Functions on (Sub)Riemannian Manifolds”, Dokl. Math., 93:3 (2016), 318–321  crossref  mathscinet  zmath  isi  scopus
    31. N. A. Evseev, “Bounded Composition Operator on Lorentz Spaces”, Math. Notes, 102:6 (2017), 763–769  mathnet  crossref  crossref  isi  elib
    32. M. B. Karmanova, “Graph surfaces on five-dimensional sub-Lorentzian structures”, Siberian Math. J., 58:1 (2017), 91–108  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    33. M. B. Karmanova, “Area formulas for classes of Hölder continuous mappings of Carnot groups”, Siberian Math. J., 58:5 (2017), 817–836  mathnet  crossref  crossref  isi  elib  elib
    34. Gol'dshtein V., Pchelintsev V., Ukhlov A., “Spectral Estimates of the P-Laplace Neumann Operator and Brennan'S Conjecture”, Boll. Unione Mat. Ital., 11:2 (2018), 245–264  crossref  mathscinet  zmath  isi  scopus
    35. M. B. Karmanova, “Three-dimensional graph surfaces on five-dimensional Carnot–Carathéodory spaces”, Siberian Math. J., 59:4 (2018), 657–676  mathnet  crossref  crossref  isi  elib
    36. S. K. Vodopyanov, “Basics of the quasiconformal analysis of a two-index scale of spatial mappings”, Siberian Math. J., 59:5 (2018), 805–834  mathnet  crossref  crossref  isi  elib
    37. S. K. Vodopyanov, “Differentiability of mappings of the Sobolev space $W^1_{n-1}$ with conditions on the distortion function”, Siberian Math. J., 59:6 (2018), 983–1005  mathnet  crossref  crossref  isi  elib
    38. S. K. Vodopyanov, “Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds”, Sb. Math., 210:1 (2019), 59–104  mathnet  crossref  crossref  adsnasa  isi  elib
    39. N. A. Evseev, A. V. Menovshchikov, “The Composition Operator on Mixed-Norm Lebesgue Spaces”, Math. Notes, 105:6 (2019), 812–817  mathnet  crossref  crossref  isi  elib
    40. Evseev N., Menovschikov A., “Bounded Operators on Mixed Norm Lebesgue Spaces”, Complex Anal. Oper. Theory, 13:5 (2019), 2239–2258  crossref  mathscinet  zmath  isi  scopus
    41. Gol'dshtein V., Pchelintsev V., Ukhlov A., “Spectral Properties of the Neumann-Laplace Operator in Quasiconformal Regular Domains”, Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemporary Mathematics, 734, eds. Kuchment P., Semenov E., Amer Mathematical Soc, 2019, 129–144  crossref  mathscinet  zmath  isi  scopus
    42. M. B. Karmanova, “Two-step sub-Lorentzian structures and graph surfaces”, Izv. Math., 84:1 (2020), 52–94  mathnet  crossref  crossref  isi  elib
    43. N. A. Evseev, A. V. Menovschikov, “O zamene peremennykh v $L^p$-prostranstvakh s izmenyayuscheisya vnutrennei strukturoi”, Izv. vuzov. Matem., 2020, no. 3, 92–97  mathnet  crossref
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