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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 5–32 (Mi izv2842)  

This article is cited in 9 scientific papers (total in 9 papers)

Spaces of differential forms and maps with controlled distortion

S. K. Vodop'yanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We study necessary and sufficient conditions for an approximately differentiable map $f\colon\mathbb M\to\mathbb M'$ between Riemannian manifolds to induce a bounded transfer operator of differential forms with respect to the norms of Lebesgue spaces. As a corollary, we see that every homeomorphism $f\colon\mathbb M\to\mathbb M'$ of class $\operatorname{ACL}(\mathbb M)$ whose transfer operator of differential forms with norm in $\mathcal L_p$ is an isomorphism must necessarily be either quasi-conformal or quasi-isometric. We give some applications of our results to the study of the functoriality of cohomology in Lebesgue spaces.

Keywords: Lebesgue space of differential forms, distortion of a map, quasi-conformal mapping, cohomology of Riemannian spaces.

DOI: https://doi.org/10.4213/im2842

Full text: PDF file (747 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:4, 663–689

Bibliographic databases:

UDC: 515.164.13+517.548.2
MSC: 46E35, 47B38, 30C65
Received: 27.06.2008

Citation: S. K. Vodop'yanov, “Spaces of differential forms and maps with controlled distortion”, Izv. RAN. Ser. Mat., 74:4 (2010), 5–32; Izv. Math., 74:4 (2010), 663–689

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

    This publication is cited in the following articles:
    1. Vodopyanov S.K., “Mappings of finite codistortion and Sobolev classes of functions”, Dokl. Math, 84:2 (2011), 640–644  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    2. S. K. Vodopyanov, “Regularity of mappings inverse to Sobolev mappings”, Sb. Math., 203:10 (2012), 1383–1410  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Terenteva Yu.V., “Issledovanie svoistv integriruemosti proizvodnykh resheniya sopryazhënnogo (nelineinogo) uravneniya beltrami v sluchae vyrozhdeniya na granichnoi duge”, Vestnik rossiiskogo universiteta druzhby narodov. seriya: matematika, informatika, fizika, 2013, no. 1, 5–18  elib
    4. 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  elib  scopus
    5. S. K. Vodop'yanov, N. A. Kudryavtseva, “On the Convergence of Mappings with $k$-Finite Distortion”, Math. Notes, 102:6 (2017), 878–883  mathnet  crossref  crossref  isi  elib
    6. N. A. Kudryavtseva, S. K. Vodopyanov, “On the convergence of mappings with $k$-finite distortion”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 88–100  mathnet  crossref  elib
    7. 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
    8. 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
    9. Vodopyanov S.K., “Foundations of Quasiconformal Analysis of a Two-Index Scale of Spatial Mappings”, Dokl. Math., 99:1 (2019), 23–27  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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