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On the Equivalence of Two Approaches to Problems of Quasiconformal Analysis

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Abstract

We prove that the two approaches to describing homeomorphisms in modern quasiconformal analysis are equivalent: A homeomorphism changes under control the capacity of the image of a condenser in terms of the weighted capacity of a condenser in the preimage if and only if the modulus of the image of a family of curves can be estimated in terms of the weighted modulus of the original family of curves.

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Notes

  1. Recall that the norm \( |x|_{p} \) of a vector \( x=(x_{1},x_{2},\dots,x_{n})\in 𝕉^{n} \) is defined as \( |x|_{p}=\big{(}\sum\nolimits_{k=1}^{n}|x_{k}|^{p}\big{)}^{\frac{1}{p}} \) for \( p\in[1,\infty) \) and \( |x|_{\infty}=\max\nolimits_{k=1,\dots,n}|x_{k}| \) is the Chebyshev norm. A ball in the norm \( |x|_{2} \) (\( |x|_{\infty} \)) is a Euclidean ball (cube). A cube \( Q(x,R) \) is a ball in the metric space \( (𝕉^{n},|\cdot|_{\infty}) \) centered at \( x \); i.e., \( Q(x,R)=\{y\in 𝕉^{n}:|y-x|_{\infty})<R\} \).

  2. Henceforth \( B_{\delta} \) is an arbitrary ball \( B(z,\delta)\subset D^{\prime} \) containing the point \( y \), while \( |B_{\delta}| \) is the Lebesgue measure of \( B_{\delta} \). Balls in this proposition can be replaced with cubes.

  3. Here we should interpret (6) as follows: A function \( u\in{L}^{1}_{p}(W;\omega)\cap\overset{\circ}{\operatorname{Lip}}_{\operatorname{loc}}(W) \), extended by zero outside \( W \), lies in \( {L}^{1}_{p}(D^{\prime};\omega)\cap\operatorname{Lip}_{\operatorname{loc}}(D^{\prime}) \).

  4. This system must contain open sets \( U\setminus F \), where \( F \) and \( U \) are elements of a condenser \( E=(F,U) \) for which (9) holds.

  5. The first inequality in claim 2 of Theorem 18 of [12] contains a misprint: Instead of \( L_{\sigma}(\varphi^{-1}(A)) \) it should read \( L_{\sigma}(\varphi^{-1}(D^{\prime})) \).

  6. See the structure property (\( \mathrm{A}_{1} \)) in [35, p. 218]; a detailed proof of this property can be found in [2, Theorem 28.2] or [9, Theorem 2.7].

  7. That is, the curves \( \gamma:[a,b]\to D^{\prime} \) such that \( \gamma((a,b))\subset U\setminus F \), \( \gamma(a)\in F \), and \( \gamma(b)\in\partial U \).

  8. The first quantity in (24) is determined by an arbitrary family \( \Gamma \) of curves \( \gamma:[a,b]\to D^{\prime} \) (cf. (20)). The second quantity in (24) is determined by the family of all cubical condensers \( E=((\overline{Q(x,r)},Q(x,R))) \) in \( D^{\prime} \) and a family \( \Gamma \) of all curves in \( Q(x,R)\setminus\overline{Q(x,r)} \) with endpoints on the boundary of the interior and exterior cubes (cf. (23)).

  9. Note that the letter \( Q \) in the term \( Q \)-homeomorphism (\( (p,Q) \)-homeomorphism) of [9] ([37]) stands for the weight function, while in the term \( \mathcal{Q}_{q,p}(D^{\prime},\omega;D) \)-homeomorphism the letter \( Q \) is the first letter of the word quasiconformal.

References

  1. Gehring F. W. and Väisälä J., “The coefficients of quasiconformality of domains in space,” Acta Math., vol. 114, no. 1, 1–70 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  2. Väisälä J., Lectures on \( n \)-Dimensional Quasiconformal Mappings, Berlin and Heidelberg, Springer (1971).

    Book  MATH  Google Scholar 

  3. Mostow G. D., “Quasi-conformal mappings in \( n \)-space and the rigidity of hyperbolic space forms,” Inst. Hautes Études Sci. Publ. Math., vol. 34, no. 1, 53–104 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  4. Vodopyanov S. K. and Goldshtein V. M., “Lattice isomorphisms of the spaces \( W_{n}^{1} \) and quasiconformal mappings,” Sib. Math. J., vol. 16, no. 2, 174–189 (1975).

    Article  Google Scholar 

  5. Hesse J., “A \( p \)-extremal length and \( p \)-capacity equality,” Ark. Mat., vol. 13, no. 1, 131–144 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  6. Miklyukov V. M., The Conformal Mapping of an Irregular Surface and Its Applications, Volgogradsk. Univ., Volgograd (2005) [Russian].

    Google Scholar 

  7. Shlyk V. A., “The equality between \( p \)-capacity and \( p \)-modulus,” Sib. Math. J., vol. 34, no. 6, 1196–1200 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  8. Aikawa H. and Ohtsuka M., “Extremal length of vector measures,” Ann. Acad. Sci. Fenn., Math., vol. 24, no. 1, 61–88 (1999).

    MathSciNet  MATH  Google Scholar 

  9. Martio O., Ryazanov V., Srebro U., and Yakubov E., Moduli in Modern Mapping Theory, Springer, New York (2008).

    MATH  Google Scholar 

  10. Vodopyanov S. K., “Composition operators on weighted Sobolev spaces and the theory of \( \mathcal{Q}_{p} \)-homeomorphisms,” Dokl. Math., vol. 102, no. 2, 371–375 (2020).

    Article  MATH  Google Scholar 

  11. Vodopyanov S. K., “On the analytic and geometric properties of mappings in the theory of \( \mathcal{Q}_{q,p} \)-homeomorphisms,” Math. Notes, vol. 108, no. 6, 889–894 (2020).

    Article  MathSciNet  Google Scholar 

  12. Vodopyanov S. K., “The regularity of inverses to Sobolev mappings and the theory of \( \mathcal{Q}_{q,p} \)-homeomorphisms,” Sib. Math. J., vol. 61, no. 6, 1002–1038 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. Vodopyanov S. K. and Tomilov A. O., “Functional and analytic properties of a class of mappings in quasi-conformal analysis,” Izv. Math., vol. 85, no. 5, 883–931 (2021).

    Article  Google Scholar 

  14. Vodopyanov S. K., “Basics of the quasiconformal analysis of a two-index scale of spatial mappings,” Dokl. Akad. Nauk, vol. 484, no. 2, 142–146 (2019).

    Google Scholar 

  15. Vodopyanov S. K., “Basics of the quasiconformal analysis of a two-index scale of spatial mappings,” Sib. Math. J., vol. 59, no. 5, 805–834 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  16. Vodopyanov S. K., “Differentiability of mappings of the Sobolev space \( W^{1}_{n-1} \) with conditions on the distortion function,” Sib. Math. J., vol. 59, no. 6, 983–1005 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  17. Vodopyanov S. K., “Regularity of mappings inverse to Sobolev mappings,” Sb. Math., vol. 203, no. 10, 1383–1410 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  18. Vodopyanov S. K. and Ukhlov A. D., “Superposition operators in Sobolev spaces,” Russian Math. (Iz. VUZ), vol. 46, no. 10, 9–31 (2002).

    MathSciNet  MATH  Google Scholar 

  19. Vodopyanov S. K., “Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds,” Sb. Math., vol. 210, no. 1, 59–104 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  20. Vodopyanov S. K., “Isomorphisms of Sobolev spaces on Riemannian manifolds and quasiconformal mappings,” Sib. Math. J., vol. 60, no. 5, 774–804 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  21. Sobolev S. L., “On some transformation groups of an \( n \)-dimensional space,” Dokl. Akad. Nauk SSSR, vol. 32, no. 6, 380–382 (1941).

    MATH  Google Scholar 

  22. Mazya V. G., Classes of Sets and Embedding Theorems of Function Classes. Some Problems of the Theory of Elliptic Operators. Extended Abstract of Cand. Sci. Dissertation, Leningrad Univ., Leningrad (1961) [Russian].

    Google Scholar 

  23. Reshetnyak Yu. G., Space Mappings with Bounded Distortion, Amer. Math. Soc., Providence (1989).

    Book  MATH  Google Scholar 

  24. Reimann H. M., “Über harmonische Kapazität und quasikonforme Abbildungen im Raum,” Comm. Math. Helv., vol. 44, 284–307 (1969).

    MATH  Google Scholar 

  25. Gehring F. W., “Lipschitz mappings and the \( p \)-capacity of rings in \( n \)-space,” in: Proc. Symp. Advances in the Theory of Riemann Surfaces. (Stony Brook, NY, 1969), Princeton Univ., Princeton (1971), 175–193.

  26. Lelong-Ferrand J., “Étude d’une classe d’applications liées à des homomorphismes d’algébres de fonctions, et généralisant les quasi conformes,” Duke Math., vol. 40, no. 1, 163–186 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  27. Molchanova A. and Vodop’yanov S., “Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity,” Calc. Var., vol. 59, no. 1 (2020) (Article number 17).

  28. Heinonen J., Kilpeläinen T., and Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon, Oxford (1993).

    MATH  Google Scholar 

  29. Radó T. and Reichelderfer P., Continuous Transformations in Analysis with an Introduction to Algebraic Topology, Springer, Berlin, Göttingen, and Heidelberg (1955).

    MATH  Google Scholar 

  30. De Guzmán M., Differentiation of Integrals in \( 𝕉^{n} \), Springer, Berlin (1975).

    Book  MATH  Google Scholar 

  31. Vodopyanov S. K. and Ukhlov A. D., “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I,” Siberian Adv. Math., vol. 14, no. 4, 78–125 (2004).

    MathSciNet  MATH  Google Scholar 

  32. Vodopyanov S. K., The Taylor Formula and Function Spaces, Novosibirsk Univ., Novosibirsk (1988) [Russian].

    Google Scholar 

  33. Ukhlov A. D., “On mappings generating the embeddings of Sobolev spaces,” Sib. Math. J., vol. 34, no. 1, 165–171 (1993).

    Article  MATH  Google Scholar 

  34. Vodopyanov S. K. and Ukhlov A. D., “Sobolev spaces and \( (P,Q) \)-quasiconformal mappings of Carnot groups,” Sib. Math. J., vol. 39, no. 4, 665–682 (1998).

    Article  MathSciNet  Google Scholar 

  35. Fuglede B., “Extremal length and functional completion,” Acta Math., vol. 98, no. 1, 171–219 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  36. Vodopyanov S. K., “Monotone functions and quasiconformal mappings on Carnot groups,” Sib. Math. J., vol. 37, no. 6, 1113–1136 (1996).

    Article  MathSciNet  Google Scholar 

  37. Salimov R. R. and Sevost’yanov E. A., “\( \operatorname{ACL} \) and differentiability of open discrete ring \( (p,Q) \)-mappings,” Mat. Stud., vol. 35, no. 1, 28–36 (2011).

    MathSciNet  MATH  Google Scholar 

  38. Kruglikov V. I., “Capacity of condensers and spatial mappings quasiconformal in the mean,” Math. USSR-Sb., vol. 58, no. 1, 185–205 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  39. Ryazanov V. I. and Sevostyanov E. A., “Equicontinuity of mean quasiconformal mappings,” Sib. Math. J., vol. 52, no. 3, 524–536 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  40. Salimov R. R., “ACL and differentiability of a generalization of quasi-conformal maps,” Izv. Math., vol. 72, no. 5, 977–984 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  41. Salimov R., “\( ACL \) and differentiability of \( Q \)-homeomorphisms,” Ann. Acad. Sci. Fenn., vol. 33, 295–301 (2008).

    MathSciNet  MATH  Google Scholar 

  42. Salimov R. R. and Sevostyanov E. A., “The theory of shell-based \( Q \)-mappings in geometric function theory,” Sb. Math., vol. 201, no. 6, 909–934 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  43. Sevost’yanov E. A. and Skvortsov S. A., On Behavior of Homeomorphisms with Inverse Modulus Conditions. arXiv:1801.01808v9 [math.MG] (2018).

    MATH  Google Scholar 

  44. Salimov R. R. and Sevostyanov E. A., “On local properties of spatial generalized quasi-isometries,” Math. Notes, vol. 101, no. 4, 704–717 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  45. Salimov R., “On \( Q \)-homeomorphisms with respect to \( p \)-modulus,” Ann. Univ. Bucharest, Ser. Math., vol. 2 (LX), no. 2, 207–213 (2011).

    MathSciNet  MATH  Google Scholar 

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Funding

The author was supported by the Mathematical Center in Akademgorodok and the Ministry of Science and Higher Education of the Russian Federation (Contract 075–15–2019–1613).

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. K. Vodopyanov

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Correspondence to S. K. Vodopyanov.

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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 6, pp. 1252–1270. https://doi.org/10.33048/smzh.2021.62.604

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Vodopyanov, S.K. On the Equivalence of Two Approaches to Problems of Quasiconformal Analysis. Sib Math J 62, 1010–1025 (2021). https://doi.org/10.1134/S0037446621060045

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