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Functional and Analytical Properties of a Class of Mappings of Quasiconformal Analysis on Carnot Groups

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Abstract

This article addresses the conceptual questions of quasiconformal analysis on Carnot groups. We prove the equivalence of the three classes of homeomorphisms: the mappings of the first class induce bounded composition operators from a weighted Sobolev space into an unweighted one; the mappings of the second class are characterized by way of estimating the capacity of the preimage of a condenser in terms of the weighted capacity of the condenser in the image; the mappings of the third class are described via a pointwise relation between the norm of the matrix of the differential, the Jacobian, and the weight function. We obtain a new proof of the absolute continuity of mappings.

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Notes

  1. A horizontal curve is a piecewise smooth path whose tangent vector at each point where it exists lies in \( V_{1} \).

  2. If \( dx \) is the volume form on \( 𝔾 \) of degree \( N \), then \( i(X_{1j})\,dx \) is a form of degree \( N-1 \), which takes the values \( (i(X_{1j})\,dx)(Y_{1},Y_{2},\dots,Y_{N-1})=dx(X_{1j},Y_{1},Y_{2},\dots,Y_{N-1}) \) on the smooth vector fields \( Y_{1},Y_{2},\dots,Y_{N-1} \) defined on \( 𝔾 \).

  3. Here \( B_{\delta} \) is an arbitrary ball \( B(x,\delta)\subset D^{\prime} \) containing the point \( y \).

  4. The measure of the image \( \varphi(A) \) vanishes for every set \( A\subset D\setminus Z \) of measure zero.

  5. By the regularity of Lebesgue measure, we can express every measurable set \( A\in D\setminus\Sigma \) as the disjoint union of a Borel set and a set of measure zero; the image of the Borel set under the homeomorphism \( \varphi \) is a Borel set, while by Luzin’s \( \mathcal{N} \)-property the image of each set of measure zero is a set of measure zero.

  6. The definition of generalized derivatives assumes that \( \frac{\partial u}{\partial y_{j}}\in L_{1,\operatorname{loc}}(D^{\prime}) \).

  7. Henceforth \( \operatorname{Lip}_{l}(D^{\prime}) \) stands for the space of locally Lipschitz functions on \( D^{\prime} \). It is obvious that \( \operatorname{Lip}_{l}(D^{\prime})=W^{1}_{\infty,\operatorname{loc}}(D^{\prime})\cap C(D^{\prime}) \).

  8. Here (3.2) should be interpreted as follows: The function \( u\in{L}^{1}_{p}(U;\omega)\cap\overset{\circ}{\operatorname{Lip}}_{l}(U) \), extended by zero beyond \( U \), lies in \( {L}^{1}_{p}(D^{\prime};\omega)\cap\operatorname{Lip}_{l}(D^{\prime}) \).

  9. Note that here the conditions \( u_{i}\in\mathcal{R}(U_{i}) \) and \( v_{i}=\varphi^{*}u_{i} \) are redundant.

  10. Here, in order to preserve the notational logic, we should write \( \mathcal{A}(E)\cap{L}^{1}_{p}(D^{\prime};\omega)\cap\operatorname{Lip}_{l}(D^{\prime}) \) for the class of admissible functions for a condenser. However, we avoid that because there is always \( u\in\mathcal{A}(E)\cap\operatorname{Lip}_{l}(D^{\prime}) \) for which the norm (3.1) is finite, and this guarantees that the capacity is finite.

  11. Henceforth \( B^{E}(a,r) \) stands for the Euclidean ball of radius \( r \) and center \( a \).

  12. Note that the right-hand side of (3.18) holds for the arbitrary constant \( K_{p} \) and quasiadditive set function \( \Psi \) in (3.17), while the left-hand side of (3.18) follows from the right-hand side on assuming that \( K_{p} \) and the quasiadditive set function \( \Psi \) are optimized using (3.21) and (3.22).

References

  1. 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 

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

    Article  Google Scholar 

  3. 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 

  4. Vodopyanov S. K. and Tomilov A. O., “Functional and analytical properties of a class of mappings in quasiconformal analysis,” Izv. Math., vol. 85, no. 5, 883–931 (2021).

    Article  MATH  Google Scholar 

  5. 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 

  6. 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 

  7. 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 

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

    Google Scholar 

  9. Vodopyanov S. K., “Mappings of homogeneous groups and imbeddings of functional spaces,” Sib. Math. J., vol. 30, no. 5, 685–698 (1989).

    Article  Google Scholar 

  10. Vodopyanov S. K., “Weighted Sobolev spaces and mapping theory,” in: Abstracts: All-Russia Mathematical School “Potential Theory” (Katsiveli, 26 June– 3 July 1991), Inst. Mat. Akad. Nauk USSR, Kiev (1991), 7.

  11. Vodopyanov S. K., Geometric Aspects of Spaces of General Differentiable Functions. Extended Abstract of Doct. (Phys.–Math.) Sci. Dissertation, Sobolev Institute of Mathematics, Novosibirsk (1992) [Russian].

    Google Scholar 

  12. 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 

  13. 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 

  14. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. 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. (Phys.–Math.) Dissertation, Leningrad University, Leningrad (1961) [Russian].

    Google Scholar 

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

    Book  MATH  Google Scholar 

  19. 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 

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

    MATH  Google Scholar 

  21. Väisälä J., Lectures on \( n \)-Dimensional Quasiconformal Mappings, Berlin and Heidelberg, Springer (1971) (Lecture Notes Math.; Vol. 229).

    Book  MATH  Google Scholar 

  22. 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 University, Princeton (1971), 175–193.

  23. 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 

  24. 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 

  25. 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 

  26. Vodopyanov S. K. and Goldshtein V. M., “Functional characteristics of quasi-isometric mappings,” Sib. Math. J., vol. 17, no. 4, 580–584 (1976).

    Article  Google Scholar 

  27. Romanov A. S., “A change of variable in the Bessel and Riesz potential spaces,” in: Functional Analysis and Mathematical Physics, Inst. Mat., Novosibirsk (1985), 117–133 [Russian].

  28. Vodopyanov S. K., “\( L_{p} \)-Potential theory and quasiconformal mappings on homogeneous groups,” in: Modern Problems of Geometry and Analysis, Nauka, Novosibirsk (1989), 45–89 [Russian].

  29. Vodopyanov S. K., “Composition operators on Sobolev spaces,” in: Complex Analysis and Dynamical Systems. II:. A conference in honor of Professor Lawrence Zalcman’s Sixtieth Birthday, June 9–12, 2003, Nahariya, Israel (M. Agranovsky, L. Karp, D. Shoikhet, eds), Amer. Math. Soc., Ann Arbor (2005), 327–342 (Contemp. Math.; Vol. 382).

  30. Vodopyanov S. K. and Evseev N. A., “Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings,” Sib. Math. J., vol. 55, no. 5, 817–848 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  31. Vodopyanov S. K. and Evseev N. A., “Isomorphisms of Sobolev spaces on Carnot groups and metric mapping properties,” Dokl. Math., vol. 82, no. 2, 232–236 (2015).

    Google Scholar 

  32. Vodopyanov S. K. and Evseev N. A., “Isomorphisms of Sobolev spaces on Carnot groups and quasiconformal mappings,” Sib. Math. J., vol. 56, no. 5, 789–821 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  33. Evseev N. A., “Composition operators in weighted Sobolev spaces on the Carnot group,” Sib. Math. J., vol. 56, no. 6, 1042–1059 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  34. Vodopyanov S. K., “On admissible changes of variables for Sobolev functions on (sub)Riemannian manifolds,” Dokl. Math., vol. 93, no. 3, 318–321 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  35. 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 

  36. 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 

  37. 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 no. 17).

  38. Vodopyanov S. K., Function-Theoretic Approach to Some Problems of the Theory of Space Quasiconformal Mappings. Extended Abstract of Cand. Sci. (Phys.–Math.) Dissertation, Sobolev Institute of Mathematics, Novosibirsk (1975) [Russian].

    Google Scholar 

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

    MATH  Google Scholar 

  40. Vodopyanov S. K., “Moduli inequalities for \( W^{1}_{n-1,\operatorname{loc}} \)-mappings with weighted bounded \( (q,p) \)-distortion,” Complex Variables and Elliptic Equations, vol. 66, no. 5, 1002–1038 (2021).

    MathSciNet  MATH  Google Scholar 

  41. Korányi A. and Reimann H. M., “Foundations for the theory of quasiconformal mappings on the Heisenberg group,” Adv. Math., vol. 111, 1–87 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  42. Pansu P., “Métriques de Carnot–Carathéodory et quasi-isométries des espaces symétriques de rang un,” Ann. Math., vol. 129, no. 1, 1–60 (1989) [French].

    Article  MathSciNet  MATH  Google Scholar 

  43. Vodopyanov S. K., “Quasiconformal mappings on Carnot groups and applications,” Dokl. Math., vol. 347, no. 4, 439–442 (1996).

    MathSciNet  Google Scholar 

  44. Folland G. B. and Stein E. M., Hardy Spaces on Homogeneous Groups, Princeton University, Princeton (1982) (Math. Notes; Vol. 28).

    MATH  Google Scholar 

  45. Bonfiglioli A., Lanconelli E., and Uguzzoni F., Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, Berlin and Heidelberg (2007).

    MATH  Google Scholar 

  46. Stein E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University, Princeton (1993).

    MATH  Google Scholar 

  47. Vodopyanov S. K., “\( \mathcal{P} \)-Differentiability on Carnot groups in various topologies and related topics,” in: Proceedings on Analysis and Geometry, Sobolev Institute of Mathematics, Novosibirsk (2000), 603–670.

  48. Vodopyanov S. K., “Differentiability of maps of Carnot groups of Sobolev classes,” Sb. Math., vol. 194, no. 6, 857–877 (2003).

    Article  MathSciNet  Google Scholar 

  49. Rado I. and Reichelderfer P. V., Continuous Transformations in Analysis with an Introduction to Algebraic Topology, Springer, Berlin (1955).

    MATH  Google Scholar 

  50. De Guzmán M., Differentiation of Integrals in \( 𝕉^{n} \), Springer, Berlin (1975) (Lecture Notes in Math.; Vol. 481).

    Book  MATH  Google Scholar 

  51. Vodop’yanov 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 

  52. Vodopyanov S. K. and Ukhlov A. D., “Superposition operators in the Lebesgue spaces and differentiability of quasiadditive set functions,” Vladikavkaz. Mat. Zh., vol. 4, no. 1, 11–33 (2002).

    MathSciNet  MATH  Google Scholar 

  53. Federer H., Geometric Measure Theory, Springer, New York (1960).

    MATH  Google Scholar 

  54. 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 

  55. Hajłasz P., “Change of variables formula under minimal assumptions,” Colloq. Math., vol. 64, no. 1, 93–101 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  56. 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 

  57. Chernikov V. M. and Vodopyanov S. K., “Sobolev spaces and hypoelliptic equations. I,” Siberian Adv. Math., vol. 6, no. 3, 27–67 (1996).

    MathSciNet  Google Scholar 

  58. Chernikov V. M. and Vodopyanov S. K., “Sobolev spaces and hypoelliptic equations. II,” Siberian Adv. Math., vol. 6, no. 4, 64–96 (1996).

    MathSciNet  Google Scholar 

  59. Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University, Princeton (1970).

    MATH  Google Scholar 

  60. Brudnyi Yu. A. and Kotlyar B. D., “A problem in combinatorial geometry,” Sib. Math. J., vol. 11, no. 5, 870–871 (1970).

    Article  Google Scholar 

  61. Reshetnyak Yu. G., “Sobolev-type classes of functions with values in a metric space,” Sib. Math. J., vol. 38, no. 3, 567–582 (1997).

    Article  MATH  Google Scholar 

  62. Halmos P. R., Measure Theory, Springer, New York (1974).

    MATH  Google Scholar 

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Funding

The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project 0314–2019–0006).

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. K. Vodopyanov & N. A. Evseev

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  1. S. K. Vodopyanov
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Correspondence to S. K. Vodopyanov.

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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 2, pp. 283–315. https://doi.org/10.33048/smzh.2022.63.204

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Vodopyanov, S.K., Evseev, N.A. Functional and Analytical Properties of a Class of Mappings of Quasiconformal Analysis on Carnot Groups. Sib Math J 63, 233–261 (2022). https://doi.org/10.1134/S0037446622020045

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