# Mappings of homogeneous groups and imbeddings of functional spaces

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## Keywords

Homogeneous Group Functional Space
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## Literature Cited

- 1.S. K. Vodop'yanov and V. M. Gol'dshtein, “Lattice isomorphisms of the spaces ℒ
_{n}^{1}and quasiconformal mappings”, Sib. Mat. Zh.,16, No. 2, 224–246 (1975).Google Scholar - 2.S. K. Vodop'yanov and V. M. Gol'dshtein, “Functional characterizations of quasiisometric mappings”, Sib. Mat. Zh.,17, No. 4, 768–773 (1976).Google Scholar
- 3.S. K. Vodop'yanov and V. M. Gol'dshtein, “A new functional invariant for quasiconformal mappings”, in: Some Problems in Modern Function Theory (Proc. Conf. Modern Problems of Geometric Theory of Functions, Novosibirsk, 1976) [in Russian], Akad. Nauk SSSR, Sibirsk. Otd., Inst. Mat., Novosibirsk (1976), pp. 18–20.Google Scholar
- 4.V. M. Gol'dshtein and A. S. Romanov, “On mappings preserving Sobolev spaces”, Sib. Mat. Zh.,25, No. 3, 55–61 (1984).Google Scholar
- 5.A. S. Romanov, “On the change of variables in the spaces of Bessel and Riesz potentials”, in: Functional Analysis in Mathematical Physics [in Russian], Academy of Sciences of the USSR, Institute of Mathematics, Novosibirsk (1985), pp. 117–133.Google Scholar
- 6.V. G. Maz'ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston (1985).Google Scholar
- 7.I. G. Markina, “Change of variables as an automorphism of Besov spaces and spaces of Bessel potentials”, in: Materials of the 24th All-Union Scientific Student Conference: Mathematics [in Russian], Novosibirsk State Univ., Novosibirsk (1986), pp. 41–46.Google Scholar
- 8.S. K. Vodop'yanov, “Geometric properties of domains satisfying the extension condition for spaces of differentiable functions”, in: Some Applications of Functional Analysis to Problems of Mathematical Physics [in Russian], Proc. Sobolev Sem., 84-2, Akad. Nauk SSSR, Sibirsk. Otd., Inst. Mat., Novosibirsk (1984), pp. 65–95.Google Scholar
- 9.S. K. Vodop'yanov, “Functional spaces and quasiconformal mappings on homogeneous groups”, in: Proceedings of the Fifth Republican Conference on Nonlinear Problems of Mathematical Physics [in Russian], L'vov (September 1985), Donetsk State University, Donetsk (1987), pp. 107–109. (Manuscript deposited at UkrNIINTI, July 16, 1987, No. 2077.)Google Scholar
- 10.S. K. Vodop'yanov, “Anisotropic spaces of differentiable functions and quasiconformal mappings”, in: Eleventh All-Union School on Operator Theory in Function Spaces, Part 2 (Chelyabinsk, May 1986), Chelyabinsk State Univ., Chelyabinsk (1986), p. 23.Google Scholar
- 11.S. Vodop'janov, “Function spaces and quasiconformal mappings on homogeneous groups”, in: Conference on Function Spaces and Applications (Lund, June 1986), Abstracts, Lund (1986), p. 11.Google Scholar
- 12.S. K. Vodop'janov, “Function spaces and quasiconformal mappings on homogeneous groups”, in: 13 Rolf Nevanlinna Colloquium, Jeonsuu (1987), pp. 79–80.Google Scholar
- 13.S. K. Vodop'yanov, “Isoperimetric relations and conditions for the extension of differentiable functions”, Dokl. Akad. Nauk SSSR,292, No. 1, 11–15 (1987).Google Scholar
- 14.S. K. Vodop'yanov, “Geometric properties of domains and estimates for the norm of an extension operator”, Dokl. Akad. Nauk SSSR,292, No. 4, 791–795 (1987).Google Scholar
- 15.S. K. Vodop'yanov, “Geometric properties of mappings and domains. Lower estimates of the norm of the extension operator”, in: Studies in Geometry and Mathematical Analysis [in Russian], Academy of Sciences of the USSR, Siberian Branch, Institute of Mathematics, Nauka, Novosibirsk (1987), pp. 70–101.Google Scholar
- 16.S. K. Vodop'yanov, “The comparison of metric and capacity characteristics in potential theory”, in: Complex Analysis and Mathematical Physics (Divnogorsk, June–July 1987), Abstracts of Communications [in Russian], Krasnoyarsk (1987), p. 20.Google Scholar
- 17.S. K. Vodop'yanov, “Quasielliptic L
_{p}-potential theory and its applications”, Dokl. Akad. Nauk SSSR,298, No. 4, 780–784 (1988).Google Scholar - 18.S. K. Vodop'yanov, “The maximum principle in potential theory and imbedding theorems for anisotropic spaces of differentiable functions”, Sib. Mat. Zh.,29, No. 2, 17–33 (1988).Google Scholar
- 19.S. K. Vodop'yanov, “Potential theory on homogeneous groups”, Dokl. Akad. Nauk SSSR,303, No. 1, 11–16 (1988).Google Scholar
- 20.S. K. Vodop'yanov, “The L
_{p}-theory of the potential and quasiconformal mappings on homogeneous groups”, in: Contemporary Problems of Geometry and Analysis [in Russian], Academy of Sciences of the USSR, Siberian Branch, Institute of Mathematics, Nauka, Novosibirsk (1989), pp. 45–89.Google Scholar - 21.G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, New Jersey (1982).Google Scholar
- 22.O. V. Besov, V. P. Il'in, and P. I. Lizorkin, “The L
_{p}-estimates of a certain class of nonisotropically singular integrals”, Dokl. Akad. Nauk SSSR,169, No. 6, 1250–1253 (1966).Google Scholar - 23.E. B. Fabes and N. M. Riviere, “Singular integrals with homogeneity”, Stud. Math.,27, 19–38 (1966).Google Scholar
- 24.E. M. Stein and S. Wainger, “Problems in harmonic analysis related with curvature”, Bull. Am. Math. Soc.,84, No. 6, 1239–1295 (1978).Google Scholar
- 25.M. de Guzman, Differentiation of Integrals in
**R**^{n}. Lecture Notes in Math., No. 481, Springer, Berlin (1975).Google Scholar - 26.Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).Google Scholar
- 27.S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).Google Scholar
- 28.O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Vols. I and II, Wiley, New York (1978 and 1979).Google Scholar
- 29.Yu. A. Brudnyi, “Spaces that can be defined by means of local approximations”, Trudy Mosk. Mat. Obshch.,24, 69–132 (1971).Google Scholar
- 30.S. K. Vodop'yanov, “Interior geometry and boundary values of differentiable functions. I”, Sib. Mat. Zh.,30, No. 2, 29–42 (1989).Google Scholar
- 31.G. A. Mamedov, “On the traces of functions from certain anisotropic spaces on the subsets of an Euclidean space”, in: Eighth Republican Conference of Young Scientists in Mathematics and Mechanics (Baku, October 1987), Elm, Baku (1988), pp. 145–146.Google Scholar
- 32.H. M. Reimann, “Functions of bounded mean oscillation and quasiconformal mappings”, Comment. Math. Helv.,49, No. 2, 260–276 (1974).Google Scholar
- 33.J. Peetre and E. Swenson, “On the generalized Hardy's inequality of McGehee, Pigno and Smith and the problem of interpolation between BMO and a Besov space”, Math. Scand.,54, No. 2, 221–241 (1984).Google Scholar
- 34.L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton (1966).Google Scholar
- 35.J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976).Google Scholar
- 36.P. I. Lizorkin, “Generalized Liouville differentiation and the multiplier method in the theory of imbeddings of differentiable functions”, Trudy MIAN SSSR,105, 89–167 (1969).Google Scholar
- 37.H. Dappa and W. Trebels, “On L
_{1}-criteria for quasiradial Fourier multipliers with applications to some anisotropic function spaces”, Anal. Math.,9, No. 4, 275–289 (1983).Google Scholar - 38.H. Dappa and W. Trebels, “A difference quotient norm for anisotropic Bessel potential spaces”, Math. Nachr.,132, 163–174 (1986).Google Scholar
- 39.P. I. Lizorkin, “Description of the spaces
*L*_{p}^{(r)}(**R**^{n}) in terms of singular difference integrals”, Mat. Sb.,81, No. 1, 79–91 (1970).Google Scholar - 40.A. A. Davtyan, “Spaces of anisotropic potentials”, Trudy Mat. Inst. Akad. Nauk SSSR,173, 113–124 (1986).Google Scholar
- 41.M. S. Alborova and S. K. Vodop'yanov, Removable singularities of solutions of quasi-eliptic equations. Novosibirsk State Univ., Novosibirsk (1987). (Manuscript deposited at VINITI, Feb. 4, 1987, No. 804-B87.)Google Scholar
- 42.P. I. Lizorkin, “(L
_{p}, L_{q})-multipliers of Fourier integrals”, Dokl. Akad. Nauk SSSR,152, No. 4, 808–811 (1963).Google Scholar - 43.V. M. Gol'dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).Google Scholar
- 44.N. G. Meyers, “A theory of capacities for potentials of functions in Lebesgue classes”, Math. Scand.,26, No. 2, 255–292 (1970).Google Scholar
- 45.P. R. Halmos, Measure Theory, Van Nostrand, New York (1950).Google Scholar
- 46.V. G. Maz'ya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).Google Scholar
- 47.D. R. Adams, “Traces of potentials. I”, Indiana Univ. Math. J.,22, 907–918 (1973).Google Scholar
- 48.L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand, Princeton (1967).Google Scholar
- 49.S. Vodop'janov, “Function spaces and quasiconformal mappings on homogeneous groups”, in: Third International Symposium on Complex Analysis and Its Applications (Herceg Novi, May 1988), Abstracts, Herceg Novi (1988), p. 36.Google Scholar
- 50.S. K. Vodop'yanov, “Function spaces and quasiconformal mappings on homogeneous groups”, in: All-Union Conference on the Geometric Theory of Functions (Novosibirsk, October 1988), Abstracts of Communications [in Russian], Novosibirsk (1988), p. 20.Google Scholar
- 51.S. K. Vodop'yanov, Taylor's Formula and Function Spaces [in Russian], Novosibirsk State Univ., Novosibirsk (1988).Google Scholar
- 52.V. G. Maz'ya and T. O. Shaposhnikova, Multipliers in Spaces of Differentiable Functions [in Russian], Leningrad State Univ. (1986).Google Scholar
- 53.S. P. Ponomarev, “Submersions and preimages of sets of measure zero”, Sib. Mat. Zh.,28, No. 1, 199–210 (1987).Google Scholar
- 54.P. W. Jones, “Quasiconformal mappings and extendability of functions in Sobolev spaces”, Acta Math.,147, 71–88 (1981).Google Scholar
- 55.B. L. Fain, “On the continuation of functions from anisotropic Sobolev spaces”, Trudy Mat. Inst. Akad. Nauk SSSR,170, 248–272 (1984).Google Scholar
- 56.P. A. Shvartsman, Extension theorems with the preservation of locally polynomial approximations. Yaroslavl' State Univ., Yaroslavl' (1986). (Manuscript deposited at VINITI, Sep. 4, 1986, No. 6457-B86.)Google Scholar

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