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Maximum principle in potential theory and imbedding theorems for anisotropic spaces of differentiable functions

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Literature Cited

  1. N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  2. M. Brelot, Foundations of Classical Potential Theory [Russian translation], Mir, Moscow (1964).

    Google Scholar 

  3. L. Carleson, Selected Problems of the Theory of Exceptional Sets [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  4. Yu. G. Reshetnyak, “Capacity in the theory of functions with generalized derivatives,” Sib. Mat. Zh.,10, No. 5, 1109–1138 (1969).

    Google Scholar 

  5. Yu. G. Reshetnyak, “Boundary behavior of functions with generalized derivatives,” Sib. Mat. Zh.,13, No. 2, 411–419 (1972).

    Google Scholar 

  6. V. G. Maz'ya and V. P. Khavin, “Nonlinear potential theory,” Usp. Mat. Nauk,27, No. 6, 67–138 (1972).

    Google Scholar 

  7. N. G. Meyers, “A theory of capacities for potentials of functions in Lebesgue classes,” Math. Scand.,26, No. 2, 255–292 (1970).

    Google Scholar 

  8. D. R. Adams and N. G. Meyers, “Thiness and Wiener criteria for nonlinear potentials,” Indiana Univ. Math. J.,22, No. 2, 169–197 (1972).

    Google Scholar 

  9. D. R. Adams and N. G. Meyers, “Bessel potentials. Inclusion relations among classes of exceptional sets,” Indiana Univ. Math. Zh.,22, No. 9, 873–905 (1973).

    Google Scholar 

  10. L. I. Hedberg, “Nonlinear potentials and approximation in the mean by analytic functions,” Math. Zh.,129, 299–319 (1972).

    Google Scholar 

  11. K. Hanson, “Imbedding theorems of Sobolev type in potential theory,” Math. Scand.,45, 77–102 (1979).

    Google Scholar 

  12. T. Ugaheri, “On the general potential and capacity,” Jpn. J. Math.,20, 37–43 (1950).

    Google Scholar 

  13. N. Ninomiya, “Sur le principle de continuité dans la theorie du potentiel,” J. Inst. Polytechn. Osaka City Univ.,8, 51–56 (1957).

    Google Scholar 

  14. S. K. Vodop'yanov, “Geometric properties of domains which satisfy the extension condition for spaces of differentiable functions,” in: Some Applications of Functional Analysis to Problems of Mathematical Physics: Proceedings of the S. L. Sobolev Seminar [in Russian], No. 2, Novosibirsk (1984), pp. 65–95.

  15. S. K. Vodop'yanov, “Isoperimetric relations and extension conditions for differentiable functions,” Dokl. Akad. Nauk SSSR,292, No. 1, 11–16 (1987).

    Google Scholar 

  16. S. K. Vodop'yanov, “Geometric properties of domains and estimates for the norm of the extension operator,” Dokl. Akad. Nauk SSSR, No. 4, 791–796 (1987).

    Google Scholar 

  17. S. K. Vodop'yanov, “Geometric properties of maps and domains lower bounds for the norm of the extension operator,” in: Studies on Geometry and Mathematical Analysis: Proceedings of the Institute of Mathematics, Siberian Section, Academy of Sciences of the USSR [in Russian], Vol. 7, Nauka, Novosibirsk (1987), pp. 70–101.

    Google Scholar 

  18. V. G. Maz'ya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).

  19. D. R. Adams, Lectures on Lp-Potential Theory, UMEA (Preprint No. 2) (1981).

  20. B. Dahlberg, “Regularity properties of Riesz potentials,” Ind. Univ. Math. J.,28, 257–268 (1979).

    Google Scholar 

  21. V. G. Maz'ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman Advanced Publ. Program, Boston-London-Melbourne (1985).

    Google Scholar 

  22. S. K. Vodop'yanov, “Maximum principle in potential theory,” in: Abstracts of Reports to the Eleventh All-Union School on Operator Theory in Function Spaces (Chelyabinsk, May, 1986) [in Russian], Part 1, Chelyabinsk (1986), p. 29.

  23. S. K. Vodop'yanov, “Anisotropic spaces of differentiable functions and quasiconformal maps,” in: Abstracts of Reports to the Eleventh All-Union School on Operator Theory in Function Spaces (Chelyabinsk, May, 1986) [in Russian], Part II, Chelyabinsk (1986), p. 23.

  24. V. G. Maz'ya, “Removable singularities of bounded solutions of quasilinear elliptic equations of any order,” J. Sov. Math.,3, No. 4 (1975).

  25. M. S. Alborova and S. K. Vodop'yanov, “Removable singularities for bounded solutions of quasielliptic equations,” Novosibirsk, 1987, Dep. in VINITI, Feb. 4, 1987, No. 804-B87.

  26. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes, No. 28, Princeton Univ. Press (1982).

  27. O. V. Besov, V. P. Il'in, and P. I. Lizorkin, “Lp-estimates of a class of nonisotropically-singular integrals,” Dokl. Akad. Nauk SSSR,169, No. 6, 1250–1253 (1966).

    Google Scholar 

  28. O. V. Besov and P. I. Lizorkin, “Singular integral operators and sequences of convolutions in Lp spaces,” Mat. Sb.,73, No. 1, 65–88 (1967).

    Google Scholar 

  29. E. B. Fabes and N. M. Riviére, “Singular integrals with homogeneity,” Stud. Math.,27, 19–38 (1966).

    Google Scholar 

  30. E. M. Stein and S. Wainger, “Problems in harmonic analysis related to the curvature,” Bull. Am. Math. Soc.,84, 1239–1295 (1978).

    Google Scholar 

  31. V. M. Gol'dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Maps [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  32. A. A. Davtyan, “Anisotropic potentials, their treatment, and some applications,” Dokl. Akad. Nauk SSSR,285, No. 3, 537–541 (1985).

    Google Scholar 

  33. D. R. Adams, “A trace inequality for generalized potentials,” Stud. Math.,48, No. 1, 99–105 (1973).

    Google Scholar 

  34. P. I. Lizorkin, “Generalized Liouville differentiation and the method of multipliers in the theory of imbeddings of classes of differentiable functions,” Tr. Mat. Inst. im. V. A. Steklova AN SSSR,105, 89–167 (1969).

    Google Scholar 

  35. P. I. Lizorkin, “Description of the spaceL (r)p (R n) in terms of differences of singular integrals,” Mat. Sb.,81, No. 1, 79–91 (1970).

    Google Scholar 

  36. H. Dappa and W. Trebels, “On hypersingular integrals and anisotropic Bessel potential spaces,” Trans. Am. Math. Soc.,286, 419–429 (1984).

    Google Scholar 

  37. H. Dappa and W. Trebels, “Pointwise multiplication on anisotropic Bessel potential spaces,” in: Abstracts A. Haar Memorial Conf. (Budapest, Aug. 1985), Budapest (1985), p. 16.

  38. H. Dappa and W. Trebels, “On L1-criteria for quasiradial Fourier multipliers with applications to some anisotropic function spaces,” Anal. Math.,9, No. 4, 275–289 (1983).

    Google Scholar 

  39. H. Dappa and W. Trebels, “A difference quotient norm for anisotropic Bessel potential spaces,” Math. Nachr.,132, 163–174 (1987).

    Google Scholar 

  40. V. I. Yudovich, “Estimates connected with integral operators and solutions of elliptic equations,” Dokl. Akad. Nauk SSSR,138, No. 4, 805–808 (1961).

    Google Scholar 

  41. S. I. Pokhozhaev, “Sobolev imbedding theorem in the case pℓ=n,” in: Reports to the Scientific-Engineering Conference of the Energy Inst., Mathematics Section [in Russian], Moscow Energy Inst., Moscow (1965), pp. 158–170.

    Google Scholar 

  42. N. S. Trudinger, “On imbedding into Orlicz spaces and some applications,” J. Math. Mech.,17, 473–483 (1967).

    Google Scholar 

  43. O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  44. V. A. Solonnikov, “Inequalities for functions from the classesW l p (R n),” J. Sov. Math.,3, No. 4 (1975).

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Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 2, pp. 17–33, March–April, 1988.

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Vodop'yanov, S.K. Maximum principle in potential theory and imbedding theorems for anisotropic spaces of differentiable functions. Sib Math J 29, 176–189 (1988). https://doi.org/10.1007/BF00969729

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