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Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations

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References

  1. S. K. Vodop'yanov, “Intrinsic geometries and boundary values of differentiable functions. I,” Sibirsk. Mat. Zh.,30, No. 2, 29–42 (1989).

    Google Scholar 

  2. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton (1982) (Math. Notes,28).

    Google Scholar 

  3. L. Hörmander, “Hypoelliptic second order differential equations,” Acta Math.,119, 141–171 (1967).

    Google Scholar 

  4. A. Nagel, E. M. Stein, and S. Wainger, “Balls and metrics defined by vector fields. I: Basic properties,” Acta Math.,155, No. 1–2, 103–147 (1985).

    Google Scholar 

  5. V. G. Maz'ya, “On continuity at a boundary point of solutions to quasilinear elliptic equations,” Vestnik Leningrad Univ. Mat.,25, 42–55 (1970).

    Google Scholar 

  6. E. B. Fabes, D. S. Jerison, and C. E. Kenig, “The Wiener test for degenerate elliptic equations,” Ann. Inst. Fourier (Grenoble),32, No. 3, 151–182 (1982).

    Google Scholar 

  7. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford-New York-Tokyo (1993).

    Google Scholar 

  8. L. Capogna, D. Danielli, and N. Garofalo, “Embedding theorems and Harnack inequality for solutions of nonlinear subelliptic equations,” C. R. Acad. Sci. Paris. Sér. I Math.,316, 809–814 (1993).

    Google Scholar 

  9. B. Franchi, S. Gallot, and R. Wheeden, “Inéqalités isopérimetriques pour des métriques dégénerees,” C. R. Acad. Sci. Paris. Sér. I Math.,317, 651–654 (1993).

    Google Scholar 

  10. D. J. Jerison, “The Poincaré inequality for vector fields satisfying Hörmander's condition,” Duke. Math. J.,53, No. 2, 503–523 (1986).

    Google Scholar 

  11. G. Lu, “Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications,” Rev. Mat. Iberoamericana,8, No. 3, 367–439 (1992).

    Google Scholar 

  12. A. Korányi and H. M. Reimann, Foundations for the Theory of Quasiconformal Mappings on the Heisenberg Group [Preprint], Inst. Math., Bern (1991).

    Google Scholar 

  13. S. K. Vodop'yanov and V. M. Chernikov, “Sobolev spaces and hypoelliptic equations,” in: Trudy Inst. Mat. (Novosibirsk). Vol. 29, Novosibirsk, 1995, pp. 3–64.

    Google Scholar 

  14. S. K. Vodop'yanov, “L p-potential theory and quasiconformal mappings on homogeneous groups,” in: Contemporary Problems on Geometry and Analysis [in Russian], Nauka, Novosibirsk, 1989, pp. 45–89.

    Google Scholar 

  15. S. K. Vodop'yanov, Geometric Aspects of the Spaces of Functions Differentiable in a Generalized Sense [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Inst. Mat. (Novosibirsk), Novosibirsk (1992).

    Google Scholar 

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

    Google Scholar 

  17. N. Aronszajn and K. T. Smith, “Functional spaces and functional completion,” Ann. Inst. Fourier (Grenoble),6, 125–185 (1956).

    Google Scholar 

  18. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Acad. Press, New York (1980).

    Google Scholar 

  19. N. Wiener, “Certain notions in potential theory,” J. Math. Phys.,3, 24–51 (1924).

    Google Scholar 

  20. S. K. Vodop'yanov, “Thin sets in weighted potential theory and degenerate elliptic equations,” Sibirsk. Mat. Zh.,36, No. 1, 28–36 (1995).

    Google Scholar 

  21. T. Bagby, “Quasi-topologies and rational approximation,” J. Funct. Anal.,10, 259–268 (1972).

    Google Scholar 

  22. J. C. Polking, “Approximation inL p by solutions of elliptic partial differential equations,” Amer. Math. J.,94, 1231–1244 (1972).

    Google Scholar 

  23. V. I. Burenkov, “On approximation of functions in the spacesW rp (Ω) by compactly-supported functions for an arbitrary open set Ω,” Trudy Mat. Inst. Steklov,131, 51–63 (1974).

    Google Scholar 

  24. L. I. Hedberg, “Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem,” Acta Math.,147, 237–264 (1981).

    Google Scholar 

  25. N. O. Belova, “Spectral synthesis in weighted Sobolev spaces,” Mat. Zametki,50, No. 2, 136–139 (1994).

    Google Scholar 

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This research was financially supported by the Russian Foundation for Fundamental Research (Grant 93-011-228).

Translated from Sibirskiĩ Matematicheskiĩ, Vol. 36, No. 2, pp. 278–300, March–April, 1995.

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Vodop'yanov, S.K. Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations. Sib Math J 36, 246–264 (1995). https://doi.org/10.1007/BF02110147

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  • DOI: https://doi.org/10.1007/BF02110147

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