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Function classes defined from local approximations by solutions to hypoelliptic equations

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Abstract

We describe anisotropic function classes of the Campanato—Morrey type in terms of local approximations by solutions to the equation P(D)f = 0 in integral metrics, where P(D) is a quasihomogeneous hypoelliptic linear differential operator with constant coefficients.

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References

  1. Fedorov V. S., “Sur la representatione des fonctions analitiques au voisinage d’un ensemble de ses points singulier,” Mat. Sb., 35, 237–250 (1928).

    Google Scholar 

  2. Kaufman R. and Wu J.-M., “Removable singularities for analytic or subharmonic functions,” Ark. Mat., 18, No. 4, 107–116 (1980).

    MathSciNet  Google Scholar 

  3. Ishchanov B. Zh., “Nonclosed singular sets for weak solutions of linear differential equations,” in: Geometric Questions in the Theory of Functions and Sets [in Russian], Kalinin Univ., Kalinin, 1989, pp. 41–49.

    Google Scholar 

  4. Ishchanov B. Zh., “The representation of solutions to equations of certain classes in terms of potentials of measures concentrated on the set of singularities of these solutions,” Dokl. Akad. Nauk Ross. Akad. Nauk, 358, No. 4, 455–458 (1998).

    MATH  MathSciNet  Google Scholar 

  5. Dolzhenko E. P., “The work of D. E. Men’shov in the theory of analytic functions and the present state of the theory of monogeneity,” Uspekhi Mat. Nauk, 47, No. 5, 67–96 (1992).

    MATH  MathSciNet  Google Scholar 

  6. Pokrovskii A. V., “Local approximations by solutions of hypoelliptic equations, and removable singularities,” Dokl. Akad. Nauk Ross. Akad. Nauk, 367, No. 1, 15–17 (1999).

    MATH  MathSciNet  Google Scholar 

  7. Hörmander L., The Analysis of Linear Partial Differential Operators. Vol. 2 [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  8. Volevich L. R., “Local properties of solutions to quasielliptic systems,” Mat. Sb., 59, No. 3, 3–50 (1962).

    MATH  MathSciNet  Google Scholar 

  9. Pokrovskii A. V., “Mean value theorems for solutions of linear partial differential equations,” Mat. Zametki, 64, No. 2, 260–272 (1998).

    MATH  MathSciNet  Google Scholar 

  10. Peetre J., “On the theory of L p,λ spaces,” J. Funct. Anal., 4, 71–87 (1968).

    MathSciNet  Google Scholar 

  11. Besov O. V., Il’in V. P., and Nikol’skii S. M., Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1996).

    Google Scholar 

  12. Adzhiev S. S., “Characterizations of the function spaces B sp,q (G), L sp, q (G), and W sp (G) and embeddings in BMO(G),” Trudy Mat. Inst. Steklov, 1997, 214, 7–24; Corrections: Trudy Mat. Inst. Steklov, 1997, 219, 463.

    MATH  Google Scholar 

  13. Adzhiev S. S., “Characterizations of B sp,q (G), L sp,q (G), W sp (G) and certain other function spaces. Applications,” Trudy Mat. Inst. Steklov., 227, 7–42 (1999).

    MATH  MathSciNet  Google Scholar 

  14. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Problems, Princeton Univ. Press, Princeton (1983).

    Google Scholar 

  15. Krein S. G., Petunin Yu. I., and Semenov E. M., Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  16. Bari N. K. and Stechkin S. B., “Best approximations and differential properties of two conjugate functions,” Trudy Moskov. Mat. Obshch., 5, 483–521 (1956).

    Google Scholar 

  17. Spanne S., “Some function spaces defined using the mean oscillation over cubes,” Ann. Scuola Norm. Sup. Pisa. Cl. Sci (3), 19, 593–608 (1965).

    MATH  MathSciNet  Google Scholar 

  18. DiBenedetto E., Degenerate Parabolic Equations, Springer-Verlag, New York (1993).

    Google Scholar 

  19. Garnett J., Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  20. Shevchuk I. A., Approximation by Polynomials and Traces of Continuous Functions on an Interval [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  21. Guseinov E. G., “Embedding theorems of classes of continuous functions,” Izv. Akad. Nauk Azerbaidžan. SSR Ser. Mat., 2, 57–60 (1979).

    MATH  MathSciNet  Google Scholar 

  22. Besov O. V., “Continuation of a function beyond the boundary of a domain with preservation of difference-differential properties in L p,” Mat. Sb., 66, No. 1, 80–96 (1965).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the National Academy of Sciences, Kiev, Ukraine

    A. V. Pokrovskii

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  1. A. V. Pokrovskii
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Original Russian Text Copyright © 2006 Pokrovski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) A. V.

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Translated from Sibirski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Matematicheski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Zhurnal, Vol. 47, No. 2, pp. 394–413, March–April, 2006.

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Pokrovskii, A.V. Function classes defined from local approximations by solutions to hypoelliptic equations. Sib Math J 47, 324–340 (2006). https://doi.org/10.1007/s11202-006-0046-1

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  • DOI: https://doi.org/10.1007/s11202-006-0046-1

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