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Mat. Sb. (N.S.), 1975, Volume 98(140), Number 4(12), Pages 591–623 (Mi msb3726)  

This article is cited in 41 scientific papers (total in 41 papers)

Analytic representation of $CR$-functions

E. M. Chirka


Abstract: The author derives a representation for functions satisfying the tangential Cauchy–Riemann equations on a real hypersurface in $\mathbf C^n$ in terms of a “jump” in the boundary values of certain analytic functions. This representation is then applied to local and global problems in holomorphic extension from the hypersurface, to the problem of polynomial approximation, and to a curvilinear “edge of the wedge” theorem.
Bibliography: 25 titles.

Full text: PDF file (3464 kB)
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English version:
Mathematics of the USSR-Sbornik, 1975, 27:4, 526–553

Bibliographic databases:

Document Type: Article
UDC: 517.53
MSC: Primary 32D15; Secondary 32E20
Received: 24.06.1975

Citation: E. M. Chirka, “Analytic representation of $CR$-functions”, Mat. Sb. (N.S.), 98(140):4(12) (1975), 591–623; Math. USSR-Sb., 27:4 (1975), 526–553

Citation in format AMSBIB
\Bibitem{Chi75}
\by E.~M.~Chirka
\paper Analytic representation of $CR$-functions
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 98(140)
\issue 4(12)
\pages 591--623
\mathnet{http://mi.mathnet.ru/msb3726}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=402118}
\zmath{https://zbmath.org/?q=an:0321.32004}
\transl
\jour Math. USSR-Sb.
\yr 1975
\vol 27
\issue 4
\pages 526--553
\crossref{https://doi.org/10.1070/SM1975v027n04ABEH002528}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. G. M. Henkin, “The Lewy equation and analysis on pseudoconvex manifolds”, Russian Math. Surveys, 32:3 (1977), 59–130  mathnet  crossref  mathscinet  zmath
    2. R. O. Uells (ml.), “Teoriya funktsii na differentsiruemykh mnogoobraziyakh v $\mathbb C^n$”, UMN, 33:1(199) (1978), 157–193  mathnet  mathscinet  zmath
    3. R. A. Airapetyan, “Analytic representations for $\operatorname{CR}$-functions satisfying Hölder's condition”, Russian Math. Surveys, 35:5 (1980), 247–248  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. R. A. Airapetyan, “Boundary properties of an integral of Cauchy–Leray type in piecewise smooth domains in $\mathbf C^n$, and some applications”, Math. USSR-Sb., 40:1 (1981), 1–20  mathnet  crossref  mathscinet  zmath  isi
    5. Khenkin G., “An Analytic Representation for Cr-Functions on Manifolds of Quasi-Dimension 2 in Cn”, 250, no. 2, 1980, 300–303  mathscinet  zmath  isi
    6. Airapetian R., Khenkin G., “Analytical Extension of Cr-Functions Through Wedge Point”, 259, no. 4, 1981, 777–781  mathscinet  isi
    7. B. Jöricke, “The relation between the solid modulus of continuity and the modulus of continuity along the Shilov boundary for analytic functions of several variables”, Math. USSR-Sb., 50:2 (1985), 495–511  mathnet  crossref  mathscinet  zmath
    8. R. A. Airapetyan, G. M. Henkin, “Integral representations of differential forms on Cauchy–Riemann manifolds and the theory of CR-functions”, Russian Math. Surveys, 39:3 (1984), 41–118  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. R. A. Airapetyan, G. M. Henkin, “Integral representations of differential forms on Cauchy–Riemann manifolds and the theory of $CR$-functions. II”, Math. USSR-Sb., 55:1 (1986), 91–111  mathnet  crossref  mathscinet  zmath
    10. R. A. Airapetyan, “Extension of CR-functions from piecewise smooth CR-manifolds”, Math. USSR-Sb., 62:1 (1989), 111–120  mathnet  crossref  mathscinet  zmath
    11. Christine Laurent-Thiebaut, “Sur l'extension des fonctions CR dans une variété de stein”, Annali di Matematica, 150:1 (1988), 141  crossref  mathscinet  zmath  isi
    12. A. M. Kytmanov, “On the removal singularities of integrable CR functions”, Math. USSR-Sb., 64:1 (1989), 177–185  mathnet  crossref  mathscinet  zmath
    13. A. M. Kytmanov, “The $\bar\partial$ Neumann problem for smooth functions and distributions”, Math. USSR-Sb., 70:1 (1991), 72–92  mathnet  crossref  mathscinet  zmath
    14. A. E. Tumanov, “Extension of CR-functions into a wedge”, Math. USSR-Sb., 70:2 (1991), 385–398  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. A. M. Kytmanov, “Holomorphic extension of CR-functions with singularities on a hypersurface”, Math. USSR-Izv., 37:3 (1991), 681–691  mathnet  crossref  mathscinet  zmath  adsnasa
    16. L. A. Aizenberg, A. M. Kytmanov, “On the possibility of holomorphic extension into a domain of function defined on a connected piece of its boundary”, Math. USSR-Sb., 72:2 (1992), 467–483  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. A. M. Kytmanov, T. N. Nikitina, “Holomorphic extension of $CR$-functions with singularities on a generic manifold”, Russian Acad. Sci. Izv. Math., 40:3 (1993), 623–635  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    18. A. B. Sukhov, “On extension of analytic sets throufh generic manifolds”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 203–212  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    19. L. A. Aizenberg, A. M. Kytmanov, “On the possibility of holomorphic extension, into a domain, of functions defined on a connected piece of its boundary. II”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 1–10  mathnet  crossref  mathscinet  zmath  isi
    20. E. M. Chirka, “Radó's theorem for CR-mappings of hypersurfaces”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 243–259  mathnet  crossref  mathscinet  zmath  isi
    21. Shlapunov A. Tarkhanov N., “Bases with Double Orthogonality in the Cauchy-Problem for Systems with Injective Symbols”, Proc. London Math. Soc., 71:Part 1 (1995), 1–52  crossref  mathscinet  zmath  isi
    22. Yarmukhamedov S., “Integral-Representation of Cr-Function and Holomorphic Continuation”, Dokl. Akad. Nauk, 341:5 (1995), 600–602  mathnet  mathscinet  zmath  isi
    23. Moulay-Youssef Barkatou, “Régularité höldérienne du
      $$\bar \partial _b $$
      dans les hypersurfaces 1-convexes-concavesdans les hypersurfaces 1-convexes-concaves”, Math Z, 221:1 (1996), 549  crossref  mathscinet  zmath  isi
    24. Coupet B., Sukhov A., “On Cr Mappings Between Pseudoconvex Hypersurfaces of Finite Type in C-2”, Duke Math. J., 88:2 (1997), 281–304  crossref  mathscinet  zmath  isi
    25. Proc. Steklov Inst. Math., 235 (2001), 64–86  mathnet  mathscinet  zmath
    26. E. M. Chirka, “Levi and Trépreau Theorems for Continuous Graphs”, Proc. Steklov Inst. Math., 235 (2001), 261–276  mathnet  mathscinet  zmath
    27. A. M. Kytmanov, M. S. Myslivets, “On $CR$-distributions defined on hypersurfaces”, Russian Math. (Iz. VUZ), 45:10 (2001), 44–48  mathnet  mathscinet  zmath  elib
    28. R. Michael Range, “Extension phenomena in multidimensional complex analysis: correction of the historical record”, Math Intelligencer, 24:2 (2002), 4  crossref  mathscinet
    29. S. G. Myslivets, “Analiticheskoe predstavlenie $CR$-funktsii na giperpoverkhnostyakh s osobennostyami”, Fundament. i prikl. matem., 8:4 (2002), 1069–1090  mathnet  mathscinet  zmath  elib
    30. Kytmanov A., Myslivets S., Tarkhanov N., “Removable Singularities of Cr Functions on Singular Boundaries”, Math. Z., 242:3 (2002), 491–515  crossref  mathscinet  zmath  isi
    31. Kytmanov A., Myslivets S., Tarkhanov N., “Analytic Representation of Cr Functions on Hypersurfaces with Singularities”, Math. Nachr., 244 (2002), 170–183  crossref  mathscinet  zmath  isi
    32. T. N. Nikitina, “$\overline\partial$-zamknutoe prodolzhenie $CR$-form s osobennostyami na porozhdayuschem mnogoobrazii”, Sib. elektron. matem. izv., 2 (2005), 264–290  mathnet  mathscinet  zmath
    33. Kytmanov A., Myslivets S., Schulze B., Tarkhanov N., “Elliptic Problems for the Dolbeault Complex”, Math. Nachr., 278:1-2 (2005), 194–215  crossref  mathscinet  zmath  isi  elib
    34. T. N. Nikitina, “Odnostoronnyaya $\overline{\partial}$- i $\overline{\partial}^*$-zamknutost $CR$-form v fiksirovannoi oblasti”, Vestn. NGU. Ser. matem., mekh., inform., 5:2 (2005), 28–57  mathnet
    35. D. P. Fedchenko, A. A. Shlapunov, “On the Cauchy problem for the multi-dimensional Cauchy-Riemann operator in the Lebesgue space $L^2$ in a domain”, Sb. Math., 199:11 (2008), 1715–1733  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    36. Chakrabarti D. Shafikov R., “Holomorphic Extension of Cr Functions From Quadratic Cones”, Math. Ann., 341:3 (2008), 543–573  crossref  mathscinet  zmath  isi  elib
    37. Forstneric F. Laurent-Thiebaut Ch., “Stein Compacts in Levi-Flat Hypersurfaces”, Trans. Am. Math. Soc., 360:1 (2008), 307–329  crossref  mathscinet  zmath  isi
    38. Ivan V. Shestakov, Alexander A. Shlapunov, “Negative Sobolev Spaces in the Cauchy Problem for the Cauchy–Riemann Operator”, Zhurn. SFU. Ser. Matem. i fiz., 2:1 (2009), 17–30  mathnet  elib
    39. Chakrabarti D., Shafikov R., “Cr Functions on Subanalytic Hypersurfaces”, Indiana Univ. Math. J., 59:2 (2010), 459–494  crossref  mathscinet  zmath  isi
    40. Alexander M. Kytmanov, “Some application of the Bochner–Martinelli integral”, Zhurn. SFU. Ser. Matem. i fiz., 4:1 (2011), 32–42  mathnet  elib
    41. R. Shamoyan, S. Kurilenko, “Traces and distances in analytic function spaces in $C^n$ and Martinelly–Bochner integrals”, Chebyshevskii sb., 16:2 (2015), 254–272  mathnet  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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