P-regularity of sets in ¢n

  • Azimba Sadullaev
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 798)


In this paper the problem of P-regularity of compacts in ¢n is considered. With the help of the notion of P-regularity a sufficient condition for P-regularity of compacts is given. An example of a nonregular Jordan domain in the real plane IR2 = {(z,w) ε ¢2: Im z = Im w = 0} is constructed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. S. LANDKCF: Fundamentals of modern potential theory [in Russian]. Izd. "Nauka", Moscow 1966.Google Scholar
  2. [2]
    J. SICKIAK: On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), no. 2, 322–357.MathSciNetCrossRefGoogle Scholar
  3. [3]
    A. SADULLAEV: A boundary uniqueness theorem in ¢n [in Russian], Mat. Sb. (N. S.) 101 (143) (1976), no. 4, 568–583.MathSciNetzbMATHGoogle Scholar
  4. [4]
    A. SADULLAEV: Defect divisors in the sense of Valiron [in Russian], Mat. Sb. (N. S.) 108 (150) (1979), no. 4, 567–580.MathSciNetGoogle Scholar
  5. [5]
    V. P. ZAHARJUTA: Spaces of analytic and harmonic functions of several variables [in Russian], Tezisy Dokladov na Vsesojuznoi Konferencii po Teorii Funkcii, Harkov 1971, pp. 74–78.Google Scholar
  6. [6]
    V. P. ZAHARJUTA: Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernštein-Walsh theorem for functions of several complex variables [in Russian], Proceedings of the Sixth Conference on Analytic Functions (Kraków, 1974), Ann. Polon. Math. 33 (1976/77), no. 1–2, 137–148.MathSciNetGoogle Scholar
  7. [7]
    J. SICIAK: Extremal plurisubharmonic functions in ¢n, Proceedings of the First Finnish-Polish Summer School in Complex Analysis at Podlesice, University of Łódź, Łódź 1977, pp. 115–152.Google Scholar
  8. [8]
    J. SICIAK: Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ¢n, Ann. Polon. Math. 22 (1969), 145–171.MathSciNetzbMATHGoogle Scholar
  9. [9]
    V. P. ZAHARJUTA: Extremal plurisubharmonic functions, Hilbert scales, and the isomorphism of spaces of analytic functions of several variables I [in Russian], Teor. Funkcii Funkcional. Anal. i Priložen. Vyp. 19 (1974), 133–157 and 161.MathSciNetGoogle Scholar
  10. [10]
    J. L. WALSH: Interpolation and approximation by ratiional functions in the complex domain, 2nd ed. (American Math. Soc. Colloq. Publ. 20), American Math. Soc., Boston 1960; Russian translation: Izdatelstvo Inostrannoi Literatury, Moscow 1961.Google Scholar
  11. [11]
    J. SICIAK: Analyticity and separate analyticity of functions defined on lower dimensional subsets of ¢n, Zeszyty Naukowe U. J. 13 (1969), 53–70.MathSciNetzbMATHGoogle Scholar
  12. [12]
    V. P. ZAHARJUTA: Separately analytic functions, generalizations of the Hartogs theorem, and envelopes of holomorphy [in Russian], Mat. Sb. (N. S.) 101 (143) (1976), no. 1, 57–76 and 159.MathSciNetGoogle Scholar
  13. [13]
    W. PLEŚNIAK: Invariance of the L-regularity of compact sets in ¢n under holomorphic mappings, Trans. Amer. Math. Soc. 246 (1978), 373–383.MathSciNetzbMATHGoogle Scholar
  14. [14]
    W. PLEŚNIAK: A criterion of the L-regularity of compact sets in ¢n, Zeszyty Naukowe U. J. 21 (1979), 97–103.MathSciNetzbMATHGoogle Scholar
  15. [15]
    R. M. DUDLEY and B. RANDOL: Implications of pointwise bounds on polynomials, Duke Math. J. 29 (1962), no. 3, 455–458.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. S. BAOUEDI and C. GAULAOUIC: Approximation of analytic functions on compact sets and Bernstein's inequality, Trans. Amer. Soc. 189 (1974), 251–261.Google Scholar
  17. [17]
    S. Ju. FAVOROV: On a problem of V. S. Vladimirov [in Russian], Funkcional. Anal. i Priložen. Tom 12 Vyp. 3 (1978), 90 pp.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Azimba Sadullaev
    • 1
  1. 1.Uнсмuмум мамемамuкu Uм. В. U. Романоьскоs Акаgемuu наук УzССРТащкенм 52Russia

Personalised recommendations