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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Duality relations in the theory of analytic capacity

Author: S. Ya. Khavinson
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 15 (2003), nomer 1.
Journal: St. Petersburg Math. J. 15 (2004), 1-40
MSC (2000): Primary 31A15
Published electronically: December 31, 2003
MathSciNet review: 1979716
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Abstract: This is a survey of duality relations arising in the theory of analytic capacity and its modifications, namely, the Cauchy capacities with various types of measures. Principal attention is paid to the material not published earlier. Also, new modifications of the capacities mentioned above are treated. Linear extremal problems (such as the problem of calculating the analytic capacity of a set) are dual to approximation processes with size constraints (the size of the approximants and the rate of approximation are measured in terms of different metrics). For the new versions of capacity introduced in this paper, approximation with size constraints is extended to the case where the approximants are taken from a fixed conical wedge (rather than from a linear subspace, as it has always been before). Extremely peculiar approximation processes arise via duality from the modifications of the analytic capacity the definition of which involves positive measures (a typical example is the so-called ``positive'' analytic capacity $ \gamma ^{+} $). More precisely, in such situations it is not even required to find an approximant within a small distance from a given element. Instead, in a fixed subspace or conical wedge we seek an addition that brings the above element to a fixed cone. Moreover, this addition must be as small as possible in a certain sense.

Extension of the collection of relations for the analytic capacity and its modifications, and comparison of various relations of this sort make it possible to better understand exceptional sets arising in various branches of holomorphic function theory. In particular, some information is obtained about possible approximation processes on null-sets in the sense of a particular capacity.

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  • 1. L. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), no. 1, 1-11. MR 9:24a
  • 2. John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
  • 3. John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
  • 4. Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
  • 5. A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838
  • 6. A. G. Vituškin, Uniform approximations by holomorphic functions, Current problems in mathematics, Vol. 4 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1975, pp. 5–12. (errata insert) (Russian). MR 0486635
  • 7. S. Ya. Khavinson, Supplementary questions of the theory of removable singularities of analytic functions, Moskov. Inzh.-Stroit. Inst. (Fak. povysheniya kvalif.), Moscow, 1982. (Russian)
  • 8. S. Ya. Khavinson, Golubev sums: a theory of extremal problems that are of the analytic capacity problem type and of accompanying approximation processes, Uspekhi Mat. Nauk 54 (1999), no. 4(328), 75–142 (Russian, with Russian summary); English transl., Russian Math. Surveys 54 (1999), no. 4, 753–818. MR 1741279,
  • 9. Lawrence Zalcmann, Analytic capacity and rational approximation, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968. MR 0227434
  • 10. A. A. Gonchar and S. N. Mergelyan, Approximation theory for functions of a complex variable, History of Mathematics in Russia. Vol. 4. Book 1, ``Naukova Dumka'', Kiev, 1970, pp. 112-193. (Russian)
  • 11. M. S. Mel'nikov and S. O. Sinanyan, Aspects of approximation theory for functions of one complex variable, Itogi Nauki i Tekhniki Ser. Sovrem. Probl. Mat., vol. 4, VINITI, Moscow, 1975, pp. 143-250; English transl., J. Soviet Math. 5 (1976), no. 5, 688-750.
  • 12. S. Ja. Havinson, Representation and approximation of functions on thin sets, Contemporary Problems in Theory Anal. Functions (Internat. Conf., Erevan, 1965) Izdat. “Nauka”, Moscow, 1966, pp. 314–318 (Russian). MR 0210923
  • 13. M. S. Mel′nikov, Analytic capacity: a discrete approach and the curvature of measure, Mat. Sb. 186 (1995), no. 6, 57–76 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 6, 827–846. MR 1349014,
  • 14. A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 0466568
  • 15. Guy David, Analytic capacity, Calderón-Zygmund operators, and rectifiability, Publ. Mat. 43 (1999), no. 1, 3–25. MR 1697514,
  • 16. X. Tolsa, Curvature of measures, Cauchy singular integral, and analytic capacity, Thesis, Univ. Autònoma Barcelona, Barcelona, 1998.
  • 17. -, The semiadditivity of analytic capacity, Preprint, 2001, pp. 1-39.
  • 18. P. R. Garabedian, Schwarz's lemma and the Szegö kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. MR 11:340f
  • 19. -, The classes $L_p$ and conformal mapping, Trans. Amer. Math. Soc. 69 (1950), 392-415. MR 12:492a
  • 20. S. Ja. Havinson, The analytic capacity of plane sets, some classes of analytic functions and the extremum function in Schwarz’s lemma for arbitrary regions, Dokl. Akad. Nauk SSSR 128 (1959), 896–898 (Russian). MR 0118836
  • 21. S. Ja. Havinson, The analytic capacity of sets as related to mass distributions, Dokl. Akad. Nauk SSSR 128 (1959), 1129–1131 (Russian). MR 0118837
  • 22. S. Ja. Havinson, The analytic capacity of sets related to the non-triviality of various classes of analytic functions, and on Schwarz’s lemma in arbitrary domains, Mat. Sb. (N.S.) 54 (96) (1961), 3–50 (Russian). MR 0136720
  • 23. S. Ja. Havinson, Approximation on sets of zero analytic capacity, Soviet Math. Dokl. 1 (1960), 205–207. MR 0123723
  • 24. V. P. Havin, On the space of bounded regular functions, Soviet Math. Dokl. 1 (1960), 202–204. MR 0120525
  • 25. V. P. Havin, On the space of bounded regular functions, Sibirsk. Mat. Ž. 2 (1961), 622–638 (Russian). MR 0138985
  • 26. R. È. Val′skiĭ, Some remarks on bounded functions representable by an integral of Cauchy-Stieltjes type, Sibirsk. Mat. Ž. 7 (1966), 252–260 (Russian). MR 0196097
  • 27. V. P. Havin and N. K. Nikolski (eds.), Linear and complex analysis. Problem book 3. Part I, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR 1334345
    V. P. Havin and N. K. Nikolski (eds.), Linear and complex analysis. Problem book 3. Part II, Lecture Notes in Mathematics, vol. 1574, Springer-Verlag, Berlin, 1994. MR 1334346
  • 28. Takafumi Murai, Construction of 𝐻¹ functions concerning the estimate of analytic capacity, Bull. London Math. Soc. 19 (1987), no. 2, 154–160. MR 872130,
  • 29. Nobuyuki Suita, On a metric induced by analytic capacity, Kōdai Math. Sem. Rep. 25 (1973), 215–218. MR 0318477
  • 30. S. Ja. Havinson, On approximation with account taken of the size of the coefficients of the approximants, Trudy Mat. Inst. Steklov. 60 (1961), 304–324 (Russian). MR 0136721
  • 31. Ph. Davis and K. Fan, Complete sequences and approximations in normed linear spaces, Duke Math. J. 24 (1957), no. 2, 183-192. MR 19:30d
  • 32. S. Ja. Havinson, Some problems concerning the completeness of systems, Soviet Math. Dokl. 2 (1961), 358–361. MR 0123722
  • 33. S. Ja. Havinson, Some approximation theorems involving the values of the coefficients of the approximating functions, Dokl. Akad. Nauk. SSSR 196 (1971), 1283–1286 (Russian). MR 0291693
  • 34. S. Ja. Havinson, A notion of completeness that takes into account the magnitude of the coefficients of the approximating polynomials, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2-3, 221–234 (Russian, with Armenian and English summaries). MR 0296582
  • 35. S. Ya. Khavinson, Complete systems in Banach spaces, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 20 (1985), no. 2, 89–111, 163 (Russian, with English and Armenian summaries). MR 804492
  • 36. N. I. Aheizer and M. Krein, Some questions in the theory of moments, translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2, American Mathematical Society, Providence, R.I., 1962. MR 0167806
  • 37. S. M. Nikol'skii, Approximation of functions in the mean by trigonometric polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 10 (1946), no. 3, 207-256. (Russian) MR 8:149b
  • 38. A. L. Garkavi, Duality theorems for approximation by elements of convex sets, Uspehi Mat. Nauk 16 (1961), no. 4 (100), 141–145 (Russian). MR 0132992
  • 39. V. M. Tikhomirov, \cyr Nekotorye voprosy teorii priblizheniĭ., Izdat. Moskov. Univ., Moscow, 1976 (Russian). MR 0487161
  • 40. N. P. Korneĭchuk, \cyrÈkstremal′nye zadachi teorii priblizheniya., Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0447934
  • 41. Pierre-Jean Laurent, Approximation et optimisation, Hermann, Paris, 1972 (French). Collection Enseignement des Sciences, No. 13. MR 0467080
  • 42. E. G. Gol′shteĭn, \cyr Teoriya dvoĭstvennosti v matematicheskom programmirovanii i ee prilozheniya., Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0322531
  • 43. S. Ya. Khavinson and E. Sh. Chatskaya, Duality relations and criteria for best approximation elements, Moskov. Inzh.-Stroit. Inst. (Fak. povysheniya kvalif.), Moscow, 1976. (Russian)
  • 44. Mahlon M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. Heft 21. Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0094675
  • 45. Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • 46. S. Ya. Khavinson, Factorization of analytic functions in finitely connected domains, Moskov. Inzh.-Stroit. Inst. (Fak. povysheniya kvalif.), Moscow, 1981. (Russian)
  • 47. I. I. Privalov, Boundary properties of analytic functions, Gos. Izdat. Tekhn.-Teor. Lit., Moscow-Leningrad, 1950. (Russian) MR 13:926h
  • 48. P. R. Halmos, Measure theory, D. Van Nostrand Co., Inc., New York, NY, 1950. MR 11:504d
  • 49. Stanisław Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. MR 0167578
  • 50. M. V. Samokhin, On the representability of the Ahlfors function by a Cauchy potential, Nauchn. Vestnik Moskov. Univ. Grazhdan. Aviatsii, Moscow, 1999, pp. 55-59. (Russian)
  • 51. M. V. Samokhin, On the Cauchy integral formula in domains of arbitrary connectivity, Mat. Sb. 191 (2000), no. 8, 113–130 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 7-8, 1215–1231. MR 1786419,
  • 52. Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
  • 53. M. V. Samokhin, Extremal problems for bounded analytic functions and for the classes $E_p$ in arbitrary domains, Sb. Trudov Moskov. Inzh.-Stroit. Inst. No. 153 (1977), 35-48. (Russian)
  • 54. Guy David, Unrectifiable 1-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), no. 2, 369–479 (English, with English and French summaries). MR 1654535,
  • 55. F. Nazarov, S. Treil, and A. Volberg, $T(b)$-theorem and analytical capacity, Preprint, 1998.
  • 56. L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 235–277. MR 0198281
  • 57. P. S. Aleksandrov and A. N. Kolmogorov, Introduction to the general theory of sets and functions, Gos. Izdat. Tekhn.-Teor. Lit., Moscow-Leningrad, 1948; German transl., VEB Deutscher Verlag Wiss., Berlin, 1956. MR 12:682f
  • 58. M. A. Naĭmark, Normed rings, Reprinting of the revised English edition, Wolters-Noordhoff Publishing, Groningen, 1970. Translated from the first Russian edition by Leo F. Boron. MR 0355601

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Keywords: Analytic capacity, Cauchy capacity, approximation with size constraints, exceptional sets
Received by editor(s): May 10, 2002
Published electronically: December 31, 2003
Additional Notes: Supported by RFBR (grant no. 01-01-00608) and by the RF Ministry of Education (grant E 00-1.0-199).
Article copyright: © Copyright 2003 American Mathematical Society