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Survey on the Best Constants in the Theory of One-dimensional Singular Integral Operators

  • Nahum Krupnik
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

A survey on the best constants in the theory of one-dimensional singular integral operators is given. Some open questions are formulated.

Keywords

Norm singular integral operators local principle matrix symbol 

Mathematics Subject Classification (2000)

Primary 47G10 Secondary 47A30 

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References

  1. [AK1]
    R. Avendanio and N. Krupnik, Local principle of computation of the factor-norms of singular integral operators, Funkts. Analiz Priloz. 22 (1988), no. 2, 57–58 (Russian). English translation: Funct. Anal. Appl. 22 (1988), 130–131.Google Scholar
  2. [AK2]
    R. Avendanio and N. Krupnik, On the connection between the norms and essential norms of SIO, Proc. of the I.N. Vekua seminar on applied math. 1 (1988), no. 2, 5–8 (Russian).Google Scholar
  3. [Ba]
    A. Baernstein, Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833–852.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [Bi]
    M.S. Birman, Re-expansion operators as objects of spectral analysis, in: Linear and Complex Analysis Problem Book, Lecture Notes in Math., Springer 1043 1984, 130–134.Google Scholar
  5. [BKS]
    A. Böttcher, N. Krupnik and B. Silbermann, A general look at local principles with special emphasis on the norm computation aspect, Integr. Equat. Oper. Th. 11 (1988), 455–479.zbMATHCrossRefGoogle Scholar
  6. [BGKKRSS]
    A. Böttcher, I. Gohberg, Yu. Karlovich, N. Krupnik, S. Roch, B. Silbermann, I. Spitkovsky, Banach algebras generated by N idempotents and applications, Singular integral operators and related topics (Tel-Aviv,1995), Oper. Theory Adv. Appl., Vol l90, Birkhäuser, Basel, 1996, 19–54.Google Scholar
  7. [Ca]
    A.P. Calderón, On the theorems of M. Riesz and Zygmund, Proc. AMS 1 (1950), 533–534.zbMATHCrossRefGoogle Scholar
  8. [Ca1]
    A.P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci, USA 74 (1977), 1324.zbMATHCrossRefGoogle Scholar
  9. [Co]
    M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Revista Mat. Cuyana 1, (1955), no. 2, 105–167.MathSciNetGoogle Scholar
  10. [DG]
    V. Dybin, S. Grudsky, The Riemann boundary value problem with discontinuities of almost periodic type in the coefficient, Soviet Math. Dokl. 18 (1977), no. 6, 1383–1387.zbMATHGoogle Scholar
  11. [DK]
    R. Duduchava and N. Krupnik, On the norms of singular integral operator on curves with cusps, Integr. Equat. Oper. Th. 20 (1994), no. 4, 377–382zbMATHCrossRefMathSciNetGoogle Scholar
  12. [DKS]
    R. Duduchava, N. Krupnik and E. Shargorodsky, An algebra of integral operators with fixed singularities in kernels, Integr. Equat Oper. Th. 33 (1999), no. 4, 406–425. See also: Preprint KCL-MTH-98-26 (King’s College, London).zbMATHCrossRefMathSciNetGoogle Scholar
  13. [FKM]
    I. Feldman, N. Krupnik and A. Markus, On the norm of polynomials of two adjoint projections, Integr. Equat. Oper. Th. 14 (1991), 70–90.CrossRefMathSciNetGoogle Scholar
  14. [FKS]
    I. Feldman, N. Krupnik and I.M. Spitkovsky, Norms of the singular integral operator with Cauchy kernel along certain contours, Integr. Equat. Oper. Th. 24 (1996), no. 1, 68–80.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Gru]
    S. Grudsky, On the compactness of the integral operator, Dep. in VINITY, 18. 11 (1980), N 4856-80, 10 pp.Google Scholar
  16. [GD]
    K.A. Georgiev and V.M. Deundyak, A criterion for the weighted Cauchy singular operators to belong to the algebra of singular integral operators with coefficients in a Sarason algebra, Func. Anal. Appl. 28 (1994), 287–288.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [GaK]
    J. Galperin and N. Krupnik, On the norms of singular integral operators along certain curves with intersections, Integr. Equat. Oper. Theory 29 (1997), 10–16.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [GoKr]
    I. Gohberg, M. Krein, Theory and Applications of Volterra Operators in Hilbert Space, American Mathematical Society, Providence, 1970, 430 pp.Google Scholar
  19. [GK1]
    I. Gohberg and N. Krupnik,On the norm of the Hilbert transform in the space Lp, Funk. Anal. Prilozhen. 2(1968), no. 2 91–92 (Russian). English translation: Functional Anal. Appl. 2 (1968), 180–181.Google Scholar
  20. [GK2]
    I. Gohberg and N. Krupnik, On the spectrum of SIO on L p, Studia Mathematica, T. XXXI (1968), 347–362.MathSciNetGoogle Scholar
  21. [GK3]
    I. Gohberg and N. Krupnik, On the quotient norm of singular integral operators, Mat. Issled. Kishinev 4 (1969), no. 3, 136–139 (Russian). English translation: Amer. Math. Soc. Transl., (2), III (1978), 117–119.MathSciNetGoogle Scholar
  22. [GK4]
    I. Gohberg and N. Krupnik, On the spectrum of SIO on weighted spaces L p, DAN SSSR 185 (1969), no. 4, 745–748.MathSciNetGoogle Scholar
  23. [GK5]
    I. Gohberg and N. Krupnik, Introduction to the theory of one-dimensional SIO, Shtiintsa, Kishinev, 1973 (Russian).Google Scholar
  24. [GK6]
    I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vol. I, Introduction, O.T. 53, Birkhäuser Verlag, Basel-Boston, 1992, 226 pp.Google Scholar
  25. [GK7]
    I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vol. II, General Theory and Applications, OT 54, Birkhäuser Verlag, Basel-Boston, 1993, 232 pp.Google Scholar
  26. [GK8]
    I. Gohberg and N. Krupnik, Singular integral operators with piecewise continuous coefficients and their symbols, Izv. Akad. Nauk SSSR, Ser. Mat. 35 (1971), 940–964.MathSciNetGoogle Scholar
  27. [HL]
    G.H. Hardy and J.E. Littlewood, Some more theorems concerning Fourier series and Fourier power series, Duke Math. J. 2 (1936), 354–382.CrossRefMathSciNetGoogle Scholar
  28. [HV]
    B. Hollenbeck, I.E. Verbitsky, Best constants for the Riesz projection, Journal of Functional Analysis 175 (2000), 370–392.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [HKV]
    B. Hollenbeck, N.J. Kalton and I.E. Verbitsky, Best constants for some operators associated with the Fourier and Hilbert transforms, Studia Mathematica 157 (2003), no. 3, 237–277.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [I]
    I. Itskovich, Integral of Cauchy type considered as operator in Hilbert space, Uchen. zap. Kishinevs. Univ. 5 (1952), 37–41.Google Scholar
  31. [K1]
    N. Krupnik, On the quotient norm of a singular integral operator, Mat. Issled. Kishinev 10 (1975), no. 2, 255–263 (Russian).zbMATHMathSciNetGoogle Scholar
  32. [K2]
    N. Krupnik, On singular integral operators with matrix coefficients, Mat. Issled. Kishinev 45 (1977), 93–100 (Russian).zbMATHMathSciNetGoogle Scholar
  33. [K3]
    N. Krupnik, Banach algebras with symbol and singular integral operators, Stiintsa, Kishinev, 1984 (Russian). English translation: Birkhäuser Verlag, Basel-Boston, 1987, 205 pp.Google Scholar
  34. [K4]
    N. Krupnik,The exact constants in Simonenko’s theorem on an envelope of a family of operators of local type, Funkt. Anal. Prilozhen. 20 (1986), no. 2, 119–120. (Russian). English translation: Funct. Anal. Appl. 20, (1986), no. 2, 144–145.MathSciNetGoogle Scholar
  35. [K5]
    N. Krupnik, The conditions of selfadjointness of the operator of singular integration, Integr. Equat. Oper. Th. 14 (1991), 760–763.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [K6]
    N. Krupnik, Symmetrization of the symbol in Banach algebras generated by idempotents, Integr. Equat. Oper. Th. 31 (1998), 470–481.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [Ka1]
    A. Yu. Karlovich, On the essential norm of the Cauchy singular operator in weighted rearrangement-invariant spaces, Integr. Equat. Oper. Th. 38 (2000), 28–50.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [Ka2]
    A.Yu. Karlovich, Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces, J. Integr. Equat. Appl. 15 (2003), 263–320.zbMATHCrossRefMathSciNetGoogle Scholar
  39. [KN]
    N. Krupnik and V.I. Neagu, On singular integral operators in weighted L p spaces, Mat. Issled. Kishinev 9 (1974), no. 3, 206–209 (Russian).zbMATHGoogle Scholar
  40. [KP]
    N. Krupnik and E.P. Polonskii, The norm of an operator of singular integration, Funkt. Anal. Prilozhen. 9 (1975), no. 3, 73–74 (Russian). English translation: Funct. Anal. App. 9, (1975), no. 4, 337–339.MathSciNetGoogle Scholar
  41. [KS]
    N. Krupnik, I.M. Spitkovsky, On the norms of singular integral operators on contour with intersections, Complex Analysis and Operator Theory 2 (2008), o. n4, 617–626.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [KRA]
    Yu.I. Karlovich and E. Ramírez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces, Integr. Equat. Oper. Theory 48 (2004), 331–363.zbMATHCrossRefGoogle Scholar
  43. [L]
    V.E. Ljance, Some properties of idempotent operators, Teor. i Prikl. Mat. 1 (1959), 16–22 (Russian).MathSciNetGoogle Scholar
  44. [N]
    V.I. Neagu, On the essential norms of SIO for some non-smooth contours, Mat. Issled. 9 (1974), no. 2, 109–125.MathSciNetGoogle Scholar
  45. [Ni]
    N.K. Nikolskii, Treatise of the shift operators: Spectral function theory, Springer Verlag, Berlin, New York, 1988.Google Scholar
  46. [NY]
    T. Nakazi and T. Yamamoto, Norms of some singular integral operators and their inverse operators, J. Operator Theory 40 (1998), 185–207.zbMATHMathSciNetGoogle Scholar
  47. [Pa]
    S. Papadopoulos, A note on the M. Riesz theorem for the conjugate function, Bull. Polish Acad. Sci. Math. 47 (1999), 283–288.zbMATHMathSciNetGoogle Scholar
  48. [Pe]
    A. Pelczyńsky, Norms of classical operators in function spaces, Colloque en l’honneur of Laurent Schwartz, Vol. I, Astérisk 131 (1985), 137–162.Google Scholar
  49. [Pi]
    S.K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia mathematica 19 (1972), no. 2, 165–170.MathSciNetGoogle Scholar
  50. [Po]
    I. Pokorny, On the norm of singular operator acting on an ellipse, Casopis pro Pestovani Matematiky 110 (1985), 158–171.zbMATHMathSciNetGoogle Scholar
  51. [R]
    M. Riesz, Sur les functions conjuguées, Math.Z. (1927), 213–244.Google Scholar
  52. [S]
    I.B. Simonenko, A new general method of investigating linear operator equations of singular integral type I. Izv. Akad. Nauk SSSR, Ser. Mat. 29 (1965), no. 3, 567–586 (Russian).MathSciNetGoogle Scholar
  53. [Sp]
    I.M. Spitkovsky, The factorization of matrix-valued functions whose Hausdorff set lies inside an angle, Soobsch. Akad. Nauk Gruzin. SSR 86 (1977), no. 3, 561–564 (Russian).Google Scholar
  54. [Sp1]
    I.M. Spitkovsky, The partial indices of continuous matrix-valued functions, Dokl. Akad. Nauk SSSR 229 (1976), no. 5, 1059–1062; Soviet Math. Dokl. 17 (1976), no. 4, 1155–1159.MathSciNetGoogle Scholar
  55. [T]
    E.C. Titchmarsh, Reciprocal formulae for series and integrals, M. Z. 25 (1926), 321–347.zbMATHCrossRefMathSciNetGoogle Scholar
  56. [V]
    I. Verbitsky, Estimate of the norm of a function in Hardy space in terms of the norms of its real and imaginary part, Math. Issled. 54 (1980), 16–20 (Russian). English translation: Amer. Math. Soc. Transl. (2) 124 (1984), 11–15.Google Scholar
  57. [VK1]
    I. Verbitsky and N. Krupnik, The exact constants in the theorems of K.I. Babenko and B.V. Khvedelidze on boundedness of singular operators, Soob. Akad. Nauk Gruz. SSR 85 (1977), no. 1, 21–24 (Russian).Google Scholar
  58. [VK2]
    I. Verbitsky and N. Krupnik, The exact constants in theorems on boundedness of singular operators in weighted spaces and their application, Mat. Issled. 54 (1980), 21–35 (Russian).Google Scholar
  59. [VK3]
    I. Verbitsky and N. Krupnik, The norm of analytical projection, Lect. Notes in Math., 1043 (1984), 325–327.Google Scholar
  60. [Z]
    A. Zygmund, Trigonometric series, Vols. I, II, Cambridge Univ. Press, 1959.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Nahum Krupnik
    • 1
  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

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