On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces

Integral Equations and Operator Theory

Volumn 38, Issue 1, 2000, Pages 28-50

  Karlovich, Alexei Yu ^a,b  

^a SOUTH UKRAINIAN NATIONAL PEDAGOGICAL UNIVERSITY   (Ukraine)

^b INSTITUTO SUPERIOR TÉCNICO   (Portugal)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0009991111     PISSN: 0378620X     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01192300     Document Type: Article

References (45)

1. R. Avendanio, N. Krupnik, A local principle for computing the quotient norm of singular integral operators, Funk. Anal. i Prilozh. 22 (1988), no. 2, 57-58 (in Russian). English transl.: Funct. Anal. Appl. 22 (1988), 130-131.
2. R. Avendanio, N. Krupnik, A local principle for computing the quotient norm of singular integral operators, Funk. Anal. i Prilozh. 22 (1988), no. 2, 57-58 (in Russian). English transl.: Funct. Anal. Appl. 22 (1988), 130-131.
3. C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, London, 1988.
4. E. I. Berezhnoi, Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 79-87.
5. A. Böttcher, Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Birkhäuser Verlag, Basel, Boston, Berlin, 1997.
6. A. Böttcher, N. Krupnik, B. Silbermann, A general look at local principles with special i emphasis on the norm computation aspect, Integr. Equat. and Oper. Theory 11 (1988), 455-479.
7. A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Akademie Verlag, Berlin, 1989, and Springer Verlag, Berlin, Heidelberg, New York, 1990.
8. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
9. R. Duduchava, N. Krupnik, On the norm of singular integral operator on curves with cusps, Integr. Equat. and Oper. Theory 20 (1994), 377-382.
10. R. Duduchava, N. Krupnik, E. Shargorodsky, An algebra of integral operators with fixed singularities in kernels, Integr. Equat. and Oper. Theory 33 (1999), 406-425.
11. E. M. Dynkin, Methods of the theory of singular integrals (Hilbert transform and Calderón-Zygmund theory), Itogi nauki i tehniki VINITI, Ser. Sovrem. probl. mat. 15 (1987), 197-292 (in Russian). English transl.: Commutative harmonic analysis I. General survey. Classical aspects, Encycl. Math. Sci. 15 (1991), 167-259.
12. E. M. Dynkin, Methods of the theory of singular integrals (Hilbert transform and Calderón-Zygmund theory), Itogi nauki i tehniki VINITI, Ser. Sovrem. probl. mat. 15 (1987), 197-292 (in Russian). English transl.: Commutative harmonic analysis I. General survey. Classical aspects, Encycl. Math. Sci. 15 (1991), 167-259.
13. I. Feldman, N. Krupnik, I. Spitkovsky, Norms of the singular integral operator with Cauchy kernel along certain contours, Integr. Equat. and Oper. Theory 24 (1996), 68-80.
14. D. Gaier, Lectures on Complex Approximation, Birkhäuser Verlag, Basel, Boston, Stuttgart, 1987.
15. I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight theory for integral transforms on spaces of homogeneous type, Pitman Monographs and Surveys in Pure and Applied Mathematics 92, Addison Wesley Longman, 1998.
16. p, Studia Math. 31 (1968), 347-362 (in Russian).
17. p, Funk. Anal. i Prilozh. 2 (1968), no. 2, 91-92 (in Russian). English transl.: Funct. Anal. Appl. 2 (1968), no. 2, 180-181.
18. p, Funk. Anal. i Prilozh. 2 (1968), no. 2, 91-92 (in Russian). English transl.: Funct. Anal. Appl. 2 (1968), no. 2, 180-181.
19. I. Gohberg, N Krupnik, One-Dimensional Linear Singular Integral Equations, Vols. 1, 2, Birkhäuser Verlag, Basel, Boston, Berlin, 1992. Russian original: Shtiintsa, Kishinev, 1973.
20. L. V. Kantorovich, G. P. Akilov, Functional Analysis, Nauka, Moscow, 3rd ed., 1984 (in Russian). English transl: Pergamon Press, Oxford, 2nd ed., 1982.
21. A. Yu. Karlovich, Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces, Math. Nachr. 179 (1996), 187-222.
22. A. Yu. Karlovich, Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces, Integr. Equat. and Oper. Theory 32 (1998), 436-481.
23. A. Yu. Karlovich, On the essential norm of the Cauchy singular integral operator in rearrangement-invariant spaces, Abstracts of the International Conference "Approximation Theory and its Applications" dedicated to the memory of V. K. Dzjadyk, Ukraine, Kyiv, May, 1999, p. 34.
24. V. Kokilashvili, M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, New Jersey, London, Hong Kong: World Scientific, 1991.
25. M. A. Krasnoselskii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958 (in Russian). English transl.: Noordhoff Ltd., Groningen, 1961.
26. S. G. Krein, Ju. I. Petunin, E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian). English transl.: AMS Translations of Mathematical Monographs 54, Providence, R.I., 1982.
27. S. G. Krein, Ju. I. Petunin, E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian). English transl.: AMS Translations of Mathematical Monographs 54, Providence, R.I., 1982.
28. N. Krupnik, Banach algebras with symbol and singular integral operators, Birkhäuser Verlag, Basel, Boston, Berlin, 1987. Russian original: Shtiintsa, Kishinev, 1984.
29. N. Krupnik, E. P. Polonsky, The norm of a singular integration operator, Funk. Anal. i Prilozh. 9 (1975), no. 4, 73-74 (in Russian). English transl.: Functional Anal. Appl. 9 (1975), no. 4, 337-339.
30. N. Krupnik, E. P. Polonsky, The norm of a singular integration operator, Funk. Anal. i Prilozh. 9 (1975), no. 4, 73-74 (in Russian). English transl.: Functional Anal. Appl. 9 (1975), no. 4, 337-339.
31. N. Krupnik, Ya. Spigel, Critical points of essential norms of singular integral operators in weighted spaces, Integr. Equat. and Oper. Theory 33 (1999), 211-220.
32. N. Krupnik, I. Verbitskii, Exact constants in the theorems of K. I. Babenko and B. V. Khvedelidze on the boundedness of singular operators, Soobshch. AN Gruz. SSR 85 (1977), no. 1, 21-24 (in Russian).
33. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I. Function Spaces, New York Berlin: Springer Verlag, 1979.
34. L. Maligranda, Indices and interpolation, Dissert. Math. 234 (1985), 1-49.
35. L. Maligranda, Orlicz Spaces and Interpolation, Sem. Math. 5, Dep. Mat., Univ. Estadual de Campinas, Campinas SP, Brazil, 1989.
36. V. I. Nyaga, On the symbol of singular integral operators in the case of piecewise Lyapunov contours, Mat. issled. 9 (1974), no. 2, 109-125 (in Russian).
37. S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165-179.
38. I. I. Privalov, Boundary Properties of Analytic Functions, GITTL, Moscow, Leningrad, 1950 (in Russian). German transl.: Randeigenschaften analytischer Funktionen, Deutscher Verlag der Wissenschaften, Berlin, 1956.
39. I. I. Privalov, Boundary Properties of Analytic Functions, GITTL, Moscow, Leningrad, 1950 (in Russian). German transl.: Randeigenschaften analytischer Funktionen, Deutscher Verlag der Wissenschaften, Berlin, 1956.
40. S. Roch, B. Silbermann, Algebras of convolution operators and their image in the Calkin algebra, Report R-Math-05/90, Karl-Weierstrass-Inst. f. Math., Berlin, 1990.
41. p with weight, Izv. AN SSSR, Ser. matem. 28, no. 2 (1964), 277-306 (in Russian).
42. I. B. Simonenko, Some general questions in the theory of the Riemann boundary value problem, Izv. AN SSSR, Ser. matem. 32, no. 5 (1968), 1138-1146 (in Russian). English transl.: Math. USSR Izv. 2 (1968), 1091-1099.
43. I. B. Simonenko, Some general questions in the theory of the Riemann boundary value problem, Izv. AN SSSR, Ser. matem. 32, no. 5 (1968), 1138-1146 (in Russian). English transl.: Math. USSR Izv. 2 (1968), 1091-1099.
44. I. Spitkovsky, Singular integral operators with PC symbols on the spaces with general weights, J. Functional Analysis 105 (1992), 129-143.
45. M. Zippin, Interpolation of operators of weak type between rearrangement invariant spaces, J. Functional Analysis 7 (1971), 267-284.