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Wiener-Hopf Operators with Oscillating Symbols on Weighted Lebesgue Spaces

  • Yu. I. Karlovich
  • J. Loreto Hernández
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)

Abstract

We establish Fredholm criteria for Wiener-Hopf operators W(a) with oscillating symbols a, continuous on ℝ and admitting mixed (slowly oscillating and semi-almost periodic) discontinuities at ±8, on weighted Lebesgue spaces L N p (ℝ+,w) where 1 < p < ∞, NN, and ±∞ belongs to a subclass of Muckenhoupt weights. For N > 1 these criteria are conditional.

Mathematics Subject Classification (2000)

Primary 47B35 Secondary 42A45 47A53 47G10 

Keywords

Wiener-Hopf operator limit operator weighted Lebesgue space Muckenhoupt weight slowly oscillating and semi-almost periodic matrix functions local principle symbol Fredholmness 

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References

  1. [1]
    M.A. Bastos, A. Bravo, and Yu.I. Karlovich, Convolution type operators with symbols generated by slowly oscillating and piecewise continuous matrix functions. Operator Theory: Advances and Applications 147 (2004), 151–174.MathSciNetGoogle Scholar
  2. [2]
    M.A. Bastos, A. Bravo, and Yu.I. Karlovich, Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data. Math. Nachrichten 269–270 (2004), 11–38.CrossRefMathSciNetGoogle Scholar
  3. [3]
    M.A. Bastos, Yu.I. Karlovich, and B. Silbermann, Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions. Proc. London Math. Soc. 89 (2004), 697–737.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Böttcher and Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkhäuser, Basel, 1997.Google Scholar
  5. [5]
    A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, The method of limit operators for one-dimensional integrals with slowly oscillating data. J. Operator Theory 43 (2000), 171–198.zbMATHMathSciNetGoogle Scholar
  6. [6]
    A. Böttcher, Yu.I. Karlovich, and I.M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Applications 131, Birkhäuser, Basel, 2002.Google Scholar
  7. [7]
    A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, 2nd edition, Springer, Berlin, 2006.zbMATHGoogle Scholar
  8. [8]
    A. Böttcher and I.M. Spitkovsky, Wiener-Hopf integral operators with PC symbols on spaces with Muckenhoupt weight. Revista Matemática Iberoamericana 9 (1993), 257–279.zbMATHGoogle Scholar
  9. [9]
    A. Böttcher and I.M. Spitkovsky, Pseudodifferential operators with heavy spectrum. Integral Equations and Operator Theory 19 (1994), 251–269.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    R.V. Duduchava, Integral Equations with Fixed Singularities. B.G. Teubner Verlagsgesellschaft, Leipzig, 1979.Google Scholar
  11. [11]
    R.V. Duduchava, On algebras generated by convolutions and discontinuous functions. Integral Equations and Operator Theory 10 (1987), 505–530.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    R.V. Duduchava and A.I. Saginashvili, Convolution integral equations on a half-line with semi-almost-periodic presymbols. Differential Equations 17 (1981), 207–216.zbMATHMathSciNetGoogle Scholar
  13. [13]
    J.B. Garnett, Bounded Analytic Functions. Academic Press, New York, 1981.zbMATHGoogle Scholar
  14. [14]
    I.M. Gelfand, D.A. Raikov, and G.E. Shilov, Commutative Normed Rings. Fizmatgiz, Moscow, 1960 [Russian]. English transl.: Chelsea, New York, 1964.Google Scholar
  15. [15]
    I. Gohberg and I. Feldman, Wiener-Hopf integro-difference equations. Soviet Math. Dokl. 9 (1968), 1312–1316.Google Scholar
  16. [16]
    I. Gohberg and I.A. Feldman, Convolution Equations and Projection Methods for Their Solutions. Transl. of Math. Monographs 41, Amer. Math. Soc., Providence, R.I., 1974. Russian original: Nauka, Moscow, 1971.Google Scholar
  17. [17]
    I. Gohberg and N. Krupnik, Singular integral operators with piecewise continuous coefficients and their symbols. Math. USSR — Izv. 5 (1971) 955–979.CrossRefGoogle Scholar
  18. [18]
    R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227–251.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Yu.I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. London Math. Soc. (3) 92 (2006), 713–761.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Yu.I. Karlovich and J. Loreto Hernández, Wiener-Hopf operators with matrix semialmost periodic symbols on weighted Lebesgue spaces. Integral Equations and Operator Theory 62 (2008), 85–128.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Yu.I. Karlovich and J. Loreto Hernández, Wiener-Hopf operators with slowly oscillating matrix symbols on weighted Lebesgue spaces. Integral Equations and Operator Theory 64 (2009), 203–237.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Yu.I. Karlovich and E. Ramírez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces. Integral Equations and Operator Theory 48 (2004), 331–363.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions. Noordhoff I.P., Leyden, 1976. Russian original: Nauka, Moscow, 1966.Google Scholar
  24. [24]
    V. Petkova, Symbole d’un multiplicateur sur L ω2(ℝ). Bull. Sci. Math. 128 (2004), 391–415.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    S.C. Power, Fredholm Toeplitz operators and slow oscillation. Can. J. Math. 32 (1980), 1058–1071.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory, Birkhäuser, Basel, 2004.zbMATHGoogle Scholar
  27. [27]
    S. Roch and B. Silbermann, Algebras of Convolution Operators and Their Image in the Calkin Algebra. Report R-Math-05/90, Akad. Wiss. DDR, Karl-Weierstrass-Institut f. Mathematik, Berlin, 1990.Google Scholar
  28. [28]
    D. Sarason, Toeplitz operators with semi-almost periodic symbols. Duke Math. J. 44 (1977) 357–364.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    R. Schneider, Integral equations with piecewise continuous coefficients in Lp spaces with weight. J. Integral Equations 9 (1985), 135–152.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Yu. I. Karlovich
    • 1
  • J. Loreto Hernández
    • 2
  1. 1.Facultad de CienciasUniversidad Autónoma del Estado de MorelosCuernavacaMéxico
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoCuernavacaMéxico

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