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Nonclassical multidimensional singular integral operators on R n+1+

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Abstract

Banach algebras of certain bounded operators acting on the half-spaceL p (R n+1+ ,x α0 ) (1<p<∞, −1<α<p−1) are defined which contain for example Wiener-Hopf operators, defined by multidimensional singular convolution integral operators, as well as certain singular integral operators with fixed singularities. Moreover the symbol may be a positive homogeneous function only piecewise continuous on the unit sphere. Actually these multidimensional singular integral operators may be not Calderón-Zygmund operators but are built up by those in lower dimensions. This paper is a continuation of a joint paper of the author together with R.V. Duduchava [10]. The purpose is to investigate invertibility or Fredholm properties of these operators, while the continuity is given by definition. This is done in [10] forp=2 and −1<α<1, and in the present paper forL p (R n+1+ ,x α0 ) with 1<p<∞ and −1<α<p−1.

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References

  1. Bergh, J.; Löfström, J.: Interpolation Spaces. Springer Verlag 1979

  2. Boutet de Monvel, L.: Boundary problems for pseudodifferential operators. Acta Mathematica 126 (1971), 11–51

    Google Scholar 

  3. Collela, P.A.; Cordes H.O.: TheC *-algebra of elliptic boundary problems. Rocky Mountain J. Math., 10 (1980), 217–238

    Google Scholar 

  4. Cordes, H.O.: Elliptic Pseudo-Differential Operators. An Abstract Theory. Lecture Notes in Math. 756, Springer Verlag 1979

  5. Costabel, M.: Inverse to Gohberg Krupnik symbol map. Proc. Royal Soc. Edinburgh 87A (1980) 153–165

    Google Scholar 

  6. Costabel, M.: Singular integral equations on curves with corners. Integral Equations and Operator Theory 3 (1980), 323–349

    Google Scholar 

  7. Duduchava, R.: Integral Equations with fixed Singularities. Teubner, Leipzig, 1979

    Google Scholar 

  8. Duduchava, R.: On multidimensional singular integral operators. I) The half-space case, J. Operator Theory 11, 1984, 41–76. II) The case of compact manifolds; J. Operator Theory 11, 1984, 199–214

    Google Scholar 

  9. Duduchava, R.: On algebras generated by convolutions and discontinuous functions. Integral Equations and Operator Theory 10 (1987), 505–530

    Google Scholar 

  10. Duduchava, R.; Schneider, R.: The Algebra of non-classical singular integral operators on R n+1+ . Integral Equations and Operator Theory 10 (1987), 531–553

    Google Scholar 

  11. Dunfond, N.; Schwartz, J.: Linear Operators, Part I. Interscience Publishers, New York- London 1963

    Google Scholar 

  12. Eskin, G.I.: Boundary Value Problems for Elliptic Pseudodifferential Equations. Translation of Mathematical Monographs 52 (1981) AMS Providence, Rhode Island

    Google Scholar 

  13. Fefferman, Ch: The multiplier theorem for the ball. Ann. Math. 94 (1971), 330–336

    Google Scholar 

  14. Gohberg, I.C.; Krupnik, N.: Einführung in die Theorie der eindimensionalen singulären Integraloperatoren. Birkhäuser-Verlag, Basel- Boston- Stuttgart, 1979

    Google Scholar 

  15. Grubb, G.: Functional Calculus of Pseudo-Differential Boundary Problems. Birkhäuser, Boston- Basel- Stuttgart, 1986

    Google Scholar 

  16. Heunemann, D.; Leiterer, J.: On the normal solvability of pseudodifferential operators in the half-space with piecewise continuous symbols. Math. Nachr. 95 (1980), 209–214

    Google Scholar 

  17. Hörmander, L.: Estimates for translation invariant operators onL p -spaces, Acta Math. 104 (1960) 93–140

    Google Scholar 

  18. Jodeit, M. Jr.: A note on Fourier multipliers. Proc. of Amer. Math. Soc. 27, 2 (1971) 423–424

    Google Scholar 

  19. Mikhlin, S.G.; Prössdorf, S.: Singular Integral Operators. Springer-Verlag, 1986

  20. Prössdorf, S.: Über eine Algebra von Pseudodifferentialoperatoren im Halbraum, Math. Nachr. 52 (1972) 113–139

    Google Scholar 

  21. Rempel, S.; Schulze, B.W.: Index Theory of Elliptic Boundary Problems. Akademie Verlag, Berlin, 1982

    Google Scholar 

  22. Rempel, S.; Schulze, B.W.: Parametrices and boundary symbolic calculus for elliptic boundary problems without the transmission property. Math. Nachr. 105 (1982) 45–149

    Google Scholar 

  23. Shamir, E.: Elliptic systems of singular integral operators. I. The half-space case. Trans. Amer. Math. Soc. 127 (1967), 107–124

    Google Scholar 

  24. Simonenko, I.B.: A new general method of investigation of singular integral equations type linear operator equations I–II. (Russian) Jzv. Akad. Nauk SSR, Sov. Math. 29 (1965) 567–586, 757–782

    Google Scholar 

  25. Speck, F.O.: General Wiener-Hopf Factorization Methods. Pitman, London, 1985

    Google Scholar 

  26. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Math. Lbr. 18, Amsterdam, New York, Oxford, 1978

  27. Schneider, R.: Integral equations with piecewise continuous coefficients inL p -spaces with weight. Journal of Integral Equations 9 (1985), 135–152

    Google Scholar 

  28. Thelen, G.: Zur Fredholmtheorie singulärer Integrodifferentialoperatoren auf der Halbachse. Dissert. Darmstadt 1985, 125 S.

  29. Thelen, G.: On singular integro-differential operators on the half-axis, Math. Nachr. 131 (1987), 235–249

    Google Scholar 

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Schneider, R. Nonclassical multidimensional singular integral operators on R n+1+ . Integr equ oper theory 13, 104–131 (1990). https://doi.org/10.1007/BF01195295

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