Skip to main content
SpringerLink
Log in

Algebras of Pseudo-differential Operators with Discontinuous Symbols

  • Chapter
Modern Trends in Pseudo-Differential Operators

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 172))

Abstract

Using the boundedness of the maximal singular integral operator related to the Carleson-Hunt theorem we prove the boundedness and study the compactness of pseudo-differential operators a(x,D) with bounded measurable V (ℝR)-valued symbols a(x, ·) on the Lebesgue spaces L p(ℝ) with 1 < p < δ, where V (ℝ) is the Banach algebra of all functions of bounded total variation on R. Replacement of absolutely continuous functions of bounded total variation by arbitrary functions of bounded total variation allows us to study pseudo-differential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols. Three different Banach algebras of pseudo-differential operators with discontinuous symbols acting on the spaces L p(ℝ) are studied. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. For the operators in the first algebra we also obtain an index formula. An application to the Haseman boundary value problem is given.

Partially supported by CONACYT (México) Project No. 49992.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. A.V. Aizenshtat, Yu.I. Karlovich and G.S. Litvinchuk, The method of conformal gluing for the Haseman boundary value problem on an open contour, Complex Variables 28 (1996), 313–346.

    MATH  MathSciNet  Google Scholar 

  2. M.A. Bastos, A. Bravo and Yu.I. Karlovich, Convolution type operators with symbols generated by slowly oscillating and piecewise continuous matrix functions, in Operator Theoretical Methods and Applications to Mathematical Physics, The Erhard Meister Memorial Volume, Editors: I. Gohberg at al., Birkhäuser, 2004, 151–174.

    Google Scholar 

  3. 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. Nachr. 269–270 (2004), 11–38.

    Article  MathSciNet  Google Scholar 

  4. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Böttcher and Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Birkhäuser, Basel, 1997.

    MATH  Google Scholar 

  6. A. Böttcher, Yu.I. Karlovich and V.S. Rabinovich, The method of limit operators for one-dimensional singular integrals with slowly oscillating data, J. Operator Theory 43 (2000), 171–198.

    MATH  MathSciNet  Google Scholar 

  7. A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989 and Springer, Berlin, 1990.

    Google Scholar 

  8. A. Böttcher and I.M. Spitkovsky, Pseudodifferential operators with heavy spectrum, Integral Equations Operator Theory 19 (1994), 251–269.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.

    Article  MATH  MathSciNet  Google Scholar 

  10. R.R. Coifman and Y. Meyer, Au delà des opérateurs pseudodifférentiels, Astérisque 57 (1978), 1–184.

    MathSciNet  Google Scholar 

  11. H.O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal. 18 (1975), 115–131.

    Article  MATH  MathSciNet  Google Scholar 

  12. H.O. Cordes, Elliptic Pseudo-differential Operators — An Abstract Theory, Springer, Berlin, 1979.

    MATH  Google Scholar 

  13. R.V. Duduchava, Integral equations of convolution type with discontinuous coefficients, Soobshch. Akad. Nauk Gruz. SSR 92 (1978), 281–284 [Russian].

    MATH  MathSciNet  Google Scholar 

  14. R.V. Duduchava, Integral Equations with Fixed Singularities, Teubner, Leipzig, 1979.

    MATH  Google Scholar 

  15. E.M. Dynkin, Methods of the theory of singular integrals: Hilbert transform and Calderón-Zygmund theory, in Commutative Harmonic Analysis I: General Surveys, Classical Results, Editors: V. P. Khavin and N.K. Nikol’skiy, Encyclopedia of Mathematical Sciences 15, Springer, Berlin, 1991; Russian original, VINITI, Moscow, 1987.

    Google Scholar 

  16. T. Finck, S. Roch and B. Silbermann, Two projection theorems and symbol calculus for operators with massive local spectra, Math. Nachr. 162 (1993), 167–185.

    Article  MATH  MathSciNet  Google Scholar 

  17. I. Gohberg and N. Krupnik, Singular integral operators with piecewise continuous coefficients and their symbols, Math. USSR-Izv. 5 (1971), 955–979 [Russian].

    Article  Google Scholar 

  18. I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vols. 1 and 2, Birkhäuser, Basel, 1992; Russian original, Shtiintsa, Kishinev, 1973.

    Google Scholar 

  19. I. Gohberg and N. Krupnik, Extension theorems for invertibility symbols in Banach algebras, Integral Equations Operator Theory 15 (1992), 991–1010.

    Article  MATH  MathSciNet  Google Scholar 

  20. N.B. Haaser and J.A. Sullivan, Real Analysis, Dover Publications, New York, 1991.

    Google Scholar 

  21. W. Hoh, A symbolic calculus for pseudo-differential operators generating Feller semigroups, Osaka J. Math. 35 (1998), 789–820.

    MATH  MathSciNet  Google Scholar 

  22. W. Hoh, Perturbations of pseudodifferential operators with negative definite symbol, Appl. Math. Optimization 45 (2002), 269–281.

    Article  MATH  MathSciNet  Google Scholar 

  23. R.A. Hunt, On the convergence of Fourier series, in Orthogonal Expansions and Their Continuous Analogues, Editor: D. Haimo, Southern Illinois Univ. Press, Carbondale, 1968, 235–255.

    Google Scholar 

  24. N. Jacob and A.G. Tokarev, A parameter-dependent symbolic calculus for pseudodifferential operators with negative-definite symbols, J. London Math. Soc. 70 (2004), 780–796.

    Article  MATH  MathSciNet  Google Scholar 

  25. O.G. Jørsboe and L. Melbro, The Carleson-Hunt Theorem on Fourier Series, Springer, Berlin, 1982.

    Google Scholar 

  26. Yu.I. Karlovich, On the Haseman problem, Demonstratio Mathematica 26 (1993), No. 3–4, 581–595.

    MATH  MathSciNet  Google Scholar 

  27. Yu.I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols, Proc. London Math. Soc. 92 (2006), 713–761.

    Article  MATH  MathSciNet  Google Scholar 

  28. Yu.I. Karlovich, Pseudodifferential operators with compound slowly oscillating symbols, in: Oper. Theory: Adv. Appl., Vol. 171, 2006, 189–224, to appear.

    Google Scholar 

  29. Yu.I. Karlovich and E. Ramírez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces, Integral Equations Operator Theory 48 (2004), 331–363.

    Article  MATH  MathSciNet  Google Scholar 

  30. C.E. Kenig and P.A. Tomas, Maximal operators defined by Fourier multipliers, Studia Math. 68 (1980), 79–83.

    MATH  MathSciNet  Google Scholar 

  31. 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 

  32. G.S. Litvinchuk, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift, Kluwer, Dordrecht, 2000.

    MATH  Google Scholar 

  33. J. Marschall, Pseudo-differential operators with nonregular symbols of the class S mρδ , Comm. Partial Differential Equations 12 (1987), 921–965.

    MATH  MathSciNet  Google Scholar 

  34. J. Marschall, On the boundedness and compactness of nonregular pseudo-differential operators, Math. Nachr. 175 (1995), 231–262.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Marschall, Nonregular pseudo-differential operators, Z. Anal. Anwendungen 15 (1996), 109–148.

    MATH  MathSciNet  Google Scholar 

  36. S.C. Power, Fredholm Toeplitz operators and slow oscillation, Canad. J. Math. 32 (1980), 1058–1071.

    MATH  MathSciNet  Google Scholar 

  37. S. Roch and 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.

    MATH  Google Scholar 

  38. L. Schwartz, Analyse Mathématique, Vol. 1, Hermann, 1967.

    Google Scholar 

  39. I.B. Simonenko and Chin Ngok Min, Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuous Coefficients, Noetherity, University Press, Rostov on Don, 1986 [Russian].

    Google Scholar 

  40. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  41. M.E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, American Mathematical Society, Providence, RI, 2000.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209, Cuernavaca, Morelos, México

    Yu. I. Karlovich

Authors
  1. Yu. I. Karlovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics and Systems Engineering, Växjö University, SE-351 95, Växjö, Sweden

    Joachim Toft

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Karlovich, Y.I. (2006). Algebras of Pseudo-differential Operators with Discontinuous Symbols. In: Toft, J. (eds) Modern Trends in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol 172. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8116-5_12

Download citation

Publish with us

Policies and ethics

Search

Navigation