Skip to main content
SpringerLink
Log in

Iterative properties of pseudo-differential operators on edge spaces

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

Pseudo-differential operators with twisted symbolic estimates play a large role in the calculus on manifolds with edge singularities. We study here aspects of the underlying abstract concept and establish a new result on iteration of quantizations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)

  2. Chang, D.-C., Habal, N., Schulze, B.-W.: Quantisation on a manifold with singular edge. J. Pseudo-Differ. Oper. Appl. 4(3), 317–343 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dines, N.: Ellipticity of a class of corner operators. In: Pseudo-differential Operators: PDE and Time-Frequency Analysis. Fields Institute Communication, vol. 52, pp. 131–169. American Mathematical Society, Providence, RI (2007)

  4. Dorschfeldt, Ch.: Algebras of pseudo-differential operators near edge and corner singularities. In: Math. Res., vol. 102. Wiley-VCH, Berlin, Weinheim (1998)

  5. Dreher, M., Witt, I.: Edge Sobolev spaces and weakly hyperbolic operators. Ann. Mat. Pura Appl. 180, 451–482 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Egorov, Ju.V., Schulze, B.-W.: Pseudo-differential operators, singularities, applications. In: Operator Theory: Advances and Applications, vol. 93. Birkh\(\ddot{\rm a}\)user Verlag, Basel (1997)

  7. Eskin, G.I.: Boundary value problems for elliptic pseudo-differential equations. Transl. Nauka, Moskva, 1973. In: Mathematical Monographs, vol. 24. American Mathematical Society (1980)

  8. Flad, H.-J., Harutyunyan, G.: Ellipticity of quantum mechanical Hamiltonians in the edge algebra. In: Proceedings of the AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden (2010)

  9. Gil, J.B., Schulze, B.-W., Seiler, J.: Cone pseudodifferential operators in the edge symbolic calculus. Osaka J. Math. 37, 219–258 (2000)

    MathSciNet  Google Scholar 

  10. Habal, N., Schulze, B.-W.: Mellin quantisation in corner operators. In: Karlovich, Y.I. et al. (eds) Operator Theory: Advances and Applications, vol. 228. Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. The Vladimir Rabinovich Anniversary Volume, Birkhäuser, Basel, 2013, pp. 151–172

  11. Harutyunyan, G., Schulze, B.-W.: Elliptic mixed. In: Transmission and Singular Crack Problems. European Mathematical Society, Zürich (2008)

  12. Hirschmann, T.: Functional analysis in cone and edge Sobolev spaces. Ann. Global Anal. Geom. 8, 167–192 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hwang, I.L.: The \(L^2\)-boundedness of pseudodifferential operators. Trans. Amer. Math. Soc. 302, 55–76 (1987)

    MATH  MathSciNet  Google Scholar 

  14. Kondratyev, V.A.: Boundary value problems for elliptic equations in domains with conical points. Trudy Mosk. Mat. Obshch. 16, 209–292 (1967)

    Google Scholar 

  15. Kondratyev, V.A., Oleynik, O.A.: Boundary problems for partial differential equations on non-smooth domains. Uspekhi Mat. Nauk 38, 3–76 (1983)

    Google Scholar 

  16. Kumano-go, H.: Pseudo-differential Operators. The MIT Press, Cambridge, Massachusetts and London, England (1981)

  17. Liu, X., Schulze, B.-W.: Ellipticity on manifolds with edges and boundary. Monatsh. Math. 146, 295–331 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Luke, G.: Pseudo-differential operators on Hilbert bundles. J. Differ. Equ. 12(566–589), 573–592 (1972)

    MathSciNet  Google Scholar 

  19. Rabinovich, V.S.: Pseudo-differential operators in non-bounded domains with conical structure at infinity. Mat. Sb. 80, 77–97 (1969)

    MathSciNet  Google Scholar 

  20. Rempel, S., Schulze, B.-W.: Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics. Ann. Global Anal. Geom. 4, 137–224 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rempel, S., Schulze, B.-W.: Asymptotics for elliptic mixed boundary problems \((\)pseudo-differential and Mellin operators in spaces with conormal singularity). In: Mathematics Research, vol. 50. Akademie, Berlin (1989)

  22. Rungrottheera, W.: Parameter-dependent corner operators. Asian-Eur. J. Math. 6, 1–29 (2013)

  23. Rungrottheera, W., Schulze, B.-W.: Weighted spaces on corner manifolds. Complex Var. Elliptic Equ. doi:10.1080/17476933.2013.876416

  24. Schulze, B.-W.: Pseudo-differential operators on manifolds with edges. In: Equations, Symp Partial Differential. Holzhau 1988, Teubner-Texte zur Mathematik, vol. 112, pp. 259–287. Teubner, Leipzig (1989)

  25. Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  26. Schulze, B.-W.: Boundary value problems and singular pseudo-differential operators. Wiley, Chichester (1998)

    MATH  Google Scholar 

  27. Schulze, B.-W.: Operators with symbol hierarchies and iterated asymptotics. Publ. Res. Inst. Math. Sci. Kyoto 38, 735–802 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schulze, B.-W.: The iterative structure of the corner calculus. In: Rodino, L., Wong, M.W., Zhu, H. (eds.) Operator Theory: Advances and Applications, vol. 213, pp. 79–103. Pseudo-Differential Operators: Analysis, Application and Computations. Birkhäuser Verlag, Basel (2011)

  29. Schulze, B.-W., Seiler, J.: The edge algebra structure of boundary value problems. Ann. Glob. Anal. Geom. 22, 197–265 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Schulze, B.-W., Wong, M.W.: Mellin operators with asymptotics on manifolds with corners. In: Rodino, L., Wong, M.W., Zhu, H. (eds.) Operator Theory Advances and Applications, vol. 213, pp. 31–78. Pseudo-Differential Operators: Analysis, Applications and Computations. Birkhäuser Verlag, Basel (2011)

  31. Schulze, B.-W., Wong, M.W.: Mellin and Green operators of the corner calculus. J. Pseudo-Differ. Oper. Appl. 2, 467–507 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Seiler, J.: Continuity of edge and corner pseudo-differential operators. Math Nachr. 205, 163–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Vishik, M.I., Grushin, V.V.: On a class of degenerate elliptic equations of higher orders. Mat. Sb. 79, 336 (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom , 73000, Thailand

    W. Rungrottheera

  2. Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 , Potsdam, Germany

    B.-W. Schulze

  3. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    M. W. Wong

Authors
  1. W. Rungrottheera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B.-W. Schulze
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. W. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. W. Wong.

Additional information

This research has been supported by Research Fund SFR-PRG-2557-04 of Faculty of Science, Silpakorn University and the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rungrottheera, W., Schulze, BW. & Wong, M.W. Iterative properties of pseudo-differential operators on edge spaces. J. Pseudo-Differ. Oper. Appl. 5, 455–479 (2014). https://doi.org/10.1007/s11868-014-0100-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-014-0100-x

Keywords

Mathematics Subject Classification

Search

Navigation