Skip to main content
Log in

The weak typeL 1 convergence of eigenfunction expansions for pseudodifferential operators

  • Published:
Inventiones mathematicae Aims and scope

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

Access this article

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

Instant access to the full article PDF.

References

  1. Avakumovič, V.G.: Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z.65, 327–344 (1956)

    Google Scholar 

  2. Berard, P.H.: Riesz means on Riemannian manifolds. Proc. Symp. Pure Math.XXXVI, 1–12 (1980)

    Google Scholar 

  3. Carleson, L., Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc. Stud. Math.44, 287–299 (1972)

    Google Scholar 

  4. Chanillo, S., Muckenhoupt, B.: Weak type estimates for Bochner-Riesz spherical summation multipliers. Trans. Am. Math. Soc.294, 693–703 (1986)

    Google Scholar 

  5. Christ, F.M.: Weak type (1, 1) bounds for rough operators. Ann. Math.128, 19–42 (1988)

    Google Scholar 

  6. Christ, F.M.: Weak type endpoint bounds for Bochner-Riesz multipliers. Rev. Mat. Ibero-Am. (to appear)

  7. Christ, F.M., Rubio de Francia, J.L.: Weak type (1, 1) bounds for rough operators. II. Invent. Math.93, 225–237 (1988)

    Google Scholar 

  8. Córdoba, A.: A note on Bochner-Riesz operators. Duke Math. J.46, 505–511 (1979)

    Google Scholar 

  9. Fefferman, C.: Inequalities for strongly singular convolution operators. Acta Math.124, 9–36 (1970)

    Google Scholar 

  10. Fefferman, C.: The multiplier problem for the ball. Ann. Math.94, 330–336 (1971)

    Google Scholar 

  11. Fefferman, C.: A note on spherical summation multipliers. Isr. J. Math.15, 44–52 (1973)

    Google Scholar 

  12. Gelfand, I.M., Shilov, G.E.: Generalized functions, Vol. I. New York: Academic Press 1964

    Google Scholar 

  13. Hörmander, L.: On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators. In: Some recent advances in the basic sciences, pp. 155–202. Yeshiva Univ., New York, 1966

    Google Scholar 

  14. Hörmander, L.: The spectral function of an elliptic operator. Acta Math.88, 341–370 (1968)

    Google Scholar 

  15. Kenig, C., Stanton, R., Tomas, P.: Divergence of eigenfunction expansions. J. Funct. Anal.46, 28–44 (1982)

    Google Scholar 

  16. Levitan, B.M.: On the asymptotic behavior of the spectral function of a self-adjont differential equation of the second order. Izv. Akad. Nauk SSSR, Ser. Mat.16, 325–352 (1952)

    Google Scholar 

  17. Mitjagin, B.S.: Divergenz von Spektralentwicklungen inL p-Räumen. Boston: Birkhäuser 1974

    Google Scholar 

  18. Seeley, R.: Complex powers of an elliptic operator. Proc. Symp. Pure Math.X, 288–307 (1968)

    Google Scholar 

  19. Sogge, C.D.: Oscillatory integrals and spherical harmonics. Duke Math. J.53, 43–65 (1986)

    Google Scholar 

  20. Sogge, C.D.: Concerning theL p norm of spectral clusters for second order elliptic operators on compact manifolds. J. Funct. Anal.77, 123–134 (1988)

    Google Scholar 

  21. Sogge, C.D.: On the convergence of Riesz means on compact manifolds. Ann. Math.126, 439–447 (1987)

    Google Scholar 

  22. Sogge, C.D., Stein, E.M.: Averages of functions over hypersurfaces inR n. Invent. Math.82, 543–556 (1985)

    Google Scholar 

  23. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, N.J.: Princeton Univ. Press 1971

    Google Scholar 

  24. Stein, E.M.: Oscillatory integrals in Fourier analysis. Beijing Lectures in Harmonic Analysis, pp. 307–356. Princeton Univ. Press 1986

  25. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press 1971

  26. Taylor, M.: Pseudodifferential operators. Princeton Univ. Press 1981

Download references

Author information

Authors and Affiliations

Authors

Additional information

Supported in part by NSF grants and a Sloan fellowship.

Supported by an NSF postdoctoral fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Christ, F.M., Sogge, C.D. The weak typeL 1 convergence of eigenfunction expansions for pseudodifferential operators. Invent Math 94, 421–453 (1988). https://doi.org/10.1007/BF01394331

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01394331

Keywords

Navigation