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Annals of Global Analysis and Geometry

, Volume 26, Issue 3, pp 231–252 | Cite as

Derivatives of the Spectral Function and Sobolev Norms of Eigenfunctions on a Closed Riemannian Manifold

Article

Abstract

Let e(x, y, λ) be the spectral function and χλ the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, λ) by Hörmander (Acta Math.88 (1968), 341–370) to that of ∂ x α y βe(x,y,λ)|x=y for any multiindices α, β in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L2,Lp) (2 ≤p≤∞) estimates of χλ by Sogge (J. Funct. Anal.77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L2, Sobolev Lp) estimates of χλ.

Laplace–Beltrami operator spectral function unit spectral projection operator Sobolev norms of eigenfunctions 

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References

  1. 1.
    Carleson, L. and Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44(1972), 287–299.Google Scholar
  2. 2.
    Grieser, D.: Uniform bounds for eigenfunctions of the laplacian on manifolds with boundary, Comm. Partial Differential Equations 27(7, 8) (2002), 1283–1299.Google Scholar
  3. 3.
    Hebey, E.: Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Math. 1365, Springer, Berlin, 1996.Google Scholar
  4. 4.
    Hörmander, L.: The spectral function of an elliptic operator, Acta Math. 88(1968), 341–370.Google Scholar
  5. 5.
    Hörmander, L.: The Analysis of Linear Partial Differential EquationsI, 2nd edn. Springer-Verlag, Berlin, 1990.Google Scholar
  6. 6.
    Hörmander, L.: The Analysis of Linear Partial Differential Equations III, Corrected second printing, Springer-Verlag, Tokyo, 1994.Google Scholar
  7. 7.
    Jost, J.: Eine geometrische Bemerkung zu Sätzen ¨ uber harmonische Abbildungen, die ein Dirich-letproblem l ösen, Manuscripta Math. 32(1980), 51–57.Google Scholar
  8. 8.
    Krätzel, E.: Lattice Points, Math. Appl. (East Europ. Ser.) 33, Kluwer Acad. Publ., Dordrecht, 1988.Google Scholar
  9. 9.
    Ozawa, S.: Asymptotic property of eigenfunction of the Laplacian at the boundary, Osaka J. Math. 30(1993), 303–314.Google Scholar
  10. 10.
    Sogge, C. D.: Oscillatory integrals and spherical harmonics, Duke Math. J. 53(1986), 43–65.Google Scholar
  11. 11.
    Sogge, C. D.: On the convergence of Riesz means of compact manifolds, Ann. Math. 126(1987), 439–447.Google Scholar
  12. 12.
    Sogge, C. D.: Concerning the L pnorm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77(1988), 123–134.Google Scholar
  13. 13.
    Sogge, C. D.: Remarks on L2 restriction theorems for Riemannian manifolds, in: E. Berkson, T. Peck and J. Uhl, Jr. (eds), Analysis at Urbana, London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, 416–422.Google Scholar
  14. 14.
    Sogge, C. D.: Eigenfunction and Bochner-Riesz estimates on manifolds with boundary, Math. Res. Lett. 9(2002), 205–216.Google Scholar
  15. 15.
    Stein, E. M.: Oscillatory integrals in Fourier analysis, in: E. M. Steyn (ed.), Beijing Lectures in Harmonic Analysis, Princeton University Press: Princeton, New Jersey, 1986; pp. 307–356.Google Scholar
  16. 16.
    Szegö, G.: Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., New York, 1939.Google Scholar
  17. 17.
    Xiangjin, X.: Gradient estimates for the eigenfunctions on compact Riemannian manifolds with boundary, Preprint.Google Scholar

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© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Xu Bin

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