Lp estimates for joint quasimodes of semiclassical pseudodifferential operators

Abstract

We develop a set of Lp estimates for functions u that are joint quasi-modes (approximate eigenfunctions) of r semiclassical pseudodifferential operators p1(x, hD),...,pr(x, hD). This work extends Sarnak [10] and Marshall’s [8] work on symmetric space to cover a more general class of manifolds/operators.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace—Beltrami eigenfunctions to submanifolds, Duke Mathematical Journal 138 (2007), 445–486.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    X. Chen and C. Sogge, A few endpoint geodesic restriction estimates for eigenfunctions, Communications in Mathematical Physics 329 (2014), 435–459.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    A. Hassell and M. Tacy, Semiclassical L p estimates of quasimodes on curved hypersurfaces, Journal of Geometric Analysis 22 (2012), 74–89.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    L. Hörmander, The spectral function of an elliptic operator, Acta Mathematica 121 (1968), 193–218.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    R. Hu, L p norm estimates of eigenfunctions restricted to submanifolds, Forum Mathematicum 21 (2009), 1021–1052.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    H. Koch, D. Tataru and M. Zworski, Semiclassical L p estimates, Annales Henri Poincaré 8 (2007), 885–916.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    B. M. Levitan, On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 16 (1952), 325–352.

    MathSciNet  Google Scholar 

  8. [8]

    S. Marshall, L p norms of higher rank eigenfunctions and bounds for spherical functions, Journal of the European Mathematical Society 18 (2016), 1437–1493.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    P. Ramacher and S. Wakatsuki, Subconvex bounds for Hecke-Maass forms on compact arithmetic quotients of semisimple lie groups, https://arxiv.org/abs/1703.06973.

  10. [10]

    P. Sarnak, Letter to Morawetz, available at https://web.math.princeton.edu/sarnak/Sarnak_Letter_to_Morawetz.pdf.

  11. [11]

    C. D. Sogge, Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds, Journal of Functioinal Analysis 77 (1988), 123–138.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    M. Tacy, Semiclassical L p estimates of quasimodes on submanifolds, Communications in Partial Differential Equations 35 (2010), 1538–1562.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    J. A. Toth and S. Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Mathematical Journal 111 (2002), 97–132.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    J. A. Toth and S. Zelditch, L p norms of eigenfunctions in the completely integrable case, Annales Henri Poincaré 4 (2003), 343–368.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    J. A. Toth and S. Zelditch, Norms of modes and quasi-modes revisited, in Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), Contemporary Mathematics, Vol. 320, American Mathematical Society, Providence, RI, 2003, pp. 435–458.

    Article  Google Scholar 

  16. [16]

    M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, Vol. 138, American Mathematical Society, Providence, RI, 2012.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Melissa Tacy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tacy, M. Lp estimates for joint quasimodes of semiclassical pseudodifferential operators. Isr. J. Math. 232, 401–425 (2019). https://doi.org/10.1007/s11856-019-1878-2

Download citation