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Approximating pointwise products of quasimodes

  • Mei Ling Jin EMAIL logo
Veröffentlicht/Copyright: 19. Dezember 2019
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 32 Heft 3

Abstract

We obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space Bn, and we prove that the size of the space dim⁡(Bn) is small. In this paper, we first study bilinear quasimode estimates of all dimensions d=2,3, d=4,5 and d≥6, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case λ=μ of bilinear quasimode estimates improves L4 quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on Lp-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when d≥8. And on this basis, we give approximation bounds in H-1-norm. We also prove approximation bounds for the products of quasimodes in L2-norm using the results of Lp-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

Keywords: Approximation pointwise products; quasimode; bilinear estimates
MSC 2010: 35A15; 35C99; 47B06; 81Q05

Communicated by Christopher D. Sogge

Acknowledgements

The research was carried out while the author was visiting Johns Hopkins University supervised by Professor C. D. Sogge. And the author would like to express her deep gratitude to Professor C. D. Sogge, for bringing this research topic to her attention, and also for the valuable guidance, helpful suggestions and comments he provided.

References

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Received: 2019-07-30
Revised: 2019-11-18
Published Online: 2019-12-19
Published in Print: 2020-05-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Approximating pointwise products of quasimodes
  3. On the group of a rational maximal bifix code
  4. Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces
  5. Newton’s method for nonlinear stochastic wave equations
  6. On the classification of Schreier extensions of monoids with non-abelian kernel
  7. Smoothness filtration of the magnitude complex
  8. Varieties of nilpotent Lie superalgebras of dimension ≤ 5
  9. The Dual Baer Criterion for non-perfect rings
  10. Spectral property of the planar self-affine measures with three-element digit sets
  11. Spectral properties of certain Moran measures with consecutive and collinear digit sets
  12. Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space
  13. A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group
  14. Well-posedness of backward stochastic partial differential equations with Lyapunov condition
  15. Free division rings of fractions of crossed products of groups with Conradian left-orders
  16. Strong uncountable cofinality for unitary groups of von Neumann algebras
  17. Irreducible holonomy groups and first integrals for holomorphic foliations
  18. Alexandroff topologies and monoid actions
https://doi.org/10.1515/forum-2019-0208
Schlagwörter für diesen Artikel
Approximation pointwise products; quasimode; bilinear estimates

Artikel in diesem Heft

  1. Frontmatter
  2. Approximating pointwise products of quasimodes
  3. On the group of a rational maximal bifix code
  4. Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces
  5. Newton’s method for nonlinear stochastic wave equations
  6. On the classification of Schreier extensions of monoids with non-abelian kernel
  7. Smoothness filtration of the magnitude complex
  8. Varieties of nilpotent Lie superalgebras of dimension ≤ 5
  9. The Dual Baer Criterion for non-perfect rings
  10. Spectral property of the planar self-affine measures with three-element digit sets
  11. Spectral properties of certain Moran measures with consecutive and collinear digit sets
  12. Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space
  13. A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group
  14. Well-posedness of backward stochastic partial differential equations with Lyapunov condition
  15. Free division rings of fractions of crossed products of groups with Conradian left-orders
  16. Strong uncountable cofinality for unitary groups of von Neumann algebras
  17. Irreducible holonomy groups and first integrals for holomorphic foliations
  18. Alexandroff topologies and monoid actions
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