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Approximating pointwise products of quasimodes

  • Mei Ling Jin EMAIL logo
From the journal Forum Mathematicum

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 d6, 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 d8. 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.


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

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