Approximating pointwise products of quasimodes

  • 1 Department of Mathematics, Harbin Institute of Technology, 150001, Harbin, P. R. China
Mei Ling Jin
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  • Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P. R. China
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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.

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    J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint (2018), https://arxiv.org/abs/1810.01024v2.

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    C. D. Sogge and S. Zelditch, A note on L p L^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville (2016), 385–397.

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