Approximating pointwise products of quasimodes

Abstract

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 B n {B_{n}} , and we prove that the size of the space dim ⁡ ( B n ) {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions d = 2 , 3 {d=2,3} , d = 4 , 5 {d=4,5} and d ≥ 6 {d\geq 6} , respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case λ = μ {\lambda=\mu} of bilinear quasimode estimates improves L 4 {L^{4}} quasimodes estimates of Sogge and Zelditch in [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] when d ≥ 8 {d\geq 8} . And on this basis, we give approximation bounds in H - 1 {H^{-1}} -norm. We also prove approximation bounds for the products of quasimodes in L 2 {L^{2}} -norm using the results of L p {L^{p}} -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.