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Approximating pointwise products of quasimodes
Forum Mathematicum ( IF 0.8 ) Pub Date : 2020-05-01 , DOI: 10.1515/forum-2019-0208
Mei Ling Jin 1
Affiliation  

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.

中文翻译:

近似准模态的逐点积

摘要 我们获得了拉普拉斯-贝尔特拉米算子在无边界的所有维度的紧凑黎曼流形上的准模态乘积的近似边界。我们通过一个低阶向量空间 B n {B_{n}} 来近似拟模 uv 的乘积,并且我们证明了空间 dim ⁡ ( B n ) {\dim(B_{n})} 的大小很小. 在本文中,我们首先分别研究了所有维度 d = 2 , 3 {d=2,3} , d = 4 , 5 {d=4,5} 和 d ≥ 6 {d\geq 6} 的双线性拟模估计, 使最高频率从右侧消失。此外,双线性拟模估计的情况 λ = μ {\lambda=\mu} 的结果改进了 [CD Sogge and S. Zelditch, A note on L 中 Sogge 和 Zelditch 的 L 4 {L^{4}} 拟模估计p L^{p} -准模态范数,谐波分析和应用中的一些主题,Adv。讲。数学。(ALM) 34, International Press, Somerville 2016, 385–397] 当 d ≥ 8 {d\geq 8} 。在此基础上,我们给出了 H - 1 {H^{-1}} -范数的近似界限。我们还证明了 L 2 {L^{2}} 中准模态乘积的近似边界,使用 L p {L^{p}} - 对 [M. Blair, Y. Sire 和 CD Sogge,准模态,具有临界奇异势的流形上薛定谔算子的特征函数和谱投影边界,预印本 2019,https://arxiv.org/abs/1904.09665]。我们将 Lu 和 Steinerberger 在 [JF Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] 中的结果扩展到拟模态。我们还证明了 L 2 {L^{2}} 中准模态乘积的近似边界,使用 L p {L^{p}} - 对 [M. Blair, Y. Sire 和 CD Sogge,准模态,具有临界奇异势的流形上薛定谔算子的特征函数和谱投影边界,预印本 2019,https://arxiv.org/abs/1904.09665]。我们将 Lu 和 Steinerberger 在 [JF Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] 中的结果扩展到拟模态。我们还证明了 L 2 {L^{2}} 中准模态乘积的近似边界,使用 L p {L^{p}} - 对 [M. Blair, Y. Sire 和 CD Sogge,准模态,具有临界奇异势的流形上薛定谔算子的特征函数和谱投影边界,预印本 2019,https://arxiv.org/abs/1904.09665]。我们将 Lu 和 Steinerberger 在 [JF Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] 中的结果扩展到拟模态。具有临界奇异势的流形上薛定谔算子的特征函数和谱投影边界,预印本 2019,https://arxiv.org/abs/1904.09665]。我们将 Lu 和 Steinerberger 在 [JF Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] 中的结果扩展到拟模态。具有临界奇异势的流形上薛定谔算子的特征函数和谱投影边界,预印本 2019,https://arxiv.org/abs/1904.09665]。我们将 Lu 和 Steinerberger 在 [JF Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] 中的结果扩展到拟模态。
更新日期:2020-05-01
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