scholarly journals Bari-Markus property for Riesz projections of Hill operators with singular potentials

2009 ◽  
pp. 59-80 ◽  
Author(s):  
Plamen Djakov ◽  
Boris Mityagin
Keyword(s):  
Singular Potentials

scholarly journals Approximating pointwise products of quasimodes

2020 ◽  
Vol 32 (3) ◽  
pp. 541-552
Author(s):  
Mei Ling Jin
Keyword(s):  
Vector Space ◽  
Harmonic Analysis ◽  
Riemannian Manifolds ◽  
Spectral Projection ◽  
Beltrami Operator ◽  
Singular Potentials ◽  
Link Type ◽  
Right Hand ◽  
Low Degree ◽  
The Right

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

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