[Paper Review] Lp bilinear quasimode estimates

Zihua Guo, Xiaolong Han+1 more|arXiv (Cornell University)|Mar 2, 2015

Advanced Mathematical Physics Problems16 references3 citations

TL;DR

This paper establishes sharp $L^p$ bilinear quasimode estimates for eigenfunctions and quasimodes on compact Riemannian manifolds of dimension $n \geq 2$, using semiclassical analysis. It proves that the $L^p$ norm of the product of two quasimodes is bounded by a function $G_{n,p}(h,\sigma)$ depending on the frequencies, with the bound improving when the higher frequency has a smaller exponent—demonstrating that bilinear estimates significantly outperform bounds derived via H"older's inequality and Sogge's linear estimates.

ABSTRACT

In this paper, we investigate the $L^p$ bilinear quasimode estimates on compact Riemannian manifolds. We obtain results in the full range $p\ge2$ on all $n$-dimensional manifolds with $n\ge2$. This in particular implies the $L^p$ bilinear eigenfunction estimates. We further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates.

Motivation & Objective

• To establish sharp $L^p$ bilinear estimates for quasimodes on compact Riemannian manifolds in the full range $p \geq 2$.
• To extend previous $L^2$ bilinear spectral cluster estimates (e.g., Burq-Gérard-Tzvetkov) to higher $L^p$ norms.
• To demonstrate that the bilinear estimates are superior to those obtainable via H"older's inequality and Sogge's linear $L^p$ eigenfunction estimates.
• To show the sharpness of the estimates by constructing quasimodes and eigenfunctions that saturate the bounds.
• To work in the semiclassical framework to treat quasimodes uniformly, enabling a unified treatment of eigenfunctions and approximate solutions.

Proposed method

• The authors use the semiclassical framework, treating eigenfunctions and quasimodes as solutions to $P(h)u(h) = 0$ with $P(h) = -h^2\Delta - 1$, where $h = \lambda^{-1}$ is the semiclassical parameter.
• They define $O_{L^2}(h)$ quasimodes as families $\{u(h)\}$ satisfying $\|P(h)u(h)\|_{L^2} \lesssim h\|u(h)\|_{L^2}$, and extend this to Laplace-like operators.
• The analysis relies on the semiclassical wavefront set and localisation properties: quasimodes are semiclassically microlocalised to the cosphere bundle $S^*\mathbb{M}$.
• They employ semiclassical Fourier integral operators and parametrix constructions for the evolution equation $hD_t + A(t)$ to model quasimode propagation along classical flow trajectories.
• The key technical tool is the use of oscillatory integral representations for the propagator, with phase functions solving Hamilton-Jacobi equations and amplitude functions solving transport equations.
• The estimates are derived by analyzing the $L^p$ norms of products of quasimodes through microlocal and oscillatory integral techniques, with sharpness verified via explicit quasimode constructions.

Experimental results

Research questions

• RQ1 Can sharp $L^p$ bilinear estimates for quasimodes be established on compact Riemannian manifolds for all $p \geq 2$ and all dimensions $n \geq 2$?
• RQ2 How do the bilinear $L^p$ estimates for quasimodes compare to those obtained via H"older's inequality and Sogge's linear $L^p$ eigenfunction estimates?
• RQ3 What is the dependence of the $L^p$ norm of the product $uv$ on the individual frequencies $\lambda$ and $\mu$ of the quasimodes $u$ and $v$?
• RQ4 Is the proposed bound $G_{n,p}(h,\sigma)$ sharp, and can it be saturated by explicit quasimodes or eigenfunctions?
• RQ5 Can the $L^2$ bilinear estimates of Burq-Gérard-Tzvetkov be generalized to $L^p$ norms with optimal dependence on the frequencies?

Key findings

• The paper establishes sharp $L^p$ bilinear quasimode estimates for all $n \geq 2$ and $p \geq 2$, with the bound depending on both frequencies $h$ and $\sigma$, where $\sigma \leq h \ll 1$.
• The bound $\|uv\|_{L^p} \lesssim G_{n,p}(h,\sigma)\|u\|_{L^2}\|v\|_{L^2}$ is strictly better than the bound obtained by combining H"older's inequality and Sogge's linear $L^p$ estimates.
• In the bilinear $L^p$ estimates, the higher frequency $\sigma$ appears with a smaller exponent than the lower frequency $h$, reflecting improved decay for higher-frequency components.
• For $n=2$, the estimate is $\|uv\|_{L^p} \lesssim \lambda^{\delta(n,p)}$ with $\delta(2,p)$ matching the sharp $L^p$ decay rate for eigenfunctions.
• In dimension $n=3$, the estimate includes a logarithmic loss $|\log \lambda|^{1/2}$, which is shown to be sharp.
• The authors construct explicit quasimodes and eigenfunctions that saturate the estimates, proving their sharpness modulo the $\log$ loss in dimension three.