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Semiclassical $L^p$ quasimode restriction estimates in two dimensions

Sewook Oh, Jaehyeon Ryu

Abstract

We establish the $L^p$ restriction estimates for quasimodes on a smooth curve in two dimensions. Our estimates are sharp for all smooth curves. As an application, we address $L^p$ eigenfunction restriction estimates for Laplace-Beltrami eigenfunctions on $2$-dimensional compact Riemannian manifolds without boundary and Hermite functions on $\mathbb R^2$. Our method involves a geometric analysis of the contact order between the curve and the bicharacteristic flow of the semiclassical pseudodifferential operator.

Semiclassical $L^p$ quasimode restriction estimates in two dimensions

Abstract

We establish the restriction estimates for quasimodes on a smooth curve in two dimensions. Our estimates are sharp for all smooth curves. As an application, we address eigenfunction restriction estimates for Laplace-Beltrami eigenfunctions on -dimensional compact Riemannian manifolds without boundary and Hermite functions on . Our method involves a geometric analysis of the contact order between the curve and the bicharacteristic flow of the semiclassical pseudodifferential operator.
Paper Structure (20 sections, 23 theorems, 256 equations, 2 figures)

This paper contains 20 sections, 23 theorems, 256 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Order of contact between curve and bicharacteristic flow
  3. Basic considerations
  4. Localization argument
  5. Order of contact
  6. Proof of Proposition \ref{['prop:main']}
  7. Reduction to estimating an oscillatory integral
  8. Dyadic decomposition in time
  9. Properties of the phase $\phi$
  10. Estimates for nonstationary part
  11. Asymptotic expansion of kernel
  12. Handling the core part
  13. Proof of Proposition \ref{['prop:phibound']}
  14. Expression for $\partial_u\partial_v \Phi$
  15. Proof of Proposition \ref{['prop:phibound']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $M$ be a $2$-dimensional compact Riemannian manifold without boundary. Assume that $\gamma$ is a smooth curve in $M$. Then there exists a constant $C(\gamma, M)>0$ such that for $\lambda\ge 1$, The estimate e:specproj is sharp for all smooth curves $\gamma$ in the sense that if the power $\rho(q,\sigma)$ is replaced by $\rho<\rho(q,\sigma)$, then e:specproj fails.

Figures (2)

  • Figure 1: The open neighborhood $U_k$ (left) and a collection of balls $B_{k,j}$ (right). The blue curve segments denote the set $\cup_{t\in (a_k,b_k)} \mathcal{Z}_{\gamma(t)}$.
  • Figure 2: The graphs of $v\mapsto \gamma(v)$ and $z_{v-u}(\gamma(u), \xi(u))$ when $\sigma$ is odd (left) and even (right). Note that $z_{v-u}(\gamma(u), \xi(u))$ intersects with $\gamma$ at a point other than $\gamma(u)$ on the right figure.

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1
  • Definition 1.2
  • Theorem 1.3: Tac10HT12
  • Theorem 1.4
  • Remark 2
  • Corollary 1.5
  • Remark 3
  • Corollary 1.6
  • Remark 4
  • ...and 34 more