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Refined $L^p$ restriction estimate for eigenfunctions on Riemannian surfaces

Chuanwei Gao, Changxing Miao, Yakun Xi

Abstract

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, Gérard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fourier extension estimate of Du and Zhang. Our results naturally unify the $L^p(M)$ estimates of Sogge and the $L^p(γ)$ restriction bounds of Burq, Gérard, and Tzvetkov, and are sharp for all $p \geq 2$, up to a $λ^\varepsilon$ loss. Second, we derive sharp estimates for the restriction of eigenfunctions to tubular neighborhoods of a curve with non-vanishing geodesic curvature. These estimates are closely related to a variable coefficient version of the Mizohata--Takeuchi conjecture, providing new insights into eigenfunction concentration phenomena.

Refined $L^p$ restriction estimate for eigenfunctions on Riemannian surfaces

Abstract

We refine the restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, Gérard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fourier extension estimate of Du and Zhang. Our results naturally unify the estimates of Sogge and the restriction bounds of Burq, Gérard, and Tzvetkov, and are sharp for all , up to a loss. Second, we derive sharp estimates for the restriction of eigenfunctions to tubular neighborhoods of a curve with non-vanishing geodesic curvature. These estimates are closely related to a variable coefficient version of the Mizohata--Takeuchi conjecture, providing new insights into eigenfunction concentration phenomena.

Paper Structure

This paper contains 22 sections, 26 theorems, 221 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Restriction of Laplacian Eigenfunctions to Fractal Measures
  3. Notation
  4. Acknowledgements.
  5. Preliminaries
  6. Reduction to Oscillatory Integral Operators with Carleson--Sjölin Phase
  7. Wave packet decomposition
  8. Locally constant property
  9. Proposition \ref{['cormain']} implies Theorem \ref{['theomain']}
  10. Proof of Proposition \ref{['theo11']}
  11. Variable-coefficient bilinear restriction estimate
  12. Bourgain--Guth argument.
  13. The Case $0 < \alpha < \frac{1}{2}$
  14. The case $\frac{1}{2}\leq\alpha<1$
  15. Proof of Proposition \ref{['theo22']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Given a smooth curve $\gamma \subset M$, there exists a constant $C$ such that where

Figures (2)

  • Figure 1: The $k^{-1/2}$-tubular neighborhood of $\gamma_0$ and the support of $\mu_k$.
  • Figure 2: $T_{\lambda^{-1+s}}(\gamma)$ and the largest $\lambda^{-1+2\beta} \times \lambda^{-1+\beta}$ region fitting inside of it.

Theorems & Definitions (43)

  • Theorem 1.1: BGT
  • Theorem 1.2: EP
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5: BGT
  • Theorem 1.6
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 33 more