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The $L^p$ restriction bounds for Neumann data on surface

Xianchao Wu

Abstract

Let $\{u_λ\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data $ λ^{-1} \partial_νu_λ\,\vline_γ$ along a unit geodesic $γ$. Using the $T$-$T^*$ argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is $O(λ^{-\frac{1}p+\frac{3}2})$. The Van De Corput theorem (Lemma 2.1) plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.

The $L^p$ restriction bounds for Neumann data on surface

Abstract

Let be a sequence of -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold . We seek to get an restriction bounds of the Neumann data along a unit geodesic . Using the - argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is . The Van De Corput theorem (Lemma 2.1) plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.
Paper Structure (3 sections, 68 equations, 1 figure)

This paper contains 3 sections, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main0']}
  3. The optimality of \ref{['main eq']}

Figures (1)

  • Figure 1:

Theorems & Definitions (3)

  • proof
  • proof
  • proof