Jump to ContentJump to Main Navigation
Advances in AnalysisThe Legacy of Elias M. Stein$
Users without a subscription are not able to see the full content.

Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger

Print publication date: 2014

Print ISBN-13: 9780691159416

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159416.001.0001

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2020. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 24 February 2020

On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

Chapter:
(p.447) Chapter Eighteen On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature
Source:
Advances in Analysis
Author(s):

Christopher D. Sogge

Steve Zelditch

, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0018

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a wave equation associated with an assigned Laplace–Beltrami operator and a bit of microlocal (wavefront) analysis remains valid as well if a certain variable is part of a periodic orbit under a set of curvature assumptions. This can be done by lifting the wave equation for a compact two-dimensional Riemannian manifold without boundary up to the corresponding one for its universal cover. By identifying solutions of wave equations for this Riemannian manifold with “periodic” ones, this chapter is able to obtain the necessary bounds using a bit of wavefront analysis and the Hadamard parametrix for the universal cover.

Keywords:   restriction theorem, eigenfunction restriction estimates, nonpositive curvature, compact surfaces, eigenfunctions, wave equation, wavefront analysis, curvature assumptions, two-dimensional Riemannian manifold

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.