# 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

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