Restriction estimates for 2D surfaces of finite type 3 and applications to dispersive equations

TL;DR

This paper establishes restriction estimates for specific 2D surfaces of finite type by leveraging existing results on perturbed paraboloids and hyperboloids, and applies these to improve analysis of discrete nonlinear Schrödinger equations.

Contribution

It extends restriction estimates to a new class of 2D surfaces of finite type and applies these results to nonlinear dispersive equations.

Findings

Restriction estimates for the specified 2D surfaces are proved.

The method reduces the problem to known results on perturbed paraboloids and hyperboloids.

Applications to discrete nonlinear Schrödinger equations are demonstrated.

Abstract

In this paper, we prove the restriction estimates for 2D surfaces S:= {(xi1, xi2, xi1^3 +/- xi2^3) : (xi1, xi2) in [0,1]^2} by reducing to Wang-Wu's result on the perturbed paraboloid and to the results on the perturbed hyperboloid obtained by Buschenhenke, M\"uller, and Vargas, as well as by Guo and Oh. The method is based on the rescaling technique developed in [LMZ21]. Besides, we will use the estimates to give a better analysis for discrete nonlinear Schr\"odinger equations.