Restriction estimates for quadratic manifolds of arbitrary codimensions

TL;DR

This paper extends restriction estimates to all quadratic manifolds of any codimension by generalizing multilinear methods and introducing algebraic geometry tools, including algorithms for algebraic computations.

Contribution

It generalizes Bourgain and Guth's multilinear restriction method to arbitrary quadratic manifolds, incorporating algebraic geometry techniques and algorithms.

Findings

Restriction estimates established for all quadratic manifolds of arbitrary codimension.

A covering lemma for varieties based on Tarski's projection theorem.

Algorithms for computing algebraic quantities using cylindrical decomposition.

Abstract

The restriction conjecture is one of the famous problems in harmonic analysis. There have been many methods developed in the study of the conjecture for the paraboloid. In this paper, we generalize the multilinear method of Bourgain and Guth for the paraboloid, and obtain restriction estimates for all quadratic manifolds of arbitrary codimensions. In particular, our theorem recovers the main theorem of Bourgain and Guth for the paraboloid. A new ingredient is a covering lemma for varieties whose proof relies on Tarski's projection theorem in real algebraic geometry. We also provide algorithms to compute several algebraic quantities that naturally appear in the argument. These algorithms rely on a cylindrical decomposition in real algebraic geometry.