Fourier restriction above rectangles

TL;DR

This paper investigates Lebesgue space inequalities for Fourier restriction operators on rectangular sections of the paraboloid, proposing a conjecture on their norms' dependence on rectangle size and connecting it to the broader restriction conjecture.

Contribution

It formulates a conjecture relating operator norms to rectangle dimensions, links it to the restriction conjecture, and proves its sharpness, advancing understanding of restriction inequalities for degenerate hypersurfaces.

Findings

Proposes a conjecture on the dependence of restriction operator norms on rectangle size.

Shows the conjecture follows from the restriction conjecture for elliptic hypersurfaces.

Uses results to establish new restriction inequalities for hypersurfaces with additive structure.

Abstract

In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure.