Linear and bilinear restriction to certain rotationally symmetric hypersurfaces

TL;DR

This paper establishes optimal Fourier restriction estimates for certain rotationally symmetric polynomial hypersurfaces, extending known results beyond the bilinear range under specific conditions.

Contribution

It proves new restriction estimates for polynomial hypersurfaces of revolution with non-negative coefficients, based on Fourier restriction estimates for elliptic hypersurfaces.

Findings

Optimal restriction estimates obtained for polynomial hypersurfaces of revolution

Uniform bounds depending only on dimension and polynomial degree

Extension beyond the bilinear range with unconditional bilinear estimates

Abstract

Conditional on Fourier restriction estimates for elliptic hypersurfaces, we prove optimal restriction estimates for polynomial hypersurfaces of revolution for which the defining polynomial has non-negative coefficients. In particular, we obtain uniform--depending only on the dimension and polynomial degree--estimates for restriction with affine surface measure, slightly beyond the bilinear range. The main step in the proof of our linear result is an (unconditional) bilinear adjoint restriction estimate for pieces at different scales.