Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

TL;DR

This paper investigates Fourier transform restriction phenomena on hypersurfaces of revolution within mixed norm spaces, providing sharp bounds, extending classical multipliers, and exploring discrete restriction conjectures.

Contribution

It offers new sharp bounds for Fourier restriction on hypersurfaces of revolution in mixed norm spaces and extends classical multiplier results.

Findings

Sharp bounds for Fourier restriction to hypersurfaces of revolution

Extension of the disc multiplier in mixed norm spaces

Results and open problems for the discrete restriction conjecture

Abstract

In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces $L^p_{rad} L^2_{ang} (\mathbb{R}^n)$. We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.