A note on the cone restriction conjecture in the cylindrically symmetric case

TL;DR

This paper proves the linear adjoint cone restriction conjecture for functions supported on the cone that are invariant under spatial rotations, using dyadic restriction estimates and Lorentz space inequalities.

Contribution

It introduces two novel arguments that establish the conjecture for cylindrically symmetric functions across all dimensions.

Findings

The conjecture holds for rotationally invariant functions on the cone.

Two different proof techniques are provided: dyadic restriction estimate and Lorentz space inequalities.

The results extend the validity of the conjecture to a broader class of functions.

Abstract

In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The first is based on a dyadic restriction estimate, while the second follows from a strengthening version of the Hausdorff-Young inequality and the H\"older inequality in the Lorentz spaces.