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

In this note, we present two arguments showing that the classical $\mathit{\text{linear}}\mathit{\text{adjoint}}\mathit{\text{cone}}\mathit{\text{restriction}}\mathit{\text{conjecture}}$ 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.