Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case

For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical $\mathit{\text{linear}}\mathit{\text{or}}\mathit{\text{bilinear}}\mathit{\text{adjoint}}\mathit{\text{restriction}}\mathit{\text{conjectures}}$ for such functions and verify the $\mathit{\text{linear}}\mathit{\text{adjoint}}\mathit{\text{restriction}}\mathit{\text{conjecture}}$ for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schr"odinger equation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere.