Adjoint restriction estimates and scaling on spheres

We test the restriction conjecture, in its adjoint form, against a class of measures $\phi_\delta d\sigma$ on the sphere ${\bf S}^{n-1}$. The densities $\phi_\delta$ are smoothed out characteristic functions of $\delta^{a_2} \times \delta^{a_3} \times \dots \times \delta^{a_n}$ rectangular caps on ${\bf S}^{n-1}$, where $a_2, a_3, \dots, a_n$are fixed nonnegative numbers.