Adjoint restriction estimates and scaling on spheres

Abstract:

We test the restriction conjecture, in its adjoint form, against a class of measures $\phi _\delta d\sigma$ on the sphere $\textbf {S}^{n-1}$. 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 $\textbf {S}^{n-1}$, where $a_2, a_3, \dots , a_n$ are fixed nonnegative numbers.