Points near the north pole are transformed stereographically to high frequencies, and points near the south pole to low frequencies. For \(v>d / 2\), the weighted distribution with density proportional to

\[
|\sin (\theta / 2)|^{2 v-d}
\]

reduces the mass on northern latitudes and increases that on southern latitudes, maintaining radial symmetry. Show that the stereographic image of the weighted distribution is inversely proportional to \(\left(1+\|\omega\|^{2}ight)^{u+d / 2}\). Find the normalizing constants for both distributions.