  For $1<p\leq q$ we show that the Poisson semigroup $e^{-t\sqrt{-\Delta}}$ on
the $n$-sphere is hypercontractive from $L^{p}$ to $L^{q}$ in dimensions $n
\leq 3$ if and only if $e^{-t\sqrt{n}} \leq \sqrt{\frac{p-1}{q-1}}$. We also
show that the equivalence fails in large dimensions.
