  In this article, we develop a linear profile decomposition for the $L^p \to
L^q$ adjoint Fourier restriction operator associated to the sphere, valid for
exponent pairs $p<q$ for which this operator is bounded. Such theorems are new
when $p \neq 2$. We apply these methods to prove new results regarding the
existence of extremizers and the behavior of extremizing sequences for the
spherical extension operator. Namely, assuming boundedness, extremizers exist
if $q>\max\{p,\tfrac{d+2}d p'\}$, or if $q=\tfrac{d+2}d p'$ and the operator
norm exceeds a certain constant times the operator norm of the parabolic
extension operator.
