On extremizing sequences for adjoint Fourier restriction to the sphere

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 $p\mathrm{\ne }2$. 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 $q>\max \{p,\frac{d+2}{d}{p}^{\mathrm{\prime }}\}$, or if $q=\frac{d+2}{d}{p}^{\mathrm{\prime }}$ and the operator norm exceeds a certain constant times the operator norm of the parabolic extension operator.