  For $\alpha\geq 2$, we investigate a class of Fourier extension operators on
fractional surfaces $(\xi,|\xi|^\alpha)$. For the corresponding
$\alpha$-Strichartz inequalities, by applying the missing mass method and
bilinear restriction theory, we characterize the precompactness of extremal
sequences. Our result is valid in any dimension. In particular for dimension
two, our result implies the existence of extremals for $\alpha \in
[2,\alpha_0)$ with some $\alpha_0>5$.
