Sharp Fourier extension on fractional surfaces

For $\alpha \ge 2$, we investigate a class of Fourier extension operators on fractional surfaces $(\xi ,\mathrm{\mid }\xi {\mathrm{\mid }}^{\alpha })$. 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 ${\alpha }_0>5$.