Sharp Fourier Extension on Fractional Surfaces

We investigate a class of Fourier extension operators on fractional surfaces \((\xi ,|\xi |^\alpha )\) $(\xi ,|\xi {|}^{\alpha })$ with \(\alpha \ge 2\) $\alpha \ge 2$ . For the corresponding \(\alpha \) $\alpha$ -Strichartz inequalities, we characterize the precompactness of extremal sequences by applying the missing mass method and bilinear restriction theory. Our result is valid in any dimension. In particular for dimension two, our result implies the existence of extremals for \(\alpha \in [2,\alpha _0)\) $\alpha \in [2,{\alpha }_0)$ with some \(\alpha _0>5\) ${\alpha }_0>5$ .