Sharp Fourier extension on fractional surfaces

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 $\alpha_0>5$.Comment: 27 pages. v2: Added a corollary on the two-dimension existence of extremals in page 3; shared to us by Quilodr\'{a}n, and the authors thanks his comments. v3: Introduction expanded; this version submitted to journa

Sharp Fourier extension to odd curves

In this paper, we characterize the precompactness of extremal sequences for the Fourier extension inequalities on odd planar curves $s=\xi^{\ell}$ with an odd integer $\ell>1$. Our results are valid for mixed-norm Strichartz inequalities. We establish the corresponding profile decomposition, where one \textit{two-profile phenomenon} appears due to some symmetries of odd curves. In addition, we prove the existence of extremals for subcritical Strichartz inequalities and obtain some analytic properties for these extremals.Comment: 37 pages. Comments are welcome

Extremals for α\alpha-Strichartz inequalities

A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities.Comment: 31 pages. Some small typos fixe