Sharp Fourier extension on fractional surfaces

Abstract

For $\alpha \ge 2$, 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 ${\alpha }_0>5$.

arXiv comment27 pages. v2: Added a corollary on the two-dimension existence of extremals in page 3; shared to us by Quilodrán, and the authors thanks his comments. v3: Introduction expanded. This version: Added detailed proof for the existence of extremals, accepted version (by JFAA)

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Cited by in corpus (1)

• On profile decomposition for Airy type equation