Sharp Fourier extension on fractional surfaces

TL;DR

This paper studies Fourier extension operators on fractional surfaces, establishing extremal existence and precompactness results for a range of dimensions and fractional powers, advancing understanding of Fourier analysis on fractional geometries.

Contribution

It introduces a new analysis of Fourier extension operators on fractional surfaces and characterizes extremal sequences using advanced restriction theory methods.

Findings

Precompactness of extremal sequences characterized for fractional surfaces.

Existence of extremals established for certain fractional powers in dimension two.

Results hold in any dimension, broadening applicability of Fourier extension theory.

Abstract

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 $\alpha_0>5$.