On $L^p$ extremals for Fourier extension estimate to fractional surface

TL;DR

The paper studies sharp $L^p\to L^q$ Fourier extension inequalities for the fractional surface $p\mapsto(\xi,|\xi|^{\alpha})$ with $\alpha\ge2$ and $d\ge2$, where the natural relation is $q=(d+\alpha)/d\,p'$. It develops an $L^2$-based profile decomposition adapted to the case of vanishing Gaussian curvature at the origin by partitioning the surface into annuli and angular sectors, then upgrades to an $L^p$ theory through refined Strichartz estimates, frequency localization, and space-time localization arguments. The main results prove the existence of extremals for all $p\in[1,2]$ and the precompactness of extremal sequences for $p\in(1,2]$, with a precise criterion linking precompactness to a comparison constant $a^*M_{2,2}$. These results extend the extremal theory beyond the paraboloid by overcoming curvature degeneracies, enabling subsequent PDE applications via profile decompositions and refined restriction theory.

Abstract

This article investigates the Fourier extension operator associated with the fractional surface $(ξ,|ξ|^α)$ for $α\geq 2$. We show that the relevant $L^p\to L^q$ Fourier extension inequality possesses extremals for all exponents $p\in[1,2]$. Moreover, for all $p\in(1,2]$, the corresponding $L^p$-extremal sequences are precompact up to symmetries.

Theorems & Definitions (31)

• Theorem 1.1
• Remark
• Lemma 2.1: Lemma 2.5 of DY2024
• Lemma 2.2: Tao's bilinear restriction estimate, see also Theorem 2.7 of DY2024
• Lemma 2.3: Proposition 2.2 of DY2024
• Lemma 2.4: Lemma 3.2 of DY2024
• Lemma 2.5: Lemma 4.1 of DY2024
• Lemma 2.6: Lemma 4.4 of DY2024
• Lemma 2.7
• Lemma 2.8: Lemma 2.3 of Stovall2020