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

TL;DR

This paper studies extremal functions for Fourier extension inequalities related to fractional surfaces, establishing existence and compactness of extremals for a range of exponents, advancing understanding of Fourier analysis on fractional surfaces.

Contribution

It proves the existence of extremals for all p in [1,2] and shows precompactness of extremal sequences for p in (1,2], a novel result in fractional Fourier extension theory.

Findings

Existence of extremals for all p in [1,2].

Precompactness of extremal sequences for p in (1,2].

Extension inequalities hold for fractional surfaces with α ≥ 2.

Abstract

This article investigates the Fourier extension operator associated with the fractional surface $(\xi,|\xi|^{\alpha})$ for $\alpha\geq 2$. 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.