On a Sharp Fourier Extension Inequality on the Circle with Lacunary Spectrum

TL;DR

This paper establishes a precise Fourier extension inequality on the circle for functions with lacunary spectra, specifically when the spectrum grows faster than three times the previous frequency, advancing understanding in harmonic analysis.

Contribution

It proves a sharp Fourier extension inequality for functions with lacunary spectra on the circle, a novel result in harmonic analysis.

Findings

Established a sharp inequality for lacunary spectra

Identified the Tomas-Stein exponent in this context

Extended the theory of Fourier extension inequalities

Abstract

We prove a sharp Fourier extension inequality on the circle for the Tomas-Stein exponent for functions whose spectrum $\{\pm \lambda_n\}$ satisfies $\lambda_{n+1}>3 \lambda_{n}$.