Stability of sharp Fourier restriction to spheres

TL;DR

This paper proves the stability of Fourier restriction inequalities on spheres in certain dimensions, showing constant functions are maximizers under perturbations and classifying all maximizers, extending previous results and deriving new inequalities.

Contribution

It establishes the stability of Fourier restriction inequalities on spheres for dimensions 3 to 7, including classification of maximizers and new sharp inequalities, extending prior work.

Findings

Constant functions maximize the inequality under perturbations.

Complete classification of maximizers in the perturbed setting.

Derivation of a new sharp adjoint restriction inequality on .

Abstract

In dimensions $d \in \{3,4,5,6,7\}$, we prove that the constant functions on the unit sphere $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ maximize the weighted adjoint Fourier restriction inequality $$ \left| \int_{\mathbb{R}^d} |\widehat{f\sigma}(x)|^4\,\big(1 + g(x)\big)\,d x\right|^{1/4} \leq {\bf C} \, \|f\|_{L^2(\mathbb{S}^{d-1})}\,,$$ where $\sigma$ is the surface measure on $\mathbb{S}^{d-1}$, for a suitable class of bounded perturbations $g:\mathbb{R}^d \to \mathbb{C}$. In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting ($g = {\bf 0}$), this was established by Foschi ($d=3$) and by the first and third authors ($d \in \{4,5,6,7\}$) in 2015. Our methods also yield a new sharp adjoint restriction inequality on $\mathbb S^7\subset \mathbb R^8$.