Stability of sharp Fourier restriction to spheres

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 $${\left|{\int }_{{\mathbb{R}}^d}|\widehat{f\ddot{I}ƒ}(x){|}^4(1+g(x))dx\right|}^{1/4}\le \mathbf{C}\Vert f{\Vert }_{L^2({\mathbb{S}}^{d-1})},$$ where $\ddot{I}ƒ$ 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 ($g=0$), 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 ${\mathbb{S}}^7\subset {\mathbb{R}}^8$.

33 pages, 2 figures; v2: examples added; v3: referee's suggestions incorporated