Stability of Sharp Fourier Restriction to Spheres

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