Stability of sharp Fourier restriction to spheres (2108.03412v3)
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 $$ \left| \int_{\mathbb{R}d} |\widehat{f\sigma}(x)|4\,\big(1 + g(x)\big)\,d x\right|{1/4} \leq {\bf C} \, |f|_{L2(\mathbb{S}{d-1})}\,,$$ 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 $\mathbb S7\subset \mathbb R8$.