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 $$ \left| \int_{\mathbb{R}^d} |\widehat{fσ}(x)|^4\,\big(1 + g(x)\big)\,d x\right|^{1/4} \leq {\bf C} \, \|f\|_{L^2(\mathbb{S}^{d-1})}\,,$$ where $σ$ 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 $\mathbb S^7\subset \mathbb R^8$.

Figures (2)

• Figure 1: Bernstein ellipses $\mathcal{E}_\rho$ in the complex plane, $\rho\in\{2,4,\frac{7+3\sqrt 5}{2}\}$, and the corresponding enveloping disks $\overline{\mathcal{D}_r}\subset\mathbb{C}$, $r\in\{\frac{5}{4},\frac{17}{8},\frac{7}{2}\}$.
• Figure 2: Graphs of $\widehat{H}_2=\widehat{H}_2(\rho)$ for $\rho\in [0, 5]$ and $\rho\in[\frac{7}{2}, 4]$, respectively.

Theorems & Definitions (38)

• Theorem 1: Sharp weighted adjoint Fourier restriction: analytic version
• Theorem 2: Sharp weighted adjoint Fourier restriction: $C^k$-version
• Theorem 3: Full classification of maximizers
• Theorem 4: Sharp inequality on $\mathbb S^7$
• Lemma 5
• proof
• Lemma 6
• proof
• Proposition 7
• proof