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 $\mathbb{S}^{d1}\subset \mathbb{R}^d$ maximize the weighted adjoint Fourier restriction inequality $$ \left \int_{\mathbb{R}^d} \widehat{f\sigma}(x)^4\,\big(1 + g(x)\big)\,d x\right^{1/4} \leq {\bf C} \, \f\_{L^2(\mathbb{S}^{d1})}\,,$$ where $\sigma$ is the surface measure on $\mathbb{S}^{d1}$, for a suitable class of bounded perturbations $g:\mathbb{R}^d \to \mathbb{C}$. In such cases we also fully classify the complexvalued 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 DOI:
 10.48550/arXiv.2108.03412
 arXiv:
 arXiv:2108.03412
 Bibcode:
 2021arXiv210803412C
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs
 EPrint:
 33 pages, 2 figures, v2: examples added