The restriction conjecture for the sphere

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The restriction conjecture for the sphere

   Let $n\geq 2$, let ${\mathbb S}^{n-1}\subset\mathbb R^n$ be the unit sphere, and let $R(p\to q)$ denote the restriction estimate

   $$\left\|\widehat{f}\big\vert_{{\mathbb S}^{n-1}}\right\|_{L^q({\mathbb S}^{n-1})}\leq C\left\|f\right\|_{L^p(\mathbb R^n)}.$$

   Here $f\in\mathcal{S}(\mathbb R^n)$ and $C$ is independent of $f$. Restriction conjecture. Determine the full range of exponents $(p,q)\in[1,\infty]^2$ for which $R(p\to q)$ holds for every smooth function $f\in\mathcal{S}(\mathbb R^n)$. The conjecture is a central problem in Fourier restriction theory; the full range remains open, although the case $q=2$ was solved by Tomas and Stein.

   source: Cristina Benea, Frederic Bernicot and Teresa Luque, “Sparse bilinear forms for Bochner Riesz multipliers and applications”, arXiv:1605.06401 (2016).

2. The restriction conjecture for the sphere

   Let $S$ be the unit sphere in ${\mathbb R}^d$, and let $\sigma_S$ denote its surface measure. For a function $f$ on ${\mathbb R}^d$, write $\widehat f$ for its Fourier transform, and let $p'$ be the conjugate exponent to $p$. Restriction conjecture. The estimate

   $$\left(\int_S |\widehat f(\xi)|^r\,d\sigma_S(\xi)\right)^{1/r} \leq C_{p,r}\left(\int_{{\mathbb R}^d}|f(x)|^p\,dx\right)^{1/p}$$

   should hold whenever

   $$p<\frac{2d}{d+1},\qquad r\leq \frac{d-1}{d+1}p'.$$

   This is the central restriction problem for the sphere; substantial partial results and recent developments are known, but the stated range remains open in general.

   source: Alex Iosevich and Azita Mayeli, “Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery”, arXiv:2311.04331 (2023).