The linear adjoint cone restriction conjecture

Let $n\ge 2$ be a fixed integer, let $S$ be a smooth compact non-empty subset of the cone $\{(\tau,\xi)\in {\mathbf{R}}\times {\mathbf{R}}^n:\,\tau=|\xi|\}$, and let $0<p,q\leq\infty$. For Schwartz functions $f$ on $S$, define

$$(fd\sigma)^{\vee}(t,x)=\int_{S} f(\tau,\xi)e^{i(x\cdot\xi+t\tau)}d\sigma(\xi).$$

Here $d\sigma$ is the pull-back of $d\xi/|\xi|$ under the projection $(\tau,\xi)\mapsto\xi$, and the linear adjoint restriction estimate is

$$\|(fd\sigma)^{\vee}\|_{L_{t,x}^q({\mathbf{R}}\times {\mathbf{R}}^n)}\leq C_{p,q,n,S}\|f\|_{L^p(S,d\sigma)}.$$

Linear adjoint cone restriction conjecture. The inequality above holds with constants depending on $S$, $n$ and $p,q$ if and only if $q>\frac{2n}{n-1}$ and $\frac{n+1}{q}\leq\frac{n-1}{p'}$. This conjecture identifies the optimal range of exponents for restricting the Fourier transform to the cone and is connected with major problems in harmonic analysis, including the Bochner–Riesz, local smoothing, Kakeya set and Kakeya maximal function conjectures. The source does not indicate whether the conjecture has been resolved.