Stein's restriction conjecture for smooth surfaces with nonzero Gaussian curvature

Let

$$\mathcal{S}=\{(\xi,\eta,h(\xi,\eta)):(\xi,\eta)\in[-1,1]^2\}$$

be the graph of a smooth function $h$, and define the extension operator

$$E_{\mathcal S}f(x,y,t)=\int_{[-1,1]^2}e^{i(x\xi+y\eta+t h(\xi,\eta))}f(\xi,\eta)\,d\xi\,d\eta.$$

Write $q'=\frac{q}{q-1}$. Stein's restriction conjecture. If the Gaussian curvature of $\mathcal S$ is nonzero at every point of $[-1,1]^2$, then there exists a constant $C$, depending only on $p$, $q$, and $\mathcal S$, such that

$$\|E_{\mathcal S}f\|_{L^{p}(\mathbb{R}^{3})}\le C\|f\|_{L^{q}([-1,1]^2)}$$

for all $p>3$ and $q'\le \frac{p}{2}$. This is the classical restriction conjecture for compact smooth surfaces with nonvanishing Gaussian curvature. The paper proves a result for surfaces with negative Gaussian curvature in the range $p>\frac{22}{7}$ and $q'<\frac{p}{2}$, but the stated conjecture itself remains unresolved in its full generality.