The bilinear adjoint restriction conjecture for transverse paraboloid pieces

Let $n\ge 3$, and let $S_1$ and $S_2$ be smooth compact non-empty subsets of the paraboloid in ${\mathbf{R}}\times{\mathbf{R}}^{n-1}$ whose unit normals are separated by at least a fixed angle $c>0$. Let $d\sigma_1$ and $d\sigma_2$ be their canonical Lebesgue measures. If

$$q\ge\frac{n}{n-1},\qquad \frac{n+2}{2q}+\frac{n}{p}\le n,\qquad \frac{n+2}{2q}+\frac{n-2}{p}\le n-1,$$

then there exists a constant $0<C<\infty$, depending on $S_1$, $S_2$, $n$, and $p,q$, such that

$$\|(fd\sigma_1)^{\vee}(gd\sigma_2)^{\vee}\|_{L^q_{t,x}({\mathbf{R}}\times{\mathbf{R}}^{n-1})}\le C\|f\|_{L^p(S_1)}\|g\|_{L^p(S_2)}$$

for all $f\in L^p(S_1)$ and $g\in L^p(S_2)$. Bilinear adjoint restriction conjecture. Under these conditions, the displayed bilinear estimate holds. The conjecture concerns the optimal exponent range for bilinear restriction estimates and is a key tool in the study of linear restriction and nonlinear dispersive equations. The supplied source gives no resolution status for the full stated range.