Sharp two-dimensional maximal estimate for the generalized Zakharov–Kuznetsov equation

Let $U(t)$ denote the linear evolution operator and let $u_0$ be initial data, with the mixed-norm estimate

$$\| U(t)u_0\|_{L^p_x L^{\infty}_{y,T}} \lesssim \|u_0\|_s.$$

Here $d$ is the spatial dimension, $p$ is the spatial integrability exponent, and $s$ is the Sobolev regularity index. The sharpness conjecture. If $d=2$ and $p=3$, then

$$\| U(t)u_0\|_{L^3_x L^{\infty}_{y,T}} \lesssim \|u_0\|_s$$

holds for all $s > \frac{2}{3}$. The estimate is known by interpolation in the larger range $s>\frac{3}{4}$, while the conjecture proposes the improved threshold $s>\frac{2}{3}$; the sharp range for these indices remains unresolved.