Sharp system-exponent range for two-dimensional ZK estimates

Determine whether the system-exponent range in the two-dimensional Zakharov–Kuznetsov Bessel-family Strichartz estimate is sharp for exponents other than the identified exceptional points $(p,q)=(7/4,7)$ and $(p,q)=(\infty,1)$.

Background

For the two-dimensional Zakharov–Kuznetsov equation, the paper establishes a Bessel-family Strichartz estimate over a specified range of time and space exponents and discrete system exponents. A separate proposition gives necessary conditions that constrain the possible value of β.

The sufficient and necessary ranges agree only at the two points (7/4,7)(7/4,7) and (∞,1)(\infty,1). Away from those points, the paper leaves unresolved whether the full stated range is optimal.

References

For $d=2$, the sharpness of the range eq:2d-ZK-beta-range for eq:2d-zk-strichartz remains unknown, except for $(p,q)=(7/4,7)$ and $(p,q)=(\infty,1)$ (the points $B_2$ and $E$), in view of the necessary condition in Proposition~\ref{prop:KP-ZK-necessary}~\textup{(2D-ZK)}.

eq:2d-ZK-beta-range:

{β≤β(q),1≤q<7,β<q/(q−3),7≤q<∞,β=1,q=∞.\begin{cases} \beta \le \beta(q), & 1\le q<7,\\[1.2ex] \beta < q/(q-3), & 7\le q<\infty,\\[1.2ex] \beta=1, & q=\infty. \end{cases}

eq:2d-zk-strichartz:

∑jλj∣UZK(t)fj∣2Ltp(;Lzq(2))≲CB(f)1−1/βλℓβ.{\sum_j\lambda_j|U_{ZK}(t)f_j|^2}_{L_t^p(;L_z^q(^2))} \lesssim C_B(f)^{1-1/\beta}{\lambda}_{\ell^\beta}.

— Almost-orthogonal Strichartz estimates and radial improvements  (2609.30726 - Ji et al., 25 Sep 2026) in Remark [Sharpness], Section 1, immediately following Theorem 1.2(i)