Sharp system-exponent range for higher-dimensional ZK estimates in the subcritical regime

Determine whether the system-exponent range in the higher-dimensional Zakharov–Kuznetsov Bessel-family Strichartz estimate is sharp for dimensions d≥3 and exponents 1<q<3, where the stated necessary condition leaves a gap.

Background

The paper proves frequency-localized Bessel-family Strichartz estimates for the Zakharov–Kuznetsov equation in dimensions d≥3. The sufficient range is inherited from the almost-orthogonal dispersive framework, while Proposition 3.1 supplies necessary conditions on the discrete system exponent β.

For 1<q<3, the sufficient and necessary bounds do not coincide. Consequently, the paper does not establish whether the admissible β-range in this part of the higher-dimensional Zakharov–Kuznetsov regime is optimal. The issue is specifically the unresolved sharpness of the system exponent, not the validity of the stated sufficient estimate.

References

However, sharpness in the regime $1<q<3$ (the segment $(B_1,E)$) remains unknown, since the necessary condition in Proposition~\ref{prop:KP-ZK-necessary}~\textup{($3{\scriptscriptstyle +}$D-ZK)} leaves a gap.

— 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(ii)