Determine the optimal isotropic Sobolev regularity threshold

Establish a matching lower bound for the infimum of admissible isotropic Sobolev indices for local well-posedness of the Benjamin–Ono–Zakharov–Kuznetsov equation, complementing the upper bound s_opt ≤ 19/16 provided by the present theory.

Background

The paper proves local well-posedness for the Benjamin–Ono–Zakharov–Kuznetsov equation on R² when the isotropic Sobolev regularity satisfies s > 19/16. It also identifies the formal scaling-critical index as s_c = −1/4 and explains that the low–high resonant structure obstructs a direct Picard iteration without, by itself, proving ill-posedness in lower-regularity spaces.

The authors distinguish the threshold produced by their particular refined Strichartz, maximal-function, local-smoothing, and modified-energy method from the genuinely optimal threshold. A matching lower bound would require a norm-inflation result, failure of continuity of the flow map, or another rigorous obstruction below the established local well-posedness range.