Sharpness conjecture for the local well-posedness Sobolev range

Let $s$ denote the Sobolev regularity exponent in Theorem~. Sharpness conjecture for the local well-posedness Sobolev range. The range of $s$ in Theorem~ is sharp for generic problems, except possibly at the endpoint. The claim concerns the small-data local theory, which the source places one-half derivative above the scaling threshold; the proposed obstruction is related to instability of non-trapping below the stated thresholds.