Low-regularity well-posedness conjecture for the modified Benjamin–Ono equation

Let $H^s$ denote the Sobolev space of order $s$, let $H^\infty$ denote the smooth-data space, and let $S_T^\infty$ be the solution map for the real-valued modified Benjamin–Ono equation. For $s>1/4$, low-regularity well-posedness conjecture. The solution map

$$S_T^\infty: H^\infty \rightarrow C([-T,T]:H^\infty)$$

can be uniquely extended to a continuous map from $H^s$ to $C([-T,T]:H^s)$ for a small $T=T(\|\phi\|_{H^s})>0$. The conjecture proposes weak well-posedness below the threshold $s=1/2$, where direct contraction methods fail; the paper's preceding theorem establishes local well-posedness at $s\geq 1/2$, while the asserted extension for $s>1/4$ remains the lower-regularity issue under study.