Ionescu and Kenig showed global wellposedness for the Benjamin-Ono equation for all (in particular, low) regularities $s\geq 0$. At $s=0$ they used modified $X^{s,b}$ spaces in order to avoid logarithmic divergences. The techniques they used seem to be classifiable as "wellposedness" techniques. My question is whether there has been more progress below the $s=0$ regularity level using wellposedness techniques.
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