On the low regularity phase space of the Benjamin-Ono equation

In this paper we prove that the Benjamin-Ono equation is globally in time $C^0$-well-posed in the Hilbert space $H^{-1\mathrm{/}2,\sqrt{\log }}(\mathbb{T},\mathbb{R})$ of periodic distributions in $H^{-1\mathrm{/}2}(\mathbb{T},\mathbb{R})$ with $\sqrt{\log }$-weights. The space $H^{-1\mathrm{/}2,\sqrt{\log }}(\mathbb{T},\mathbb{R})$ can thus be considered as a maximal low regularity phase space for the Benjamin-Ono equation corresponding to the scale $H^s(\mathbb{T},\mathbb{R})$, $s>-1\mathrm{/}2$.