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

Abstract

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