$H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ 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 be considered as the largest (low regularity) phase space corresponding to the scale $H^s(\mathbb{T},\mathbb{R})$, $s>-1/2$. (This is a joint work with P. Gérard.)