Low regularity well-posedness for generalized Benjamin–Ono equations on the circle

New low regularity well-posedness results for the generalized Benjamin-Ono equations with quartic or higher nonlinearity and periodic boundary conditions are shown. We use the short-time Fourier transform restriction method and modified energies to overcome the derivative loss. Previously, Molinet–Ribaud established local well-posedness in $H^1(\mathbb{T},\mathbb{R})$ via gauge transforms. We show local existence and a priori estimates in $H^s(\mathbb{T},\mathbb{R}),$ $s>1/2$, and local well-posedness in $H^s(\mathbb{T},\mathbb{R})$, $s\ge3/4$ without using gauge transforms. In case of quartic nonlinearity we prove global existence of solutions conditional upon small initial data.