Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $i |\xi|^\alpha \xi$, with $1\le \alpha \le 2$, and that the nonlinear term is of the form $\partial_x f(u)$ where $f$ is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in $H^{s}(\mathbb{T})$ for $s\ge 1-\frac{\alpha}{2(\alpha+1)}$. This leads to some global existence results above the energy space $H^{\alpha/2}(\mathbb{T})$, for $\alpha \in [\sqrt{2},2]$.