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 $i\mathrm{\mid }\xi {\mathrm{\mid }}^{\alpha }\xi$, with $1\le \alpha \le 2$, and that the nonlinear term is of the form ${\mathrm{\partial }}_{x}f(u)$ 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 $H^{\alpha \mathrm{/}2}(\mathbb{T})$, for $\alpha \in [\sqrt{2},2]$.