Gevrey and analytic hypoellipticity on the torus for non-linear operators constructed from rigid vector fields

We consider non-linear operators constructed from rigid vector fields. In particular, we study (global) Gevrey and analytic regularity on the torus; this is particularly interesting since even in the linear case we have a different behaviour on the torus and locally in R^n. To this aim we compute the "transposed" of a non-linear operator constructed from rigid vector fields, giving then a result of global Gevrey and analytic regularity on the torus, by the method of majorant series