Global hypoellipticity for a class of periodic Cauchy operators

This note presents an investigation on the global hypoellipticity problem for Cauchy operators on $\mathbb{T}^{n+1}$ belonging to the class \linebreak $L = \prod_{j=1}^{m}\left(D_t + c_j(t) P_j(D_x)\right)$, where $P_j(D_x)$ is a pseudo-differential operator on $\mathbb{T}^n$ and $c_j = c_j(t)$, a smooth, complex valued function on $\mathbb{T}$. The main goal of this investigation consists in establishing connections between the global hypoellipticity of the operators $L$ and its normal form $L_0 = \prod_{j=1}^m \left( D_t + c_{0,j}P_j(D_x)\right)$. In order to do so, the problem is approached by combining H\"{o}rmander's and Siegel's conditions on the symbols of the operators $L_j = D_t + c_j(t) P_j(D_x)$.