Global pseudo-differential operators on the Lie group $G=(-1,1{)}^n$

In this work we characterise the Hörmander classes $\text{symbClassOn}m\rho \delta \text{group},\text{H}\ddot{\text{o}}\text{r}$ on the open manifold $\text{group}=(-1,1{)}^n$. We show that by endowing the open manifold $\text{group}=(-1,1{)}^n$ with a group structure, the corresponding global Fourier analysis on the group allows one to define a global notion of symbol on the phase space $\text{group}\times {\mathbb{R}}^n$. Then, the class of pseudo-differential operators associated to the global Hörmander classes $\text{symbClassOn}m\rho \delta \text{group}\times {\mathbb{R}}^n$ recovers the Hörmander classes $\text{symbClassOn}m\rho \delta \text{group},\text{loc}$ defined by local coordinate systems. The analytic and qualitative properties of the classes $\text{symbClassOn}m\rho \delta \text{group}\times {\mathbb{R}}^n$ are presented in terms of the corresponding global symbols. In particular, $L^p$-Fefferman type estimates and Calderón-Vaillancourt theorems are analysed, as well as the spectral properties of the operators.