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

In this work we characterise the H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{H\"or}}$ on the open manifold $\group = (-1,1)^n$. We show that by endowing the open manifold $\group = (-1,1)^n$ 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 $\group \times \R^n$. Then, the class of pseudo-differential operators associated to the global H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ recovers the H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{loc}}$ defined by local coordinate systems. The analytic and qualitative properties of the classes $\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ are presented in terms of the corresponding global symbols. In particular, $L^p$-Fefferman type estimates and Calder\'on-Vaillancourt theorems are analysed, as well as the spectral properties of the operators.