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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 onthe group allows one to define a global notion of symbol on the phase space$\group \times \R^n$. Then, the class of pseudo-differential operatorsassociated to the global H\"ormander classes$\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ recovers the H\"ormanderclasses $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{loc}}$ defined bylocal coordinate systems. The analytic and qualitative properties of theclasses $\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ are presented interms of the corresponding global symbols. In particular, $L^p$-Fefferman typeestimates and Calder\'on-Vaillancourt theorems are analysed, as well as thespectral properties of the operators.

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