Several integro-differential equations from mathematical physics on the interval \((-1,1)\) , studied by V. E. Petrov, can be reformulated as convolution equations with respect to a natural group structure on \((-1,1)\) . In this work, we generalize the tools developed in this one-dimensional setting to higher dimensions and establish a global pseudo-differential calculus on the group \(I^n = (-1,1)^n\) . We study boundedness properties for the Hörmander classes \(\Psi ^{m}_{\rho ,\delta }(I^n \times \mathbb {R}^n)\) such as the \(\mathbb {L}_p\) -Fefferman theorem, the sharp Gårding inequality and the Fefferman-Phong inequality. Furthermore, we investigate \(\mathbb {L}_2\) -compactness and the Atiyah-Singer-Fedosov index theorem for pseudo-differential operators with symbols in certain Shubin-type classes.
