On a class of hyperbolic equations in Weyl-Hörmander calculus

In this paper we study the Cauchy problem for hyperbolic equations in the setting of Hörmander S(m,g) classes. We provide regularity estimates, existence and uniqueness in the scale of Sobolev spaces H(m,g) adapted to the Weyl-Hörmander calculus. We also obtain estimates for some parabolic evolution equations.