Lp$L^p$‐bounds in Safarov pseudo‐differential calculus on manifolds with bounded geometry

Abstract Given a smooth complete Riemannian manifold with bounded geometry and a linear connection on it (not necessarily a metric one), we prove the ‐boundedness of operators belonging to the global pseudo‐differential classes constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: and symmetric; and flat with any values of and . Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some ‐ boundedness. Different examples and applications are presented.