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

Given a smooth complete Riemannian manifold with bounded geometry (M,g) and a linear connection del on it (not necessarily a metric one), we prove the L-p-boundedness of operators belonging to the global pseudo-differential classes Psi(m )(rho,delta)(Omega(kappa), del, tau) constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: rho>1/3 and del symmetric; and del flat with any values of rho and delta. Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some L-p-L-q boundedness. Different examples and applications are presented.