Lᵖ-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 ∇ on it (not necessarily a metric one), we prove the Lᵖ-boundedness of operators belonging to the global pseudo-differential classes Ψρ, δᵐ(Ω^κ, ∇, τ) constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: ρ>1/3 and ∇ symmetric; and ∇ flat with any values of ρ and δ. Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some Lᵖ-Lq boundedness. Different examples and applications are presented.