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

Abstract

Given a smooth complete Riemannian manifold with bounded geometry $(M,g)$ and a linear connection $\nabla$ on it (not necessarily a metric one), we prove the $L^p$-boundedness of operators belonging to the global pseudo-differential classes ${\Psi }_{\rho ,\delta }^m\left({\Omega }^{\kappa },,,\nabla ,,,\tau \right)$ constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: $\rho >1/3$ and $\nabla$ symmetric; and $\nabla$ 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.