A strong maximum principle for the Paneitz operator and a non-local flow for the Q-curvature

In this paper we consider Riemannian manifolds (M-n, g) of dimension n >= 5 with semi-positive Q-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green's function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive Q-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive Q-curvature.