Resolvent at low energy and Riesz transform for schrödinger operators on asymptotically conic manifolds. II

Let M^o be a complete noncompact manifold of dimension at least 3 and g an asymptotically conic metric on M^o, in the sense that M ^o compactifies to a manifold with boundary M so that g becomes a scattering metric on M. We study the resolvent kernel (P + k²) ^-1 and Riesz transform T of the operator P - δ_g + V, where δ_g is the positive Laplacian associated to g and V is a real potential function smooth on M and vanishing at the boundary. In our first paper we assumed that P has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is polyhomogeneous conormal on a blown up version of M² × [0, k₀], and (ii) T is bounded on L ^p(M^p) for 1 < p < n, which range is sharp unless V = 0 and M^o has only one end. In the present paper, we perform a similar analysis allowing zero modes and zeroresonances. We show once again that (unless n = 4 and there is a zero-resonance) the resolvent kernel is polyhomogeneous on the same space, and we find the precise range of p (generically n/(n - 2) < p < n/3) for which T is bounded on L ^P(M) when zero modes are present.