RESOLVENT AT LOW ENERGY AND RIESZ TRANSFORM FOR SCHRODINGER OPERATORS ON ASYMPTOTICALLY CONIC MANIFOLDS. II

摘要

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 + k(2))(-1) and Riesz transform T of the operator P = Delta(g) + V, where Delta(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 M-2 x [0, k(0)], and (ii) T is bounded on L-p(M-o) for 1 < p < n, which range is sharp unless V equivalent to 0 and M-o leas only one end. In the present paper; we perform a similar analysis allowing zero modes and zero-resonances. We show once again that (unless n = 4 and there is a zero-resonance) the resolvent kernel is polyhomogeneous oil the swine 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.