Resolvent at low energy and Riesz transform for Schroedinger operators on asymptotically conic manifolds. II
Let $(M^\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\partial M = S^{n-1}$ and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel $(P + k^2)^{-1}$ and Riesz transform of the operator $P = Δ_g + V$, where $Δ_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function $V$ that is smooth on $M$ and vanishes to some finite order at the boundary. In the first paper in this series we made the assumption that $n \geq 3$ and that $P$ has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of $M^2 \times [0, k_0]$, and (ii) the Riesz transform of $P$ is bounded on $L^p(M^\circ)$ for $1 < p < n$, and that this range is optimal unless $V \equiv 0$ and $M^\circ$ has only one end. In the present paper, we perform a similar analysis assuming again $n \geq 3$ but allowing zero modes and zero-resonances. We find the precise range of $p$ for which the Riesz transform (suitably defined) of $P$ is bounded on $L^p(M)$ when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for $p$ precisely in the range $(n/(n-2), n/3)$, with a bigger range possible if the zero modes have extra decay at infinity.