Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

Abstract

Let $$M^\circ$$ be a complete noncompact manifold and g an asymptotically conic manifold on $$M^\circ$$ , 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 that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S n−1; such manifolds have an end that can be identified with $${\mathbb{R}}^n \backslash B(R,0)$$ in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k 2)−1 where $$P = \Delta_g + V$$ is the sum of the positive Laplacian associated to g and a real potential function $$V\in C^{\infty}(M)$$ which vanishes to second order at the boundary (i.e. decays to second order at infinity on $$M^\circ$$ ) and such that $$\Delta_{\partial M}+(n-2)^2/4+V_0 > 0$$ if $$V_0:=(x^{-2}V)|_{\partial M}$$ . Then we show that on a blown up version of $$M^2 \times [0, k_0]$$ the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on $$L^p(M^\circ)$$ for 1 < p < n if $$V_0\equiv 0$$ , and that this range is optimal if $$V \not\equiv 0$$ or if M has more than one end. The result with $$V\not\equiv 0$$ is new even when $$M^\circ = {\mathbb{R}}^n$$ , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 < p < p max where p max > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary $$\partial M$$ . Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In the follow-up paper Guillarmou and Hassell (Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, preprint) [7] we analyze the same situation in the presence of zero modes and zero-resonances.