2007/03/12 by Colin Guillarmou, Guillarmou, Colin, Andrew Hassell +1
Computer Science · Mathematics · #35J10 #58J37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.math/0703316
openalex publication_date 2007/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M^∘, g) be an asymptotically conic manifold, in the sense that M^∘ 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 ∂ M = Sn-1 and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel (P + k2)-1 and Riesz transform of the operator P = Δg + V, where Δg 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 ≥ 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 M2 × [0, k0], and (ii) the Riesz transform of P is bounded on Lp(M^∘) for 1 < p < n, and that this range is optimal unless V ≡ 0 and M^∘ has only one end. In the present paper, we perform a similar analysis assuming again n ≥ 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 Lp(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.