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Colin Guillarmou; Andrew Hassell
Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II
(Résolvante à basse énergie et transformée de Riesz pour l’opérateur de Schrödinger sur des variétés asymptotiquement coniques. II)
Annales de l'institut Fourier, 59 no. 4 (2009), p. 1553-1610, doi: 10.5802/aif.2471
Article PDF | Reviews MR 2566968 | Zbl 1175.58011
Class. Math.: 58J50, 42B20, 35J10
Keywords: Asymptotically conic manifold, scattering metric, resolvent kernel, low energy asymptotics, Riesz transform, zero-resonance

Résumé - Abstract

Let $M^\circ $ be a complete noncompact manifold of dimension at least 3 and $g$ an asymptotically conic metric on $M^\circ $, in the sense that $M^\circ $ 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 $M^2 \times [0, k_0]$, and (ii) $T$ is bounded on $L^p(M^\circ )$ for $1 < p < n$, which range is sharp unless $V \equiv 0$ and $M^\circ $ has 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 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.


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