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, Tome 59 (2009) no. 4, pp. 1553-1610.

Soit M une variété complète de dimension n3 et g une métrique asymptotiquement conique sur M , au sens où M se compactifie en une variété à bord M telle que g soit une métrique de type “scattering” sur M. On étudie le noyau intégral de la résolvante (P+k 2 ) -1 et la transformée de Riesz T de l’opérateur P=Δ g +V, où Δ g est le laplacien positif associé à g et V un potentiel réel, lisse sur M et s’annulant au bord.

Dans le premier article nous avons supposé que 0 n’est ni résonance ni valeur propre pour P et montré (i) que le noyau de la résolvante est conormal polyhomogène sur une version éclatée de M 2 ×[0,k 0 ], et (ii) que T est borné sur L p (M ) pour 1<p<n, ce qui optimal sauf si V0 ou bien M a seulement un bout.

Dans le présent article, on effectue une analyse similaire tout en autorisant les cas où 0 est résonance ou valeur propre. On montre là encore (sauf si n=4 et 0 est résonance) que le noyau de la résolvante est polyhomogène sur le même espace, et on donne l’intervalle de p (génériquement n/(n-2)<p<n/3) pour lequel T est borné sur L p (M) quand 0 est valeur propre.

Let M be a complete noncompact manifold of dimension at least 3 and g an asymptotically conic metric on M , in the sense that M 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=Δ g +V, where Δ 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 ×[0,k 0 ], and (ii) T is bounded on L p (M ) for 1<p<n, which range is sharp unless V0 and M 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.

DOI : 10.5802/aif.2471
Classification : 58J50, 42B20, 35J10
Keywords: Asymptotically conic manifold, scattering metric, resolvent kernel, low energy asymptotics, Riesz transform, zero-resonance
Mot clés : variété asymptotiquement conique, métrique scattering, noyau de la résolvante, asymptotique à basse énergie, transformée de Riesz, zéro-résonance
Guillarmou, Colin 1 ; Hassell, Andrew 2

1 Université de Nice Laboratoire J. Dieudonné Parc Valrose 06100 Nice(FRANCE)
2 Australian National University Department of Mathematics Canberra ACT 0200 (AUSTRALIA)
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Guillarmou, Colin; Hassell, Andrew. Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II. Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1553-1610. doi : 10.5802/aif.2471. http://archive.numdam.org/articles/10.5802/aif.2471/

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