Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II
Annales de l'Institut Fourier, Volume 59 (2009) no. 4, p. 1553-1610
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.
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.
DOI : https://doi.org/10.5802/aif.2471
Classification:  58J50,  42B20,  35J10
Keywords: Asymptotically conic manifold, scattering metric, resolvent kernel, low energy asymptotics, Riesz transform, zero-resonance
@article{AIF_2009__59_4_1553_0,
     author = {Guillarmou, Colin and Hassell, Andrew},
     title = {Resolvent at low energy and Riesz transform for Schr\"odinger operators on asymptotically conic manifolds. II},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     pages = {1553-1610},
     doi = {10.5802/aif.2471},
     zbl = {1175.58011},
     mrnumber = {2566968},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2009__59_4_1553_0}
}
Guillarmou, Colin; Hassell, Andrew. Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1553-1610. doi : 10.5802/aif.2471. https://aif.centre-mersenne.org/item/AIF_2009__59_4_1553_0/

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