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 ${M}^{\circ }$ 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 ${\left(P+{k}^{2}\right)}^{-1}$ 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 ${M}^{2}×\left[0,{k}_{0}\right]$, and (ii) $T$ is bounded on ${L}^{p}\left({M}^{\circ }\right)$ for $1, 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/\left(n-2\right)) for which $T$ is bounded on ${L}^{p}\left(M\right)$ when zero modes are present.

Soit ${M}^{\circ }$ une variété complète de dimension $n\ge 3$ et $g$ une métrique asymptotiquement conique sur ${M}^{\circ }$, au sens où ${M}^{\circ }$ 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 ${\left(P+{k}^{2}\right)}^{-1}$ et la transformée de Riesz $T$ de l’opérateur $P={\Delta }_{g}+V$, où ${\Delta }_{g}$ 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 ${M}^{2}×\left[0,{k}_{0}\right]$, et (ii) que $T$ est borné sur ${L}^{p}\left({M}^{\circ }\right)$ pour $1, ce qui optimal sauf si $V\equiv 0$ ou bien ${M}^{\circ }$ 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/\left(n-2\right)) pour lequel $T$ est borné sur ${L}^{p}\left(M\right)$ 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},
mrnumber = {2566968},
zbl = {1175.58011},
language = {en},
url = {http://www.numdam.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. http://www.numdam.org/item/AIF_2009__59_4_1553_0/

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