# Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II

Colin Guillarmou^{[1]}; Andrew Hassell^{[2]}

- [1] Université de Nice Laboratoire J. Dieudonné Parc Valrose 06100 Nice(FRANCE)
- [2] Australian National University Department of Mathematics Canberra ACT 0200 (AUSTRALIA)

Annales de l’institut Fourier (2009)

- Volume: 59, Issue: 4, page 1553-1610
- ISSN: 0373-0956

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topGuillarmou, Colin, and Hassell, Andrew. "Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II." Annales de l’institut Fourier 59.4 (2009): 1553-1610. <http://eudml.org/doc/10435>.

@article{Guillarmou2009,

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.},

affiliation = {Université de Nice Laboratoire J. Dieudonné Parc Valrose 06100 Nice(FRANCE); Australian National University Department of Mathematics Canberra ACT 0200 (AUSTRALIA)},

author = {Guillarmou, Colin, Hassell, Andrew},

journal = {Annales de l’institut Fourier},

keywords = {Asymptotically conic manifold; scattering metric; resolvent kernel; low energy asymptotics; Riesz transform; zero-resonance; asymptotically conic manifold},

language = {eng},

number = {4},

pages = {1553-1610},

publisher = {Association des Annales de l’institut Fourier},

title = {Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II},

url = {http://eudml.org/doc/10435},

volume = {59},

year = {2009},

}

TY - JOUR

AU - Guillarmou, Colin

AU - Hassell, Andrew

TI - Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II

JO - Annales de l’institut Fourier

PY - 2009

PB - Association des Annales de l’institut Fourier

VL - 59

IS - 4

SP - 1553

EP - 1610

AB - 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.

LA - eng

KW - Asymptotically conic manifold; scattering metric; resolvent kernel; low energy asymptotics; Riesz transform; zero-resonance; asymptotically conic manifold

UR - http://eudml.org/doc/10435

ER -

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