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

Guillarmou, 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 $1 &lt; p &lt; n$, 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 $n/(n-2) &lt; p &lt; n/3$) 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 $1 &lt; p &lt; n$, 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 $n/(n-2) &lt; p &lt; n/3$) 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 -