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**Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II.**
*(English)*
Zbl 1175.58011

Summary: 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 [Math. Ann. 341, No. 4, 859–896 (2008; Zbl 1141.58017)], 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.

In our first paper [Math. Ann. 341, No. 4, 859–896 (2008; Zbl 1141.58017)], 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.

### MSC:

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

35J10 | Schrödinger operator, Schrödinger equation |

47F05 | General theory of partial differential operators |

### Keywords:

asymptotically conic manifold; scattering metric; resolvent kernel; low energy asymptotics; Riesz transform; zero-resonance### Citations:

Zbl 1141.58017
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\textit{C. Guillarmou} and \textit{A. Hassell}, Ann. Inst. Fourier 59, No. 4, 1553--1610 (2009; Zbl 1175.58011)

### References:

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