## 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.

### 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

Zbl 1141.58017
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