Abstract. Let M • be a complete noncompact manifold and g an asymptotically conic manifold on M • , in the sense that M • compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M . A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S n−1 ; such manifolds have an end that can be identified with R n \ B(R, 0) in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k 2 ) −1 where P = ∆g + V is the sum of the positive Laplacian associated to g and a real potential function V that is smooth on M and vanishes to third order at the boundary (i.e. decays to third order at infinity on M • ). We show that on a blown up version of M 2 × [0, k 0 ] the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on L p (M • ) for 1 < p < n, and that this range is optimal if V ≡ 0 or if M has more than one end. The result with V ≡ 0 is new even when M • = R n , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 < p < pmax where pmax > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary ∂M .Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In a follow-up paper we shall analyze the same situation in the presence of zero modes and zero-resonances.

Abstract. The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure dE(λ) of the square root of the Laplacian on R n is bounded fromis the conjugate exponent to p, with operator norm scaling as λ n(1/p−1/p ′ )−1 .We prove a geometric generalization in which the Laplacian on R n is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete L 2 restriction theorem, which is anthe L p → L p ′ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for p in the range above. The paper is in three parts. In the first part, we show at an abstract level that restriction estimates imply spectral multiplier estimates, and are implied by certain pointwise bounds on the Schwartz kernel of λ-derivatives of the spectral measure. In the second part, we prove such pointwise estimates for the spectral measure of the square root of Laplace-type operators on asymptotically conic manifolds. These are valid for all λ > 0 if the asymptotically conic manifold is nontrapping, and for small λ in general. In the third part, we observe that Sogge's estimate on spectral projections is valid for any complete manifold with C ∞ bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on dE(λ) may blow up exponentially as λ → ∞ when trapping is present. This justifies the statement that the estimate on dE(λ) is strictly stronger than Sogge's estimate.

Let M • be a complete noncompact manifold and g an asymptotically conic metric on M • , in the sense that M • compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M . A special case of particular interest is that of asymptotically Euclidean manifolds, where ∂M = S n−1 and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel (P + k 2 ) −1 and Riesz transform of the operator P = ∆g + V , where ∆g is the positive Laplacian associated to g and V is a real potential function V that is smooth on M and vanishes to some finite order at the boundary.In the first paper in this series we made the assumption that n ≥ 3 and that P has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of M 2 × [0, k 0 ], and (ii) the Riesz transform of P is bounded on L p (M • ) for 1 < p < n, and that this range is optimal unless V ≡ 0 and M • has only one end.In the present paper, we perform a similar analysis assuming again n ≥ 3 but allowing zero modes and zero-resonances. We show that once again that (unless n = 4 and there is a zero-resonance) the resolvent kernel is polyhomogeneous on the same space and compute its leading asymptotics. This generalizes results of Jensen-Kato and Murata to the variable coefficient setting. We also find the precise range of p for which the Riesz transform (suitably defined) of P is bounded on L p (M ) when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for p precisely in the range (n/(n − 2), n/3), with a bigger range possible if the zero modes have extra decay at infinity.

Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R > 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M ) → L p (M ; T * M ) for 1 < p < n and unbounded for p ≥ n if there is more than one end. It follows from known results that in such a case the Riesz transform on M is bounded for 1 < p ≤ 2 and unbounded for p > n; the result is new for 2 < p ≤ n. We also give some heat kernel estimates on such manifolds.We then consider the implications of boundedness of the Riesz transform in L p for some p > 2 for a more general class of manifolds. Assume that M is a n-dimensional complete manifold satisfying the Nash inequality and with an O(r n ) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on L p for some p > 2 implies a Hodge-de Rham interpretation of the L p cohomology in degree 1, and that the map from L 2 to L p cohomology in this degree is injective.

The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on R 2 which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at noncritical energies are shown to originate both at minima and maxima, although the latter are not germane to the L 2 spectral theory. Asymptotic completeness is shown, both in the traditional L 2 sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.1991 Mathematics Subject Classification. 35P25, 81Uxx.