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.

On an asymptotically hyperbolic manifold (X n+1 , g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ) := (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 2 (n − N) that they did not deal with. We show that the points of n 2 − N are at most some poles of finite multiplicity, and that the same property holds for the points of n+1 2 − N if and only if the metric is 'even'. On the other hand, there exist some metrics for which R(λ) has an essential singularity on n+1 2 − N and these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity. p i=1 rank M i the total polar rank of M (λ) at λ 0 . If now the total polar rank is finite at each pole of M (λ) we say that M (λ) is finite-meromorphic and Mer f (U, B 0 ) denotes the space of finite-meromorphic functions in U with values in B 0 . At last, if λ 0 ∈ U and M (λ) is meromorphic in U \ {λ 0 } but not in U , we will say that λ 0 is an essential singularity of M (λ).At last, note that all these definitions extend to locally convex vector spaces (see for instance Bunke-Olbrich [5]).Here is an interpretation of the result of Mazzeo and Melrose [17, Th. 7.1] :Theorem 1.1. Let (X, g) be an asymptotically hyperbolic manifold, ∆ g its Laplacian acting on functions and x a boundary defining function on X. The modified resolventwith poles at points λ ∈ O 0 such that λ(n − λ) ∈ σ pp (P ), extends to a finite-meromorphic family

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.

We define Pollicott-Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane. 1 arXiv:1410.5516v2 [math.DS] 3 Apr 2016Theorem 3. Assume that the stable/unstable foliations E u , E s are orientable. Then the function ζ V (λ) admits a meromorphic continuation to λ ∈ C.Theorem 3 was established in [GLP] in the special case of Anosov flows when additionally V = 0. Another argument based on microlocal methods was presented in [DyZw13] and served as the starting point of our proof. The singularities (zeroes and poles) or ζ V are Pollicott-Ruelle resonances for certain operators on the bundle of differential forms, see §5.2. Our methods actually prove meromorphic continuation of more general dynamical traces -see Theorem 4 in §5.1. The orientability condition can be relaxed, see (5.11) and the remarks following it.In Theorem 3 we assumed that the potential V is smooth. However it is likely that this statement also holds for certain nonsmooth potentials arising from (un)stable Jacobians by passing to the Grassmanian bundle of M as in [FaTs13b, §2]. The framework of the present paper appears convenient for that goal since the lifted flow on a neighborhood of the unstable bundle in the Grassmanian bundle of an open hyperbolic system produces another open hyperbolic system.Applications to boundary value problems. A useful corollary of our work is the well-posedness (up to a finite dimensional space corresponding to resonant states) of the two boundary value problems for the transport equationfor u, f in certain anisotropic Sobolev spaces, where ∂ ± U := {x ∈ ∂U | ∓Xρ > 0} and V is a potential; see for instance Proposition 6.1 and particularly [G, §4]. The microlocal description of solutions is crucial in the proof of lens rigidity of surfaces with hyperbolic trapped sets and no conjugate points in [G].Motivation and discussion. We call a resonance λ 0 the first resonance of X if λ 0 is simple (that is, rank Π λ 0 = 1), λ 0 ∈ R, and there are no other resonances with Re λ ≥ Re λ 0 . We say that there is a spectral gap of size ν > 0 if there are no resonances with Re λ ≥ Re λ 0 − ν, and an essential spectral gap if the number of resonances with Re λ ≥ Re λ 0 −ν is finite. The size of an essential spectral gap on compact manifolds is bounded from above, see Jin-Zworski [JiZw]; a combination of the techniques of [JiZw] with those of the present paper could potentially lead to a similar result in our more general setting.The first resonances and the corresponding resonant states capture key dynamical features of the flow. For Anosov flows in the scalar case E = C, X = X, zero is always a resonance since the function 1 is a resonant sta...