SCATTERING THEORY FOR THE LAPLACIAN ON MANIFOLDS WITH BOUNDED CURVATURE

WERNER MÜLLER AND GORM SALOMONSEN.

Abstract. In this paper we study the behaviour of the continuous spectrum of the Laplacian on a complete Riemannian manifold of bounded curvature under perturbations of the metric. The perturbations that we consider are such that its covariant derivatives up to some order decay with some rate in the geodesic distance from a fixed point. Especially we impose no conditions on the injectivity radius. One of the main results are conditions on the rate of decay, depending on geometric properties of the underlying manifold, that guarantee the existence and completeness of the wave operators.

0. Introduction.

The basic objects of the time-dependent approach to scattering theory are the wave operators. They are attached to a pair H_{0} and H of self-adjoint operators, acting in Hilbert spaces H_{0} and H, respectively, and a unitary operator J: H_{0} → H. Let P_{ac}(H_{0}) be the orthogonal projection onto the absolutely continuous subspace of H_{0}. Then the wave operators W_{±}(H, H_{0}; J) are said to exist, if the strong limit

(0.1)  W_{±}(H, H_{0}; J) = s-lim_{t→±∞} e^{itH}Je^{−itH_{0}}P_{ac}(H_{0}),

exists. If the wave operators exist and are complete, they give rise to a unitary equivalence of the absolutely continuous parts H_{0,ac} and H_{ac} of H_{0} and H, respectively. In this case the scattering operator is defined by S = W_{+}^{∗} ∘ W_{−}. There exist several general techniques to establish the existence and completeness of the wave operators. We will use the Kato-Birman theory, especially the invariance principle. Scattering theory is intimately connected with quantum mechanics and there is a vast literature dealing with the existence and completeness of the wave operators in this case. For a comprehensive account of mathematical scattering theory we refer to [BW], [RS], [Ya].

In this paper we study scattering theory in the geometric context. The basic setup is as follows. Let (M, g) be a complete Riemannian manifold and let Δ_{g} be the Laplacian on functions attached to g. Since M is complete, Δ_{g} is an essentially self-adjoint operator in L^{2}(M) [Cn]. If M is non-compact, then Δ_{g} may have a nonempty continuous spectrum. We will consider perturbations h of the metric g which decay with a certain rate in the geodesic distance from a fixed point. Especially the metrics will be quasi-isometric so that the Hilbert spaces L^{2}(M, dμ_{g}) and L^{2}(M, dμ_{h}) are equivalent. Let J be the corresponding identification operator. The main purpose of this paper is to study the conditions on the perturbation h which imply the existence and completeness of the wave operators W_{±}(Δ_{h}, Δ_{g}; J).

Date: June 1, 2007.

1991 Mathematics Subject Classification. Primary: 58J50; Secondary: 35P25.

Key words and phrases. Scattering theory, Laplace operator, manifolds of bounded curvature.

Scattering theory for the Laplacian on manifolds has been studied in a number of cases. In particular, it has been studied for manifolds with a special structure at infinity. So, for example, the case of manifolds with asymptotically cylindrical ends has been treated in [Me3]. Asymptotically Euclidean spaces were studied in [Me2]. In [Mu1], [Mu3] generalizations of locally symmetric manifolds of finite volume and ℚ-rank one have been considered. See [Me1] for an overview and a discussion of some of these examples.

Our goal is to study non-compactly supported perturbations of the metric on arbitrary complete Riemannian manifolds with some restrictions on the curvature. To this end we introduce a certain class of functions, called functions of moderate decay, which describe the rate of decay of the perturbation of a given metric. Let β: [1, ∞) → ℝ^{+} be a function of moderate decay (see Definition 1.4). Then two complete metrics g and h are said to be equivalent up to order k ∈ ℕ, if there exist C > 0 and p ∈ M such that

(0.2)  |g − h|_{g}(x) + ∑_{j=0}^{k−1} |(∇^{g})^{j}(∇^{g} − ∇^{h})|_{g} (x) ≤ Cβ(1 + d_{g}(x, p)), x ∈ M,

where d_{g}(x, p) is the geodesic distance of x and p with respect to g, and ∇^{g} (resp. ∇^{h}) the Levi-Civita connection with respect to g (resp. h). Note that ∇^{g} − ∇^{h} is a tensor and therefore, (∇^{g})^{j}(∇^{g} − ∇^{h}) is a tensor field. Condition (0.2) turns out to be an equivalence relation in the set of complete metrics on M. We denote this equivalence relation by g ∼_{β}^{k} h. It implies, in particular, that the two metrics are quasi-isometric.

To develop scattering theory for the Laplacian we need to impose additional assumptions on the metrics. In this paper we restrict attention to the class of complete metrics with bounded sectional curvature. In some cases we will also demand that higher derivatives of the curvature tensor are bounded. The assumption that the metric has bounded sectional curvature allows us to control the behavior of the injectivity radius ı(x) sufficiently well. Let ĩ(x) be the modified injectivity radius, defined by (2.1) which is bounded from above by a constant that depends on the bound of the sectional curvature. Then one of our main results is the following theorem.

Theorem 0.1. Assume g and h be complete metrics on M with bounded curvature up to order 2. Let β be a function of moderate decay. Suppose that g ∼_{β}^{2} h. Assume that there exist real numbers a, b satisfying

i) b ≥ 1 and a + b = 2, ii) β^{\frac{b}{3}} ∈ L^{1}(M), iii) β^{\frac{a}{3}} ĩ^{− \frac{n(n+2)}{2}} ∈ L^{∞}(M).

Then e^{−tΔ_{g}} − e^{−tΔ_{h}} is a trace class operator.

By the invariance principle for wave operators [Ka], Theorem 0.1 implies that the wave operators W_{±}(Δ_{h}, Δ_{g}; J) exist and are complete (see Theorem 7.1). To demonstrate this result, we discuss three examples in section 7, namely manifolds with cylindrical ends, manifolds with bounded geometry, and manifolds with cusps. Under additional assumptions on (M, g), the conditions on β can be relaxed. This is, for example, the case for manifolds with cusps and manifolds with cylindrical ends. In either case, the method of Enss can be used to prove the existence and completeness of the wave operators.

The time-independent approach to scattering theory is based on the study of the resolvent. An important problem in this context is the question of the existence of an analytic continuation of the resolvent as operator in appropriate weighted L^{2}-spaces. We study this problem in the geometric setting described above. Assuming that the resolvent R_{g}(λ) = (Δ_{g} − λ Id)^{−1}, regarded as an operator in certain weighted L^{2}-spaces, admits a meromorphic continuation to some ramified covering of a domain in Ω ⊂ ℂ, we show that the same is true for the resolvent of the perturbed Laplacian Δ_{h} under suitable decay conditions on h−g. See Theorem 8.4 for details. The existence of an analytic continuation of the resolvent has been studied for several classes of metrics with special structures at infinity. Examples are: manifolds which outside a compact set are isometric to a neighborhood of a cusp of a locally symmetric space of ℚ-rank one [Mu2], asymptotically flat metrics [Me2], cylindrical end metrics [Me3], and metrics with asymptotically constant negative curvature [MM]. [Me1] contains a discussion of these examples. In [MV] and [Sm], the analytic continuation of the resolvent of the Laplacian on a non-compact Riemannian symmetric space has been studied. As shown in [MV], there is a certain analogy between spectral theory of the Laplacian on symmetric spaces and N-body quantum scattering.

The structure of the paper is as follows. In section 1 we introduce our class of functions of moderate decay and study some of its elementary properties. Then we set up the equivalence relation mentioned above and prove some facts about equivalent metrics. In section 2 we study the behavior of the injectivity radius on manifolds with bounded sectional curvature. Then we introduce and study weighted Sobolev spaces in section 3. In section 4 we show that certain functions of the Laplacian including the heat kernel and the resolvent extend to bounded operators in weighted L^{2}-spaces. Section 5 deals with the comparison of weighted Sobolev spaces with respect to equivalent metrics. Then we prove Theorem 0.1 in section 6. In section 7 we deal with the existence and completeness of wave operators. First we prove a general result which is based on Theorem 0.1 and we discuss some examples. Then we consider the case of a manifold with cusps and use the method of Enss to establish the existence and completeness of the wave operators under weaker assumptions on β. The final section 8 deals with the analytic continuation of the resolvent, regarded as operator in weighted L^{2}-spaces.

1. Equivalence of Riemannian Metrics.

Let M be an open, connected C^{∞}–manifold of dimension n and let M = M(M) be the space of all complete Riemannian metrics on M. Eichhorn [Ei1] has shown that M can be endowed with a canonical topology given by a metrizable uniform structure. We briefly recall its definition.

For a given Riemannian metric g on M, denote by ∇^{g} the Levi–Civita connection of g and by | · |_{g} the norm induced by g in the fibers of ⊕_{p,q≥0}(T M^{⊗p} ⊗ T^{∗}M^{⊗q}). Let h be any other Riemannian metric on M. For k ≥ 0 set

(1.1)  ^{k}|g − h|_{g}(x) = |g − h|_{g}(x) + ∑_{j=0}^{k−1} |(∇^{g})^{j}(∇^{g} − ∇^{h})|_{g} (x), x ∈ M.

and

(1.2)  ^{k}||g − h ∥_{g} = sup_{x∈M} ^{k}|g − h|_{g}(x).

Recall that two metrics g, h are said to be quasi-isometric if there exist C_{1}, C_{2} > 0 such that

(1.3)  C_{1}g(x) ≤ h(x) ≤ C_{2}g(x), for all x ∈ M,

in the sense of positive definite forms. We shall write g ∼ h for quasi–isometric metrics g and h. If g and h are quasi–isometric, then (1.3) implies that for all p, q ≥ 0, there exist A_{p,q}, B_{p,q} > 0 such that for every tensor field T on M of bidegree (p, q), we have

(1.4)  A_{p,q}|T |_{g}(x) ≤ |T |_{h}(x) ≤ B_{p,q}|T |_{g}(x), x ∈ M.

Put ∇ := ∇^{g} and ∇′ := ∇^{h}. Let ∇^{p,q} and ∇′^{p,q} be the canonical extension of ∇ and ∇′, respectively, to the tensor bundle T^{p,q}(M). Then for all p, q ∈ ℕ there exists C_{p,q} > 0 such that

(1.5)  |∇^{p,q} − ∇′^{p,q}|_{g}(x) ≤ C_{p,q}|∇ − ∇′|_{g}(x), x ∈ M.

For k ≥ 1 and δ > 0, set

V_{δ} = {(g, g′) ∈ M × M | g ∼ g′ and ^{k}||g − g′ ∥_{g}< δ}.

It is proved in [Ei1], Proposition 2.1, that {V_{δ}}_{δ>0} is a basis for a metrizable uniform structure on M.

Lemma 1.1. Let g, h ∈ M. Assume that there exists a compact subset K ⊂ M and 0 < δ < 1 such that |g − h|_{g}(x) ≤ δ for all x ∈ M \ K. Then g and h are quasi-isometric.

Proof: Let x ∈ M \ K. Choose geodesic coordinates w.r.t. g, centered at x. Then g_{ij}(x) = δ_{ij}. Let H = (h_{ij}(x)) be the matrix representing h(x) in these coordinates. Denote by ∥ · ∥ the supremum norm of linear maps in ℝ^{n}. Then by assumption, we have ∥ H − Id ∥≤ δ < 1. Hence the Neumann series for H^{−1} = (Id − (Id − H))^{−1} converges in norm which implies that ∥ H^{−1} ∥≤ 1/(1 − δ). Thus for all ξ ∈ ℝ^{n}, we get

(1 − δ) ∥ ξ ∥^{2}≤ (∥ H^{−1} ∥)^{−1} ∥ ξ ∥^{2}≤ ⟨Hξ, ξ⟩ ≤∥ H ∥ ∥ ξ ∥^{2}≤ (1 + δ) ∥ ξ ∥^{2} .

This implies that

(1 − δ)g(x) ≤ h(x) ≤ (1 + δ)g(x), for all x ∈ M \ K.

Since K is compact, it follows that g and h are quasi-isometric. □

We need two results from the proof of Proposition 2.1 in [Ei1] which we state as lemmas. For the convenience of the reader we repeat the proofs.

Lemma 1.2. Let g, h ∈ M be quasi–isometric. For every k ≥ 0, there exists a polynomial P_{k}(X_{1}, ..., X_{k}), depending on the quasi-isometry constants, with nonnegative coefficients and vanishing constant term, such that

k|g − h|_{h}(x) ≤ P_{k}(|g − h|_{g}(x), |∇^{g} − ∇^{h}|_{g}(x), ..., |(∇^{g})^{k−1}(∇^{g} − ∇^{h})|_{g}(x)), x ∈ M.

Proof: From (1.4) follows that

(1.6)  |g − h|_{h}(x) ≤ C_{3}|g − h|_{g}(x)

and

(1.7)  |∇^{g} − ∇^{h}|_{h}(x) ≤ C_{4}|∇^{g} − ∇^{h}|_{g}(x), x ∈ M.

This takes care of the first two terms in (1.1) and settles the question for k = 0, 1. Now we shall proceed by induction. Let k ≥ 2 and suppose that the lemma holds for l ≤ k − 1. For each p ≥ 0, we have

(1.8)  (∇^{h})^{p}(∇^{h} − ∇^{g}) = ∇^{g}(∇^{h})^{p−1}(∇^{h} − ∇^{g}) + (∇^{h} − ∇^{g})(∇^{h})^{p−1}(∇^{h} − ∇^{g}).

Let p ≤ k. Using (1.7), (1.5) and the induction hypothesis, we can estimate the pointwise h-norm of the second term on the right hand side of (1.8) in the desired way. To deal with the first term, we use the formula

\begin{matrix}(∇^{g})^{p}(∇^{h})^{l}(∇^{h} − ∇^{g}) = & (∇^{g})^{p+1}(∇^{h})^{l−1}(∇^{h} − ∇^{g}) \\ + (∇^{g})^{p}(∇^{h} − ∇^{g})(∇^{h})^{l−1}(∇^{h} − ∇^{g}). \end{matrix} ; (1.9)

Applying the Leibniz rule, we get

\begin{matrix}|(∇^{g})^{p}(∇^{h}− & ∇^{g})(∇^{h})^{l−1}(∇^{h} − ∇^{g})|_{g}(x) \\ ≤ C ∑_{i=0}^{p} |((∇^{g})^{i}(∇^{h} − ∇^{g}))|_{g}(x) · |((∇^{g})^{p−i}(∇^{h})^{l−1}(∇^{h} − ∇^{g}))|_{g}(x) \end{matrix}

for some C > 0 and all x ∈ M. Inserting (1.8) and iterating these formulas reduces everything to the induction hypothesis. □

Lemma 1.3. Let g_{i} ∈ M, i = 1, 2, 3, and suppose that g_{1} ∼ g_{2} ∼ g_{3}. For every k ≥ 0, there exists a polynomial Q_{k}, depending on the quasi-isometry constants, in the variables ^{i}|g_{1} − g_{2}|_{g_{1}}(x) and ^{j}|g_{2} − g_{3}|_{g_{2}}(x), i, j = 0, …, k, with nonnegative coefficients and vanishing constant term, such that

k|g_{1} − g_{3}|_{g_{1}}(x) ≤ Q_{k}(^{i}|g_{1} − g_{2}|_{g_{1}}(x), ^{j}|g_{2} − g_{3}|_{g_{2}}(x)), x ∈ M.

If there exists δ < 1 such that ∥g_{1} − g_{2}∥_{g_{1}} ≤ δ and ∥g_{2} − g_{3}∥_{g_{2}} ≤ δ, the dependence on the quasi-isometry constants can be removed.

Proof: Since g_{1} ∼ g_{2}, it follows from (1.4) that

|g_{1} − g_{3}|_{g_{1}}(x) ≤ |g_{1} − g_{2}|_{g_{1}}(x) + C_{1}|g_{2} − g_{3}|_{g_{2}}(x).

Set ∇_{i} = ∇^{g_{i}}, i = 1, 2, 3. By the same argument, we get

|∇_{1} − ∇_{3}|_{g_{1}}(x) ≤ |∇_{1} − ∇_{2}|_{g_{1}}(x) + C_{2}|∇_{2} − ∇_{3}|_{g_{2}}(x).

Thus, the lemma holds for k = 0, 1, and we can use induction to prove the lemma. First observe that for p ≥ 0,

∇_{1}^{p}(∇_{1} − ∇_{3}) = ∇_{1}^{p}(∇_{1} − ∇_{2}) + ∇_{1}^{p}(∇_{2} − ∇_{3}).

The pointwise g_{1}–norm of the first term on the right hand side gives already what we want. The second term can be written as

∇_{1}^{p}(∇_{2} − ∇_{3}) = (∇_{1} − ∇_{2})∇_{1}^{p−1}(∇_{2} − ∇_{3}) + ∇_{2}∇_{1}^{p−1}(∇_{2} − ∇_{3}).

Iteration of this formula and application of the Leibniz rule reduces again everything to the induction hypothesis. The last statement again follows from Lemma 1.1. □

To set up our equivalence relation in M, we introduce an appropriate class of functions.

Definition 1.4. Let β : [1, ∞) → ℝ be a positive, continuous, non-increasing function. Then β is called a function of moderate decay, if it satisfies the following conditions

\begin{matrix}1) sup_{x∈[1,∞)} xβ(x) < ∞; \\ 2) ∃ C_{β} > 0 : β(x + y) ≥ C_{β} β(x)β(y), x, y ≥ 1. \end{matrix} ; (1.10)

Furthermore, β is called of sub-exponential decay if for any c > 0, e^{cx}β(x) → ∞ as x → ∞.

Remark 1. The class of functions which are of moderate or sub-exponential decay are closed under multiplication, and also under raising to positive powers. The function e^{−tx}, t ≥ 0, is of moderate decay and the functions x^{−1} and exp(−x^{α}), 0 < α < 1, are of subexponential decay. Thus the class of functions introduced in Definition 1.4 is not empty.

Next we establish some elementary properties of β.

Lemma 1.5. Let β be of moderate decay. Then there exist constants C > 0 and c ≥ 0 such that

(1.11)  β(x) ≥ Ce^{−cx}, x ∈ [1, ∞).

Proof: Given x ∈ [1, ∞), write x as x = y + n, where y ∈ [1, 2) and n ∈ ℕ. Applying condition 2) of (1.10) repeatedly, we get

(1.12)  β(x) ≥ β(y)(C_{β}β(1))^{n}.

By assumption, β is continuous. Hence there exists C > 0 such that β(y) ≥ C for y ∈ [1, 2]. Since β is non-increasing, it follows that C_{β}β(1) ≤ 1. Thus there exists c ≥ 0 such that C_{β}β(1) = e^{−c}. Together with (1.12) the claim follows. □

Thus for a function β of moderate decay there exist constants c, C_{1}, C_{2} > 0 such that

C_{1}e^{−cx} ≤ β(x) ≤ C_{2}x^{−1}, x ≥ 1.

Lemma 1.6. Let β be a function of moderate decay. Then for all x, y, q ∈ M, we have

(1.13)  C_{β} β(1 + d(x, y)) ≤ \frac{β(1 + d(x, q))}{β(1 + d(y, q))} ≤ \frac{1}{C_{β}β(1 + d(x, y))}.

Moreover, for every q′ ∈ M there exists a constant C > 0, depending only on q and q′, such that

(1.14)  C^{−1} β(1 + d(x, q′)) ≤ β(1 + d(x, q)) ≤ C β(1 + d(x, q′)).

Proof: Since β is non-increasing, it follows from (1.10) that

\frac{β(1 + d(x, q))}{β(1 + d(y, q))} ≤ \frac{β(1 + d(x, q))}{β(1 + d(x, q) + 1 + d(x, y))}

≤ \frac{β(1 + d(x, q))}{C_{β} β(1 + d(x, q))β(1 + d(x, y))}

= \frac{1}{C_{β}β(1 + d(x, y))}.

Switching the roles of x and y, we obtain the other inequality in (1.13). Furthermore, switching the roles of x and q and putting y = q′ in (1.13) gives (1.14). □

Lemma 1.7. Let β be a function of moderate decay. Let g, h ∈ M, q ∈ M, and suppose that

(1.15)  |g − h|_{g}(x) ≤ β(1 + d_{g}(x, q)), x ∈ M.

Then g and h are quasi-isometric and there exist constants C_{1}, C_{2} > 0 such that

(1.16)  C_{1}d_{g}(x, y) ≤ d_{h}(x, y) ≤ C_{2}d_{g}(x, y), x, y ∈ M,

and

(1.17)  C_{1}β(1 + d_{g}(x, q)) ≤ β(1 + d_{h}(x, q)) ≤ C_{2}β(1 + d_{g}(x, q)), x ∈ M.

Proof: Let 0 < δ < 1. From condition 1) of (1.10) follows that there exists r_{0} such that β(1 + r) ≤ δ for r ≥ r_{0}. Thus by Lemma 1.1, g and h are quasi-isometric and this implies (1.16). To prove the second part, we first note that it follows from the proof of Lemma 1.1 that

d_{h}(x, q) ≤ (1 + β(1 + d_{g}(x, q))d_{g}(x, q), d_{g}(x, q) ≥ r_{0}.

Moreover, by condition 1) of (1.10) there exists C > 0 such that

β(1 + d_{g}(x, q))d_{g}(x, q) ≤ C, x ∈ M.

Then using (1.10), (1.16) and the assumption that β is non-increasing, we get

β(1 + d_{h}(x, q)) ≥ β(1 + (1 + β(1 + d_{g}(x, q))d_{g}(x, q)) ≥ C_{β}β(C)β(1 + d_{g}(x, q)).

Switching the roles of g and h, we obtain the other inequality. □

Let k ≥ 0, and consider the following relation for metrics g, h ∈ M:

There exist q ∈ M and C > 0 such that for all x ∈ M we have

k|g − h|_{g}(x) ≤ Cβ(1 + d_{g}(x, q)). (1.18)

Proposition 1.8. The relation (1.18) defines an equivalence relation in M.

Proof: Let g, h ∈ M and suppose that (1.18) holds. Then by Lemma 1.7, g, h are quasi– isometric. Then Lemma 1.2 combined with (1.17) implies that

k|g − h|_{h}(x) ≤ C_{3}β(1 + d_{g}(x, y)) ≤ C_{4}β(1 + d_{h}(x, q)).

Thus the relation (1.18) is symmetric. The transitivity follows from Lemma 1.3 and (1.17). By Lemma 1.6, the relation is independent of q. □

This justifies the following definition.

Definition 1.9. Let β be a function of moderate decay. Two metrics g, h ∈ M are said to be β–equivalent up to order k if (1.18) holds. In this case we write g ∼_{β}^{k} h.

Example. Let (M, g) be a complete Riemannian manifold which is Euclidean at infinity, that is, there exists a compact subset K ⊂ M such that (M \K, g) is isometric to ℝ^{n}\B_{r}(0) for some r > 0, where ℝ^{n} is equipped with its standard metric. Let β(r) = r^{−a}, a > 1, and let h be a complete Riemannian metric on M such that h ∼_{β}^{k} g for some k ∈ ℕ. Then h|_{M\K} may be regarded as metric on ℝ^{n} \ B_{r}(0) and if h_{ij} are the components of h|_{M\K} with respect to the standard coordinates x_{1}, ..., x_{n} ∈ ℝ^{n}, then the condition h ∼_{β}^{k} g is equivalent to

(1.19)  |\frac{∂^{α}}{∂x_{1}^{α_{1}} ⋯ ∂x_{n}^{α_{n}}} (h_{ij}(x) − δ_{ij}) | ≤ C(1+ ∥ x ∥)^{−a}

for all multi-indices α with |α| ≤ k and all x ∈ ℝ^{n} \ B_{r}(0). Such metrics are called asymptotically Euclidean.

To simplify notation, we will write β(x) in place of β(1 + d_{g}(x, q)). If g ∼_{β}^{k} h, it follows from Lemma 1.7, that we may use both d_{g} and d_{h} in (1.18).

Next we show that the β-equivalence can also be defined in a different manner. Namely we have the following proposition.

Proposition 1.10. Let k ≥ 0 and let g, h ∈ M. Then g ∼_{β}^{k} h holds if and only if there exists C_{1} > 0 such that

∑_{i=0}^{k} |(∇^{g})^{i}(g − h)|_{g}(x) ≤ C_{1}β(x), x ∈ M.

Proof: Let g, h ∈ M. The lemma holds obviously for k = 0. Let k ≥ 1. Recall that ∇^{g}g = 0 and ∇^{h}h = 0. Using this fact, we get

(∇^{g})^{k}(g − h) = −(∇^{g})^{k}h = −(∇^{g})^{k−1}(∇^{g} − ∇^{h})h

Using the Leibniz rule it follows that there exists C_{1} > 0 such that

(1.20) |(∇^{g})^{k}(g − h)|_{g}(x) ≤ C_{1} ∑_{i=0}^{k−1} |(∇^{g})^{i}(∇^{g} − ∇^{h})|_{g}(x) · |(∇^{g})^{k−1−i}(h)|_{g}(x), x ∈ M.

Suppose that ^{k}|g − h|_{g}(x) ≤ Cβ(x), x ∈ M, for some constant C > 0. Then |h|_{g}(x) ≤ C′ for some constant C′ > 0. By induction it follows from (1.5) and (1.20) that

(1.21)  ∑_{i=0}^{k} |(∇^{g})^{i}(g − h)|_{g}(x) ≤ C_{2}β(x), x ∈ M,

for some constant C_{2} > 0, depending on C and k.

Now assume that (1.21) holds. We observe that for any smooth vector fields X, Y, Z, the following formula holds

\begin{aligned}h((∇_{X}^{g} − ∇_{X}^{h})Y, Z) = \frac{1}{2} \{∇_{X}^{g}(g − h)(Y, Z) & + ∇_{Y}^{g} (g − h)(X, Z) \\ & − ∇_{Z}^{g}(g − h)(X, Y)\}. \end{aligned} ; (1.22)

From this formula we get

|∇^{h} − ∇^{g}|_{h} ≤ C |∇^{g}(g − h)|_{h}.

Taking covariant derivatives of (1.22) and using induction, we obtain

k|h − g|_{h}(x) ≤ C ∑_{i=0}^{k} |(∇^{g})^{i}(g − h)|_{h}(x).

By (1.4) and (1.21), we get

k|h − g|_{h}(x) ≤ C β(x),

and Lemma 1.2 implies that

k|g − h|_{g}(x) ≤ C_{1} β(x), x ∈ M,

for some constant C_{1} > 0. □

Thus, we may define β–equivalence also by requiring that (1.21) holds for some constant C_{1}. It follows from the previous proposition that this gives rise to an equivalence relation.

Finally, we study the behavior of the curvature tensor and its covariant derivatives under β–equivalence. Given g ∈ M, denote by R^{g} the curvature tensor of g.

Lemma 1.11. Let k ≥ 2 and let g, h ∈ M. Suppose that g ∼_{β}^{k} h. Then there exists C_{k} > 0 such that

|(∇^{g})^{i}(R^{g} − R^{h})|_{g}(x) ≤ C_{k}β(x), x ∈ M, i = 0, …, k − 2.

Proof: Set ∇ = ∇^{g}, ∇′ = ∇^{h}. We define the exterior differential

d^{∇} : C^{∞}(Λ^{p}(T^{∗}M) ⊗ T M) → C^{∞}(Λ^{p+1}(T^{∗}M) ⊗ T M)

associated with ∇ by the following formula

\begin{aligned}(d^{∇}α)(X_{0}, ..., X_{p}) = & ∑_{i=0}^{p}(−1)^{i}∇_{X_{i}}(α(X_{0}, ..., X̂_{i}, ..., X_{p})) \\ & − ∑_{i<j}(−1)^{i+j}α([X_{i}, X_{j}], X_{0}, ..., X̂_{i}, ..., X̂_{j}, ..., X_{p}). \end{aligned}

Then, regarded as operators C^{∞}(T M) → C^{∞}(Λ^{2}(T^{∗}M) ⊗ T M), we have

R^{∇} = d^{∇} ∘ d^{∇},

and a corresponding formula holds for ∇′. Set A = ∇′ − ∇ and let X, Y be smooth vector fields on M. Then we have [Be, p. 25]

R^{∇′}(X, Y) − R^{∇}(X, Y) = ∇_{X}(A(Y)) − ∇_{Y} (A(X)) − A([X, Y])
− A(X) ∘ A(Y) + A(Y) ∘ A(X)
= (∇A)(X, Y) − (∇A)(Y, X)
− A(X) ∘ A(Y) + A(Y) ∘ A(X).

Differentiating this equality and using induction gives the desired result. □

Recall that a Riemannian manifold (M, g) is said to have bounded curvature of order k, if the covariant derivatives ∇^{i}R, 0 ≤ i ≤ k, of the curvature tensor R are uniformly bounded on M, i.e., there exists C > 0 such that |∇^{i}R|(x) ≤ C, x ∈ M, 0 ≤ i ≤ k.

Corollary 1.12. Let k ≥ 2 and let g, h ∈ M. Suppose that g ∼_{β}^{k} h. Then

1) (M, g) has bounded curvature of order k − 2 if and only if (M, h) has bounded curvature of order k − 2.

2) The sectional curvature of (M, g) is bounded from below (resp. from above) if and only if the sectional curvature of (M, h) is bounded from below (resp. above).

3) The Ricci curvature of (M, g) is bounded from below (resp. from above) if and only if the Ricci curvature of (M, h) is bounded from below (resp. above).

2. Injectivity radius and bounded curvature.

In this section we establish some properties of the injectivity radius on a manifold with bounded sectional curvature. Let (M, g) be a complete, n–dimensional Riemannian manifold with bounded sectional curvature, say |K_{M}| ≤ K. Let p ∈ M. Recall that the injectivity radius i(p) at p equals the minimal distance from p to its cut locus C(p) (see [CE], [Kl]). Also note that i(p) is a continuous function of p ∈ M [Kl, Proposition 2.1.10].

Proposition 2.1. Let h be another complete Riemannian metric on M with bounded sectional curvature |K_{M}^{h}| ≤ K and assume that g and h are equivalent. Given p ∈ M, let i_{g}(p) and i_{h}(p) denote the injectivity radii at p with respect to g and h, respectively. Then there exist constants c, c′ > 0 such that

i_{h}(p) ≥ min{ci_{g}(p), c′}, p ∈ M.

Proof: Since g and h are assumed to be equivalent, there exists ε > 0 such that

e^{−ε}g ≤ h ≤ e^{ε}g.

Let x ∈ M and suppose that i_{h}(x) < min \{e^{−2ϵ}π/(2\sqrt{K}), e^{−ϵ}i_{g}(x)/2\}. It follows from [CE, Corollary 1.30] that distinct conjugate points along a geodesic (with respect to h) have distance ≥ π/\sqrt{K}. Therefore, by [CE, Lemma 5.6], there exists a closed geodesic loop γ^{h} at x with respect to the metric h, with

h -length(γ^{h}) < min \{e^{−2ϵ}π/\sqrt{K}, e^{−ϵ}i_{g}(x)\} .

Hence, we have

g -length(γ^{h}) < min \{e^{−ϵ}π/\sqrt{K}, i_{g}(x)\} .

In particular, g-length(γ^{h}) < π/\sqrt{K}. Let r_{max}(x) be the maximal rank radius of exp_{x} with respect to g. Then we obtain g–length(γ^{h}) < π/\sqrt{K} ≤ r_{max}(x). By [BK], Proposition 2.2.2, there exists a unique g-geodesic loop γ̃: [0, 1] ⟶ M at x with g–length(γ̃) < r_{max}(x), which is obtained from γ^{h} by a length decreasing homotopy H: [0, 1] × [0, 1] ⟶ M (cf. [BK], 2.1.2). Hence, we have

g -length(γ̃) ≤ g -length(γ^{h}) < min \{e^{−ϵ} \frac{π}{\sqrt{K}} , i_{g}(x)\} .

Since h-length(H(·, s)) ≤ e^{ϵ}g-length(γ^{h}) < 2π/\sqrt{K} for s ∈ [0, 1] , it follows from [Kl, Lemma 2.6.4], that g-length(γ̃) > 0. Parameterize γ̃ by g–arc length. Then either γ̃(t) or γ̃(length(γ̃) − t) belongs to the cut locus of x for some t ≤ \frac{1}{2}g-length(γ̃). Therefore i_{g}(x) < i_{g}(x), a contradiction. □

Let β be a function of moderate decay. Suppose that g ∼_{β}^{0} h. Then by Lemma 1.7, g and h are quasi-isometric. Therefore, if h has bounded sectional curvature, then Proposition 2.1 can be applied to g, h. For x ∈ M set

(2.1)  ĩ(x) := min \{\frac{π}{12\sqrt{K}} , i(x)\} .

Then it follows, that under the assumptions of Proposition 2.1, there exists c_{2} > 0 such that

ĩ_{h}(p) ≥ c_{2}ĩ_{g}(p), p ∈ M.

Next recall the Bishop–Günther inequalities [Gra, Theorem 3.17], [Gro, Lemma 5.3], which give estimates of the volume of small balls from above and below.

Lemma 2.2. For r ≤ ĩ(x_{0}),

\frac{2π^{n/2}}{Γ (\frac{n}{2})} ∫_{0}^{r} (\frac{sin t\sqrt{K}}{\sqrt{K}} )^{n−1} dt ≤ Vol(B_{r}(x_{0})) ≤ \frac{2π^{n/2}}{Γ (\frac{n}{2})} ∫_{0}^{r} (\frac{sinh t\sqrt{K}}{\sqrt{K}} )^{n−1} dt.

We note that the inequality on the right hand side holds for all r ∈ ℝ_{+}. In particular

(2.2)  Vol(B_{r}(x_{0})) = O (e^{(n−1)\sqrt{K}r})

as r → ∞.

It is also important to know the maximal possible decay of the injectivity radius.

Lemma 2.3. There exists a constant C > 0, depending only on K, such that

(2.3)  ĩ(x) ≥ C ĩ(p)^{n}e^{−(n−1)\sqrt{K} d(x,p)}

for all x, p ∈ M.

Proof: Let p ∈ M and fix r, r_{0}, s, with r_{0}+2s < π/\sqrt{K}, r_{0} ≤ π/4\sqrt{K}. By [CGT, Theorem 4.7] we get

(2.4)  ĩ(x) ≥ \frac{r_{0}}{2} · \frac{1}{1 + (V_{r_{0}+s}^{K}/Vol(B_{r}(p))) (V_{d(x,p)+r}^{K}/V_{s}^{K})},

where V_{s}^{K} denotes the volume of a ball of radius s in the n–dimensional hyperbolic space of curvature −K. Set r_{0} = s = \frac{π}{5\sqrt{K}} , r = ĩ(p) and apply Lemma 2.2 to estimate Vol (B_{ĩ(p)}(p)) from below. Then (2.4) implies

\begin{aligned}ĩ(x) & ≥ C_{1} ĩ(p)^{n}e^{−(n−1)\sqrt{K}(d(x,p)+ĩ(p))} \\ & ≥ C ĩ(p)^{n}e^{−(n−1)\sqrt{K}d(x,p)}. \end{aligned}

□

Corollary 2.4. Given p ∈ M, there exists a constant C = C(p) > 0 such that

ĩ(x) ≥ Ce^{−(n−1)\sqrt{K} d(x,p)}, x ∈ M.
□

Lemma 2.5. There exists a constant C, depending only on K, such that for each x, y ∈ M we have the inequality

(2.5)  ĩ(y) ≥ Cĩ(x)e^{− \frac{(n−1)π}{12} \frac{d(x,y)}{ĩ(x)}} .

Proof: Let λ = max\{1, \frac{π^{2}}{144Ki(x)^{2}} \}. Then the injectivity radius i_{λ} at x with respect to λg is given by

i_{λ}(x) = λ^{\frac{1}{2}} i(x) = \{ \begin{matrix}i(x) & , if i(x) > \frac{π}{12\sqrt{K}}; \\ \frac{π}{12\sqrt{K}} & , if i(x) ≤ \frac{π}{12\sqrt{K}} . \end{matrix}

Since λ^{−1} ≤ 1, the sectional curvature K_{M}^{λg} with respect to λg also satisfies |K_{M}^{λg}| ≤ K.

Let r = \frac{π}{\sqrt{K}}, r_{0} = s = \frac{r}{12} = \frac{π}{12\sqrt{K}} and set d = λ^{\frac{1}{2}} d_{g}(x, y). Then d is the distance between x and y with respect to λg.

Let V_{s}(y) be the volume of the geodesic ball of radius s and center y with respect to λg and let V_{s}^{K} denote the volume of a ball of radius s in the n-dimensional simply connected space of constant curvature −K. Then by [CGT, Theorem 4.3] we get

(2.6)  i_{λ}(y) ≥ \frac{r_{0}}{2} \frac{1}{1 + \frac{V_{r_{0}+s}^{K}}{V_{s}(y)}} ≥ \frac{r_{0}}{4} \frac{V_{s}(y)}{V_{r_{0}+s}^{K}}.

Now, [CGT, Proposition 4.1, i)] states that

\frac{V_{s}(y)}{V_{s}^{K}} ≥ \frac{V_{d+s}(y)}{V_{d+s}^{K}}.

Together with (2.6) this gives

i_{λ}(y) ≥ \frac{r_{0}}{4} \frac{V_{d+s}(y)V_{s}^{K}}{V_{d+s}^{K}V_{r_{0}+s}^{K}} .

From the definition of d it follows that, with respect to the metric λg, the ball of radius d + s around y contains the ball of radius s around x. Hence V_{d+s}(y) ≥ V_{s}(x). Since s = \frac{π}{12\sqrt{K}} = ĩ_{λ}(x), it follows from Lemma 2.2 that there exists c > 0 such that V_{s}(x) ≥ c for all x ∈ M. Hence, we get

\begin{aligned}i_{λ}(y) & ≥ \frac{r_{0}}{4} \frac{V_{s}(x)V_{s}^{K}}{V_{d+s}^{K}V_{r_{0}+s}^{K}} ≥ C \frac{V_{s}^{K}}{V_{d+s}^{K}} ≥ Ce^{−(n−1)\sqrt{K}d} \\ & ≥ Ce^{−(n−1) max\{\sqrt{K}, \frac{π}{12i(x)} \}d(x,y)} = Ce^{− \frac{(n−1)π}{12} \frac{d(x,y)}{ĩ(x)}} , \end{aligned}

for some constant C > 0. Now the lemma follows by dividing both sides of this inequality by λ^{\frac{1}{2}} . □

We can now establish the following basic result about the existence of uniformly locally finite coverings on manifolds with bounded curvature.

Theorem 2.6. Assume that M is non-compact. Let h be a continuous real valued function on M such that

i) ∀x : 0 < h(x) ≤ ĩ(x).

ii) There exists constants C_{1}, C_{2} > 0 such that

h(x) ≥ C_{1}h(x_{0})e^{−C_{2} \frac{d(x,x_{0})}{h(x_{0})}}

for all x, x_{0} ∈ M.

Then for each a ≥ 1, there exists a sequence {x_{i}}_{i=0}^{∞} ⊂ M and a constant C_{3} < ∞, depending only on K, a, C_{1} and C_{2} such that

1)

⋃_{i=0}^{∞} B_{h(x_{i})}(x_{i}) = M.

2) ∀i ∈ ℕ : #{j | B_{ah(x_{i})}(x_{i}) ∩ B_{ah(x_{j})}(x_{j}) ≠ ∅} ≤ C_{3}.

Proof: Let x_{0} ∈ M. For k ∈ ℕ define recursively

m(k) = min{m ∈ ℕ | B_{m}(x_{0}) \ ∪_{i<k}B_{h(x_{i})}(x_{i}) ≠ ∅}

and pick x_{k} ∈ B_{m(k)} \ ∪_{i<k}B_{h(x_{i})}(x_{i}). In this way we get a sequence {x_{i}}_{i=0}^{∞} of points of M. From the construction it follows that this sequence satisfies the following condition:

(2.7)  ∀i, j ∈ ℕ : d(x_{i}, x_{j}) ≥ min{h(x_{i}), h(x_{j})}.

Let m ∈ ℕ. Then by ii), there exists c > 0 such that h(x) ≥ c for all x ∈ B_{m}(x_{0}). Hence it follows from (2.7) that d(x_{i}, x_{j}) ≥ c if x_{i}, x_{j} ∈ B_{m}(x_{0}). Since B_{m}(x_{0}) is compact, this implies that only finitely many of the x_{i}’s, say x_{1}, ..., x_{r_{m}}, are contained in B_{m}(x_{0}). Hence

B_{m}(x_{0}) ⊂ ⋃_{i=0}^{r_{m}} B_{h(x_{i})}(x_{i})

which implies that

M = ⋃_{i=0}^{∞} B_{h(x_{i})}(x_{i}).

It remains to prove 2). Let a ≥ 1. For j ∈ ℕ put B_{j} = B_{ah(x_{j})}(x_{j}). Let i ∈ ℕ be given and put

Ω_{i} = {x_{j} | B_{i} ∩ B_{j} ≠ ∅}.

Since h is bounded from above, Ω_{i} is contained in a compact subset Y of M. By ii) there exists c > 0 such that h(x) ≥ c for all x ∈ Y. Using (2.7), it follows that Ω_{i} is a discrete subset of Y and hence, Ω_{i} is a finite set. Let x_{j_{1}} ∈ Ω_{i} be such that

h(x_{j_{1}}) = max{h(x_{j}) | x_{j} ∈ Ω_{i}}.

Since B_{i} ∩ B_{j_{1}} ≠ ∅, it follows that B_{i} ⊂ B_{3ah(x_{j_{1}})}(x_{j_{1}}) which in turn implies that

B_{\frac{h(x_{j})}{2}} (x_{j}) ⊂ B_{(4a+1)h(x_{j_{1}})}(x_{j_{1}})

for all x_{j} ∈ Ω_{i}. Therefore by ii) we get

h(x_{j}) ≥ C_{1}h(x_{j_{1}})e^{−C_{2} \frac{d(x_{j_{1}} ,x_{j})}{h(x_{j_{1}} )}} ≥ C_{1}h(x_{j_{1}})e^{−4aC_{2}}.

Thus there exists C_{4} > 0 such that

(2.8)  h(x_{j}) ≥ C_{4}h(x_{j_{1}})

for all x_{j} ∈ Ω_{i}. Obviously C_{4} ≤ 1. Hence by i), we obtain

(2.9)  \frac{C_{4}h(x_{j_{1}})}{2} ≤ \frac{ĩ(x_{j_{1}})}{2} ≤ \frac{π}{24\sqrt{K}} .

Moreover, by (2.7) and (2.8) we have d(x_{k}, x_{l}) ≥ C_{4}h(x_{j_{1}}) for all x_{k}, x_{l} ∈ Ω_{i}. Therefore, the balls B_{\frac{C_{4}}{2} h(x_{j_{1}})}(x_{j}), x_{j} ∈ Ω_{i}, are pairwise disjoint. Using Lemma 2.2, we get

(2.10)  #\{x_{j} | B_{i} ∩ B_{j} ≠ ∅\} ≤ \frac{∫_{0}^{(4a+1)h(x_{j_{1}})} (\frac{sinh t\sqrt{K}}{\sqrt{K}} )^{n−1} dt}{∫0^{\frac{C_{4}h(x_{j_{1}} )}{2}} (\frac{sin t\sqrt{K}}{\sqrt{K}} )^{n−1} dt}.

There exist constants c_{1} > 0 and c_{2} > 0 such that

sinh t\sqrt{K} ≤ c_{1}t, 0 ≤ t ≤ \frac{(4a + 1)π}{12C_{4}\sqrt{K}};

sin t\sqrt{K} ≥ c_{2}t, 0 ≤ t ≤ \frac{π}{24\sqrt{K}} .

Hence by (2.9), it follows that the right hand side of (2.10) is bounded by \frac{c_{2}}{c_{1}} (\frac{(8a+2)c_{1}}{C_{4}c_{2}} )^{n} . This proves the lemma. □

Finally we will define and estimate some global invariants of (M, g).

Definition 2.7. Let s > 0. For s > ε ≥ 0, let κ_{ε}(M, g; s) ∈ ℕ∪{∞} be the smallest number such that there exists a sequence {x_{i}}_{i=1}^{∞} such that {B_{s−ε}(x_{i})}_{i=1}^{∞} is an open covering of M and

(2.11)  sup_{x∈M} #{i ∈ ℕ | x ∈ B_{3s+ε}(x_{i})} ≤ κ_{ε}(M, g; s).

Further, let κ(M, g; s) = κ_{0}(M, g, s). Put κ(M, g, 0) = 1.

Lemma 2.8. κ_{ε}(M, g; s) is finite for all s > ε. Moreover, there exist constants C, c > 0, which depend only on K, such that for s > \frac{2π}{\sqrt{K}} + ε, we have

κ_{ε}(M, g; s) ≤ Ce^{cs}.

Proof: We may proceed as in the proof of Theorem 2.6 and construct a sequence {x_{i}}_{i=1}^{∞} ⊂ M such that d(x_{i}, x_{j}) ≥ s − ε for all i, j ∈ ℕ and {B_{s−ε}(x_{i})}_{i=1}^{∞} is a covering of M. Let x ∈ M. If x ∈ B_{3s+ε}(x_{i}), it follows that B_{\frac{s−ε}{2}} (x_{i}) ⊂ B_{5s}(x). Moreover, B_{\frac{s−ε}{2}} (x_{i}) ∩ B_{\frac{s−ε}{2}} (x_{j}) = ∅ if i ≠ j. Hence, we get

(2.12)  #\{i | x ∈ B_{3s+ε}(x_{i})\} ≤ \frac{Vol(B_{5s}(x))}{min_{i} Vol(B_{\frac{s−ε}{2}} (x_{i}))}.

Next observe that for any x_{i} with d(x, x_{i}) ≤ 5s we have B_{5s}(x) ⊂ B_{10s}(x_{i}). Moreover, by Lemma 5.3 of [Gro], we have

\frac{Vol(B_{10s}(x_{i}))}{Vol(B_{\frac{s−ε}{2}} (x_{i}))} ≤ \frac{∫_{0}^{10s} (sinh t\sqrt{K})^{n−1} dt}{∫0^{\frac{s−ε}{2}} (sinh t\sqrt{K})^{n−1} dt}.

Then combined with (2.12) we obtain

#\{i | x ∈ B_{3s+ε}(x_{i})\} ≤ \frac{∫_{0}^{10s} (sinh t\sqrt{K})^{n−1} dt}{∫0^{\frac{s−ε}{2}} (sinh t\sqrt{K})^{n−1} dt}.

If (s − ε)/2 ≥ π/\sqrt{K}, the right hand side can be estimated by Ce^{cs} for certain constants C, c > 0 depending on K. □

3. Weighted Sobolev Spaces

In this section we introduce weighted Sobolev spaces on manifolds with bounded curvature.

Let (M, g) be a Riemannian manifold. Let ∇ be the Levi-Civita connection of g and let Δ = d^{∗}d be the Laplacian on functions with respect to g. Let ξ be a positive, measurable function on M, which is finite a.e. Given m ∈ N_{0}, and p ∈ N, we define the weighted L^{p}-space L_{ξ}^{p}(M, T M^{⊗m}) by

L_{ξ}^{p}(M, T M^{⊗m}) = {φ ∈ Lp_{loc}(M, T M^{⊗m}) | ξ^{1/p}φ ∈ L^{p}(M, T M^{⊗m})}.

Then for k ∈ ℕ we define the weighted Sobolev space W_{ξ}^{p,k}(M) by

(3.1)  W_{ξ}^{p,k}(M) = {f ∈ L_{ξ}^{p}(M) | ∇^{m}f ∈ L_{ξ}^{p}(M, T M^{⊗m}) for all m = 1, ..., k},

where ∇ is applied iteratively in the distributional sense and the norm of f ∈ W_{ξ}^{p,k}(M) is given by

(3.2)  ∥ f ∥_{W_{ξ}^{p,k}}= (∑_{i=0}^{k} ∫_{M} |∇^{i}f(x)|_{g}^{p}ξ(x) dv_{g}(x))^{1/p} .

Then W_{ξ}^{p,k}(M) is a Banach space. In this paper we will only consider the case p = 2. To simplify notation we shall write W_{ξ}^{k}(M) in place of W_{ξ}^{2,k}(M). The closure of C_{0}^{∞}(M) in W_{ξ}^{k}(M) will be denoted by W_{0,ξ}^{k}(M). We shall write W^{k}(M) for W_{1}^{k}(M) and W_{0}^{k}(M) for W_{0,1}^{k}(M). Since 0 is not a weight, this cannot lead to any confusion. Note that W_{ξ}^{k}(M) and W_{0,ξ}^{k}(M) are Hilbert spaces. The weighted Sobolev space H_{ξ}^{l}(M) is defined for even integers l. Let k ∈ ℕ. Then

(3.3)  H_{ξ}^{2k}(M) = {f ∈ L_{ξ}^{2}(M) | Δ^{l}f ∈ L_{ξ}^{2}(M) for all l = 1, ..., k}.

The norm is given by

∥ f ∥_{H_{ξ}^{2k}}^{2}= ∑_{j=0}^{k} ∫_{M} |Δ^{j}f(x)|^{2}ξ(x) dv_{g}(x).

As an equivalent norm one can use the norm defined by

(3.4)  ∥ f ∥_{H_{ξ}^{2k}}=∥ (Δ + Id)^{k}f ∥_{L_{ξ}^{2}} .

The closure of C_{0}^{∞}(M) in H_{ξ}^{2k}(M) will be denoted by H_{0,ξ}^{2k}(M). If ξ ≡ 1, the Sobolev space H_{ξ}^{2k}(M) will be denoted by H^{2k}(M) and H_{0,ξ}^{2k} by H_{0}^{2k}(M). Note that the Laplacian Δ induces a bounded operator

(3.5)  Δ_{ξ}: H_{ξ}^{2}(M) → L_{ξ}^{2}(M)

which is defined in the obvious way.

Next we establish some elementary properties of weighted Sobolev spaces. Lemma 3.1. Assume that ξ is continuous. Let p, k ∈ ℕ. Then C^{∞}(M) ∩ W_{ξ}^{p,k}(M) is dense in W_{ξ}^{p,k}(M) and C^{∞}(M) ∩ H_{ξ}^{2k}(M) is dense in H_{ξ}^{2k}(M).

Proof: We proceed as in the proof of Theorem 1 in [Ma, 1.1.5]. Let {U_{i}: i ∈ I} be a locally finite covering of M such that for each i ∈ I there exists an open subset V_{i} with \overline{U}_{i} ⊂ V_{i} and V_{i} is diffeomorphic to the unit ball in ℝ^{n}. Let {φ_{i}: i ∈ I} be an associated partition of unity. Let u ∈ W_{ξ}^{p,k}(M) and let ε ∈ (0, 1/2). For each i ∈ I let u_{i} = φ_{i}u. Then u_{i} belongs to W_{ξ}^{p,k}(M) with supp u_{i} ⊂ U_{i}. Since ξ is continuous, it follows that u_{i} ∈ W^{p,k}(U_{i}) and supp u_{i} is contained in the interior of U_{i}. Hence there exists a mollification g_{i} ∈ C_{c}^{∞}(U_{i}) of u_{i} such that

∥ g_{i} − u_{i} ∥_{W^{p,k}}≤ \frac{ε^{i}}{max_{x∈\overline{U}_{i}} ξ(x)}.

[Ev, Section 5.3]. Then

∥ g_{i} − u_{i} ∥_{W_{ξ}^{p,k}}≤ ε^{i}.

Clearly g = ∑_{i} g_{i} belongs to C^{∞}(M). Let ω ⊂ M be a relatively compact open subset. Then we have

u|_{ω} = ∑_{i} u_{i}|_{ω},

and the sum is finite. Hence

∥ g − u ∥_{W_{ξ}^{p,k}(ω)}≤ ∑_{i} ∥ g_{i} − u_{i} ∥_{W_{ξ}^{p,k}}≤ ε(1 − ε)^{−1} ≤ 2ε.

This implies that ∥ u ∥_{W_{ξ}^{p,k}(ω)}≤∥ u ∥_{W^{p,k}} +2ϵ for all relatively compact open subsets

ω ⊂ M. Hence by the theorem of Beppo-Levi, we have g ∈ C^{∞} ∩ W_{ξ}^{p,k}(M) and

∥ g − u ∥_{W_{ξ}^{p,k}}≤ 2ε.

The proof that C^{∞}(M) ∩ H_{ξ}^{2k}(M) is dense in H_{ξ}^{2k}(M) is similar. □

Therefore we can use the following alternative definition of the Sobolev spaces. Let C_{k}^{∞}(M) denote the space of all f ∈ C^{∞}(M) such that |∇^{j}f| ∈ L_{ξ}^{p}(M) for j = 0, ..., k. Then W_{ξ}^{p,k}(M) is the completion of C_{k}^{∞}(M) with respect to the norm (3.2). Similarly let C̃_{k}^{∞}(M) the space of all f ∈ C^{∞}(M) such that (Δ + Id)^{k}f ∈ L_{ξ}^{2}(M). Then H_{ξ}^{2k}(M) is the completion of C̃_{k}^{∞}(M) with respect to the norm (3.4). This implies that we can define H_{ξ}^{s}(M) for all s ∈ ℝ. Let (Δ + Id)^{s/2} be defined by the spectral theorem. Let C̃_{s}^{∞}(M) be the space of all f ∈ C^{∞}(M) such that (Δ + Id)^{s/2}f ∈ L_{ξ}^{2}(M). Let H_{ξ}^{s}(M) be the completion of C̃_{s}^{∞}(M) with respect to the norm

∥ f ∥_{H_{ξ}^{s}(M)}:=∥ (Δ + Id)^{s/2}f ∥_{L_{ξ}^{2}} .

In general the Sobolev spaces W_{ξ}^{k}(M) and W_{0,ξ}^{k}(M) (resp. H_{ξ}^{2k}(M) and H_{0,ξ}^{2k}(M)) will not coincide. If (M, g) is complete and ξ ≡ 1, the following is known [Sa] .

Lemma 3.2. Assume that (M, g) is complete. Then for all k ∈ N we have

W^{k}(M) = W_{0}^{k}(M), H^{2k}(M) = H_{0}^{2k}(M), and W^{2k}(M) = H^{2k}(M).

Proof: For the proof we refer to [Sa]. The fact that C_{0}^{∞}(M) is dense in H^{2k}(M) is an immediate consequence of [Cn]. Indeed by [Cn], (Δ + Id)^{k} is essentially self-adjoint on C_{0}^{∞}(M) for all k ∈ ℕ. Thus

(3.6)  \overline{(Δ + Id)^{k}(C_{c}^{∞}(M))} = L^{2}(M).

Let f ∈ H^{2k}(M). Then (Δ + Id)^{k}f ∈ L^{2}(M) and hence, by (3.6) there exists a sequence {φ_{j}} ⊂ C_{c}^{∞}(M) such that

∥ f − φ_{j} ∥_{H^{2k}}=∥ (Δ + Id)^{k}(f − φ_{j}) ∥_{L^{2}}→ 0

as j → ∞. □

Under additional assumptions on ξ, similar results hold for weighted Sobolev spaces [Sa]. In general the following weaker results hold.

Lemma 3.3. For all k ∈ ℕ, the natural inclusion W_{ξ}^{2k}(M) ↪ H_{ξ}^{2k}(M) is bounded.

Proof: Let k ∈ ℕ. Let f ∈ W_{ξ}^{2k}(M). Then we have ∇^{j}f ∈ L_{ξ}^{2}(M) for j = 0, ..., 2k. Recall that

(3.7)  Δ = − Tr(∇^{2}f)

and ∇ Tr = 0. Hence it follows that there exists C > 0 such that

|Δ^{j}f|(x) ≤ C|∇^{2j}f|_{g}(x)

for all j = 0, ..., k and x ∈ M. This implies Δ^{j}f ∈ L_{ξ}^{2}(M) for j = 0, ..., k, and

∥ f ∥_{H_{ξ}^{2k}}≤ C ∥ f ∥_{W_{ξ}^{2k}} .
□

In order to deal with the inclusion in the other direction, we need some preparation. Let B_{s} ⊂ ℝ^{n} denote the ball of radius s > 0 around the origin in ℝ^{n}. Given m ∈ ℕ and r, K, λ > 0, denote by Ell^{m}(r, K, λ) the set of elliptic differential operators

P = ∑_{|α|≤m} a_{α}(x)D^{α}

of order m in B_{r} such that the coefficients of P satisfy:

(1) a_{α} ∈ C^{m}(B_{r}).

(2) ∑_{|α|<m} ∥ a_{α} ∥_{C^{0}(B_{r})}≤ K, ∑_{|α|=m} ∥ a_{α} ∥_{C^{1}(B_{r})}≤ K.

(3) λ^{−1} ∥ ξ ∥^{m}≤ ∑_{|α|=m} a_{α}(x)ξ^{α} ≤ λ ∥ ξ ∥^{m} for all ξ ∈ ℝ^{n} and x ∈ B_{r}.

Given an open subset Ω ⊂ ℝ^{n} and k ∈ ℕ, W^{k}(Ω) is the usual Sobolev space.

Lemma 3.4. Let K, λ > 0 be given. There exists r_{0} = r_{0}(K, λ) > 0 and C = C(λ) > 0 such that for all r ≤ r_{0}, P ∈ Ell^{m}(r, K, λ) and x_{0} ∈ B_{r}:

∥ u ∥_{W^{m}(B_{r})}≤ C (∥ P u ∥_{L^{2}(B_{r})} + ∥ u ∥_{L^{2}(B_{r})})

for all u ∈ C_{0}^{∞}(B_{r})

Proof: Let 1 ≥ r > 0 and let P ∈ Ell^{m}(r, K, λ). Put

P_{0} = ∑_{|α|=m} a_{α}(0)D^{α}.

By Lemma 17.1.2 of [H] there exists C_{1} > 0 which depends only on λ such that for all u ∈ C_{0}^{∞}(B_{r}):

∥ u ∥_{W^{m}(B_{r})}≤ C (∥ P_{0}u ∥_{L^{2}(B_{r})} + ∥ u ∥_{L^{2}(B_{r})}) .

Now P u = P_{0}u + (P − P_{0})u. Thus

(3.8)  ∥ u ∥_{W^{m}(B_{r})}≤ C (∥ P u ∥_{L^{2}(B_{r})} + ∥ (P − P_{0})u ∥_{L^{2}(B_{r})} + ∥ u ∥_{L^{2}(B_{r})}) .

Next observe that

(P − P_{0})u = ∑_{|α|=m}(a_{α}(x) − a_{α}(0))D^{α}u + ∑_{|α|<m} a_{α}(x)D^{α}u.

Hence by 2):

\begin{matrix}∥ (P − P_{0})u ∥_{L^{2}(B_{r})}≤ & r ∑_{|α|=m} ∥ a_{α} ∥_{C^{1}(B_{r})}∥ u ∥_{W^{m}(B_{r})} \\ + ∑_{|α|<m} ∥ a_{α} ∥_{C^{0}(B_{r})}∥ u ∥_{W^{m−1}(B_{r})} \\ ≤ K (r ∥ u ∥_{W^{m}(B_{r})} + ∥ u ∥_{W^{m−1}(B_{r})}) . \end{matrix} ; (3.9)

By the Poincaré inequality there exists C_{2} > 0 which is independent of r ≤ 1 such that for all u ∈ C_{0}^{∞}(B_{r}):

∥ u ∥_{W^{m−1}(B_{r})}≤ r C_{2} ∥ u ∥_{W^{m}(B_{r})} .

Using this inequality, it follows from (3.9) that

∥ (P − P_{0})u ∥_{L^{2}(B_{r})}≤ r C(K) ∥ u ∥_{W^{m}(B_{r})} .

Together with (3.8) we get

(1 − rCC(K)) ∥ u ∥_{W^{m}(B_{r})}≤ C (∥ P u ∥_{L^{2}(B_{r})} + ∥ u ∥_{L^{2}(B_{r})}) .

Set

r_{0} = min\{1, \frac{1}{2CC(K)}\}.

Then it follows that for all r ≤ r_{0} and u ∈ C_{0}^{∞}(B_{r}):

∥ u ∥_{W^{m}(B_{r})}≤ 2C (∥ P u ∥_{L^{2}(B_{r})} + ∥ u ∥_{L^{2}(B_{r})}) .
□

Lemma 3.5. Let k ≥ 1 be even. Assume that M has bounded curvature of order k. Let K > 0 be such that sup_{x∈M} |∇^{l}R(x)| ≤ K, l = 0, ..., 2k. There exist constants r_{0} = r_{0}(K) > 0 and C = C(K) > 0 such that for all x_{0} ∈ M and r ≤ min{r_{0}, ĩ(x_{0})} one has

∥ u ∥_{W^{2k}(B_{r}(x_{0}))}≤ C ∥ u ∥_{H^{2k}(B_{r}(x_{0}))}

for all u ∈ C_{0}^{∞}(B_{r}(x_{0})).

Proof: By [Ei2, Corollary 2.6 and 2.7] there exists a constant C_{1} > 0, which depends only on K, such that for every x_{0} ∈ M, every r ≤ ĩ(x_{0}), and all i, j, k = 1, ..., n, one has

(3.10) sup_{x∈B_{r}(x_{0})} |D^{α}g_{ij}(x)| ≤ C_{1}, |α| ≤ 2k, sup_{x∈B_{r}(x_{0})} |D^{β}Γ_{jk}^{i}(x)| ≤ C_{1}, |β| ≤ 2k − 1,

where the g_{ij} and Γ_{jk}^{i} denote the coefficients of g and ∇, respectively, with respect to normal coordinates on the geodesic ball B_{r}(x_{0}) of radius r with center x_{0}.

Let x_{0} ∈ M and r ≤ ĩ(x_{0}). Let B_{r} ⊂ T_{x_{0}}M denote the ball of radius r around the origin. Let W^{2k}(B_{r}) be the Sobolev space with respect to the flat connection. Then it follows from (3.10) that there exists C_{2} = C_{2}(K) > 0 such that

(3.11)  C_{2}^{−1} ∥ u ∘ exp_{x_{0}} ∥_{W^{2k}(B_{r})}≤∥ u ∥_{W^{2k}(B_{r}(x_{0}))}≤ C_{2} ∥ u ∘ exp_{x_{0}} ∥_{W^{2k}(B_{r})}

for all x_{0} ∈ M, r ≤ ĩ(x_{0}), and u ∈ C_{c}^{∞}(B_{r}(x_{0})). Let g̃ be the metric on B_{r} which is the pull-back of g ↾ B_{r}(x_{0}) with respect to exp_{x_{0}} : B_{r} → B_{r}(x_{0}). Let Δ̃ be the Laplacian on B_{r} with respect to g̃. Then by (3.11) it is sufficient to show that there exists C_{3} = C_{3}(K) > 0 such that

(3.12)  ∥ f ∥_{W^{2k}(B_{r})}≤ C_{3} ∥ (Δ̃ + Id)^{k}f ∥_{L^{2}(B_{r})}

for all x_{0} ∈ M, r ≤ ĩ(x_{0}), and f ∈ C_{0}^{∞}(B_{r}). Set P = (Δ̃ + Id)^{k}. By (3.10) there exists C_{4} > 0, which depends only on K, such that P ∈ Ell^{2k}(r, 1, C_{4}). Then by Lemma 3.4, there exist r_{0} > 0 and C_{3} > 0 such that (3.12) holds for all x_{0} ∈ M and r ≤ min{r_{0}, ĩ(x_{0})}. This completes the proof of the lemma. □

Lemma 3.6. Let k ∈ ℕ be even. Suppose that (M, g) has bounded curvature of order 2k. Let β : M → ℝ^{+} be a function of moderate decay. Then there exists a canonical bounded inclusions

(3.13)  H_{βĩ^{−2kn}}^{k}(M) ↪ W_{β}^{k}(M) and H_{β}^{k}(M) ↪ W_{βĩ^{2kn}}^{k}(M).

Proof: By Theorem 2.6, there exists a covering \{B_{\frac{ĩ}{2^{k}} (x_{i})}(x_{i})\}_{i=1}^{∞} of M by balls and a constant C > 0 such that

(3.14)  ∀x ∈ M : #{x_{i} | x ∈ B_{ĩ(x_{i})}(x_{i})} ≤ C.

Let φ ∈ C^{∞}(ℝ) be such that φ = 1 on [0, 1] and φ = 0 on [2, ∞). For x ∈ M and 1 ≤ j ≤ k, we define

(3.15)  φ_{j,x}(y) = \{ \begin{matrix}φ(2^{j} \frac{d(x,y)}{ĩ(x)}), & y ∈ B_{ĩ(x)}(x); \\ 0, & otherwise. \end{matrix}

Then φ_{j,x} ∈ C_{0}^{∞}(M). Let f ∈ H^{k}(M). Using Lemma 3.1, it follows that φ_{j,x}f ∈ H^{k}(B_{ĩ(x)}(x)). Then by Lemma 3.5 we get φ_{j,x}f ∈ W^{k}(B_{ĩ(x)}(x)), and by the Leibniz rule there is C > 0 such that

|∇^{j}(φ_{k,x}f)|_{g}(y) ≤ C ∑_{p=0}^{j} |∇^{p}φ_{k,x}|_{g}(y) · |∇^{j−p}f)|_{g}(y), y ∈ M, j = 0, ..., k.

By estimating the supremum-norm of the derivatives of φ_{k,x} and using Lemma 3.5 , we get

\begin{aligned}∥φ_{k,x}f∥_{W^{k}} & ≤ C∥f∥_{W^{k}(B_{\frac{ĩ}{2^{k−1}} (x)}(x))} + C′ ∑_{p=1}^{k} \binom{k}{p}ĩ^{−p}(x)∥φ_{k−1,x}f∥_{W^{k−p}} \\ & ≤ C∥f∥_{H^{k}(B_{\frac{ĩ}{2^{k−1}} (x)}(x))} + C′′ ∑_{p=1}^{k} \binom{k}{p}ĩ^{−p}(x)∥φ_{k−1,x}f∥_{H^{k−p}}. \end{aligned} ; (3.16)

By induction, this yields

(3.17)  ∥φ_{k,x_{i}}f∥_{W^{k}} ≤ Cĩ^{−k}(x_{i})∥f∥_{H^{k}(B_{ĩ(x_{i})}(x_{i}))}.

Let f ∈ H_{β}^{k}. By Lemma 1.6, Lemma 3.5, (3.14) and (3.17) we get

\begin{aligned}∥f∥_{W k ; β̃} ≤ C ∑_{i=1}^{∞} β^{\frac{1}{2}} (x_{i})∥φ_{k,x_{i}}f∥_{W^{k}}ĩ^{k}(x_{i}) & ≤ C ∑_{i=1}^{∞} β^{\frac{1}{2}} (x_{i})∥φ_{k,x_{i}}f∥_{H^{k}} \\ & ≤ C ∑_{i=1}^{∞} β^{\frac{1}{2}} (x_{i})ĩ^{−k}(x_{i})∥f∥_{H^{k}(B_{ĩ(x_{i})}(x_{i}))}. \end{aligned} ; (3.18)

By (2.3) there exists a constant C_{1} > 0 such that

ĩ(x_{i})^{−k}ĩ(x)^{kn} ≤ C_{1}

for all i ∈ ℕ and x ∈ B_{ĩ(x_{i})}(x_{i}). This implies

∑_{i=1}^{∞} β^{\frac{1}{2}} (x_{i})ĩ^{−k}(x_{i})∥f∥_{H^{k}(B_{ĩ(x_{i})}(x_{i}))} ≤ C_{2} ∥ f ∥_{H_{ĩ^{−2kn}β}^{k}},

which together with (3.18) gives the first inclusion. The proof of the second inclusion is analogous. □

Remark 2. Lemma 3.6 is not optimal. Under additional assumptions on β one can show that W_{β}^{2k}(M) = H_{β}^{2k}(M) [Sa].

4. Functions of the Laplacian.

Assume that (M, g) is complete. Then Δ: C_{c}^{∞}(M) → L^{2}(M) is essentially self-adjoint and functions f(\sqrt{Δ}) can be defined by the spectral theorem for unbounded self-adjoint operators by

f(\sqrt{Δ}) = ∫_{0}^{∞} f(λ)dE_{λ},

where dE_{λ} is the projection spectral measure associated with \sqrt{Δ}. Let f ∈ L^{1}(ℝ) be even and let

f̂(λ) = ∫_{−∞}^{∞} f(x) cos(λx) dx.

Then f(\sqrt{Δ}) can also be defined by

(4.1)  f(\sqrt{Δ}) = \frac{1}{2π} ∫_{−∞}^{∞} f̂(λ) cos(λ\sqrt{Δ}) dλ.

This representation has been used in [CGT] to study the kernel of f(\sqrt{Δ}). We will use (4.1) to study f(\sqrt{Δ}) as operator in weighted L^{2}-spaces. To this end we need to study cos(λ\sqrt{Δ}) as operator in L_{β}^{2}(M). Given s > 0, let κ(M, g, s) be the constant introduced in Definition 2.7.

Theorem 4.1. Assume that (M, g) has bounded curvature. Let β be a function of moderate decay. Then cos(s\sqrt{Δ}) extends to a bounded operator in L_{β}^{2}(M) for all s ∈ ℝ and there exist C, c > 0 such that

(4.2)  ∥ cos(s\sqrt{Δ})∥_{L_{β}^{2},L_{β}^{2}} ≤ Ce^{c|s|}, s ∈ ℝ.

Moreover cos(s\sqrt{Δ}): L_{β}^{2}(M) → L_{β}^{2}(M) is strongly continuous in s.

Proof: Let s > 0. Choose a sequence {x_{k}}_{k=1}^{∞} ⊂ M which minimizes κ(M, g; s). For k ∈ ℕ let P_{k} denote the multiplication by the characteristic function of B_{s}(x_{k}) \ ⋃_{i=0}^{k−1} B_{s}(x_{j}). Then each P_{k} is an orthogonal projection in L^{2}(M) and L_{β}^{2}(M), respectively. Moreover the projections satisfy P_{k}P_{k′} = 0 for k ≠ k′ and ∑_{k=1}^{∞} P_{k} = 1, where the series is strongly convergent. Obviously the image of P_{k} consists of functions with support in B_{s}(x_{k}). Now recall that cos(t\sqrt{Δ}) has unit propagation speed [CGT, p.19], i.e.,

supp cos(s\sqrt{Δ})δ_{x} ⊂ \overline{B_{|t|}(x)}

for all x ∈ M and t ∈ ℝ. Let f ∈ L^{2}(M). Then it follows that

supp cos(s\sqrt{Δ})P_{k}f ⊂ B_{2s}(x_{k})

and

supp cos(s\sqrt{Δ}) ((1 − χ_{B_{3s}(x_{k})})f) ⊂ M − B_{2s}(x_{k}).

Hence

∥ cos(s\sqrt{Δ})f ∥_{β}^{2} = ∑_{k=1}^{∞}⟨cos(s\sqrt{Δ})P_{k}f, cos(s\sqrt{Δ})f⟩_{β}

= ∑_{k=1}^{∞}⟨cos(s\sqrt{Δ})P_{k}f, cos(s\sqrt{Δ})(χ_{B_{3s}(x_{k})}f)⟩_{β}.

(4.3)

Now observe that the norm of cos(s\sqrt{Δ}) as an operator in L^{2}(M) is bounded by 1. This implies

|⟨cos(s\sqrt{Δ})P_{k}f, cos(s\sqrt{Δ})(χ_{B_{3s}(x_{k})}f)⟩_{β}|

≤ sup_{y∈B_{3s}(x_{k})} β(y) ∥ P_{k}f ∥_{L^{2}} · ∥ χ_{B_{3s}(x_{k})}f ∥_{L^{2}} .

To estimate the right hand side, we write

sup_{y∈B_{3s}(x_{k})} β(y) ∥ P_{k}f ∥_{L^{2}}^{2}= ∫_{M} |P_{k}f(x)|^{2} sup_{y∈B_{3s}(x_{k})} (\frac{β(y)}{β(x)}) β(x) dx.

Since the support of P_{k}f is contained in B_{s}(x_{k}), we can use (1.13) to estimate the right hand side. This gives

sup_{y∈B_{3s}(x_{k})} β(y) ∥ P_{k}f ∥_{L^{2}}^{2}≤ C_{β}^{−1} \frac{1}{β(1 + 4s)} ∥ P_{k}f ∥_{L_{β}^{2}}^{2} .

A similar inequality holds with respect to ∥ χ_{B_{3s}(x_{k})}f ∥_{L^{2}}. Putting the estimations together, we get

|⟨cos(s\sqrt{Δ})P_{k}f, cos(s\sqrt{Δ})(χ_{B_{3s}(x_{k})}f)⟩_{β}|

≤ C_{β}^{−1} \frac{1}{β(1 + 6s)} ∥ P_{k}f ∥_{L_{β}^{2}} · ∥ χ_{B_{3s}(x_{k})}f ∥_{L_{β}^{2}} . (4.4)

Now recall that by Lemma 2.8 we have κ(M, g; s) < ∞. Hence we get

∑_{k=1}^{∞} ∥χ_{B_{3s}(x_{k})}f∥_{L_{β}^{2}}^{2} ≤ κ(M, g; s)∥f∥_{L_{β}^{2}}^{2} < ∞.

Together with 4.3 and (4.4) we obtain

\begin{aligned}∥ cos(s\sqrt{Δ})f ∥_{L_{β}^{2}}^{2} & ≤ C_{β}^{−1} \frac{1}{β(1 + 6s)} ∥ f ∥_{L_{β}^{2}} ∑_{k=1}^{∞} ∥ χ_{B_{3s}(x_{k})}f ∥_{L_{β}^{2}} \\ & ≤ C_{β}^{−1} \frac{1}{β(1 + 6s)}κ(M, g, s)^{1/2} ∥ f ∥_{L_{β}^{2}}^{2} . \end{aligned} ; (4.5)

Recall that by (1.10) we have β(x) ≤ C(1+d(x, p))^{−1}, x ∈ M. Therefore L^{2}(M) ⊂ L_{β}^{2}(M), and L^{2}(M) is a dense subspace of L_{β}^{2}(M). This implies that cos(s\sqrt{Δ}) extends to a bounded operator in L_{β}^{2}(M). Moreover by (1.11) and Lemma 2.8 it follows that there exist constants C, c > 0 such that

∥ cos(s\sqrt{Δ}) ∥_{L_{β}^{2},L_{β}^{2}}^{2}≤ Ce^{cs}, s ∈ [0, ∞).

Since cos(−s\sqrt{Δ}) = cos(s\sqrt{Δ}), this extends to all s ∈ ℝ such that (4.2) holds. The strong continuity is a consequence of the local bound of the norm and the strong continuity on the dense subspace L^{2}(M) ⊆ L_{β}^{2}(M). □

Using Theorem 4.1, we can study f(\sqrt{Δ}) as an operator in L_{β}^{2}(M). Given c ≥ 0, let

F^{1}(c) = {f ∈ L^{1}(ℝ): ∫_{−∞}^{∞} |f̂(λ)|e^{c|λ|}dλ < ∞}.

Lemma 4.2. Assume (M, g) has bounded curvature and let β be a function of moderate decay. Then there exists a constant c = c(M, g, β), such that for all even functions f ∈ F^{1}(c), the operator f(\sqrt{Δ}) extends to a bounded operator in L_{β}^{2}(M). Moreover, there exists a constant C_{1} = C_{1}(M, g, β) > 0 such that

(4.6)  ∥f(\sqrt{Δ})∥_{L_{β}^{2},L_{β}^{2}} ≤ C_{1}∥f̂∥_{L_{e^{c|·|}}^{1}}

for all f as above. If κ(M, g; s) is at most sub-exponentially increasing, then c(M, g, β) > 0 can be chosen arbitrarily.

Proof: By Theorem 4.1 there exist constants C, c > 0, depending on (M, g, β), such that

∥ cos(\sqrt{Δ})∥_{L_{β}^{2},L_{β}^{2}} ≤ Ce^{c|s|},

for all s ∈ ℝ. Let φ ∈ L^{2}(M). Using (4.1), it follows that

(4.7)  ∥f(\sqrt{Δ})φ∥_{L_{β}^{2}} ≤ \frac{C}{\sqrt{2π}} ∥f̂∥_{L_{e^{C|·|}}^{1}}∥φ∥_{L_{β}^{2}}.

Since L^{2}(M) is dense in L_{β}^{2}(M), it follows from (4.7) that f(\sqrt{Δ}) extends to a bounded operator in L_{β}^{2}(M). The last statement is obvious. □

Remark 3. It is not difficult to see, that (4.1) is in fact strongly convergent in L_{β}^{2}.

Corollary 4.3. Suppose that (M, g) has bounded curvature and let β be a function of moderate decay. Then the following holds:

a) For every t > 0, the heat operator e^{−tΔ} extends to bounded operator in L_{β}^{2}(M). Its norm is uniformly bounded in t on compact intervals of ℝ^{+}.

b) In the region \{λ ∈ ℂ: Re(\sqrt{−λ}) > c(M, g, β)\} the resolvent (Δ − λ)^{−1} extends to a bounded operator in L_{β}^{2}(M). The function of λ ↦ (Δ − λ)^{−1} is locally bounded and holomorphic on this domain.

c) If β is of sub-exponential decay and κ(M, g; s) is at most sub-exponentially increasing for s > s_{0}, then (Δ−λ)^{−1} : L_{β}^{2}(M) ↦ L_{β}^{2}(M) is defined and bounded for all λ ∈ ℂ\[0, ∞).

Proof: This follows from Lemma 4.2 and

(4.8)  ∫_{−∞}^{∞} e^{−tx^{2}} cos(xy)dx = \sqrt{\frac{π}{t}} e^{− \frac{y^{2}}{4t}} , ∫_{−∞}^{∞} \frac{1}{λ + x^{2}} cos(xy)dx = \frac{π}{\sqrt{λ}}e^{−\sqrt{λ} |y|}.
□

Let β be of moderate decay. There is a canonical pairing (·, ·) between L_{β}^{2}(M) and L_{β^{−1}}^{2}(M) given by

(f, g) = ∫_{M} f(x)g(x) dx, f ∈ L_{β}^{2}(M), g ∈ L_{β^{−1}}^{2}(M).

This pairing is non-degenerate so that L_{β^{−1}}^{2}(M) is canonically isomorphic to the dual of L_{β}^{2}(M). Moreover, we have the following inclusions

L_{β^{−1}}^{2}(M) ⊂ L^{2}(M) ⊂ L_{β}^{2}(M).

By duality it follows that Theorem 4.1, Lemma 4.2 and Corollary 4.3 also hold w.r.t. β^{−1}. Especially, it follows that f(\sqrt{Δ}) defined on L_{β^{−1}}^{2}(M) is the restriction of f(\sqrt{Δ})_{|L^{2}}. Moreover, we have the identity

(4.9)  f(\sqrt{Δ})_{|L_{β^{−1}}^{2}} = (f̄(\sqrt{Δ})_{|L_{β}^{2}})^{∗} .

Lemma 4.4. Let β be a function of moderate decay. If λ and λ̄ satisfy condition b) of Corollary 4.3, then

H_{β}^{2}(M) = (Δ − λ)^{−1}(L_{β}^{2}(M)).

Proof: First note that C_{0}^{∞}(M) is dense in L_{β}^{2}(M). Indeed C_{0}^{∞}(M) is dense in L^{2}(M), and L^{2}(M) is dense in L_{β}^{2}(M). Let f = (Δ − λ)^{−1}g, g ∈ L_{β}^{2}(M). Then there exists a sequence {φ_{i}}_{i∈ℕ} ⊂ C_{0}^{∞}(M) which converges to g in L_{β}^{2}(M) and (Δ − λ)^{−1}φ_{i} converges to f in L^{2}(M). Let φ ∈ C_{0}^{∞}(M). Then

⟨f, Δφ⟩ = lim_{i→∞}⟨(Δ − λ)^{−1}φ_{i}, Δφ⟩ = lim_{i→∞}⟨φ_{i} + λ(Δ − λ)^{−1}φ_{i}, φ⟩ = ⟨g + λf, φ⟩.

Thus Δf = g + λf ∈ L_{β}^{2}(M) and hence f ∈ H_{β}^{2}(M). Now suppose that f ∈ H_{β}^{2}(M) and set g = (Δ − λ)f. Then g ∈ L_{β}^{2}(M) and we need to show that f = (Δ − λ)^{−1}g. Let φ ∈ C_{0}^{∞}(M). By definition of (Δ − λ)^{−1}g, there exists a sequence {g_{i}}_{i∈ℕ} ⊂ L^{2}(M) such that (Δ − λ)^{−1}g_{i} converges to (Δ − λ)^{−1}g in L_{β}^{2}(M) as i → ∞. Using this fact, we get

(4.10)  ⟨(Δ − λ)^{−1}g, φ⟩ = ⟨g, (Δ − λ̄)^{−1}φ⟩ = ⟨(Δ − λ)f, (Δ − λ̄)^{−1}φ⟩.

Now observe that (Δ − λ̄)^{−1}φ belongs to H^{2}(M). By Lemma 3.1, there exists a sequence {φ_{i}}_{i∈ℕ} ⊂ C_{0}^{∞}(M) which converges to (Δ − λ̄)^{−1}φ in H^{2}(M). Thus

⟨(Δ − λ)f, (Δ − λ̄)^{−1}φ⟩ = lim_{i→∞}⟨(Δ − λ)f, φ_{i}⟩ = ⟨f, (Δ − λ̄)φ_{i}⟩ = ⟨f, φ⟩.

Together with (4.10) this implies that f = (Δ − λ)^{−1}g. □

Lemma 4.5. Let β be a function of moderate decay. Then (Δ + λ)(C_{0}^{∞}(M)) is dense in L_{β}^{2}(M) for every λ ∈ ℝ^{+}.

Proof: As in (3.6) it follows from the essential self-adjointness of Δ + λ Id that (Δ + λ)(C_{c}^{∞}(M)) is dense in L^{2}(M). Moreover since β is monotonically decreasing, we have that L^{2}(M) ⊂ L_{β}^{2}(M) is dense and ∥ f ∥_{β}≤ C ∥ f ∥ for f ∈ L^{2}(M). This implies that (Δ + λ)(C_{c}^{∞}(M)) is also dense in L_{β}^{2}(M). □

Corollary 4.6. Let β be of moderate decay. Then C_{0}^{∞}(M) is dense in H_{β}^{2}(M).

Proof: Let f ∈ H_{β}^{2}(M). Let λ ≫ 0. By Lemma 4.4 there exists g ∈ L_{β}^{2}(M) such that f = (Δ + λ)^{−1}g. By Lemma 4.5 there exists a sequence {φ_{i}}_{i∈ℕ} ⊂ C_{c}^{∞}(M) such that (Δ + λ)φ_{i} converges to g in L_{β}^{2}(M) as i → ∞. Thus φ_{i} → f in L_{β}^{2}(M) and (Δ + λ)φ_{i} converges to g = (Δ + λ)f as i → ∞. This implies that φ_{i} converges to f in H_{β}^{2}(M). □

5. Equivalent Metrics and Sobolev Spaces.

In this section we study the dependence of the Sobolev spaces on the metric. We will prove, that if g ∼_{β}^{k} h for an appropriate β, then the Sobolev spaces defined with respect to g and h are equivalent up to order k. We assume that all metrics have bounded sectional curvature. To indicate the dependence of the corresponding Sobolev space on the Riemannian metric g, we will write W_{ξ}^{k}(M; g) and H_{ξ}^{2k}(M; g), respectively.

Lemma 5.1. Let β be of moderate decay. Assume that g ∼_{β}^{k} h. Then the Sobolev spaces W_{ξ}^{k}(M; g) and W_{ξ}^{k}(M; h) are equivalent.

Proof: First note that by Lemma 1.7 the metrics g and h are quasi-isometric. This implies that L_{ξ}^{2}(M, g) and L_{ξ}^{2}(M; h) are equivalent. So the statement of the lemma holds for k = 0. Let f ∈ C^{∞}(M). Let k ∈ ℕ. By induction we will prove that for l ≤ k there exists C_{l} > 0 such that for a, b ∈ ℕ_{0}, a + b = l,

(5.1)  |(∇^{g})^{a}(∇^{h})^{b}f|_{h}(x) ≤ C_{l} ∑_{i=0}^{a+b} |(∇^{g})^{i}f|_{g}(x), x ∈ M.

Let l = 1. Since on functions the connections equal d, (5.1) follows from quasi-isometry of g and h.

Next suppose that (5.1) holds for 1 ≤ l < k. To establish (5.1) for l + 1, we proceed by induction with respect to a. Let a, b ∈ N_{0} with a + b = l + 1. We may assume that a < l + 1. Using

(5.2)  (∇^{g})^{a}(∇^{h})^{b}f = (∇^{g})^{a}(∇^{h} − ∇^{g})(∇^{h})^{b−1}f + (∇^{g})^{a+1}(∇^{h})^{b−1}f,

and g ∼_{β}^{k} h, it follows that (5.1) holds for l + 1.

Especially, putting a = 0 we get

(5.3)  |(∇^{h})^{l}f|_{h}(x) ≤ C_{l} ∑_{i=0}^{l} |(∇^{g})^{i}f|_{g}(x), x ∈ M, l ≤ k.

Suppose that f ∈ C^{∞}(M) ∩ W_{ξ}^{k}(M; g). Then (5.3) implies that f ∈ C^{∞}(M) ∩ W_{ξ}^{k}(M; h) and

∥ f ∥_{W_{ξ}^{k}(M;h)}≤ C ∥ f ∥_{W_{ξ}^{k}(M;g)} .

By Lemma 3.1, C^{∞}(M) ∩ W_{ξ}^{k}(M; g) is dense in W_{ξ}^{k}(M; g). Therefore this inequality holds for all f ∈ W_{ξ}^{k}(M, g). By symmetry, a similar inequality holds with the roles of g and h interchanged. This concludes the proof. □

Next we compare the Sobolev spaces H_{ξ}^{2k}(M; g) and H_{ξ}^{2k}(M; h). Let Δ_{g} denote the Laplace operator with respect to the metric g. Recall, that

Δ_{g} = (∇^{g})^{∗}∇^{g},

and that the formal adjoint (∇^{g})^{∗} of ∇^{g} is given by

(5.4)  (∇^{g})^{∗} = − Tr(g^{−1}∇^{g}),

where g^{−1} : T^{∗}M → T M is the isomorphism induced by the metric and Tr: T^{∗}M ⊗T M → ℝ denotes the contraction. Since contraction commutes with covariant differentiation and ∇^{g}g^{−1} = 0, we get the well known formula

Δ = − Tr(g^{−1}∇^{2}).

This can be iterated. For ω_{1} ⊗ ⋯ ⊗ ω_{k} ∈ (T^{∗}M)^{⊗k} define

g_{j}^{−1}(ω_{1} ⊗ ⋯ ⊗ ω_{k}) := ω_{1} ⊗ ⋯ ⊗ ω_{j−1} ⊗ g^{−1}(ω_{j}) ⊗ ω_{j+1} ⊗ ⋯ ⊗ ω_{k},

and let Tr_{i,j}(g_{j}^{−1}) denote g_{j}^{−1} followed by the contraction of the i-th and j-th component. Using that contraction commutes with covariant differentiation and ∇^{g}g^{−1} = 0, we get

(5.5)  Δ_{g}^{k} = (−1)^{k} Tr_{1,2}(g_{2}^{−1}) ∘ ⋯ ∘ Tr_{2k−1,2k}(g_{2k}^{−1})(∇^{g})^{2k}.

In a more traditional notation this means

Δ_{g}^{k}f = (−1)^{k} ∑_{i_{1},...,i_{k}} f_{;i_{1}i_{1}i_{2}i_{2}...i_{k}i_{k}}.

For short notation we will write

Tr ((g^{−1})^{⊗k}) := Tr_{1,2}(g_{2}^{−1}) ∘ ⋯ ∘ Tr_{2k−1,2k}(g_{2k}^{−1}).

Lemma 5.2. Assume that g ∼_{β}^{2k} h. Then for each l, 0 ≤ l ≤ 2k and j, 0 ≤ j ≤ 2l, there exist sections ξ_{jl}^{g}, ξ_{jl}^{h} ∈ C^{∞}(Hom((T^{∗}M)^{⊗j}, ℝ)) such that

(5.6)  Δ_{g}^{l} − Δ_{h}^{l} = ∑_{j=0}^{2l} ξ_{jl}^{g} ∘ (∇^{g})^{j} = ∑_{j=0}^{2l} ξ_{jl}^{h} ∘ (∇^{h})^{j}

and there exists C > 0 such that for 0 ≤ p ≤ l

(5.7)  |(∇^{g})^{p}ξ_{jl}^{g}|_{g}(x) ≤ Cβ(x), |(∇^{h})^{p}ξ_{jl}^{h}|_{h}(x) ≤ Cβ(x), x ∈ M.

Proof: Using (5.5) we get

(−1)^{l}(Δ_{g}^{l} − Δ_{h}^{l}) = Tr ((g^{−1})^{⊗l}) (∇^{g})^{2l} − Tr ((h^{−1})^{⊗l}) (∇^{h})^{2l}
= Tr ((g^{−1})^{⊗l}) ((∇^{g})^{2l} − (∇^{h})^{2l})
+ (Tr ((g^{−1})^{⊗l}) − Tr ((h^{−1})^{⊗l})) (∇^{h})^{2l}.

(5.8)

First consider the second term. Note that there exists C > 0 such that

\begin{aligned}& |(∇^{g})^{p} (Tr ((g^{−1})^{⊗l}) − Tr ((h^{−1})^{⊗l})) |_{g}(x) ≤ C|(∇^{g})^{p}(g − h)|_{g}(x). \\ (5.9) \end{aligned}

Since g ∼_{β}^{2k} h, the right hand side is bounded by C_{1}β(x). By symmetry, the same estimation holds with respect to h.

To deal with the first term on the right hand side of (5.8), we use

(∇^{g})^{j} − (∇^{h})^{j} = (∇^{g})^{j−1}(∇^{g} − ∇^{h}) + ((∇^{g})^{j−1} − (∇^{h})^{j−1}) ∇^{h}

and proceed by induction with respect to j. □

Corollary 5.3. Let β be of moderate decay. Assume that βĩ^{−2kn} is bounded, g ∼_{β}^{2k} h and (M, g) and (M, h) have both bounded curvature of order 2k. Then H_{ρ}^{2k}(M, g) and H_{ρ}^{2k}(M, h) are equivalent for all functions ρ of moderate decay.

Proof: Let f ∈ C^{∞}(M) ∩ H_{ρ}^{2k}(M; g). Using Lemma 3.6 and Lemma 5.1 we get

∥ f ∥_{H_{ρ}^{2k}(M;g)}≥ C_{1} ∥ f ∥_{W_{ĩ^{4k}ρ}^{2k}(M;g)}≥ C_{2} ∥ f ∥_{W_{ĩ^{4kn}ρ}^{2k}(M;h)}≥ C_{3} ∥ f ∥_{W_{β^{2}ρ}^{2k}(M;h)} .

By Lemma 5.2 it follows that f ∈ C^{∞}(M) ∩ H_{ρ}^{2k}(M, h) and there exists a constant C > 0, which is independent of f, such that

∥ f ∥_{H_{ρ}^{2k}(M;h)}≤ C ∥ f ∥_{H_{ρ}^{2k}(M;g)} .

By symmetry, a similar inequality holds with g and h interchanged. □

6. Trace class estimates

Let (M, g) be an n-dimensional Riemannian manifold with bounded sectional curvature, |K_{M}| < K. Let e^{−tΔ_{g}}(x, y) denote the heat kernel of Δ_{g}. Let 0 < a_{1} < a_{2} < ∞. Let ĩ be the modified injectivity radius defined by (2.1). It follows from [CGT, Proposition 1.3], that there exist C_{1}, c_{1} > 0 such that

(6.1)  e^{−tΔ_{g}}(x, y) ≤ C_{1}ĩ(x)^{− \frac{n}{2}} ĩ(y)^{− \frac{n}{2}} e^{−c_{1}d^{2}(x,y)}, t ∈ [a_{1}, a_{2}].

Let c < c_{1}. Then by (6.1) and (2.3) there exists C > 0 such that

(6.2)  e^{−tΔ_{g}}(x, y) ≤ Cĩ(x)^{− \frac{n(n+1)}{2}} e^{−cd^{2}(x,y)}, t ∈ [a_{1}, a_{2}].

Lemma 6.1. Let β be a function of moderate decay. Assume that there exist real numbers a, b such that

i) a + b = 2, ii) β^{b} ∈ L^{1}(M), iii) β^{a}ĩ^{− \frac{n(n+1)}{2}} ∈ L^{∞}(M).

Let M_{β} the operator of multiplication by β. Then for every p ∈ ℕ_{0}, the operator M_{β}Δ_{g}^{p}e^{−tΔ_{g}} is Hilbert-Schmidt. For t in a compact interval in ℝ^{+}, the Hilbert-Schmidt norm is bounded.

Proof: we have

(6.3)  M_{β}Δ^{p}e^{−tΔ} = (M_{β}e^{− \frac{t}{2} Δ}) (Δ^{p}e^{− \frac{t}{2} Δ}) .

Note that the operator norm of Δ^{p}e^{− \frac{t}{2} Δ} is bounded on compact subsets of ℝ^{+}. Hence we may assume that p = 0. By Corollary 4.3, 1), it follows that e^{−tΔ} extends to a bounded operator in L_{β^{b}}^{2}(M) and its norm is uniformly bounded for 0 < a ≤ t ≤ b. The condition β^{b} ∈ L^{1}(M) implies that 1 ∈ L_{β^{b}}^{2}. Hence e^{−tΔ}1 ∈ L_{β^{b}}^{2}(M). Let e^{−tΔ}(x, y) be the kernel of e^{−tΔ}. Then

⟨1, e^{−tΔ}1⟩_{L_{β^{b}}^{2}} = ∫_{M} ∫_{M} β^{b}(x)e^{−tΔ_{g}}(x, y)dydx.

The integral converges since e^{−tΔ}(x, y) is positive. Thus we get

∫_{M} ∫_{M} |β(x)e^{−tΔ_{g}}(x, y)|^{2}dydx = ∫_{M} ∫_{M} β^{2}(x) (e^{−tΔ_{g}}(x, y))^{2} dydx

≤ sup_{z,w∈M} |β^{a}(z)e^{−tΔ_{g}}(z, w)| ∫_{M} ∫_{M} β^{b}(x)e^{−tΔ_{g}}(x, y)dydx

≤ C sup_{z∈M} |β^{a}(z)ĩ^{− \frac{n(n+1)}{2}} (z)| ∫_{M} β^{b}(x) (e^{−tΔ}(1)) (x)dx

≤ C_{1} ∥ e^{−tΔ}(1) ∥_{L_{β^{b}}^{2}} .

This proves the lemma. □

Lemma 6.2. Assume β is a function of moderate decay and that there exist real numbers a, b such that

i) b ≥ 1 and a + b = 2, ii) β^{\frac{b}{3}} ∈ L^{1}(M), iii) β^{\frac{a}{3}} ĩ^{− \frac{n(n+2)}{2}} ∈ L^{∞}(M).

Let M_{β} be the operator of multiplication by β. Then the operator M_{}_{ĩ^{−2n}}M_{β}Δ^{p}e^{−tΔ} is a trace-class operator for p ∈ ℕ. For t in a compact interval, the trace-class norm is bounded.

Proof: We decompose the operator as

(6.4)  M_{}_{ĩ^{−2n}}M_{β}Δ^{p}e^{−tΔ} = \{M_{}_{ĩ^{−2n}}M_{β}e^{− \frac{t}{2} Δ}M_{β^{− \frac{1}{3}}} \} · \{M_{β^{\frac{1}{3}}} Δ^{p}e^{− \frac{t}{2} Δ^{2}}\} .

Since β is non-increasing and β(x) ≤ 1/2 outside a compact set, it follows that β^{\frac{1}{3}} ≤ Cβ^{\frac{b}{3}} for b ≥ 1. Hence by ii) we get β^{\frac{1}{3}} ∈ L^{1}(M). Moreover by iii) it follows that β^{\frac{a}{3}} ĩ^{− \frac{n(n+1)}{2}} ∈ L^{∞}(M). Hence by Lemma 6.1, the second factor on the right hand side of (6.4) is a Hilbert-Schmidt operator and its Hilbert-Schmidt norm is bounded for t in a compact interval in ℝ^{+}. It remains to show that the first factor is Hilbert-Schmidt and that the Hilbert-Schmidt norm is bounded on compact intervals. By iii) we have

β^{a}ĩ^{− \frac{n(n+1)}{2} −2n} ∈ L^{∞}(M).

Using this observation together with (6.2), we get

\begin{matrix}∫_{M} ∫_{M} |ĩ^{−2n}(x) & β(x)e^{−tΔ}(x, y)β^{− \frac{1}{3}} (y)|^{2}dxdy \\ ≤ C sup_{z∈M} |ĩ^{− \frac{n(n+1)}{2} −2n}(z)β^{a}(z)| ∫_{M} ∫_{M} β^{b}(x)e^{−tΔ}(x, y)β^{− \frac{2}{3}} (y)dxdy. \end{matrix} ; (6.5)

Now observe that by ii), β^{− \frac{2}{3}} belongs to L_{β^{\frac{b+4}{3}}}^{2} (M). Since β^{\frac{b+4}{3}} ≤ Cβ^{\frac{b}{3}} , it follows from ii)

that β^{\frac{b+4}{3}} is integrable. Hence by Corollary 4.3, e^{−tΔ} extends to a bounded operator in L_{β^{\frac{b+4}{3}}}^{2} (M). Therefore ∫_{M} e^{−Δ}(x, y)β^{− \frac{2}{3}} (y)dy ∈ L_{β^{\frac{b+4}{3}}}^{2} , and the norm is uniformly bounded

for t in a compact interval of ℝ^{+}. Next note that β^{b} ∈ L_{β^{− \frac{b+4}{3}}}^{2} . Hence

(6.6)  ∫_{M} ∫_{M} β^{b}(x)e^{−tΔ}(x, y)β^{− \frac{2}{3}} (y)dxdy = ⟨β^{b}, e^{−tΔ}β^{− \frac{2}{3}} ⟩ < ∞.

This implies the lemma. □

Lemma 6.3. Let β be a function of moderate decay, satisfying the conditions of Lemma 6.2. Let g, h be two complete metrics on M such that g ∼_{β}^{2} h. Let Δ_{g} and Δ_{h} be the Laplacians of g and h, respectively. Then

(Δ_{g} − Δ_{h})e^{−tΔ_{g}} and e^{−tΔ_{g}}(Δ_{g} − Δ_{h})

are trace class operators, and the trace norm is uniformly bounded for t in a compact subset of (0, ∞).

Proof: We decompose e^{−tΔ_{g}} as

(6.7)  e^{−tΔ_{g}} = (e^{− \frac{t}{2} Δ_{g}}M_{β^{− \frac{1}{3}}} ) · (M_{β^{\frac{1}{3}}} e^{− \frac{t}{2} Δ_{g}}) .

By Lemma 6.1, the second factor is a Hilbert-Schmidt operator and it suffices to show that (Δ_{g} − Δ_{h})e^{−tΔ_{g}}M_{β^{− \frac{1}{3}}} is Hilbert-Schmidt and that the Hilbert-Schmidt norm is bounded for t in a compact interval. Using Lemma 5.2 and Lemma 3.6,it follows that the Hilbert-Schmidt norm can be estimated by

∥ (Δ_{g} − Δ_{h})e^{−tΔ_{g}}M_{β^{− \frac{1}{3}}} ∥_{2}^{2} ≤ C ∑_{i=0}^{2} ∫_{M} ∫_{M} |(∇^{g})^{i}e^{−tΔ_{g}}(x, y)β^{− \frac{1}{3}} (y)|_{g}^{2}β^{2}(x) dxdy

= C ∫_{M} ∥ e^{−tΔ_{g}}(·, y)β^{− \frac{1}{3}} (y) ∥_{W_{β^{2}}^{2}}^{2} dy

≤ C_{1} ∫_{M} ∥ e^{−tΔ_{g}}(·, y)β^{− \frac{1}{3}} (y) ∥_{H_{β^{2}ĩ^{−4n}}^{2}}^{2} dy

≤ C_{2} ∑_{q=0}^{1} ∫_{M} ∥β(·)ĩ^{−2n}(·)Δ_{g}^{q}e^{−tΔ_{g}}(·, y)β^{− \frac{1}{3}} (y)∥_{2}^{2}dy

= C_{2} ∑_{q=0}^{1} ∥M_{β}M_{}_{ĩ^{−2n}}Δ_{g}^{q}e^{−tΔ_{g}}M_{β^{− \frac{1}{3}}} ∥_{2}^{2}.

By Lemma 6.2 the right hand side is finite and bounded for t in a compact interval of ℝ^{+}. To prove that e^{−tΔ_{g}}(Δ_{g} − Δ_{h}) is a trace class operator, it suffices to establish it for its adjoint (Δ_{g} − (Δ_{h})^{∗_{g}})e^{−tΔ_{g}} with respect to g. By (5.6) and (5.4) we have

(6.8)  Δ_{g} − (Δ_{h})^{∗_{g}} = (ξ_{01}^{g})^{∗_{g}} + (∇^{g})^{∗_{g}} ∘ (ξ_{11}^{g})^{∗_{g}} + [(∇^{g})^{∗_{g}}]^{2} ∘ (ξ_{21}^{g})^{∗_{g}}.

Using (5.4) and (5.7), it follows that there exist η_{j} ∈ C^{∞}(Hom((T^{∗}M)^{⊗j}, ℝ)) such that

(6.9)  Δ_{g} − (Δ_{h})^{∗_{g}} = η_{0} + η_{1} ∘ ∇^{g} + η_{2} ∘ (∇^{g})^{2}

and these section satisfy

(6.10)  |η_{j}|_{g}(x) ≤ Cβ(x), 0 ≤ j ≤ 2, x ∈ M.

Using (6.9) and (6.10) we can proceed as above and prove that (Δ_{g} − (Δ_{h})^{∗_{g}})e^{−tΔ_{g}} is a trace class operator. □

We are now ready to prove Theorem 0.1. We note that for equivalent metrics, the Hilbert spaces L^{2}(M, g) and L^{2}(M, h) are equivalent. Hence we may regard e^{−tΔ_{h}} as bounded operator in L^{2}(M, g).

Proof of Theorem 0.1: By Duhamel’s principle we have

e^{−tΔ_{g}} − e^{−tΔ_{h}} = ∫_{0}^{t} e^{−sΔ_{g}}(Δ_{h} − Δ_{g})e^{−(t−s)Δ_{h}} ds
= ∫_{0}^{t/2} e^{−sΔ_{g}}(Δ_{h} − Δ_{g})e^{−(t−s)Δ_{h}} ds
+ ∫_{t/2}^{t} e^{−sΔ_{g}}(Δ_{h} − Δ_{g})e^{−(t−s)Δ_{h}} ds.

(6.11)

The integrals converge in the strong operator topology. By Lemma 6.3 the first integral is a trace class operator. In order to prove that the second integral is a trace class operator, it is sufficient to prove, that its adjoint with respect to h is of the trace class. This adjoint can be written as the strong integral

(6.12)  ∫ t_{\frac{t}{2}} (e^{−(t−s)Δ_{g}})^{∗_{h}} (Δ_{h} − (Δ_{g})^{∗_{h}}) e^{−sΔ_{h}}ds.

Since (e^{−(t−s)Δ_{g}})^{∗_{h}} is uniformly bounded in s, it follows again from Lemma 6.3 that (6.12) is a trace class operator. □

7. Existence and completeness of wave operators

In this section we study the wave operators associated to (Δ_{g}, Δ_{h}) for equivalent metrics g and h. Let J: L^{2}(M, dμ_{g}) → L^{2}(M, dμ_{h}) be the identification operator.

Theorem 7.1. Let g and h be two complete metrics of bounded curvature on M which satisfy the assumptions of Theorem 0.1. Let P_{ac}(Δ_{g}) be the orthogonal projection onto the absolutely continuous subspace of Δ_{g}. Then the strong wave operators

W_{±}(Δ_{h}, Δ_{g}; J) = s − lim_{t→±∞} e^{itΔ_{h}}Je^{−itΔ_{g}}P_{ac}(Δ_{g})

exist and are complete. In particular, the absolutely continuous parts of Δ_{g} and Δ_{h} are unitarily equivalent.

Proof: By Theorem 0.1, e^{−tΔ_{g}} − e^{−tΔ_{h}} is trace class. Then the existence and completeness of the wave operators follows from the invariance principle of Birman and Kato [Ka, Chapter X, Theorem 4.7]. □

Examples. We give some examples to demonstrate Theorem 0.1:

1) Let M be a manifold with cylindrical ends. Then ĩ is bounded from below, and we may take b = 2, a = 0. The condition β^{\frac{2}{3}} ∈ L^{1}(M) is satisfied for β(t) = t^{− \frac{3}{2} −ε} for any ε > 0. We note that scattering theory on manifolds with asymptotically cylindrical ends has been studied by Melrose [Me3], [Me1, Section 7]. A metric with “asymptotically cylindrical ends” in the sense of [Me3] means a perturbation of a metric with cylindrical ends such that all derivatives of the perturbation decay exponentially. This is a much stronger condition than the decay condition we impose. On the other hand, it gives the existence of an analytic continuation of the resolvent and the existence of generalized eigenfunctions.

2) More generally, let M be a manifold with bounded geometry of order 2. (i.e. there is a lower bound for the injectivity radius and the covariant derivatives of the curvature of order ≤ 2 are bounded). Then we may choose x_{0} ∈ M arbitrary and let β(t) ≤ vol(B_{t}(x_{0}))^{− \frac{3}{2} −ε} for any ε > 0. To see this we first notice that if M is non-compact, the volume of such a manifold is infinite. This follows from Günther’s inequality because we may find infinitely many disjoint balls of the same radius. Let a(r) := \frac{∂}{∂r} vol(B_{r}(x_{0})). Then ∫_{0}^{1} a(r)β(1 + r)^{\frac{2}{3}} dr < ∞ and

\begin{aligned}∫_{1}^{∞} a(r)β(1 + r)^{\frac{2}{3}} dr ≤ ∫_{1}^{∞} a(r)β(r)^{\frac{2}{3}} dr ≤ & ∫_{1}^{∞} a(r) (∫_{0}^{r} a(s)ds)^{−1− \frac{2ε}{3}} dr \\ & = ∫_{vol(B_{1}(x_{0}))}^{∞} t^{−1− \frac{2ε}{3}} dt < ∞. \end{aligned}

3) Let M be a Riemannian manifold with cusps in the sense of [Mu1]. Assume that M has bounded curvature. Then the injectivity radius is exponentially decreasing in the distance and the volume of M is finite. Thus we may take b = 1. It follows a = 1, and we may take β(t) = e^{−( \frac{n(n+1)}{2} +4n)ct}, where c is chosen such that ĩ(x) ≥ Ce^{−cd(x,q)}. Scattering theory on manifolds with asymptotically cusp metrics has also been studied by Melrose [Me1, Section 8]. Again the decay conditions are stronger than in our case.

□

The assumptions on β in Theorem 7.1 that guarantee the existence of the wave operators are not optimal. Under additional assumptions on (M, g), the conditions on β can be relaxed. For example, let (M, g) be a complete manifold which is Euclidean at infinity and let h be a metric on M which satisfies (1.19), that is (M, h) is an asymptotically Euclidean manifold. Then Cotta-Ramusino, Krüger, and Schrader [CKS] proved that the wave operators W_{±}(Δ_{g}, Δ_{h}) exist. The condition (1.19) is weaker than the assumption which is necessary in Theorem 7.1 in this case. The proof is based on Enss’s method [Si], which applies to this scattering system. An abstract version of Enss’s method has been developed by Amrein, Pearson and Wollenberg [APW], [BW, 16,IV,§15]. This method can be applied in cases where the structure of the continuous spectrum of the “free Hamiltonian” is sufficiently well known. To explain this in more detail we need to introduce some notation.

Let C_{∞}(ℝ) be the space of all continuous functions on ℝ that vanish at infinity. For any closed countable subset I ⊂ ℝ let C_{∞}(ℝ − I) of all functions f ∈ C_{∞}(ℝ) satisfying f(x) = 0 for x ∈ I. A subset A_{I} of the space C(ℝ) of all bounded continuous functions on ℝ is called multiplicative generating for C_{∞}(ℝ − I), if the linear span of the set

{f | f = hg, h ∈ A_{I}, g ∈ C_{c}^{∞}(ℝ − I)}

is dense in C_{∞}(ℝ − I) with respect to the norm ∥ f ∥= sup_{x∈ℝ} |f(x)|. The main result of [APW] can be stated as follows.

Theorem 7.2. Let H and H_{0} be two self-adjoint operators in a Hilbert space H. Let R_{H}(λ) and R_{H_{0}}(λ) denote the resolvents of H and H_{0}, respectively. Assume that there exist selfadjoint operators P_{+} and P_{−} in H and a set A_{I} of multiplicative generating functions with respect to some closed countable subset I ⊂ ℝ satisfying the following properties

(1) P_{ac}(H_{0}) = P_{+} + P_{−} and s-lim_{t→±∞} e^{itH_{0}}P_{∓}e^{−itH_{0}}P_{ac}(H_{0}) = 0.

(2) (Id −P_{ac}(H_{0}))α(H_{0}) is compact for all α ∈ A_{I}.

(3) R_{H}(i) − R_{H_{0}}(i) is compact.

(4) ∫_{0}^{±∞} ∥ (R_{H}(i) − R_{H_{0}}(i))e^{−itH_{0}}α(H_{0})P_{±} ∥ dt < ∞ for all α ∈ A_{I}.

Then the wave operators W_{±}(H, H_{0}) exist and are complete. Moreover H and H_{0} have no singularly continuous spectrum and each eigenvalue of H and H_{0} in ℝ − I is of finite multiplicity. These eigenvalues accumulate at most at points of I ∪ {±∞}.

For the proof see Corollary 19 in [BW, 16,IV,§15].

As example, we consider a manifold X with cusps as defined in [Mu1]. For simplicity we assume that X has a single cusp. Then X is a complete Riemannian manifold of dimension n + 1 that admits a decomposition

X = M ∪_{Y} Z

in a compact Riemannian manifold M with boundary Y and a half-cylinder Z = [1, ∞)×Y, and M and Z are glued along their common boundary Y. The metric g on X is such that its restriction to Z is given by

(7.1)  g^{Z} = u^{−2}(du^{2} + g^{Y}),

where g^{Y} denotes the metric of Y. The metric g is the fixed background metric and we consider perturbations h of g. As free Hamiltonian H_{0} we are taking a modification of the Laplacian Δ_{g} which is defined as follows. We regard Y as a hypersurface in X that separates X into M and Z. Let C_{0}^{∞}(X − Y) be the subspace of all f ∈ C_{c}^{∞}(X) that vanish in a neighborhood of Y. Let Δ_{0} denote Friedrichs’s extension of

Δ_{g}: C_{0}^{∞}(X − Y) → L^{2}(X).

To begin with we need to study the spectrum of Δ_{0}. With respect to the decomposition L^{2}(X) = L^{2}(M) ⊕ L^{2}(Z) we have

(7.2)  Δ_{0} = Δ_{M,0} ⊕ Δ_{Z,0},

where Δ_{M,0} and Δ_{Z,0} are the Dirichlet Laplacians on M and Z, respectively. Since M is compact, Δ_{M,0} has pure point spectrum. Let

L_{0}^{2}(Z) := {f ∈ L^{2}(Z): ∫_{Y} f(u, y) dy = 0 for almost all u ∈ [1, ∞)}.

The orthogonal complement L_{0}^{2}(Z)^{⊥} of L_{0}^{2}(Z) in L^{2}(Z) consists of functions which are independent of y ∈ Y and therefore, can be identified with L^{2}([1, ∞), u^{−(n+1)}du). The decomposition

(7.3)  L^{2}(Z) = L_{0}^{2}(Z) ⊕ L_{0}^{2}(Z)^{⊥}

is invariant under Δ_{Z,0}.

Lemma 7.3. The restriction of Δ_{Z,0} to L_{0}^{2}(Z) has a compact resolvent. In particular, Δ_{Z,0} has pure point spectrum.

Proof: Let Δ_{Y} be the Laplacian of Y. Let {ϕ_{j}}_{j=0}^{∞} be an orthonormal basis of eigenfunctions of Δ_{Y} with eigenvalues 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ · · ·. Let f ∈ C_{c}^{∞}(Z) ∩ L_{0}^{2}(Z). Then f has an expansion of the form

f(u, y) = ∑_{k=1}^{∞} a_{k}(u)ϕ_{k}(y),

where the series converges in the C^{∞}-topology. Let b > 1 and put Z_{b} = [b, ∞) × Y. Let C = λ_{1}^{−1}. Then we have

(7.4)  ∥ f ∥_{L^{2}(Z_{b})}^{2}= ∑_{k=1}^{∞} ∫_{b}^{∞} |a_{k}(u)|^{2} \frac{du}{u^{n+1}} ≤ \frac{C}{b^{2}} ∑_{k=1}^{∞} λ_{k} ∫_{b}^{∞} |a_{k}(u)|^{2} \frac{du}{u^{n−1}} .

Now observe that the Laplacian Δ_{Z} with respect to the metric (7.1) equals

(7.5)  −u^{2} \frac{∂^{2}}{∂u^{2}} + (n − 1)u\frac{∂}{∂u} + u^{2}Δ_{Y} .

Moreover, since a_{k} ∈ C_{c}^{∞}((1, ∞)), we have

∫_{1}^{∞} (−u^{2}a′′_{k}(u) + (n − 1)ua′_{k}(u)) \overline{a_{k}(u)} \frac{du}{u^{n+1}} = ∫_{1}^{∞} |a′_{k}(u)|^{2}u^{1−n} du ≥ 0.

This together with (7.4) implies

(7.6)  ∥ f ∥_{L^{2}(Z_{b})}^{2}≤ \frac{C}{b^{2}} ⟨Δ_{Z}f, f⟩_{L^{2}(Z)} = \frac{C}{b^{2}} ∥ ∇f ∥_{L^{2}(Z)}^{2}≤ \frac{C}{b^{2}} ∥ f ∥_{H^{1}(Z)}^{2} .

Let H_{0}^{1}(Z) := H^{1}(Z) ∩ L_{0}^{2}(Z). By continuity, (7.4) holds for all f ∈ H_{0}^{1}(Z). By Rellich’s lemma, the embedding

i_{b}: H^{1}(Z − Z_{b}) ∩ L_{0}^{2}(Z − Z_{b}) → L^{2}(Z)

is compact. It follows from (7.6) that as b → ∞, i_{b} converges strongly to the embedding

i: H_{0}^{1}(Z) → L^{2}(Z).

Hence i is compact which implies the lemma. □

Let

D_{0} := −u^{2} \frac{d^{2}}{du^{2}} + (n − 1)u\frac{d}{du}: C_{c}^{∞}((1, ∞)) → L^{2}([1, ∞), u^{−(n+1)}du)

and let L_{0} be the self-adjoint extension of D_{0} with respect to Dirichlet boundary conditions at 1. By (7.5), the restriction of Δ_{Z,0} to L_{0}^{2}(Z)^{⊥} ≅ L^{2}([1, ∞), u^{−(n+1)}du) is equivalent to L_{0}. The spectrum of L_{0} is absolutely continuous and equals [n^{2}/4, ∞). Thus we get the following lemma.

Lemma 7.4. The spectrum of Δ_{0} is the union of a pure point point spectrum and an absolutely continuous spectrum. The point spectrum consists of eigenvalues of finite multiplicity 0 < λ_{1} < λ_{2} < ⋯ → ∞. The absolutely continuous spectrum is equal to [n^{2}/4, ∞) and the absolutely continuous part Δ_{0,ac} of Δ_{0} is equivalent to L_{0}.

Let ε > 0 and let β(t) = e^{−εt}. Let h be a complete metric on X. We put

H := Δ_{h} and H_{0} := Δ_{0}.

Since H and H_{0} are positive operators, we can replace i by −1 in Theorem 7.2. So let

(7.7)  R_{g} := (Δ_{g} + Id)^{−1}, R_{h} := (Δ_{h} + Id)^{−1} and R_{0} := (Δ_{0} + Id)^{−1}.

First we have the following lemma.

Lemma 7.5. Suppose that h ∼_{β}^{2} g. Then R_{h} − R_{0} is a compact operator.

Proof: First we show that R_{g} − R_{0} is a compact operator. Let f_{1}, f_{2} ∈ C^{∞}(ℝ^{+}) be such that f_{1}(u) = 1 for u ≥ 3, f_{1}(u) = 0 for u ≤ 2, and f_{2}(u) = 1 for u ≥ 2, f_{2}(u) = 0 for u ≤ 1. Put ϕ(u, y) = f_{1}(u) and ψ(u, y) = f_{2}(u), (u, y) ∈ ℝ^{+} × Y. Let M_{ϕ} and M_{ψ} denote the multiplication operator by ϕ and ψ, respectively. By [Mu1, Remark 4.29], R_{g}−M_{ϕ}(L_{0}+1)^{−1}M_{ψ} is a compact operator. So it suffices to show that R_{0}−M_{ϕ}(L_{0}+1)^{−1}M_{ψ} is compact. By (7.2) we have

(7.8)  R_{0} = (Δ_{M,0} + 1)^{−1} ⊕ (Δ_{Z,0} + 1)^{−1}.

Since M is compact, (Δ_{M,0} + 1)^{−1} is compact. By Lemma 7.3 it suffices to prove that

(L_{0} + 1)^{−1} − M_{ϕ}(L_{0} + 1)^{−1}M_{ψ} = M_{1−ϕ}(L_{0} + 1)^{−1}M_{ψ} + (L_{0} + 1)^{−1}M_{1−ψ}

is compact as operator inL^{2}([1, ∞), u^{−(n+1)}du). To this end consider the kernel g(u, u′) of (L_{0} + 1)^{−1}. It equals

(7.9)  g(u, u′) = \frac{(uu′)^{n/2}}{\sqrt{n^{2}/4 + 1}} \begin{cases}(u′/u)^{\sqrt{n^{2}/4+1}} − (uu′)^{−\sqrt{n^{2}/4+1}}, u > u′; \\ (u/u′)^{\sqrt{n^{2}/4+1}} − (uu′)^{−\sqrt{n^{2}/4+1}}, u′ > u. \end{cases}

From this formula follows that g(u, u′) is bounded on [1, 3] × [1, ∞) and [1, ∞) × [1, 3] and therefore, the kernels of the operators M_{1−ϕ}(L_{0} + 1)^{−1}M_{ψ} and (L_{0} + 1)^{−1}M_{1−ψ} are square integrable w.r.t. the measure u^{−(n+1)}du (u′)^{−(n+1)}du′. Hence (L_{0} + 1)^{−1} − M_{ϕ}(L_{0} + 1)^{−1}M_{ψ} is compact.

So in order to prove the lemma, it suffices to show that R_{h} − R_{g} is compact. We have

(7.10)  R_{h} − R_{g} = −R_{g}(Δ_{h} − Δ_{g})R_{h}.

By Lemma 5.2 we have

(7.11)  Δ_{h} − Δ_{g} = ∑_{j=0}^{2} ξ_{j} ∘ (∇^{h})^{j}

and ξ_{j} satisfies

(7.12)  |ξ_{j}(x)| ≤ Ce^{−εd(x,x_{0})}, x ∈ X.

Now R_{h}: L^{2}(X) → W^{2}(X) is continuous. Therefore by (7.11) and (7.12) it follows that

(Δ_{h} − Δ_{g})R_{h}: L^{2}(X) → L^{2}(X)

is a bounded operator. Using again that R_{g} − R_{0} is compact, it follows from (7.10) that it suffices to show that R_{0}(Δ_{h} − Δ_{g})R_{h} is a compact operator.

For a > 1 let

X_{a} = M ∪_{Y} ([1, a] × Y).

Denote by χ_{a} the characteristic function of X_{a} in X. We claim that R_{0}χ_{a} is a compact operator. Let χ_{[1,a]} be the characteristic function of the interval [1, a] in [1, ∞). Using (7.8) and (7.9), this follows in the same way as above. Let M_{(1−χ_{a})β} denote the multiplication operator by (1 − χ_{a})β. By (7.11) and (7.12), we get

\begin{aligned}∥ R_{0} & (1 − χ_{a})(Δ_{h} − Δ_{g})R_{h} ∥ \\ & ≤ C (∑_{j=0}^{2} ∥ (∇^{h})^{j}R_{h} ∥) · ∥ R_{0} ∥ · ∥ M_{(1−χ_{a})β} ∥ . \end{aligned} ; (7.13)

Let Z_{a} = [a, ∞) × Y. Then

∥ M_{(1−χ_{a})β} ∥≤ sup_{x∈Z_{a}} β(x) = sup_{x∈Z_{a}} e^{−εd(x,x_{0})}.

Now observe that there exists C_{1} > 0 such that for all (u, y) ∈ Z_{a} we have

d((u, y), x_{0}) ≥ d((u, y), (1, y)) − C = log u − C_{1}.

Hence together with (7.13) we get

∥ R_{0}(1 − χ_{a})(Δ_{h} − Δ_{g})R_{h} ∥≤ C_{2}a^{−ε}.

Thus R_{0}(Δ_{h} − Δ_{g})R_{h} can be approximated in the operator norm by compact operators and hence, is a compact operator. □

Next we construct self-adjoint projections P_{±} which satisfy the conditions of Theorem 7.2. Let

(7.14)  e(u, λ) := u^{n/2+iλ} − u^{n/2−iλ}, u ∈ [1, ∞), λ ∈ ℝ.

Then e(u, λ) satisfies

D_{0}e(u, λ) = (n^{2}/4 + λ^{2})e(u, λ), e(1, λ) = 0.

Thus e(u, λ) is the generalized eigenfunction for L_{0}. For φ ∈ C_{c}^{∞}(1, ∞)) set

φ̂(λ) := \frac{1}{2π} ∫_{1}^{∞} e(u, λ)φ(u) \frac{du}{u^{n+1}} .

The map φ ↦ φ̂ extends to an isometry

F: L^{2}([1, ∞), u^{−(n+1)}du) → L^{2}(ℝ^{+})

such that

F ∘ L_{0} ∘ F^{∗} = L̃_{0},

where L̃_{0} is the multiplication operator by (n^{2}/4 + λ^{2}). Let

U: L^{2}(ℝ^{+}) → L^{2}([n^{2}/4, ∞))

be defined by

(Uf)(λ) = \frac{f(\sqrt{λ − n^{2}/4})}{\sqrt{2}(λ − n^{2}/4)^{1/4}} .

Then U is an isometry such that U ∘ L̃_{0} ∘ U^{∗} = L̂_{0}, where L̂_{0} is the multiplication operator by λ. Thus U ∘ F provides the spectral resolution of L_{0} = Δ_{0,ac}. Let

J: L^{2}([n^{2}/4, ∞)) → L^{2}(ℝ)

denote the inclusion, let F: L^{2}(ℝ) → L^{2}(ℝ) be the Fourier transform, and let χ_{±} denote the characteristic function of [0, ∞) and (−∞, 0], respectively. Set

P̃_{±} := J^{∗}Fχ_{±}F^{∗}J.

Then P̃_{+} + P̃_{−} is the identity of L^{2}([n^{2}/4, ∞)). Let A = −id/du, regarded as self-adjoint operator in L^{2}(ℝ). Then

P̃_{±}e^{−itL̂_{0}} = J^{∗}Fχ_{±}e^{−itA}F^{∗}J.

Let f ∈ L^{2}(ℝ). Using the Fourier transformation, it follows that (e^{−itA}f)(u) = f(u − t). Thus we get

∥ χ_{±}e^{−itA}f ∥^{2}= ± ∫_{−t}^{±∞} |f(u)|^{2} du → 0

as t → ∓∞. Hence we get

(7.15)  s-lim_{t→±∞} e^{itL̂_{0}} P̃_{∓}e^{−itL̂_{0}} = 0.

Now put

P_{±} := F^{∗}U^{∗}P̃_{±}UF

on L^{2}([1, ∞), u^{−(n+1)}du) and set P_{±} := 0 on the orthogonal complement of L_{0}^{2}(Z)^{⊥} = L^{2}([1, ∞), u^{−(n+1)}du) in L^{2}(X). Then P_{±} are self-adjoint projections that satisfy

P_{+} + P_{−} = P_{ac}(Δ_{0}).

Furthermore we have

e^{itH_{0}}P_{±}e^{−itΔ_{0}}P_{ac}(Δ_{0}) = F^{∗}U^{∗}e^{itL̂_{0}} P̃_{±}e^{−itL̂_{0}}UF.

So it follows from (7.15) that

s-lim_{t→±∞} e^{itH_{0}}P_{∓}e^{−itH_{0}}P_{ac}(H_{0}) = 0.

Thus condition (1) of Theorem 7.2 is satisfied. Let I = {n^{2}/4}. and put

A_{I} := C_{c}^{∞}(ℝ − I).

Then it is clear that A_{I} is multiplicative generating for C_{∞}(ℝ − I). By Lemma 7.4, Δ_{0} has pure point spectrum in the subspace (Id −P_{ac}(Δ_{0}))L^{2}(X) consisting of eigenvalues of finite multiplicity with no finite points of accumulation. Let α ∈ A_{I}. Then (Id −P_{ac}(Δ_{0}))α(Δ_{0}) is a finite rank operator. This is condition (2) of Theorem 7.2. Condition (3) holds by Lemma 7.5. It remains to verify condition (4).

Given t > 0, let χ_{t} be the characteristic function of [e^{t}, ∞) × Y in X. Let δ > 0. We have

\begin{aligned}∥ (R_{h} & − R_{0})e^{itΔ_{0}}α(Δ_{0})P_{±} ∥ \\ & ≤∥ R_{h} − R_{0} ∥ · ∥ (1 − χ_{δt})e^{itΔ_{0}}α(Δ_{0})P_{±} ∥ \\ & + ∥ (R_{h} − R_{0})χ_{δt} ∥ · ∥ α(Δ_{0}) ∥ . \end{aligned} ; (7.16)

We will prove that for each α ∈ C_{c}^{∞}(ℝ − {n^{2}/4}) there exists δ > 0 such that the right hand side is an integrable function of t ∈ ℝ^{+}. To estimate the first term on the right hand side we need the following auxiliary result.

Lemma 7.6. Let a ∈ ℝ and let f ∈ C_{c}^{∞}(ℝ − {a}). Let ε > 0 such that f(λ^{2} + a) = 0 for |λ| < ε. Then for every m ∈ ℕ there exists C > 0 such that for t ∈ ℝ−{0} and |u| < ε|t|/2 one has

| ∫_{0}^{∞} e^{2iuλ+itλ^{2}}f(λ^{2} + a) dλ| ≤ C|t|^{−m}.

Proof: Let t ≠ 0 and set x = u/t. Then the left hand side of the inequality equals

\begin{aligned}| ∫_{0}^{∞} e^{it(λ+x)^{2}} & f(λ^{2} + a) dλ| \\ & = (2t)^{−m}| ∫_{0}^{∞} e^{it(λ+x)^{2}} (\frac{1}{λ + x} \frac{d}{dλ} − \frac{1}{(λ + x)^{2}} )^{m} f(λ^{2} + a) dλ|. \end{aligned}

Now assume that |u| < ε|t|/2. Then |x| < ε/2. On the other hand, we have f(λ^{2} + a) = 0 for |λ| < ε. Thus if f(λ^{2} + a) ≠ 0, then we have |λ + x| ≥ |λ| − |x| > ε/2. Hence the right hand side can be estimated by C|t|^{−m}. □

Let φ ∈ L^{2}([1, ∞), u^{−(n+1)}du) = P_{ac}(Δ_{0})(L^{2}(X)). Then

\begin{aligned}(e^{−itΔ_{0}} & α(Δ_{0})φ)(u) \\ & = \frac{1}{2π} ∫_{0}^{∞} e(u, n/2 − iλ)e^{−it(λ^{2}+n^{2}/4)}α(λ^{2} + n^{2}/4)(F φ)(λ) dλ. \end{aligned} ; (7.17)

Let v ∈ C_{c}^{∞}((1, ∞)). Put φ = P_{+}v and w = F^{∗}JUF v. Then w ∈ L^{1}(ℝ) and F φ = U^{∗}J^{∗}F(χ_{+}w). Using the definition of U, J and F, we get

(F φ)(λ) = \sqrt{2λ} ∫_{0}^{∞} e^{−is(λ^{2}+n^{2}/4)}w(s) ds.

Assume that t > 0. If we insert this expression into the right hand side of (7.17) and switch the order of integration, we obtain

\begin{matrix}( & e^{−itΔ_{0}}α(Δ_{0})P_{+}v)(u) \\ = \frac{1}{\sqrt{2}π} ∫_{0}^{∞} w(s) ∫_{0}^{∞} e(u, n/2 − iλ)e^{−i(t+s)(λ^{2}+n^{2}/4)}α(λ^{2} + n^{2}/4)\sqrt{λ} dλ ds. \end{matrix} ; (7.18)

Now there exists ε > 0 such that α(λ^{2}+n^{2}/4) = 0 for |λ| < ε. Assume that | log(u)| < εt/2. Using the definition (7.14) of e(u, λ) and Lemma 7.6, it follows that there exists C > 0 such that

\begin{aligned}& |e^{−itΔ_{0}} & α(Δ_{0})P_{+}v(u)|^{2} ≤ C ∥ w ∥^{2} u^{n}t^{−3} ≤ C ∥ v ∥^{2} u^{n}t^{−3}. \\ (7.19) \end{aligned}

Thus for every α ∈ C_{c}^{∞}(ℝ − {n^{2}/4}) there exist C > 0 and δ > 0 such that for t > δ^{−1} one has

∥ (1 − χ_{δt})e^{−itΔ_{0}}α(Δ_{0})P_{+} ∥≤ C t^{−3} ∫_{1}^{e^{δt}} \frac{du}{u} = Cδt^{−2}.

Similarly one can show that

∥ (1 − χ_{δt})e^{−itΔ_{0}}α(Δ_{0})P_{−} ∥≤ C t^{−3} ∫_{1}^{e^{δt}} \frac{du}{u} = Cδt^{−2}, t > δ^{−1}.

Hence for this choice of δ, the first term on the right hand side of (7.16) is an integrable function of t ∈ ℝ^{+}.

Now consider the second term on the right hand side of (7.16). We have

\begin{aligned}& ∥ (R_{h} − R_{0})χ_{δt} ∥≤∥ (R_{h} − R_{g})χ_{δt} ∥ + ∥ (R_{g} − R_{Δ_{0}})χ_{δt} ∥ . \\ (7.20) \end{aligned}

Let M_{χ_{δt}β} denote the multiplication operator by χ_{δt}β. By (7.10) - (7.12) we get

\begin{aligned}∥ (R_{h} − R_{g})χ_{δt} ∥ & ≤∥ R_{g} ∥ · ∥ χ_{δt}(Δ_{h} − Δ_{g})R_{h} ∥ \\ & ≤ C ∥ M_{χ_{δt}β} ∥ (∑_{j=0}^{2} ∥ (∇^{h})^{j}R_{h} ∥) ≤ C_{1}e^{−εδt}. \end{aligned} ; (7.21)

It remains to estimate the second term on the right of (7.20). Let ψ ∈ C^{∞}(ℝ) such that f(u) = 0, if u ≤ 2, and f(u) = 1, if u ≥ 3. Define f ∈ C^{∞}(Z) by f(u, y) = ψ(u) and extend f by zero to a smooth function on X. Then we have

R_{g} − R_{0} = (f − 1)R_{0} − R_{g}((Δ_{g} + Id)(fR_{0}) − Id).

Observe that

(Δ_{g} + Id)(fR_{0}) − Id = f − 1 + 2∇f · ∇R_{0} + Δf · R_{0}.

Moreover note that (f − 1)χ_{δt} = 0 if t ≫ 0. Thus

\begin{aligned}& (R_{g} − R_{0}) · χ_{δt} = (f − 1) · R_{0} · χ_{δt} − R_{g}(2∇f · ∇R_{0} · χ_{δt} + Δf · R_{0} · χ_{δt}) \\ (7.22) \end{aligned}

for t ≫ 0. It follows from (7.2) that R_{0} · χ_{δt} acts in L^{2}(Z) and preserves the decomposition (7.3). Moreover ∥ R_{0} · χ_{δt}|_{L_{0}^{2}(Z)} ∥=∥ χ_{δt} · R_{0}|_{L_{0}^{2}(Z)} ∥. Let φ ∈ L_{0}^{2}(Z). Then R_{0}φ ∈ L_{0}^{2}(Z) ∩ H^{2}(Z) and by (7.4) we obtain

\begin{aligned}& ∥ χ_{δt}R_{0}φ ∥≤ Ce^{−2δt} ∥ R_{0}φ ∥_{1}≤ Ce^{−2δt} ∥ φ ∥ . \\ (7.23) \end{aligned}

On the orthogonal complement L_{0}^{2}(Z)^{⊥}, the kernel of R_{0} is given by (7.9). Let h ∈ C_{c}^{∞}(Z). Then it follows from (7.9) that

\begin{aligned}& ∥ h · R_{0} · χ_{δt}|_{L_{0}^{2}(Z)^{⊥}} ∥≤ Ce^{−δt\sqrt{n^{2}/4+1}} ≤ Ce^{−δt}. \\ (7.24) \end{aligned}

Combining (7.23) and (7.24) we obtain

∥ h · R_{0} · χ_{δt} ∥≤ Ce^{−δt}.

Similar estimations hold for ∇R_{0}. This proves that the second term on the right hand side of (7.16) is an integrable function of t ∈ ℝ^{+}. This is condition (4) of Theorem 7.2. Summarizing we have proved the following theorem.

Theorem 7.7. Let (X, g) be a manifold with cusps and let Δ_{0} be defined by (7.2). Let ε > 0 and put β(u) = e^{−εu}, u ∈ ℝ. Let h be a complete metric on X such that h ∼_{β}^{2} g. Then we have

(1) The wave operators W_{±}(Δ_{h}, Δ_{0}; J) exist and are complete.

(2) Δ_{h} has no singularly continuous spectrum.

Corollary 7.8. Let g and h be as above. Then the wave operators W_{±}(Δ_{h}, Δ_{g}; J) exist and are complete.

This is a considerable improvement of the result that we get from Theorem 7.1 in this case.

Remark. Other cases of complete manifolds (M, g) with a sufficiently explicit structure at infinity can be treated in the same way. This includes, for example, manifolds with cylindrical ends and asymptotically Euclidean manifolds.

8. β-Equivalence and Analytic Continuations of the Resolvent

In this section we study the existence of an analytic continuation of the resolvent in weighted L^{2}-spaces. Provided that such a continuation exists, we are able to study the behavior of the absolutely continuous spectrum under perturbation in more detail. The method is a modification of the method used in [Mu2].

Definition 8.1. Let B be a Banach space, Ω ⊂ ℂ a domain and F : Ω ↦ B a meromorphic function. Let Σ be a Riemann surface and let π : Σ → ℂ be a ramified covering. A meromorphic continuation of F to Σ is a meromorphic function F̃ : Σ → B such that

a) There exists Ω̃ ⊆ Σ such that π: Ω̃ → Ω is biholomorphic.

b) F ∘ π = F̃ on Ω̃.

Definition 8.2. Let δ be a function of moderate decay and let p ∈ ℕ. By H_{δ^{−1}}^{−p} we denote the dual space of H_{δ}^{p}, with respect to the extension of the L^{2}-pairing.

Lemma 8.3. Let ζ(u) be a non-increasing continuous function on [1, ∞) with ζ(u) → 0 as u → ∞ and let δ be a weight function. Then the canonical inclusion j: L_{δζ^{−1}}^{2}(M) → H_{δ}^{−2}(M) is compact.

Proof: It is enough to prove, that the adjoint ȷ^{∗} : H_{δ^{−1}}^{2}(M) → L_{δ^{−1}ζ}^{2}(M) is compact. For k ∈ ℕ let

Ω_{k} = {x ∈ M | ζ(1 + d(x, x_{0})) ≥ 1/k}.

Then each Ω_{k} is a compact subset of M. Let P_{k} be the multiplication operator by the characteristic function of Ω_{k}. By Rellich’s lemma, ȷ^{∗}P_{k} is compact. For f ∈ H_{δ^{−1}}^{2}(M) we have

∫_{M−Ω_{k}} |f(x)|^{2}δ^{−1}(x)ζ(x) dx ≤ \frac{1}{k} ∥ f ∥_{H_{δ^{−1}}^{2}}^{2} .

Thus ȷ^{∗}P_{k} converges to ȷ^{∗} in the operator topology. Hence ȷ^{∗} is compact. □

Let δ, ρ be functions of moderate decay. Then L_{δ^{−1}}^{2}(M) ⊂ L^{2}(M) and H^{2}(M) ⊂ H_{ρ}^{2}(M). Thus for λ ∈ ℂ − [0, ∞), the resolvent (Δ − λ)^{−1} : L^{2}(M) → H^{2}(M) may be regarded as a bounded operator

(Δ − λ)^{−1} : L_{δ^{−1}}^{2}(M) → H_{ρ}^{2}(M).

Denote by L(L_{δ^{−1}}^{2}(M), H_{ρ}^{2}(M)) the Banach space of all bounded operators from L_{δ^{−1}}^{2}(M) into H_{ρ}^{2}(M), equipped with the strong operator norm.

Theorem 8.4. Let g, h be complete Riemannian metrics on M with bounded curvature of order 2. Let β, δ, ζ and ρ be functions of moderate decay on M such that

(8.1)  β^{2}(x) ≤ Cĩ_{g}^{4n}(x)ρ(x)δ(x)ζ(x), x ∈ M,

and g ∼_{β}^{2} h. Let Ω ⊂ ℂ − [0, ∞) be open. Assume that there is a Riemann surface Σ and a covering Σ → Ω such that the operator valued function

λ ∈ Ω ↦ (Δ_{g} − λ)^{−1} ∈ L(L_{δ^{−1}}^{2}(M, g), H_{ρ}^{2}(M, g))

admits an analytic continuation to a meromorphic function

λ ∈ Σ → R_{g}(λ) ∈ L(L_{δ^{−1}}^{2}(M, g), H_{ρ}^{2}(M, g))

with finite rank residues. Then

λ ∈ Ω ↦ (Δ_{h} − λ)^{−1} ∈ L(L_{δ^{−1}}^{2}(M, h), H_{ρ}^{2}(M, h))

also admits a meromorphic continuation to Σ with finite rank residues.

Proof: By assumption, βĩ^{−2n} is bounded. Hence by Corollary 5.3, H^{2}(M, g) and H^{2}(M, h) are equivalent and therefore, by duality, H^{−2}(M, g) and H^{−2}(M, h) are also equivalent. Let λ ∈ ℂ − [0, ∞). Then

K(λ) := (Δ_{g} − λ)^{−1}(Δ_{h} − Δ_{g})

is a bounded operator in L^{2}(M). Moreover Id +K(λ) = (Δ_{g} −λ)^{−1}(Δ_{h} −λ) has a bounded inverse in L^{2}(M) which is given by

(Id +K(λ))^{−1} = (Δ_{h} − λ)^{−1}(Δ_{g} − λ).

Thus for λ ∈ ℂ − [0, ∞) we have

(8.2)  (Δ_{h} − λ)^{−1} = (Id +K(λ))^{−1}(Δ_{g} − λ)^{−1}.

By Corollary 4.3 there exists λ ∈ ℂ−[0, ∞) such that (Δ_{h} −λ)^{−1} extends to a bounded operator in L_{ρ}^{2}(M). By Lemma 4.4 it follows that (Δ_{h}−λ)^{−1} maps L_{ρ}^{2}(M) into H_{ρ}^{2}(M). Moreover by definition Δ_{g} − λ is a bounded operator of H_{ρ}^{2}(M) to L_{ρ}^{2}(M). Hence (Id +K(λ))^{−1} extends to a bounded operator in H_{ρ}^{2}(M). Let μ ∈ Ω. Then

Id +K(μ) = (Id +K(λ)) − {K(λ) − K(μ)}

= (Id +K(λ)) − (λ − μ)(Δ_{g} − μ)^{−1}(Δ_{g} − λ)^{−1}(Δ_{h} − Δ_{g}). (8.3)

By Corollary 4.3 we may choose λ such that (Δ_{g} − λ)^{−1} extends to a bounded operator in L_{δζ}^{2}(M). By duality, and Lemma 4.4, it defines a bounded operator

(Δ_{g} − λ)^{−1} : L_{δ^{−1}ζ^{−1}}^{2}(M) → H_{δ^{−1}ζ^{−1}}^{2}(M).

Using Lemma 3.6, Lemma 5.2 and the assumption on β, it follows that the operator (Δ_{g} − λ)^{−1}(Δ_{h} − Δ_{g}) is the composition of the following chain of bounded operators

\begin{matrix}H_{ρ}^{2}(M) → W_{ĩ^{4n}ρ}^{2}(M) & ⟶^{Δ_{h}−Δ_{g}} L_{β^{−2}ĩ^{4n}ρ}^{2}(M) → \\ L_{δ^{−1}ζ^{−1}}^{2}(M) ⟶^{(Δ_{g}−λ)^{−1}} H_{δ^{−1}ζ^{−1}}^{2}(M) ⟶^{j} L_{δ^{−1}}^{2}(M). \end{matrix} ; (8.4)

By Lemma 8.3, the inclusion j is a compact. Hence

(Δ_{g} − λ)^{−1}(Δ_{h} − Δ_{g}): H_{ρ}^{2}(M) → L_{δ^{−1}}^{2}(M)

is compact operator. Set

(8.5)  H_{λ}(μ) = (λ − μ)R_{g}(μ) ∘ (Δ_{g} − λ)^{−1}(Δ_{h} − Δ_{g}), μ ∈ Σ.

Then H_{λ}(μ), μ ∈ Σ, is a meromorphic family of compact operators and

(8.6)  Id +K(μ) = (Id +K(λ)) {Id − (Id +K(λ))^{−1} H_{λ}(μ)} .

It then follows from [St], that (Id +K(μ))^{−1} exists except for on a discrete set and is meromorphic in μ. Thus, we may define

(8.7)  R_{h}(μ) = (Id +K(μ))^{−1} ∘ R_{g}(μ).

By (8.2) this is the desired meromorphic continuation of the resolvent (Δ_{h} − λ)^{−1}. □

Examples.

1) Let M be a surface with cusps. Here by a cusp we mean a half-cylinder [a, ∞) × S^{1}, a > 0, equipped with the Poincaré metric y^{−2}(dx^{2} + dy^{2}), and M is a surface with a complete metric g which in the complement of compact set is isometric to the disjoint union of finitely many cusps. Let c > 0 and let x_{0} ∈ M. Set

(8.8)  δ(x) := e^{−cd(x,x_{0})}, x ∈ M,

and ρ = ζ = δ. Then δ, ρ, and ζ are functions of moderate decay. Let

Ω = {s ∈ ℂ | Re(s) > 1/2, s ∉ (1/2, 1]}.

We consider the resolvent R_{g}(s) = (Δ_{g} − s(1 − s))^{−1} as a function of s ∈ Ω. Then it follows from [Mu2, Theorem 1] that R(s) admits an analytic continuation to a meromorphic function on ℂ with values in L(L_{δ^{−1}}^{2}(M), L_{δ}^{2}(M)). Using the same method, one can show that the R_{g}(s) takes values in H_{δ}^{2}(M). Now observe that the injectivity radius satisfies ı(x) ∼ e^{−d(x,x_{0})}. Let ϵ > 0 and set β(x) = e^{−(4+ϵ)d(x,x_{0})}. Choose the constant c > 0 in (8.8) such that c < ϵ/4. Then β is a function of moderate decay which satisfies (8.1) with respect to our choice of the functions δ, ρ, and ζ. Now note that the metric g has bounded curvature of all orders. Let h be complete metric on M with bounded curvature of order 2 which satisfies g ∼_{β}^{2} h. Then it follows from Theorem 8.4 that the resolvent R_{h}(s) = (Δ_{h} − s(1 − s))^{−1}, s ∈ Ω, also admits a meromorphic extension to ℂ with values in L(L_{δ^{−1}}^{2}(M), H_{δ}^{2}(M)). We think that the condition on β can be weakened.

2) Let M be a manifold with a cylindrical end. This means that M is a complete Riemannian manifold that admits a decomposition M = M_{0} ∪_{Y} (ℝ^{+} × Y) into a compact manifold M_{0} with boundary Y and a half-cylinder (ℝ^{+} ×Y) which is glued to M_{0} along the common boundary Y. The restriction of the metric g of M to the half-cylinder is assumed to be the product metric. Then g is a metric with bounded geometry, that is, g has bounded curvature of all orders and the injectivity radius has a positive lower bound. Let Δ_{Y} be the Laplacian of Y and let 0 = μ_{1} < μ_{2} ≤ μ_{3} ≤ ⋯ be the eigenvalues of Δ_{Y}. Let Σ → ℂ be the Riemann surface to which the square roots λ ↦ \sqrt{λ − μ_{j}}, j ∈ ℕ, extend holomorphically. Define δ, ρ, and ζ as in example 1. Then it follows as in [Mu2, Theorem 5] that the resolvent (Δ_{g} − λ)^{−1} extends from ℂ − [0, ∞) to a meromorphic function λ ∈ Σ ↦ R_{g}(λ) with values in L(L_{δ^{−1}}^{2}(M), H_{δ}^{2}(M)). Now let ϵ > 0, x_{0} ∈ M, and set

β(x) = e^{−ϵd(x,x_{0})}, x ∈ M.

Choose c in the definition of δ such that c < ϵ/2. Then β satisfies (8.1) with respect to our choice of the functions δ, ρ, and ζ. Let h be a complete metric on M with bounded curvature of order 2, and suppose that g ∼_{β}^{2} h. Then it follows from Theorem 8.4 that the resolvent (Δ_{h} − λ)^{−1} also admits a extension from ℂ − [0, ∞) to a meromorphic function λ ∈ Σ ↦ R_{g}(λ) with values in L(L_{δ^{−1}}^{2}(M), H_{δ}^{2}(M)).

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Universität Bonn, Mathematisches Institut, Beringstrasse 1, D – 53115 Bonn, Germany

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