L^{p}-SPECTRUM OF THE DIRAC OPERATOR ON PRODUCTS WITH HYPERBOLIC SPACES

BERND AMMANN AND NADINE GROSSE

Abstract. We study the L^{p}-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on p. As a first example where p-independence fails we compute explicitly the L^{p}-spectrum for the hyperbolic space and its product with compact spaces.

1. Introduction

The L^{p}-spectrum of the Laplacian and its p-(in)dependence was and still is studied by many authors, e.g. in [17], [18], [22]. On closed manifolds one easily sees that the spectrum is independent of p ∈ [1, ∞]. For open manifolds, independence only holds under additional geometric conditions. Hempel and Voigt [22], [23] proved such results for Schrödinger operators in ℝ^{n} with potentials admitting certain singularities. Then Kordyukov [27] generalized this result to uniformly elliptic operators with uniformly bounded smooth coefficients on a manifold of bounded geometry with subexponential volume growth. Independently, Sturm [34] showed the independence of the L^{p}-spectrum for a class of uniformly elliptic operators in divergence form on manifolds with uniformly subexponential volume growth and Ricci curvature bounded from below. Both results include the Laplacian acting on functions. Later the Hodge-Laplacian acting on k-forms was considered. E.g. under the assumptions of the result by Sturm from above, Charalambous proved the L^{p}-independence for the Hodge-Laplacian in [14, Proposition 9]. The machinery used to obtain these independence results uses estimates for the heat kernel as in [32]. In contrast, the L^{p}-spectrum of the Laplacian on the hyperbolic space does depend on p [16, Theorem 5.7.1]. Its L^{p}-spectrum is the convex hull of a parabola in the complex plane, and this spectrum degenerates only for p = 2 to a ray on the real axis, cf. Remark 10.1. In addition to the intrinsic interest of the p-independence of the L^{p}-spectrum, such results were used to get information on the L^{2}-spectrum by considering the L^{1}-spectrum, as in particular examples the L^{1}-spectrum can be easier to control. The result of Sturm was used for example by Wang [37, Theorem 3] to prove that the spectrum of the Laplacian acting on functions on complete manifolds with asymptotically non-negative Ricci curvature is [0, ∞). Some explicit calculations for the L^{p} spectrum of the Laplace-Beltrami operator on symmetric spaces were already carried by Lahoué and Rychener in the 1980’s [29], and Taylor [36] was able to determine the L^{p}-spectrum of this operator for all symmetric spaces of noncompact type. For locally symmetric spaces important progress was made recently by Ji and Weber, see e.g. [24, 25, 26, 39]. About the L^{p}-spectrum of the Dirac operator much less is known. As before, on closed manifolds the spectrum is independent on p ∈ [0, ∞]. Kordyukov’s methods [27] do not apply directly to the Dirac operator D, but following a remark of [27, Page 224] his methods generalize to suitable systems, and thus also to the square D^{2}. Unfortunately, the system case is not completely worked out, but it seems to us, that the case of systems is completely analogous to the case of operators on functions. Assuming this, Kordyukov has shown that the spectrum of D^{2} is p-independent for 1 ≤ p < ∞ on manifolds with bounded geometry and subexponential volume growth. For many such manifolds (e.g. for all such manifolds of even dimension or all manifolds of dimension 4k + 1), this already implies the p-independence of the L^{p}-spectrum of D, see our Lemma B.8 together with the following symmetry considerations. Many of the results and techniques that were constructed up for Laplace operators are not yet developed for Dirac operators. It is interesting to compare the properties of the Dirac and Laplace operator. Furthermore, the L^{p}-spectrum of the Dirac operator is important for non-linear variational problems based on the Dirac operator, e.g., for (classical) Dirac operators certain L^{p}-spaces and L^{p}-spectral gaps naturally occur when considering a spinorial Yamabe-type problem which was our motivation to enter into this subject, see [5] and below.

Date: September 16, 2019.

2010 Mathematics Subject Classification. 58J50, 34B27.

Key words and phrases. Dirac operator, L^{p}-spectrum, Green function, hyperbolic space, product spaces.

In this paper we determine explicitly the L^{p}-spectrum for a special class of complete manifolds – products of compact spaces with hyperbolic spaces. More precisely, we study the following manifolds: Let (N^{n}, g_{N}) be a closed Riemannian spin manifold. Let M = 𝕄_{c} be the product manifold (𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N^{n}, g_{M} = g_{ℍ_{c}^{k+1}} + g_{N}) where ℍ_{c}^{k+1} is the (k + 1)-dimensional hyperbolic space scaled such that its scalar curvature is −c^{2}k(k + 1) for c ≠ 0 and ℍ_{0}^{k+1} is the (k + 1)- dimensional Euclidean space. For those manifolds we obtain the following result which is also illustrated in Figure 1:

Theorem 1.1. We use the notions from above. Let p ∈ [1, ∞], c ≥ 0, and k ∈ ℕ ∪ {0}. The L^{p}-spectrum of the Dirac operator on 𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N^{n} is given by the set

σ_{p} := \{μ ∈ ℂ | μ^{2} = λ_{0}^{2} + κ^{2}, |Im κ| ≤ ck |\frac{1}{p} − \frac{1}{2}|\}

where λ_{0}^{2} is the lowest eigenvalue of (D^{N})^{2}, λ_{0} ≥ 0, and D^{N} is the Dirac operator on (N, g_{N}). In particular, the Dirac operator D: H_{1}^{p} → L^{p} on 𝕄_{c}^{m,k} has a bounded inverse if and only if λ_{0} > ck |\frac{1}{p} − \frac{1}{2} |.

Note that we also allow the case n = 0, i.e. the case that N is a finite set. In this case we define λ_{0} := 0. For an overview of the structure of the proof, see the end of the introduction. From the Theorem 1.1 one can directly read off the L^{p}-spectrum of D^{2} and compare it to the known spectrum of the Laplacian acting on functions which is done in Remark 10.1. As mentioned above our interest in the L^{p}-spectrum for the Dirac operator on such spaces comes from an application in conformal spin geometry which we want to sketch very briefly: In the article [3] the first named author studied the behavior of a spinorial analogue of the Yamabe invariant under surgeries. In these Gromov-Lawson type constructions an important role was played by spinorial Yamabe-type invariants on the spaces ℍ_{c}^{k+1} × 𝕊^{n} where c is in [0, 1] and where 𝕊^{n} is the standard sphere. When these invariants were studied in [21], the important role of the L^{p}-spectrum of the Dirac operator on such spaces became apparent. The results of the present article are preliminaries for some theorems about spinorial Yamabe-type invariants in [5]. The paper is structured as follows: Notations and preliminaries are collected in Section 2. Results on the Green function of the Dirac operator acting on L^{2}-spinors can be found in Section 3. General remarks and results for the Dirac operator acting on L^{p}-sections are given in Appendix B. In Section 4, the Dirac operator on the model spaces 𝕄_{c}^{m,k} is written in polar coordinates and the action of Spin(k +1) on 𝕄_{c}^{m,k} is studied. This is used in Section 5 to prove a certain symmetry property of the Green function on 𝕄_{c}^{m,k} and in Section 6 to study its decay. After these preparations we are ready to prove the main theorem (Theorem 1.1). In Section 7 we decompose the Green function into a singular part and the integral kernel of a smoothing

Re μ ; Im μ ; x_{L} ; Re μ ; Im μ ; Re μ ; Im μ ; x_{M} ; x_{R}

Figure 1. The shaded region (including the boundary) illustrates the L^{p}- spectrum of the Dirac operator on 𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N^{n}, k ≥ 1, cf. Theorem 1.1. Left: λ_{0} = 0 (x_{L} = ck |\frac{1}{p} − \frac{1}{2} |).

Middle: 0 < λ_{0} < ck |\frac{1}{p} − \frac{1}{2} | (x_{M} = (c^{2}k^{2} (\frac{1}{p} − \frac{1}{2})^{2} − λ_{0}^{2})^{\frac{1}{2}} ).

Right: λ_{0} > ck |\frac{1}{p} − \frac{1}{2} | (x_{R} = (λ_{0}^{2} − c^{2}k^{2} (\frac{1}{p} − \frac{1}{2})^{2})^{\frac{1}{2}} ).

operator. Using the homogeneity of the hyperbolic space we show in Proposition 7.1 that the singular part gives rise to a bounded operator from L^{p} to itself for all p ∈ [1, ∞]. In Proposition 7.2 we show that under certain assumptions on the decay of the Green function also the smoothing part gives rise to a bounded operator from L^{p} to L^{p} for certain p. Using the decay estimate obtained in Section 6 we then see in Section 8 that the L^{p}-spectrum of 𝕄_{c}^{m,k} is contained in the set σ_{p} given in Theorem 1.1. Thus, it only remains to show that each element of σ_{p} is already in the L^{p}-spectrum of 𝕄_{c}^{m,k}. For that we construct test spinors on ℍ_{c}^{k+1} in Section 9 and finish the proof for product spaces in Section 10.

Acknowledgements. We thank an anonymous referee for many helpful comments.

2. Preliminaries

2.1. Notations and conventions. In the article we will use the convention that a spin manifold is a manifold which admits a spin structure together with a fixed choice of spin structure. Let (M, g) be a spin manifold and Σ_{M} the corresponding spinor bundle, see Section 2.3.

Γ(Σ_{M}) denotes the space of spinors, i.e., sections of Σ_{M}. The space of smooth compactly supported sections is denoted by C_{c}^{∞}(M, Σ_{M}), or shortly C_{c}^{∞}(Σ_{M}). The hermitian metric on fibers of Σ_{M} is denoted by ⟨., .⟩, the corresponding norm by |.|. For s_{1}, s_{2} ∈ Γ(M, Σ_{M}) we define the L^{2}-scalar product

(s_{1}, s_{2})_{L^{2}(g)} := ∫_{M}⟨s_{1}, s_{2}⟩ dvol_{g}.

For s ∈ [1, ∞] ∥.∥_{L^{s}(g)} is the L^{s}-norm on (M^{n}, g). In case the underlying metric is clear from the context we abbreviate shortly by ∥.∥_{s}. Spec_{L^{s}}^{M}(D) denotes the spectrum of the Dirac operator on M viewed as an operator from L^{s} to L^{s}, cf. Appendix B. We denote by π_{i}: M × M → M, i = 1, 2, the projection to the i-th component. Moreover, we set Σ_{M} ⊠ Σ_{M}^{∗} := π_{1}^{∗}(Σ_{M}) ⊗ π_{2}^{∗}(Σ_{M}^{∗}). C^{i}(M) denotes the space of i-times continuously differentiable functions on M. B_{ε}(x) ⊂ M is the ball around x ∈ M of radius ε w.r.t. the metric given on M.

A Riemannian manifold is of bounded geometry, if its injectivity radius is positive and the curvature tensor and all derivatives are bounded. The metric on the k-dimensional sphere 𝕊^{k} with constant sectional curvature 1 will be denoted by σ^{k}. For 𝕊^{k} with metric r^{2}σ^{k} we write 𝕊_{r}^{k}.

2.2. Coordinates and notations for ℍ_{c}^{k+1} and its product spaces. We use polar coordinates on ℍ_{c}^{k+1}. Namely, we fix a point p_{0} ∈ ℍ_{c}^{k+1} and an identification of T_{p_{0}}ℍ_{c}^{k+1} with ℝ^{k+1}. Often we also use the exponential map to identify T_{p_{0}}ℍ_{c}^{k+1} ≅ ℝ^{k+1} with ℍ_{c}^{k+1}. In polar coordinates the metric is then g_{ℍ_{c}^{k+1}} = dr^{2} + f(r)^{2}σ^{k} where σ^{k} is the standard metric on 𝕊^{k} and

f(r) := sinh_{c}(r) := \begin{cases}\frac{1}{c} sinh(cr) & if c ≠ 0 \\ r & if c = 0. \end{cases}

In particular, the distance r = dist_{ℍ_{c}^{k+1}}(y, p_{0}) of y to p_{0} ≅ 0 w.r.t. g_{ℍ_{c}^{k+1}} coincides with the euclidean one on ℝ^{k+1}. The subset {y ∈ ℍ_{c}^{k+1} | dist_{ℍ_{c}^{k+1}}(y, 0) = r} is isometric to 𝕊_{f(r)}^{k} and its (unnormalized) mean curvature is given by

H⃗_{𝕊_{f(r)}^{k}} = −k \frac{∂_{r}f(r)}{f(r)}∂_{r} = −k coth_{c}(r)∂_{r} where coth_{c} r := \begin{cases}c coth(cr) & if c ≠ 0 \\ \frac{1}{r} & if c = 0. \end{cases}

Let N be a closed Riemannian spin manifold. Note that we include the case where N is just a point. Set 𝕄_{c}^{m,k} := ℍ_{c}^{k+1} × N, and π_{ℍ} shall denote the projection of 𝕄_{c}^{m,k} onto its ℍ_{c}^{k+1}-coordinates.

2.3. General preliminaries about spin geometry. The following can e.g. be found in [19]. A spin structure on M^{m} is a pair (P_{Spin}(M), α) where P_{Spin}(M) is a principal Spin(m)-bundle and where α: P_{Spin}(M) → P_{SO}(M) is a fiber map over the identity of M that is compatible with the double covering Θ: Spin(m) → SO(m) and the corresponding group actions, i.e., the following diagram commutes

\xymatrix{Spin(m) × P_{Spin}(M) \ar[dd]^{Θ×α} \ar[r] & P_{Spin}(M) \ar[dd]^{α} \ar[dr] \\  &  & M \\ SO(m) × P_{SO}(M) \ar[r] & P_{SO}(M) \ar@{-->}[ur]}

Let Σ_{m} be an irreducible representation of Cl_{m}. In case m is odd there are two such irreducible representations. Both of them coincide if considered as Spin(m)-representations. For m odd we fix one of the two representations for the whole article. If m is even, there is only one irreducible Cl_{m}-representation of Σ_{m}, but it splits into non-equivalent subrepresentations Σ_{m}^{(+)} and Σ_{m}^{(−)} as Spin(m)-representations. Define ω_{M} = i^{[ \frac{m+1}{2} ]}e_{1} · e_{2} · . . . · e_{m} with (e_{i})_{i} being a positively oriented orthonormal frame on M. We have ω_{M}^{2} = 1. If m is odd, then ω_{M} commutes with the Clifford multiplication and thus acts as +1 or −1 on Σ_{m}. If m is even, the it acts as ±1 on Σ_{m}^{(±)}. Sections of Σ_{m}^{(+)} (resp. Σ_{m}^{(−)}) are called positive (resp. negative) spinors. In the article it will be convenient to use the notation Σ_{m}^{(ε)}, ε ∈ {0+, −} also in the case m odd as well, by setting Σ_{m}^{(ε)} = Σ_{m} for both choices of ε. The spinor bundle Σ_{M} is defined as Σ_{M} = P_{Spin}(M) ×_{ρ_{m}} Σ_{m} where ρ_{m}: Spin(m) → End(Σ_{m}) is the complex spinor representation. Moreover, the spinor bundle is endowed with a Clifford multiplication, denoted by ’·’, ·: T M → End(Σ_{M}). Then, the Dirac operator acting on the space of smooth sections of Σ_{M} is defined as the composition of the connection ∇ on Σ_{M} (obtained as a lift of the Levi-Civita connection on T M) and the Clifford multiplication. Thus, in local coordinates this reads as

D = ∑_{i=1}^{m} e_{i} · ∇_{e_{i}}

where (e_{i})_{i=1,...,m} is a local orthonormal basis of T M. The Dirac operator is formally selfadjoint as an operator on L^{2}, i.e., for ψ ∈ C^{∞}(M, Σ_{M}) and φ ∈ C_{c}^{∞}(M, Σ_{M}) we have (φ, Dψ) = (Dφ, ψ). As M is complete, the Dirac operator is not only formally self-adjoint, but actually has a unique self-adjoint extension that is a densely defined unbounded operator D: H_{1}^{2}(Σ_{M}) ⊂ L^{2}(Σ_{M}) → L^{2}(Σ_{M}), see [41]. Here H_{1}^{2}(Σ_{M}) is the subset of all φ ∈ L^{2}(Σ_{M}) such that ∇φ, defined in the distributional sense, is again in L^{2}. The notation above indicates that H_{1}^{2}(Σ_{M}) is the domain of the operator, and that “densely defined” and “self-adjoint” should be understood as an unbounded operator from L^{2}(Σ_{M}) to L^{2}(Σ_{M}). From the spectral theorem it follows that D − μ: H_{1}^{2}(Σ_{M}) ⊂ L^{2}(Σ_{M}) → L^{2}(Σ_{M}) is invertible for all μ ∉ ℝ, in the sense that the inverse is a bounded operator from L^{2}(Σ_{M}) to L^{2}(Σ_{M}). See also Appendices A and B for more details.

2.4. Dual spinors. The hermitian metric induces a natural isomorphism from Σ_{M}^{∗} to Σ̄_{M}. In this way we obtain a metric connection and a Clifford multiplication on Σ_{M}^{∗} and this allows us to define a Dirac operator D^{t} : C^{∞}(Σ_{M}^{∗}) → C^{∞}(Σ_{M}^{∗}). Locally D^{t}f = ∑_{i} e_{i} · ∇_{e_{i}}f where f ∈ C^{∞}(Σ_{M}^{∗}) and e_{i} is a local orthonormal frame on M. Completely analogously to the proof that the usual Dirac operator is formally self-adjoint, one proves that for f ∈ C^{∞}(Σ_{M}^{∗}), φ ∈ C^{∞}(Σ_{M}) such that supp f ∩ supp φ is relatively compact we have

∫ D^{t}f(φ)dvol_{g} = ∫ f(Dφ)dvol_{g}.

2.5. Spinors on product manifolds. In this subsection our notation is close to [8]. Let (P^{m+n} = M^{m} × N^{n}, g_{P} = g_{M} + g_{N}) be a product of Riemannian spin manifolds (M, g_{M}) and (N, g_{N}). We have

P_{Spin}(M × N) = (P_{Spin}(M) × P_{Spin}(N)) ×_{ξ} Σ_{m+n}

where ξ: Spin(m) × Spin(n) → Spin(m + n) is the Lie group homomorphism lifting the standard embedding SO(m) × SO(n) → SO(m + n). Note that ξ is not an embedding, its kernel is (−1, −1), where −1 denotes the non-trivial element in the kernel of Spin(m) → SO(m) resp. Spin(n) → SO(n). The spinor bundle can be identified with

Σ_{P} = \begin{cases}Σ_{M} ⊗ (Σ_{N} ⊕ Σ_{N}) & if both m and n are odd \\ Σ_{M} ⊗ Σ_{N} & else, \end{cases}

and the Levi-Civita connection acts as ∇^{Σ_{M} ⊗Σ_{N}} = ∇^{Σ_{M}} ⊗ Id_{Σ_{N}} + Id_{Σ_{M}} ⊗∇^{Σ_{N}}. This identification can be chosen such that for X ∈ T M, Y ∈ T N, φ ∈ Γ(Σ_{M}), and ψ = (ψ_{1}, ψ_{2}) ∈ Σ_{N} ⊕ Σ_{N} for both n and m odd and ψ ∈ Γ(Σ_{N}) otherwise, we have

(X, Y) ·_{P} (φ ⊗ ψ) = (X ·_{M} φ) ⊗ (ω_{N} ·_{N} ψ) + φ ⊗ (Y ·_{N} ψ)

where for both n and m odd we set ω_{N} ·_{N} (ψ_{1}, ψ_{2}) := i(ψ_{2}, −ψ_{1}) and Y ·_{N} (ψ_{1}, ψ_{2}) := (Y ·_{N} ψ_{2}, Y ·_{N} ψ_{1}). The Dirac operator is then given by

D^{P} (φ ⊗ ψ) = (D^{M}φ ⊗ ω_{N} ·_{N} ψ) + (φ ⊗ D̃^{N}ψ)

where D̃^{N} = diag(D^{N}, −D^{N}) if both m and n are odd and D̃^{N} = D^{N} otherwise. Since ω_{N}· and D̃^{N} anticommute, D^{M} ⊗ ω_{N} and id ⊗D̃^{N} anticommute as well. Thus

(D^{P})^{2} = (D^{M})^{2} ⊗ id + id ⊗(D̃^{N})^{2}.  (1)

If we additionally assume that M and N are compact, then this equation implies together with the spectral theorem that the L^{2}-spectrum of the square of the Dirac operator on the product space M × N is given by

{λ^{2} + μ^{2} | λ^{2} ∈ Spec_{L^{2}}^{M}((D^{M})^{2}), μ^{2} ∈ Spec_{L^{2}}^{N}((D^{N})^{2})}.

2.6. The L^{2}-spectrum of the Dirac operator. The simplest case of Theorem 1.1 is when p = 2 and when N is a point. In this case we have by definition λ_{0} = 0. The theorem then states that the L^{2}-spectrum of the Dirac operator on ℍ_{c}^{k+1} is the whole real line ℝ ⊂ ℂ. This fact was already known before, by scaling it suffices to consider the Euclidean case c = 0 and the hyperbolic case c = 1. On Euclidean space ℝ^{k+1} = ℍ_{0}^{k+1} it is an easy calculation. The spinor bundle is trivialized by parallel orthonormal sections s_{1}, …, s_{t} with t the rank of the spinor bundle. The Dirac operator preserves V_{i} := C^{∞}(ℝ^{k+1}) · s_{i}, anticommutes with the Clifford action by parallel vector fields, and its square maps f · s_{i} to (Δf) · s_{i}. As Δ has L^{2} spectrum [0, ∞) on ℝ^{k+1}, the statement follows. On hyperbolic space, the L^{2}-spectrum of the Dirac operator was calculated by representation theoretical methods in [11]. Unfortunately [11] contains a small error, and the value 0 is not an eigenvalue of the Dirac operator in contrast to what is claimed in [11]. The argument can be repaired, and it follows that the spectrum is as claimed. A complete proof was given in [13], using similar techniques as in the present article. We will not give a proof of this special case of Theorem 1.1, instead we will use this special case for the proof of the full version of Theorem 1.1. Using the results of the previous subsection, we also get the L^{2}-spectrum of the Dirac operator on 𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N: With Lemma B.8 we get

Spec_{L^{2}}^{𝕄_{c}}(D^{2}) = [λ_{0}^{2}, ∞).

Together with Lemma B.11, Example B.12 and Lemma B.8 this yields

Spec_{L^{2}}^{𝕄_{c}}(D) = (−∞, −λ_{0}] ∪ [λ_{0}, ∞).

2.7. A covering lemma.

Lemma 2.1 (Covering lemma). Let (M, g) be a Riemannian manifold of bounded geometry, and let R > 0. Then there are points (x_{i})_{i∈I} ⊂ M where I is a countable index set such that

(i) the balls B_{R}(x_{i}) are pairwise disjoint and (ii) (B_{2R}(x_{i}))_{i∈I} and (B_{3R}(x_{i}))_{i∈I} are both uniformly locally finite covers of M.

Proof. Choose a maximal family of points (x_{i})_{i∈I} in M such that the sets B_{R}(x_{i}) are pairwise disjoint. Then ⋃_{i∈I} B_{2R}(x_{i}) = M. For y ∈ M let L(y) = {i ∈ I | y ∈ B_{3R}(x_{i})}. For i ∈ L(y) we have B_{R}(x_{i}) ⊂ B_{4R}(y) and, thus,

⨆_{i∈L(y)} B_{R}(x_{i}) ⊂ B_{4R}(y),

where ⊔ denotes disjoint union. Comparing the volumes of both sides and using the bounded geometry of M we see that there exists a number L_{R} such that |L(y)| ≤ L_{R} for all y ∈ M. Thus, the covering by sets B_{3R}(x_{i}), and hence the one by B_{2R}(x_{i}), is uniformly locally finite. □

2.8. Interpolation theorems.

Theorem 2.2 (Riesz-Thorin Interpolation Theorem, [40, Theorem II.4.2]). Let T be an operator defined on a domain D that is dense in both L^{q} and L^{p}. Assume that T f ∈ L^{q} ∩L^{p} for all f ∈ D and that T is bounded in both norms. Then, for any r between p and q the operator T is a bounded operator from L^{r} to L^{r}.

Theorem 2.3 (Stein Interpolation Theorem, [16, Section 1.1.6], [30, Theorem IX.21]). Let p_{0}, q_{0}, p_{1}, q_{1} ∈ [1, ∞], 0 < t < 1, and S = {z ∈ ℂ | 0 ≤ Re z ≤ 1}. Let A_{z} be linear operators from L^{p_{0}} ∩ L^{p_{1}} to L^{q_{0}} + L^{q_{1}} for all z ∈ S with the following properties

(i) z ↦ ⟨A_{z}f, g⟩ is uniformly bounded and continuous on S and analytic in the interior of S whenever f ∈ L^{p_{0}} ∩ L^{p_{1}} and g ∈ L^{r_{0}} ∩ L^{r_{1}} where r_{i} is the conjugate exponent to q_{i}.

(ii) There is M_{0} > 0 such that ∥A_{iy}f∥_{q_{0}} ≤ M_{0}∥f∥_{p_{0}} for all f ∈ L^{p_{0}} ∩ L^{p_{1}} and y ∈ ℝ. (iii) There is M_{1} > 0 such that ∥A_{1+iy}f∥_{q_{1}} ≤ M_{1}∥f∥_{p_{1}} for all f ∈ L^{p_{0}} ∩ L^{p_{1}} and y ∈ ℝ.

Then, for 1/p = t/p_{1} + (1 − t)/p_{0} and 1/q = t/q_{1} + (1 − t)/q_{0}

∥A_{t}f∥_{q} ≤ M_{1}^{t}M_{0}^{1−t}∥f∥_{p}

for all f ∈ L^{p_{0}} ∩ L^{p_{1}}. Hence, A_{t} can be extended to a bounded operator from L^{p} to L^{q} with norm at most M_{1}^{t}M_{0}^{1−t}.

3. The Green function

In this section, we collect results on existence and properties of the Green function of the Dirac operator D and its shifts D − μ, μ ∈ ℂ. They are obvious applications of standard methods, but a suitable reference does not exist yet. Unless otherwise stated we only assume in this section that the Riemannian spin manifold (M, g) is complete. Let Δ := {(x, x) | x ∈ M} ⊂ M × M be the diagonal.

Definition 3.1. [6, Definition 2.1] A smooth section G_{D−μ}: M × M \ Δ → Σ_{M} ⊠ Σ_{M}^{∗} that is locally integrable on M ×M is called a Green function of the shifted Dirac operator D −μ if

(D_{x} − μ)(G_{D−μ}(x, y)) = δ_{y} Id_{Σ_{M} |_{y}}  (2)

in the sense of distributions, i.e., for any y ∈ M, ψ_{0} ∈ Σ_{M}|_{y}, and φ ∈ C_{c}^{∞}(Σ_{M})

∫_{M}⟨G_{D−μ}(x, y)ψ_{0}, (D − μ̄)φ(x)⟩dx = ⟨ψ_{0}, φ(y)⟩

and G_{D−μ}(., y) ∈ L^{2}(M \ B_{r}(y)) for any r > 0.

In case that the operator D − μ is clear from the context, we shortly write G = G_{D−μ}.

Proposition 3.2. If M is a closed Riemannian spin manifold with invertible operator D − μ: H_{1}^{2}(Σ_{M}) ⊂ L^{2}(Σ_{M}) → L^{2}(Σ_{M}), then a unique Green function exists. This Green function is the integral kernel of the inverse of D − μ: H_{1}^{2}(Σ_{M}) ⊂ L^{2}(Σ_{M}) → L^{2}(Σ_{M}).

To prove the well-known proposition, one usually starts by showing the existence of a parametrix.

Lemma 3.3. [28, III.§4] Let M be a closed Riemannian spin manifold. Then there is a smooth section P_{D−μ}: M × M \ Δ → Σ_{M} ⊠ Σ_{M}^{∗}, called parametrix, which is L^{1} on M × M and which satisfies

(D_{x} − μ)(P_{D−μ}(x, y)) = δ_{y}Id_{Σ_{M} |_{y}} + R(x, y)

in the distributional sense for a smooth section R: M × M → Σ_{M} ⊠ Σ_{M}^{∗}.

Let ε > 0 be smaller than the injectivity radius. We choose χ: [0, ∞) → [0, 1] smooth with support in [0, ε] and constant to 1 in a neighborhood of 0. The statement of the lemma is still true is we replace P_{D−μ}(x, y) by χ(dist(x, y))P_{D−μ}(x, y) and an appropriately modified R. We thus can assume without loss of generality that the support P_{D−μ}(x, y) and thus the support of R is in (M × M)_{ε} := {(x, y) ∈ M × M | dist(x, y) < ε}. Convolution with P_{D−μ} defines an operator P_{D−μ} by

(P_{D−μ}ψ, φ) = ∫_{M} ∫_{M}⟨P_{D−μ}(x, y)ψ(y), φ(x)⟩ dx dy

for all ψ, φ ∈ C_{c}^{∞}(Σ_{M}). Then, P_{D−μ} is a right inverse to D − μ up to infinitely smoothing operators. We thus call it a right parametrix. The existence of such a right parametrix follows using the symbol calculus from the fact that D is an elliptic operator. An efficient and very readable overview over how to construct a right parametrix for an elliptic differential operator on a compact manifold can be found e.g. in [28, III.§4], although the reader should pay attention to the fact that it is not so obvious that the different notions of infinitely smoothing operators used in there are in fact all equivalent. The latter fact follows from standard techniques used in the theory of pseudo differential operators, see e.g. [1] or [35] for textbooks on this subject.

Proof of Proposition 3.2. From the last Lemma we have the existence of a parametrix P_{D−μ}(x, y). We will use the notations of that Lemma. Since D − μ is assumed to be invertible, there is a section P′_{D−μ}: M × M → Σ_{M} ⊠ Σ_{M}^{∗} with (D_{x} − μ)P′_{D−μ}(x, y) = R(x, y). By elliptic regularity P′_{D−μ}(x, y) is smooth in x for each y ∈ M. Again by elliptic regularity (D − μ)^{−1} is a continuous linear operator C_{c}^{∞}(Σ_{M}) → C^{ℓ}(Σ_{M}) for any ℓ ∈ ℕ. Further P′_{D−μ}( · , y) is the composition

Σ_{M} ⟶ C_{c}^{∞}(Σ_{M}) ⟶^{(D−μ)^{−1}} C^{ℓ}(Σ_{M})

where the first map is the smooth map that maps φ ∈ Σ_{M}|_{y} to R( · , y)φ. This implies that P′_{D−μ}(x, y) is smooth as function in both x and y. We set G_{D−μ}(x, y) = P_{D−μ}(x, y) − P′_{D−μ}(x, y) and obtain (D_{x} − μ)(G_{D−μ}(x, y)) = δ_{y} Id_{Σ_{M} |_{y}}. Moreover, since P_{D−μ} is L^{1} on M × M and P′_{D−μ} is smooth in both entries the Green function G_{D−μ} is L^{1} as well. Furthermore, P_{D−μ}(., y) is smooth on M \ B_{r}(y)) for any r > 0 and, hence, the same is true for G_{D−μ}(., y). In particular, G_{D−μ}(., y) ∈ L^{2}(M \ B_{r}(y)). If G̃_{D−μ} is a possibly different Green function of D − μ then (D − μ)(G_{D−μ}(., y) − G̃_{D−μ}(., y)) = 0 for all y ∈ M. As D − μ is invertible we have G_{D−μ} = G̃_{D−μ}. □

As for P_{D−μ}, convolution with G_{D−μ} defines an operator G_{D−μ} by

(G_{D−μ}ψ, φ) = ∫_{M} ∫_{M}⟨G_{D−μ}(x, y)ψ(y), φ(x)⟩dxdy

for all ψ, φ ∈ C_{c}^{∞}(Σ_{M}). By construction G_{D−μ} is the right inverse of D − μ, and is thus even defined on L^{2}. Since the inverse of D − μ exists by assumption, G_{D−μ} = (D − μ)^{−1}, and G_{D−μ} is in particular also a left inverse of D − μ.

Lemma 3.4. Let M be a closed Riemannian spin manifold, and let D − μ be invertible. Then G_{D−μ}(x, y) is the adjoint of G_{D−μ̄}(y, x), i.e. G_{D−μ̄}(y, x) is the integral kernel of the adjoint operator of G_{D−μ}.

Proof. Using the definitions and discussions from above and Lemma B.3(ii) we have G_{D−μ}^{∗} = ((D − μ)^{−1})^{∗} = (D − μ̄)^{−1} = G_{D−μ̄}. In particular, we get for all ψ, φ ∈ L^{2}(Σ_{M}) that

(ψ, G_{D−μ}^{∗}φ) =(G_{D−μ}ψ, φ) = ((D − μ)^{−1}ψ, φ) = (ψ, (D − μ̄)^{−1}φ)
= ∫ ∫ ⟨ψ(y), G_{D−μ̄}(y, x)φ(x)⟩dydx.
□

Moreover, we have

Lemma 3.5. In the situation of Lemma 3.4 we have (D_{y}^{t} − μ)G_{D−μ}(x, y) = δ_{x} Id_{Σ_{M}^{∗} |_{x}}, i.e., for f_{0} ∈ Γ(Σ_{M}^{∗}|_{x}), φ ∈ C_{c}^{∞}(Σ_{M})

∫ ((D_{y}^{t} − μ)G_{D−μ}(x, y)f_{0})(φ(y))dy = f_{0}(φ(x)).

Proof.

∫ ((D_{y}^{t} − μ)G_{D−μ}(x, y)f_{0})(φ(y))dy = ∫ (G_{D−μ}(x, y)f_{0})((D_{y} − μ)φ(y))dy
= ∫ f_{0}(G_{D−μ}(x, y)(D_{y} − μ)φ(y))dy
=f_{0}(φ(x)).

where the last step uses that G_{D−μ} is also the left inverse of D − μ.

□

Now, M has no longer to be closed, but we assume bounded geometry.

Proposition 3.6. Let (M, g) be a Riemannian spin manifold of bounded geometry. Let D − μ: H_{1}^{2}(Σ_{M}) ⊂ L^{2}(Σ_{M}) → L^{2}(Σ_{M}) be invertible. Then there exists a unique Green function.

Proof. We choose R > 0 such that 3R is smaller than the injectivity radius. Let (x_{i})_{i∈I} be as in the Covering Lemma 2.1. Recall that (M × M)_{ε} := {(x, y) ∈ M × M | dist(x, y) < ε}. Because of M = ⋃_{i∈I} B_{2R}(x_{i}) we have

(M × M)_{R} ⊂ ⋃_{i∈I} B_{3R}(x_{i}) × B_{3R}(x_{i}).

We embed each ball B_{3R}(x_{i}) isometrically into a closed connected manifold M_{x_{i}}, which is diffeomorphic to a sphere and D^{M_{x_{i}}} − μ is invertible. This can always be achieved by local metric deformation on M_{x_{i}} \ B_{3R}(x_{i}), see Proposition C.1. Thus, by Proposition 3.2 the operator D^{M_{x_{i}}} − μ possesses a Green function G^{x_{i}}(x, y) with (D_{x}^{M_{x_{i}}} − μ)G^{x_{i}}(x, y) = δ_{y} Id_{Σ_{y}}. By abuse of notation we will view G^{x_{i}}(x, y) for x, y ∈ B_{3R}(x_{i}) also as a partially defined section of Σ_{M} ⊠ Σ_{M}^{∗} → M × M, which is defined on B_{3R}(x_{i}) × B_{3R}(x_{i}). Now we choose smooth functions a_{i} on M × M such that supp a_{i} ⊂ B_{3R}(x_{i}) × B_{3R}(x_{i}) ⊂ (M × M)_{6R} and such that ∑_{i∈I} a_{i} equals to 1 on (M × M)_{R/2}. Now we set

H(x, y) = ∑_{i∈I} a_{i}(x, y)G^{x_{i}}(x, y).

This implies supp H ⊂ (M × M)_{6R}. Moreover, H(., y) ∈ L^{2}(M \ B_{r}(y)) for all r > 0 since this is true for each summand. Our next goal is to prove that (D_{x} − μ)H(x, y) − δ_{y} Id_{Σ_{y}} is smooth. Note that G^{x_{i}}(x, y) and G^{x_{j}}(x, y) are both defined for (x, y) ∈ (B_{3R}(x_{i}) × B_{3R}(x_{i})) ∩ (B_{3R}(x_{j}) × B_{3R}(x_{j})), but they will not coincide in general. On the other hand their defining property and the locality of the differential operator D (cp. Lemma 3.5) imply that

(D_{x} − μ) (G^{x_{i}}(x, y) − G^{x_{j}}(x, y)) = (D_{y}^{t} − μ) (G^{x_{i}}(x, y) − G^{x_{j}}(x, y)) = 0.

Thus,

\underbrace{((D_{x} − μ)^{2} + (D_{y}^{t} − μ)^{2})}_{=:P} (G^{x_{i}}(x, y) − G^{x_{j}}(x, y)) = 0.

Since P is an elliptic operator, elliptic regularity implies that G^{x_{i}}(x, y)−G^{x_{j}}(x, y) viewed as a difference of distributions is a smooth function on (B_{3R}(x_{i})×B_{3R}(x_{i}))∩(B_{3R}(x_{j})×B_{3R}(x_{j})), and thus a_{j}(x, y) (G^{x_{i}}(x, y) − G^{x_{j}}(x, y)) as well. On B_{3R}(x_{j}) × B_{3R}(x_{j}) we rewrite

H(x, y) = G^{x_{j}}(x, y) + ∑_{i∈I\{j}} a_{i}(x, y) (G^{x_{i}}(x, y) − G^{x_{j}}(x, y)) ,

and we conclude that (D_{x} − μ)H(x, y) = δ_{y} Id_{Σ_{y}} +F (x, y) where F (x, y) is a smooth section of Σ_{M} ⊠ Σ_{M}^{∗} with support in (M × M)_{6R}. There is a unique section H′ of Σ_{M} ⊠Σ_{M}^{∗} such that (D_{x}−μ)H′(x, y) = F (x, y) and such that H′(., y) is L^{2} for all y. This follows for each y from the assumption that D − μ is invertible.

As D −μ is a linear operator with continuous inverse and by elliptic regularity H′ is smooth in x and y. We set G(x, y) = H(x, y) − H′(x, y), and this gives a Green function for D − μ. Smoothness of G follows by smoothness of all G^{x_{i}}, and smoothness of F and H′. Assume that G and G̃ are two Green functions for D. We set ψ_{y}(x) := (G − G̃)(x, y)). For almost all y ∈ M the properties of the Green function imply (D − μ)ψ_{y} = 0 in the sense of distributions, ψ_{y} ∈ L_{loc}^{1}(M) and ψ_{y}|_{M\B_{r}(y)} ∈ L^{2}(M \ B_{r}(y)). Local regularity implies ψ_{y} ∈ L^{2}(M). By the invertibility of D − μ we btain ψ_{y} = 0 for almost all y ∈ M and thus G = G̃. □

Note that due to the last Proposition Lemmata 3.4 and 3.5 also hold true for manifolds M of bounded geometry.

We finish this section by stating another property of the Green function:

Lemma 3.7. Let (M, g) be a Riemannian spin manifold of bounded geometry, and let D − μ be invertible. Then the Green function also decays in L^{2} in the second entry, i.e., G_{D−μ}(x, .) ∈ L^{2}(M \ B_{r}(x)) for all r > 0.

Proof. The Green function G_{D−μ̄}(., x) is in L^{2}(M \ B_{r}(x)) in the first component. Then the claim follows from Lemma 3.4 in the extended version to manifolds M of bounded geometry. □

4. The Dirac operator on hyperbolic space and its products

In this section we examine the Dirac operator on the model spaces 𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N. Note that we also allow the case where N is zero dimensional. First, we introduce polar coordinates on ℍ_{c}^{k+1} and write the Dirac operator in these coordinates. For this purpose it is convenient to introduce three foliations on the space M := (ℍ_{c}^{k+1} × N) \ ({p_{0}} × N), and to relate the associated Dirac operator with the Dirac operator on the total space. Then, we study the canonical action of Spin(k + 1) on 𝕄_{c}^{m,k} and its spinor bundle.

4.1. The Dirac operator on a foliation. In this subsection we want to keep the notation rather general to allow other applications in future work. So we recall some well-known facts about Dirac operators on foliations, and describe them in a way suitable for our needs. We assume that M is a Riemannian spin manifold with the Levi-Civita connection ∇^{M}. To define foliations on M, we will choose our terminology close to Warner’s book, see [38, 1.56–1.64]. We assume that F = ⋃_{x∈M} F_{x} is an ℓ-dimensional distribution on M, so for each x ∈ M, F_{x} is an ℓ-dimensional subspace of T_{x}M, depending smoothly on x. In particular a distribution F on M defines a vector bundle over M, denoted by F → M. A vector field X ∈ C^{∞}(M, T M) is called tangential to F if for all x ∈ M we have X_{x} ∈ F_{x}, and we then simply write X ∈ C^{∞}(M, F). The distribution F is called involutive if X, Y ∈ C^{∞}(M, F) implies [X, Y] ∈ C^{∞}(M, F). A foliation is defined as an involutive distribution. It follows from Frobenius’ Theorem that there exists a family of disjoint connected ℓdimensional submanifolds (Z_{b})_{b∈B}, the leaves of the foliations, such that M = ⋃_{b∈B} Z_{b} and such that T_{x}Z_{b} = F_{x} for all x ∈ Z_{b} ⊂ M and all b ∈ B. Note that in general B is just a set. In the cases considered in the following subsections, we can achieve that B is a smooth manifold with Z_{b} depending smoothly on b and such that all manifolds Z_{b} are diffeomorphic to each other, but we will not assume this in this subsection. For each x ∈ M let ν_{x} be the orthogonal complement of F_{x} in T_{x}M. We obtain a vector bundle ν := ⋃_{x∈M} ν_{x} over M, the normal bundle. As vector bundles with scalar product we have T M = F ⊕ ν. Each leaf will be equipped with the Riemannian metric induced from M. Then the Levi-Civita connection provides a connection, called the intrisic connection ∇^{int}, on the bundle T Z_{b} → Z_{b} for every b ∈ B. Also the restriction of ν to a leaf Z_{b} carries a natural connection, also denoted by ∇^{int}; for X ∈ C^{∞}(Z_{b}, T Z_{b}) and W ∈ C^{∞}(Z_{b}, ν|_{Z_{b}}) we define ∇_{X}^{int}W as the normal component of ∇_{X}^{M}W.

Remark 4.1. These connections actually yield a partial connections on M, i.e. a bilinear maps ∇^{int} : C^{∞}(M, F) × C^{∞}(M, F) → C^{∞}(M, F) resp. ∇^{int} : C^{∞}(M, F) × C^{∞}(M, ν) → C^{∞}(M, ν) which are C^{∞}(M)-linear in the first argument and which satisfy the usual product rule for C^{∞}(M)-multiplication in the second argument. The partial connection on C^{∞}(M, F) is characterized by being torsionfree and metric. However both partial connections are not a connections in the ordinary sense on M as ∇_{X}^{int} is not defined if X is not tangential to F.

Although T M = F ⊕ ν holds in the sense of vector bundles, it does not hold for vector bundles with partial connection. The difference between the (partial) connection on T M and the partial connection of the sum F ⊕ν is the second fundamental form II. More exactly for X ∈ T_{x}Z_{b}, Y ∈ C^{∞}(T Z_{B}) and W ∈ C^{∞}(ν_{B}) we have

∇_{X}^{M}Y − ∇_{X}^{int}Y = II_{Z_{b}}(X, Y), ⟨\underbrace{∇_{X}^{M}W − ∇_{X}^{int}W}_{∈T_{x}Z_{b}}, Y ⟩ = −⟨II_{Z_{b}}(X, Y), W ⟩.

The bundle of positively oriented orthornormal frames of M will be denoted by P_{SO}(M), and let P_{Spin}(M) → P_{SO}(M) be a spin structure on M. We restrict these principal bundles to Z_{b} and obtain principal bundles P_{Spin}(M)|_{Z_{b}} and P_{SO}(M)|_{Z_{b}} over Z_{b}. On T M|_{Z_{b}} = T Z_{b} ⊕ ν|_{Z_{b}} we have a connection defined by ∇^{M} and another connection defined by ∇^{int}. Both connections define connection-1-forms on P_{SO}(M)|_{Z_{b}} and they lift to connection-1-forms on P_{Spin}(M)|_{Z_{b}}. Finally both connection-1-forms yield a connection on the associated spinor bundle again denoted by ∇^{M} and ∇^{int}. Note that the associated spinor bundle coincides with Σ_{M}|_{Z_{b}}, and ∇^{M} is just the restriction of the standard connection on Σ_{M} to Z_{b}. These connections are in fact partial connections on the bundle Σ_{M}. A calculation shows for all X ∈ C^{∞}(M, F) and all spinors φ ∈ C^{∞}(M, Σ_{M})

∇_{X}^{M}φ = ∇_{X}^{int}φ + \frac{1}{2} ∑_{i} e_{i} · II_{Z_{b}}(X, e_{i}) · φ  (3)

where (e_{i})_{i} is a local positively oriented orthonormal frame on F, cp. [8, around (9)].

Remark 4.2. In [8] a slightly different notation is used, as can be seen in the following dictionary of notations

Bär [8] Q M ∇^{Q} and ∇^{M} ⊕ ∇^{N} and D̂ D̃
∇^{ΣQ} ∇^{ΣM} ⊗ id + id ⊗∇^{ΣN}
Our article M ⊂ ℍ_{c}^{k+1} × N Z  ∇^{M}  ≈ ∇^{int}  D_{∂}^{Z} D_{int}^{Z}

Furthermore, in [8] one single submanifold is considered whereas we foliated the space. But this is not an essential difference. In order to avoid missunderstandings, let us mention that a slightly different formalism is used in [9], it introduces an internal and an ambient Clifford multiplication, which differ by multiplication with a unit normal field. This is convenient if the submanifold is a hypersurface. However as we need arbitrary codimensions this is not helpful for our purpose. In fact, in [9] the Clifford multiplication of the ambient manifold coincides with the Clifford multiplication on the hypersurface only up to Clifford multiplication with the normal vector field. In contrast to this in our notation the Clifford multiplication of the ambient space M coincides with the one on the submanifolds Z_{b}.

The partial Dirac operator D_{∂}^{F} is now defined as D_{∂}^{F} = ∑_{i=1}^{ℓ} e_{i} · ∇_{e_{i}}^{M}, and the intrinsic Dirac operators are given by D_{int}^{F} = ∑_{i=1}^{ℓ} e_{i} · ∇_{e_{i}}^{int}. As this definition does not depend on the choice of frame, it yields a global definition. Note that (D_{∂}^{F}φ)|_{Z_{b}} and (D_{int}^{F})|_{Z_{b}} only depend on φ|_{Z_{b}}, each of them defines a family of Dirac-type operators on the leaves. On each leaf the intrinsic Dirac operator is locally a twisted Dirac operator on the submanifold N, the twist bundle is either the normal bundle if ℓ(dim M − ℓ) is even, or two copies of the normal bundle ℓ(dim M − ℓ) is odd. In the applications in the following subsections all normal bundles have a parallel trivialization, hence, in this case the intrinsic Dirac operator coincides on each leaf with several copies of the Dirac operator on this leaf. As multiplicities are irrelevant for our discussion we have chosen the name ’intrinsic Dirac operator’ for D_{int}, slightly abusing the language. By (3), the intrinsic Dirac operator D_{int}^{F} is related to the partial Dirac operator D_{∂}^{F} via

D_{∂}^{F}φ = D_{int}^{F}φ − \frac{1}{2}H⃗_{F} · φ,

where H⃗_{F} = tr II_{F} ∈ C^{∞}(M, ν) is the unnormalised mean curvature vector field of the leaves Z_{b} in M, see [8, Lemma 2.1].

4.2. The Dirac operator in polar coordinates. The goal of this subsection is to study spinors in suitable polar coordinates on 𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N. We express the hyperbolic metric in polar normal coordinates centered in a fixed point p_{0} which will be sometimes identified with 0. In these polar coordinates M = (ℍ_{c}^{k+1} \{p_{0}})×N is parametrized by ℝ^{+} × 𝕊^{k} × N. Define the following foliations on M:

• The radial foliation F_{r} where the leaves are the 1-dimensional submanifolds ℝ^{+} × {x} × {y}. The radial vector field ∂_{r} on hyperbolic space generates F_{r} pointwise.

• The spherical foliation F_{𝕊^{k}} where the leaves are the k-dimensional manifolds {r} × 𝕊^{k} × {y}.

• The N-folation F_{N}, where the leaves are the n-dimensional manifolds {r}×{x}×N^{n}. On an open subset of M we choose an orthonormal frame e_{1}, …, e_{m}, m = n+k+1 = dim M, such that e_{k+2}, …, e_{m} is an orthonormal frame for F_{N}, and e_{2}, …, e_{k+1} is an orthonormal frame for F_{𝕊^{k}} and where e_{1} := ∂_{r}. The notation should be read such that \frac{∇}{dr} and ∂_{r} denote essentially the same (radial) vector, but ∂_{r} is viewed as a vector which acts via Clifford multiplication whereas \frac{∇}{dr} acts as a covariant derivative. The three foliations provide three partial Dirac operators ∂_{r} · \frac{∇}{dr} = D_{∂}^{F_{r}}, D_{∂}^{𝕊^{k}} := D_{∂}^{F_{𝕊^{k}}} and D_{∂}^{N} := D_{∂}^{F_{N}}. The partial Dirac operators along N and 𝕊^{k} are locally defined as

D_{∂}^{N}φ := ∑_{i=1}^{n} e_{i} · ∇_{e_{i}}^{M}φ, D_{∂}^{𝕊^{k}}φ := ∑_{i=n+1}^{n+k} e_{i} · ∇_{e_{i}}^{M}φ,

for φ ∈ C^{∞}(Σ_{M}). Note that we allow k = 0 which simply gives D_{∂}^{𝕊^{0}} = 0. The Dirac operator D on (r_{0}, ∞) × 𝕊^{k} × N is the sum of partial Dirac operators

D = ∂_{r} · \frac{∇}{dr} + D_{∂}^{𝕊^{k}} + D_{∂}^{N}.

The intrinsic Dirac operators along N and 𝕊^{k} are given by

D_{int}^{N}φ := ∑_{i=1}^{n} e_{i} · ∇_{e_{i}}^{int}φ, D_{int}^{𝕊^{k}}φ := ∑_{i=n+1}^{n+k} e_{i} · ∇_{e_{i}}^{int}φ.

We denote the second fundamental form of 𝕊^{k} in ℍ_{c}^{k+1} as II_{𝕊^{k}} and set H⃗_{𝕊^{k}} := tr II_{𝕊^{k}}. Then using natural identifications II_{𝕊^{k}} and H⃗_{𝕊^{k}} do not depend on whether they represent the second fundamental form and the mean curvature field of 𝕊^{k} in ℍ_{c}^{k+1}, or of 𝕊^{k} in ℍ_{c}^{k+1} × N or of 𝕊^{k} × N in ℍ_{c}^{k+1} × N. Using H⃗_{N} = 0 and f(r) = sinh_{c}(r), cp. Section 2.2,

H⃗_{𝕊^{k}×N} = H⃗_{𝕊^{k}} = −k \frac{∂_{r}f(r)}{f(r)}∂_{r} = −k coth_{c}(r)

we obtain D^{N} := D_{∂}^{N} = D_{int}^{N} and D_{∂}^{𝕊^{k}} = D_{int}^{𝕊^{k}} + \frac{k}{2} coth_{c}(r)∂_{r}·. We set D^{𝕊^{k}} := f(r)D_{int}^{𝕊^{k}} which is on each spherical submanifold up to multiplicity the standard Dirac operator on 𝕊^{k} and obtain

D = \frac{1}{sinh_{c}(r)}D^{𝕊^{k}} + ∂_{r} · \frac{∇}{dr} + \frac{k}{2} coth_{c}(r)∂_{r} · +D^{N}.  (4)

Lemma 4.3. The following operators anticommute: D^{N} with D^{𝕊^{k}}, D^{N} with ∂_{r}·, D^{N} with ∂_{r} · \frac{∇}{dr}, D^{𝕊^{k}} with ∂_{r}·, and D^{𝕊^{k}} with ∂_{r} · \frac{∇}{dr}. However ∂_{r}· commutes with ∂_{r} · \frac{∇}{dr}, and (D^{𝕊^{k}})^{2} commutes with D.

Proof. Let P_{Spin}(ℍ_{c}^{k+1}) → P_{SO}(ℍ_{c}^{k+1}) and P_{Spin}(N) → P_{SO}(N) be the fixed spin structures on ℍ_{c}^{k+1} and N. Then we write as in Subsection 2.5

Σ_{ℍ_{c}^{k+1}×N} = (\underbrace{P_{Spin}(ℍ_{c}^{k+1}) × P_{Spin}(N)}_{P}) ×_{ζ} Σ_{m}  (5)

where ζ is the composition Spin(k + 1) × Spin(n) →^{ξ} Spin(m) →^{ρ_{m}} End(Σ_{m}). The bundle P carries the Levi-Civita connection-1-form α_{𝕄_{c}}^{LC} and another connection-1-form α^{int} as explained before. We obtain a connection preserving bundle homomorphism I_{c}, which is fiberwise an isometric isomorphism, and

\begin{matrix}Σ_{ℍ_{c}^{k+1}\\{p_{0}\}×N}, ∇^{int} & ⟶^{I_{c}} & Σ_{ℝ^{+}×𝕊^{k}×N}, ∇^{LC} \\ ↓ & & ↓ \\ ℍ_{c}^{k+1} \ \{p_{0}\} × N & ⟶^{id} & ℝ^{+} × 𝕊^{k} × N \end{matrix}  (6)

commutes. Note that I_{c} is also compatible with the Clifford multiplication in the sense that for X ∈ T Z we have

I_{c}(X · φ) = \begin{cases}X · I_{c}(φ) & for Z = ℝ^{+} × \{x\} × \{y\} or \{r\} × \{x\} × N \\ \frac{f(r)}{r}X · I_{c}(φ) & for Z = \{r\} × 𝕊^{k} × \{y\}. \end{cases}

Then the lemma follows immediately by the corresponding statements for Σ_{ℝ^{+}×𝕊^{k}×N}. □

We will also use the map Î_{c} := I_{0}^{−1} ∘ I_{c} : Σ_{ℍ_{c}^{k+1}\{p_{0}}×N} → Σ_{ℝ^{k+1}\{0}×N} which allows to identify Σ_{ℍ_{c}^{k+1}×N}|_{(x,y)} with Σ_{ℝ^{k+1}×N}|_{(x,y)} and thus with Σ_{ℝ^{k+1}×N}|_{(0,y)}, 0 ≅ p_{0}.

4.3. The action of Spin(k + 1) on 𝕄_{c}^{m,k} = ℍ_{c}^{k+1} × N. We identify T_{p_{0}}ℍ_{c}^{k+1} with ℝ^{k+1}. The left action a_{1} of the spin group Spin(k + 1) on ℝ^{k+1} obtained by composing the double covering Spin(k + 1) → SO(k + 1) with the standard representation yields a left action on ℍ_{c}^{k+1} via the exponential map exp_{p_{0}} : ℝ^{k+1} → ℍ_{c}^{k+1} which is a diffeomorphism. As this action is isometric it yields a left action on P_{Spin}(ℍ_{c}^{k+1}) – also called a_{1}. Thus, we obtain a Spin(k + 1)-action on P_{Spin}(ℍ_{c}^{k+1}) × P_{Spin}(N) × Σ_{m} as â_{1} = a_{1} × id × id. Since a_{1} and the principal Spin(k + 1)-action which acts from the right commute, the â_{1}-action descends to a Spin(k + 1)-action from the left – denoted by a_{2} – on the spinor bundle Σ_{ℍ_{c}^{k+1}×N} = (P_{Spin}(ℍ_{c}^{k+1}) × P_{Spin}(N)) ×_{ζ} Σ_{m} (for ζ as in (5)) such that

Σ_{ℍ_{c}^{k+1}×N} ⟶^{a_{2}(γ)} Σ_{ℍ_{c}^{k+1}×N}
↓  ↓
ℍ_{c}^{k+1} × N ⟶^{a_{1}(γ)×id} ℍ_{c}^{k+1} × N

commutes. By construction, the action a_{1} does not depend on c. Thus, Diagram (6) commutes with this Spin(k + 1)-action.

Moreover, note that a_{1} preserves the spheres 𝕊_{r,y}^{k} := {r} × 𝕊^{k} × {y} ⊂ ℍ_{c}^{k+1} × N; for k > 0 this action is even transitive on 𝕊_{r,y}^{k}. Hence, the diagram above can be restricted to this submanifold. Furthermore, (p_{0}, y) is a fixed point of a_{1} × id for all y ∈ N. Thus, the a_{2}-action can be restricted to an action that maps Σ_{ℍ_{c}^{k+1}×N}|_{(p_{0},y)} to itself.

4.4. Spinors on 𝕊^{k} ⊂ ℝ^{k+1}. We will now analyse the special case N = {y} and c = 0, thus ℍ_{c}^{k+1} = ℝ^{k+1}. This well-known case is not only important as an example, but will also be used to derive consequences for the general case. We obtain immediately from (3) and II_{𝕊_{r}^{k}} = −\frac{1}{r}g|_{𝕊_{r}^{k}}∂_{r} where 𝕊_{r}^{k} is the sphere of radius r canonically embedded in ℝ^{k+1}:

Lemma 4.4. Assume that φ is a parallel spinor on ℝ^{k+1}. Then for any X ∈ T 𝕊_{r}^{k} we have

∇_{X}^{int}φ = −\frac{1}{2r} ∂_{r} · X · φ and ∇_{X}^{int}(∂_{r} · φ) = \frac{1}{2r} ∂_{r} · X · (∂_{r} · φ).

In particular, we have

D^{𝕊^{k}}φ = rD_{int}^{𝕊^{k}}φ = −\frac{k}{2} ∂_{r} · φ and D^{𝕊^{k}}(∂_{r} · φ) = −\frac{k}{2} ∂_{r} · (∂_{r} · φ).

Using Lemma 4.3 and ∇_{X}^{int}∂_{r} = 0 this implies

(D^{𝕊^{k}})^{2}φ = \frac{k^{2}}{4} φ and (D^{𝕊^{k}})^{2}(∂_{r} · φ) = \frac{k^{2}}{4} (∂_{r} · φ).

5. Modes of Spin(k + 1)-equivariant maps

In this section we assume k ≥ 1. We now have a Spin(k + 1)-action on Σ_{ℝ^{k+1}}|_{0} ≅ Σ_{k+1}, {r} × 𝕊^{k} and Σ_{ℝ^{k+1}}|_{{r}×𝕊^{k}}, and thus one on C^{∞}(𝕊^{k}, Σ_{ℝ^{k+1}}|_{{r}×𝕊^{k}}) given by (γ · f)(x) = a_{2}(γ)f(a_{1}(γ)^{−1}x). To simplify notations we mostly write 𝕊^{k} for {r} × 𝕊^{k}. We now have to classify Spin(k + 1)-equivariant functions Σ_{ℝ^{k+1}}|_{0} → C^{∞}(𝕊^{k}, Σ_{ℝ^{k+1}}|_{𝕊^{k}}). For ψ_{0} ∈ Σ_{ℝ^{k+1}}|_{0} let the parallel spinor on ℝ^{k+1} with value ψ_{0} at 0 be denoted by Ψ_{0}. For k odd, the positive and negative parts of Ψ_{0} are denoted by Ψ_{0}^{(±)}.

Lemma 5.1. Let F: Σ_{k+1} → C^{∞}(𝕊^{k}, Σ_{ℝ^{k+1}}|_{𝕊^{k}}) be a linear Spin(k + 1)-equivariant map. Then for k even F has the form

ψ_{0} ↦ (a_{1}Ψ_{0} + a_{2}∂_{r} · Ψ_{0})|_{𝕊^{k}}

and for k odd F has the form

ψ_{0} ↦ (a_{11}Ψ_{0}^{(+)} + a_{22}Ψ_{0}^{(−)} + a_{21}∂_{r} · Ψ_{0}^{(+)} + a_{12}∂_{r} · Ψ_{0}^{(−)})|_{𝕊^{k}}

for suitable constants a_{i}, a_{ij} ∈ ℂ.

The lemma is in fact a special case of Frobenius reciprocity as partially explained in the proof.

Proof. In the case k even, one easily checks that ψ_{0} ↦ Ψ_{0}|_{𝕊^{k}} and ψ_{0} ↦ ∂_{r} · Ψ_{0}|_{𝕊^{k}} are linear Spin(k + 1)-equivariant maps. The same statement holds for k odd for ψ_{0} ↦ Ψ_{0}^{(+)}|_{𝕊^{k}}, ψ_{0} ↦ ∂_{r} · Ψ_{0}^{(+)}|_{𝕊^{k}} ψ_{0} ↦ Ψ_{0}^{(−)}|_{𝕊^{k}} and ψ_{0} ↦ ∂_{r} · Ψ_{0}^{(−)}|_{𝕊^{k}}. The converse will be checked using Frobenius reciprocity. Since ∂_{r} is a Spin(k+1)-equivariant vector field, we obtain a Spin(k+1)-equivariant inclusion P_{SO}(𝕊^{k}) ↪ P_{SO}(ℝ^{k+1})|_{𝕊^{k}}, (e_{1}, ⋯, e_{k}) ↦ (e_{1}, ⋯, e_{k}, ∂_{r}). Via the inclusion SO(k) → SO(k+1) we can view P_{SO}(ℝ^{k+1})|_{𝕊^{k}} as an SO(k)- principal bundle, then the inclusion commutes with the SO(k) action of the SO(k)-principle bundles. It follows from the construction of the spin structure on 𝕊^{k} that this inclusion lifts to a Spin(k + 1)-equivariant inclusion P_{Spin}(𝕊^{k}) ↪ P_{Spin}(ℝ^{k+1})|_{𝕊^{k}} of Spin(k)-principal bundles. Note that the Spin(k + 1) action is the one described above which commutes with the Spin(k)-actions from the principal bundles. Thus we have the Spin(k + 1)-equivariant isomorphism of Spin(k + 1)-principal bundles.

P_{Spin}(𝕊^{k}) ×_{Spin(k)} Spin(k + 1) ⟶^{≅} P_{Spin}(ℝ^{k+1})|_{𝕊^{k}}.

We apply Frobenius reciprocity [33, Theorem 7.47] for the homogeneous space 𝕊^{k} = G/H, G = Spin(k + 1), H = Spin(k) which states that

Hom_{G}(W, Ind_{H}^{G}(V)) ≅ Hom_{H}(W |_{H}, V)

for any G-representation W and any H-representation V. Here W |_{H} denotes the restriction of the G action on W to an H action on W. Further, Ind_{H}^{G}(V) is the induced representation, defined as

Ind_{H}^{G}(V) := {smooth maps G → V | f(gh) = h^{−1}f(g) ∀g ∈ G, h ∈ H}

and the action of G on f ∈ Ind_{H}^{G}(V) is given by

(g̃f)(g) := f(g̃^{−1}g) ∀g, g̃ ∈ G.

So let W = V = Σ_{k+1} be the standard spinor representation with the standard action of G, resp. its restriction to H. Then

Σ_{ℝ^{k+1}}|_{𝕊^{k}} = P_{Spin}(ℝ^{k+1})|_{𝕊^{k}} ×_{Spin(k+1)} Σ_{k+1} ≅ P_{Spin}(𝕊^{k}) ×_{Spin(k)} Σ_{k+1} ≅ G ×_{H} V

as a G-equivariant bundle over 𝕊^{k} = G/H. Thus its space of sections C^{∞}(𝕊^{k}, Σ_{ℝ^{k+1}}|_{𝕊^{k}}) equals Ind_{H}^{G}(Σ_{k+1}), see also [33, Theorem 7.46]. Hence, Hom_{G}(W, Ind_{H}^{G}(V)) is the space of all F satisfying the assumptions of the lemma. The lemma follows by calculating

dim Hom_{H}(W |_{H}, V) = dim Hom_{Spin(k)}(Σ_{k+1}, Σ_{k+1}) = \begin{cases}2 & for k even, \\ 4 & for k odd. \end{cases}

This follows from Schur’s theorem as V is the sum of two non-isomorphic irreducible representations if k is even and as V is the sum of two isomorphic irreducible representations if k is odd. □

Then using Lemma 4.4 we obtain immediately

Corollary 5.2. Let F: Σ_{k+1} → C^{∞}(𝕊^{k}, Σ_{ℝ^{k+1}}|_{𝕊^{k}}) be a linear Spin(k + 1)-equivariant map. Let ψ_{0} ∈ Σ_{k+1} and φ = F ψ_{0}. Then (D^{𝕊^{k}})^{2}φ = \frac{k^{2}}{4} φ.

We say that φ is in the spherical mode \frac{k^{2}}{4} , and thus φ is in the mode of lowest energy on the sphere.

Now we want to carry over the last result to 𝕄_{c}^{m,k}. In the following p_{0} ∈ ℍ_{c}^{k+1} denotes again the fixed point of the Spin(k + 1)-action, and let y_{0}, y ∈ N.

Lemma 5.3. Let F: Σ_{ℍ_{c}^{k+1}×N}|_{(p_{0},y_{0})} → C^{∞}(𝕊^{k}, Σ_{ℍ_{c}^{k+1}×N}|_{𝕊^{k}×{y}}) be a linear Spin(k + 1)- equivariant map. Let ψ_{0} ∈ Σ_{ℍ_{c}^{k+1}×N}|_{(p_{0},y_{0})} and φ = F ψ_{0}. Then (D^{𝕊^{k}})^{2}φ = \frac{k^{2}}{4} φ.

Proof. Note that the composition Î_{c} := I_{0}^{−1} ∘ I_{c} where I_{c} is defined as in (6) maps the spinor bundle over (ℍ_{c}^{k+1} \ {p_{0}}) × N to the spinor bundle over (ℝ^{k+1} \ {0}) × N. This map preserves the intrinsic connection ∇^{int} and uniquely extends into p_{0} ≅ 0. Via pullback we then obtain a Spin(k + 1)-equivariant vector space isomorphism

C^{∞}({r} × 𝕊^{k}, Σ_{ℍ_{c}^{k+1}×N}|_{{r}×𝕊^{k}×{y}}) →^{J_{r,y}} C^{∞}({r} × 𝕊^{k}, Σ_{ℝ^{k+1}×N}|_{{r}×𝕊^{k}×{y}}), ψ ↦ Î_{c} ∘ ψ.

Moreover, we can write in the sense of Spin(k +1)-modules Σ_{ℝ^{k+1}×N}|_{(x,y)} ≅ Σ_{m} ≅ Σ_{k+1} ⊗V if k is even or Σ_{ℝ^{k+1}×N}|_{(x,y)} ≅ Σ_{k+1}^{(+)} ⊗ V^{(+)} ⊕ Σ_{k+1}^{(−)} ⊗ V^{(−)} if k is odd, where V^{(ε)} := Hom_{Spin(k+1)}(Σ_{k+1}^{(ε)}, Σ_{ℝ^{k+1}×N}|_{(x,y)}) is a vector space which is independent of x ∈ ℝ^{k+1}.

Let now k be odd. Then any α ∈ (V^{(ε)})^{∗} defines a map Σ_{ℍ_{c}^{k+1}×N}|_{(x,y)} → Σ_{k+1}^{(ε)}.

Let A: Σ_{k+1}^{(δ)} → Σ_{ℍ_{c}^{k+1}×N}|_{(p_{0},y_{0})} be a Spin(k+1)-equivariant map. By composition we obtain for fixed A, α and δ, ε ∈ {+, −} a Spin(k + 1)-equivariant map

Σ_{k+1}^{(δ)} →^{A} Σ_{ℍ_{c}^{k+1}×N}|_{(p_{0},y_{0})} →^{F} C^{∞}(𝕊^{k}, Σ_{ℍ_{c}^{k+1}×N}|_{𝕊^{k}×{y}})
→^{J_{r,y}}C^{∞}(𝕊^{k}, Σ_{ℝ_{c}^{k+1}×N}|_{𝕊^{k}×{y}}) ≅ C^{∞}(𝕊^{k}, Σ_{ℝ^{k+1}}|_{𝕊^{k}} ⊗ V) →^{α} C^{∞}(𝕊^{k}, Σ_{k+1}^{(ε)}).

Let now k be even. Then the argumentation is analogous to the one above when replacing V^{(ε)} by V and Σ_{k+1}^{(ε)} by Σ_{k+1}. Then the Lemma follows from Corollary 5.2 together with the identification by J_{r,y}. □

Corollary 5.4. Let G(q, p) be the Green function of the operator D − μ, μ ∉ Spec_{L^{2}}^{𝕄_{c}}(D). Let q = (r, x, y) ∈ 𝕄_{c}^{m,k} be the polar coordinates when using p_{0} as the origin, r > 0. Let ψ_{0} ∈ Σ_{𝕄_{c}^{m,k}}|_{(p_{0},y_{0})}, y_{0} ∈ N. Set φ(q) := G(q, (p_{0}, y_{0}))ψ_{0}. Then

(D^{𝕊^{k}})^{2}φ|_{\{r\}×𝕊^{k}×\{y\}} = \frac{k^{2}}{4} φ|_{\{r\}×𝕊^{k}×\{y\}}.

Proof. Now we consider the Green function of the shifted Dirac operator D − μ for μ ∉ Spec_{L^{2}}^{𝕄_{c}}(D) as a map

G( · , (p_{0}, y_{0})): Σ_{ℍ_{c}^{k+1}×N}|_{(p_{0},y_{0})} → Γ(Σ_{ℍ_{c}^{k+1}×N\{(p_{0},y_{0})}}).

It follows from the definition of G, in particular from its uniqueness, and from the equivariance of D under Spin(k + 1) that G( · , (p_{0}, y_{0})) is Spin(k + 1)-equivariant as well. In particular, G( · , (p_{0}, y_{0}))|_{𝕊^{k}×{y}} is a Spin(k + 1)-equivariant map as considered in Lemma 5.3. Thus, together with Lemma 5.3 the corollary follows. □

6. Decay estimates for a fixed mode

Let μ ∉ Spec_{L^{2}}^{𝕄_{c}^{m.k}}(D). Then, by Theorem 3.6 there exists a unique Green function for D − μ. The goal of this section is to estimate the decay of this Green function at infinity. For that, let y = (p_{0}, y_{N}) ∈ ℍ_{c}^{k+1} × N and ψ_{0} ∈ Σ_{𝕄_{c}}|_{y} be fixed. Set φ(x) := G(x, y)ψ_{0}. The Definition of the Green function, cf. (2), implies that φ is an L^{2}-eigenspinor of D to the eigenvalue μ outside a neighbourhood of y. Moreover, by Corollary 5.4 we know that φ is in the spherical mode \frac{k^{2}}{4} . Recall from Subsection 2.6 that

Spec_{L^{2}}^{𝕄_{c}}(D) = (−∞, −λ_{0}] ∪ [λ_{0}, ∞).

In the following the complement of this spectrum is denoted by I_{λ_{0}} := (ℂ \ ℝ) ∪ (−λ_{0}, λ_{0}). Now we decompose the space of spinors restricted to {r_{1}} × 𝕊^{k} × N into complex subspaces of minimal dimensions which are invariant under D^{N}, ∂_{r}·, D^{𝕊^{k}}. For k > 0 such spaces have a basis of the form ψ, ∂_{r} · ψ, P ψ, and ∂_{r} · P ψ, where ψ satisfies D^{N}ψ = λψ, (D^{𝕊^{k}})^{2}ψ = ρ^{2}ψ, ρ ∈ \frac{k}{2} + ℕ_{0}, and P := D^{𝕊^{k}}/ρ. All these operations commute with parallel transport in r-direction, so by applying parallel transport in r-direction we obtain spinors ψ, ∂_{r} · ψ, P ψ, and ∂_{r} · P ψ on ℝ^{+} × 𝕊^{k} × N with similar relations, and the space of all spinors of the form

φ = φ_{1}(r)ψ + φ_{2}(r)∂_{r} · ψ + φ_{3}(r)P ψ + φ_{4}(r)∂_{r} · P ψ  (7)

is preserved under the Dirac operator D on 𝕄_{c}^{m,k} because of (4). Then the operators discussed above restricted to such a minimal subspace are represented by the matrices, cp. Lemma 4.3,

D^{N} = (\begin{matrix}λ & 0 & 0 & 0 \\ 0 & −λ & 0 & 0 \\ 0 & 0 & −λ & 0 \\ 0 & 0 & 0 & λ \end{matrix})  D^{𝕊^{k}} = (\begin{matrix}0 & 0 & ρ & 0 \\ 0 & 0 & 0 & −ρ \\ ρ & 0 & 0 & 0 \\ 0 & −ρ & 0 & 0 \end{matrix})  ∂_{r}· = (\begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix})

For k = 0 we have D^{𝕊^{0}} = 0 and one can proceed analogously with a basis ψ, ∂_{r} · ψ and obtains

D^{N} = (\begin{matrix}λ & 0 \\ 0 & −λ \end{matrix}) ∂_{r}· = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Proposition 6.1. Assume that φ is an L^{2}-solution to the equation Dφ = μφ, μ ∈ I_{λ_{0}} on (𝕄_{c}^{m,k})_{>r_{0}} := (ℍ_{c}^{k+1} \ B_{r_{0}}(p_{0})) × N. Assume that φ has the form given in (7) with parameters ρ and λ. Let κ satisfy κ^{2} = λ^{2} − μ^{2}, Re κ ≥ 0. Then Re κ > 0. Moreover, let κ_{λ_{0}}^{2} = λ_{0}^{2} − μ^{2}. If Re κ_{λ_{0}} > 0, then there is are positive constants C and r_{1} such that

|φ(x)| ≤ C∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r_{0}} )}e^{(−ck/2−Re κ_{λ_{0}} )d(x_{1},p_{0})} for all x = (x_{1}, x_{2}) ∈ (ℍ_{c}^{k+1}\B_{r_{1}}(p_{0}))×N

where C is a constant that only depends on c, k, r_{1}, λ_{0}, μ and ρ but not on λ. For c = 0 an analogous estimate holds when replacing e^{−(ck/2)d(x_{1},p_{0})} by r^{−k/2} where r = d(x_{1}, p_{0}).

Proof. We prove first the case k > 0: By assumption φ can be written as in (7). We view the components of φ as a vector in ℂ^{4}, i.e., Φ(r) := (φ_{1}(r), φ_{2}(r), φ_{3}(r), φ_{4}(r)). So by (4) the following equation is equivalent to Dφ = μφ:

0 = (\begin{matrix}λ − μ & −\frac{k}{2} coth_{c} r & \frac{ρ}{sinh_{c} r} & 0 \\ \frac{k}{2} coth_{c} r & −λ − μ & 0 & −\frac{ρ}{sinh_{c} r} \\ \frac{ρ}{sinh_{c} r} & 0 & −λ − μ & −\frac{k}{2} coth_{c} r \\ 0 & −\frac{ρ}{sinh_{c} r} & \frac{k}{2} coth_{c} r & λ − μ \end{matrix}) Φ(r) + (\begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix}) Φ′(r).

Thus using 𝟙 for the identity matrix and setting

A := (\begin{matrix}0 & λ + μ & 0 & 0 \\ λ − μ & 0 & 0 & 0 \\ 0 & 0 & 0 & −λ + μ \\ 0 & 0 & −λ − μ & 0 \end{matrix}) , B := (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) ,

we obtain

Φ′(r) = (A − \frac{k coth_{c} r}{2} 𝟙 + \frac{ρ}{sinh_{c} r} B) Φ(r).

We start with the case c ≠ 0: We now substitute t = e^{−cr}, Φ̃(t) = Φ(−c^{−1} log t). Then

\frac{dΦ̃}{dt} = (−\frac{1}{ct}A + \frac{k(1 + t^{2})}{2(t − t^{3})}𝟙 + \frac{2ρ}{t^{2} − 1}B) Φ̃.

Such singular ordinary differential equations are well understood, see [15, Chap. 4, Sec. 1–3]. In particular, t = 0 is a singular point of first kind, and [15, Chap. 4 Thm 2,1] yields that t = 0 is a so-called “regular singular point”, and the associated theory applies. However, in our situation it is more efficient to analyse the equation directly. We set h(t) := (log t − log(t + 1) − log(1 − t))k/2, then h′(t) = \frac{k(1+t^{2})}{2(t−t^{3})}. We define

Φ̂(t) := e^{−h(t)}t^{A/c}Φ̃(t),

and we calculate

\frac{dΦ̂}{dt} = −\frac{2ρ}{1 − t^{2}} t^{A/c}Bt^{−A/c}Φ̂.

As B anticommutes with A, we have t^{A/c}Bt^{−A/c} = t^{2A/c}B, and as B is an isometry of ℂ^{4}, we see that

∥t^{A/c}Bt^{−A/c}∥ = t^{2|Re κ_{+}|/c}

where ∥ . ∥ denotes the operator norm and where

κ_{±} := ±\sqrt{λ^{2} − μ^{2}}.

are the (complex) eigenvalues of A. It follows that for 0 < t < 1/2

|\frac{d}{dt} log |Φ̂(t)|| ≤ \frac{|\frac{d}{dt} Φ̂|}{|Φ̂|} ≤ \frac{2ρ}{1 − t^{2}} ∥t^{A/c}Bt^{−A/c}∥ ≤ 3ρt^{2|Re κ_{+}|/c}.

Thus the solution extends to t = 0, and

|Φ̂(0)|e^{−3ρt^{2|Re κ_{+}|/c}} ≤ |Φ̂(t)| ≤ |Φ̂(0)|e^{3ρt^{2|Re κ_{+}|/c}}.

This estimate yields explicit asymptotic control for Φ̂(t), and thus for φ. Namely, assume cr_{0} ≥ 1 > log 2, there are two fundamental solutions φ_{±} of Dφ_{±} = μφ_{±} such that Φ̂_{±}(0) is an eigenvector of A to the eigenvalue κ_{±} and such that

e^{−3ρe^{−2|Re κ_{+}|r}}e^{Re κ_{±}r}e^{h(e^{−cr})} ≤ \frac{|φ_{±}(x)|}{|Φ̂_{±}(0)|} ≤ e^{3ρe^{−2|Re κ_{+}|r}}e^{Re κ_{±}r}e^{h(e^{−cr})} r := d(x_{1}, p_{0}) > r_{0}.

This implies that for every δ ∈ (0, 1) there is r̃_{0} such that

(1 − δ)e^{(−(ck/2)+Re κ_{±})r} ≤ \frac{|φ_{±}(x)|}{|Φ̂_{±}(0)|} ≤ (1 + δ)e^{(−(ck/2)+Re κ_{±})r}  r := d(x_{1}, p_{0}) > r̃_{0}. (8)

From

∫ ∞ ; r̃_{0} |Φ(r)|^{2}(sinh_{c} r)^{k} dr ≤ \frac{∥φ∥_{L^{2}}^{2}}{vol(𝕊^{k})vol(N)}

and the left inequality of (8) we see that φ_{±} is in L^{2}((𝕄_{c}^{m,k})_{>r̃_{0}}) if and only of Re κ_{±} < 0. In the following we call this κ_{±} just κ_{λ} and also replace the ± index by λ in all other occurrences. We note that |Re κ_{λ}| is increasing in |λ|. Thus, δ and r̃_{0} from above can be chosen independent on λ. Next, we multiply the first inequality of (8) by |Φ̂_{λ}(0)| and then integrate its square:

\frac{∥φ∥_{L^{2}}^{2}}{vol(𝕊^{k} × N)} ≥(1 − δ)^{2}|Φ̂_{λ}(0)|^{2} ∫ ∞ ; r̃_{0} e^{(−ck+2Re κ_{λ})r}(sinh_{c} r)^{k} dr.

Hence, we obtain an upper bound

|Φ̂_{λ}(0)|^{2} ≤ C_{1}^{2}(1 − δ)^{−2}∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r̃_{0}} )}^{2} (\frac{e^{2Re κ_{λ}r̃_{0}}}{−2Re κ_{λ}} )^{−1}

where C_{1} is a constant independent on λ. Using this again with the right inequality of (8) we get for all x with r = dist(x, p_{0}) > r̃_{0} that

|φ(x)| ≤\frac{1 + δ}{1 − δ} C_{1}∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r̃_{0}} )}e^{(−ck/2+Re κ_{λ})r} (\frac{e^{2Re κ_{λ}r̃_{0}}}{−2Re κ_{λ}} )^{−1/2}
≤C_{1}(−2Re κ_{λ})^{\frac{1}{2}} ∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r̃_{0}} )}e^{−ckr/2+Re κ_{λ}(r−r̃_{0})}.  (9)

For r > r̃_{0} we see that (−2Re κ_{λ})e^{2Re κ_{λ}(r−r̃_{0})} is monotonically decreasing in |Re κ_{λ}|, and we obtain from (9)

|φ(x)| ≤C_{1}(−2Re κ_{λ_{0}})^{\frac{1}{2}} ∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r̃_{0}} )}e^{−ckr/2+Re κ_{λ_{0}} (r−r̃_{0})}
≤C∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r̃_{0}} )}e^{(−ck/2+Re κ_{λ_{0}} )r}

for all x with r = d(x_{1}, p_{0}) > r̃_{0}. Here, C can be chosen such that it only depends on c, k, r̃_{0}, λ_{0}, μ and ρ but not on λ. Note that the κ in the claim is simply −κ_{λ_{0}}. Next we consider the case c = 0, k > 0:

Φ′(r) = (A − \frac{k}{2r} 𝟙 + \frac{ρ}{r}B) Φ(r).

Set Φ̂(r) = r^{\frac{k}{2}} e^{−Ar}Φ(r). Then, Φ̂′(r) = \frac{ρ}{r}e^{−Ar}Be^{Ar}Φ̂ = \frac{ρ}{r}e^{−2Ar}BΦ̂. Then we can proceed as above and obtain the claim.

It remains the case k = 0: Then

\binom{φ_{1}(r)}{φ_{2}(r)}′ = (\begin{matrix}0 & λ + μ \\ λ − μ & 0 \end{matrix}) \binom{φ_{1}(r)}{φ_{2}(r)}

and direct calculation gives the claim. □

In order to estimate the decay of φ(x) = G(x, y)ψ_{0}, ψ_{0} ∈ Σ_{𝕄_{c}^{m,k}}|_{y} at infinity we will decompose φ into its modes in 𝕊^{k} and N direction, respectively. Lemma 6.1 provides an estimate of the decay of each mode which is independent of the mode in direction of N. Moreover, from Corollary 5.4 we know that φ has spherical mode \frac{k^{2}}{4} . Thus, we obtain a decay estimate for φ:

Lemma 6.2. Let μ ∉ Spec_{L^{2}}^{𝕄_{c}}(D), and let G be the unique Green function of D − μ. We set M_{y}(r) := {x ∈ 𝕄_{c} | dist(x, N^{y}) = r} where N^{y} = {p_{0}} × N and y = (p_{0}, y_{N}) ∈ ℍ_{c}^{k+1} × N. Let κ satisfy κ^{2} = λ_{0}^{2} − μ^{2} and Re κ ≥ 0. Then for all ε > 0 and r_{0} sufficiently large there is a constant C > 0 independent on y such that

∫_{M_{y}(r)} |G(x, y)|^{2}dx ≤ Ce^{−2r Reκ} for all r > r_{0}.

Proof. Let ψ_{0} ∈ Σ_{𝕄_{c}}|_{y}. Set φ(x) := G(x, y)ψ_{0}. Then, for k > 0 the spinor φ decomposes into a sum of spinors φ_{ρ^{2},λ} of the form (7) with (D^{𝕊^{k}})^{2}φ_{ρ^{2},λ} = ρ^{2}φ_{ρ^{2},λ} and D^{N}φ_{ρ^{2},λ} = λφ_{ρ^{2},λ}, respectively. By Corollary 5.4 ρ^{2} may only take the value \frac{k^{2}}{4} . For k = 0 the spinor φ decomposes into a sum of spinors φ_{0,λ} with D^{N}φ_{0,λ} = λφ_{0,λ}. Thus, for all k ∫_{M_{y}(r)} |φ(x)|^{2}dx = ∑_{λ} ∥φ_{k^{2}/4,λ}∥_{L^{2}(M_{y}(r))}^{2}. Together with Proposition 6.1 we obtain for c ≠ 0

∫_{M_{y}(r)} |φ(x)|^{2}dx ≤ ∑_{λ} C∥φ_{k^{2}/4,λ}∥_{L^{2}((𝕄_{c}^{m,k})_{>r_{0}} )}^{2}e^{(−ck−2Re κ)r} sinh_{c}^{k}(r)

≤ C′e^{−2rRe κ} ∑_{λ} ∥φ_{k^{2}/4,λ}∥_{L^{2}((𝕄_{c}^{m,k})_{>r_{0}} )}^{2}

≤ C′e^{−2r Reκ}∥φ∥_{L^{2}((𝕄_{c}^{m,k})_{>r_{0}} )}^{2}.

The case c = 0 follows analogously. □

7. Decomposition of the Green function

We decompose the Green function G of the shifted Dirac operator D − μ on M = 𝕄_{c}^{m,k} into a singular part and a smoothing operator. Both operators will be shown to be bounded operators from L^{p} to L^{p} for all p ∈ [1, ∞]. At first we choose a smooth cut-off function χ: ℝ → [0, 1] with supp χ ⊂ [−R, R] and χ_{|_{(−R/2,R/2)}} ≡ 1. Let ρ: M × M → [0, 1] be given by ρ(x, y) = χ(dist_{ℍ_{c}^{k+1}}(π_{ℍ}(x), π_{ℍ}(y))). Let now

G_{1}(x, y) := ρ(x, y)G(x, y) and G_{2}(x, y) := G(x, y) − G_{1}(x, y).

Then G_{2} is zero on a neighbourhood of the diagonal, and thus smooth everywhere. The singular part is only contained in G_{1}.

Proposition 7.1. Let M = 𝕄_{c}^{m,k} and G_{1} be as defined above. Then, for all 1 ≤ p ≤ ∞ the map P_{1} : φ ↦ ∫_{M} G_{1}(., y)φ(y)dy defines a bounded operator from L^{p} to L^{p}.

Proof. We start with a smooth spinor φ compactly supported in B_{2R}(0)×N ⊂ M. For such a φ the spinor P_{1}φ is supported in B_{3R}(0) × N ⊂ M. We embed B_{3R}(0) isometrically into a closed Riemannian manifold M_{R}. Let M_{R} × N. The metric on M_{R} can be chosen such that D^{M_{R}×N} − μ is invertible, cf. Proposition C.1. The norm of (D^{M_{R}×N} − μ)^{−1} : L^{p} → L^{p} is denoted by C_{R}(p).

For p < ∞ we estimate

∫_{M} |P_{1}φ|^{p} dx = ∫_{M} |∫_{M} G_{1}(x, y)φ(y)dy|^{p} dx

≤ ∫_{B_{3R}(p_{0})×N} |∫_{B_{2R}(p_{0})×N} G(x, y)φ(y) dy|^{p} dx ≤ ∫_{M_{R}×N} |(D^{M_{R}×N} − μ)^{−1}φ|^{p} dx

≤ C_{R}(p)^{p} ∫_{M_{R}×N} |φ|^{p} dx = C_{R}(p)^{p} ∥φ∥_{L^{p}}^{p}.

Next we want to consider arbitrary φ ∈ L^{p}(M, Σ_{M}), p < ∞. Then C_{c}^{∞}(M, Σ_{M}) is dense in L^{p}(M, Σ_{M}), and it suffices to consider φ ∈ C_{c}^{∞}(M, Σ_{M}). Choose points (x_{i})_{i∈I} ⊂ ℍ_{c}^{k+1} as in Lemma 2.1. Then (B_{2R}(x_{i}) × N)_{i∈I} and (B_{3R}(x_{i}) × N)_{i∈I} both cover 𝕄_{c}^{m,k} uniformly locally finite. We denote the multiplicity of the second cover by L and choose a partition of unity η_{i} subordinated to (B_{2R}(x_{i}) × N)_{i∈I}. Let φ = ∑ φ_{i} where φ_{i} = η_{i}φ ∈ C_{c}^{∞}(B_{2R}(x_{i}) × N, Σ_{M}). Hence, P_{1}φ_{i} ∈ C_{c}^{∞}(B_{3R}(x_{i}) × N, Σ_{M}). Moreover, let f̄_{i}: M → M be given by f̄_{i} = (Id, f_{i}) where f_{i} is an isometry of ℍ_{c}^{k+1} that maps x_{i} to p_{0}. We choose a lift of f̃_{i} to an isometry on the spinor bundle. Due to the homogeneity of ℍ_{c}^{k+1} we have P_{1}(f̃_{i} ∘ φ ∘ f̄_{i}^{−1}) = f̃_{i} ∘ (P_{1}φ) ∘ f̄_{i}^{−1}. Then, by triangle inequality and Hölder inequality and since for fixed x the value P_{1}φ_{i}(x) is nonzero for at most L spinors φ_{i}, we have

|P_{1}φ(x)|^{p} = |∑_{i} P_{1}φ_{i}(x)|^{p} ≤ |∑_{i} |P_{1}φ_{i}(x)||^{p} ≤ L^{p−1} ∑_{i} |P_{1}φ_{i}(x)|^{p}.

Thus, we obtain

∥P_{1}φ∥_{L^{p}(M)}^{p} ≤ L^{p−1} ∑_{i} ∥P_{1}φ_{i}∥_{L^{p}(B_{3R}(x_{i})×N)}^{p} = L^{p−1} ∑_{i} ∥P_{1}(f̃_{i} ∘ φ_{i} ∘ f̄_{i}^{−1})∥_{L^{p}(B_{3R}(p_{0})×N)}^{p}

≤ L^{p−1}C_{R}(p)^{p} ∑_{i} ∥f̃_{i} ∘ φ_{i} ∘ f̄_{i}^{−1}∥_{L^{p}(B_{3R}(p_{0})×N)}^{p}

= L^{p−1}C_{R}(p)^{p} ∑_{i} ∥φ_{i}∥_{L^{p}(B_{3R}(x_{i})×N)}^{p}

≤ L^{p}C_{R}(p)^{p}∥φ∥_{L^{p}(M)}^{p}.

It remains the case p = ∞. Let η_{i} as above, and let φ ∈ L^{∞}. We decompose again φ = ∑ φ_{i} where φ_{i} = η_{i}φ is compactly supported. Then, we obtain as above that

∥P_{1}φ∥_{L^{∞}(M)} ≤ ∑_{i} ∥P_{1}φ_{i}∥_{L^{∞}(B_{3R}(x_{i})×N)} ≤ C ∑_{i} ∥φ_{i}∥_{L^{∞}(B_{3R}(x_{i})×N)} ≤ CL∥φ∥_{L^{∞}(M)}.

□

We now turn to the off-diagonal part G_{2}.

Note that ℍ_{c}^{k+1} is homogeneous for all c. In particular, the representation of the metric in polar coordinates – dr^{2} + sinh_{c}^{2}(r) σ^{k} (cf. Section 2.2) – is independent of the chosen origin of the polar coordinates on ℍ_{c}^{k+1}. We set M_{y}(r) := {x ∈ 𝕄_{c}^{m,k} | dist(x, N^{y}) = r} where N^{y} = {y_{1}} × N where y = (y_{1}, y_{2}) ∈ ℍ_{c}^{k+1} × N. Then, the volume vol(M_{y}(r)) = f(r)^{k}vol(N)vol(S^{k}) = sinh_{c}^{k}(r)vol(N)vol(S^{k}) is independent of y. We will subsequently leave out the y in the notation and write vol(M(r)).

Proposition 7.2. Using the notations from above, assume that there are constants C, ρ > 0 with

∫_{M_{y}(r)} |G_{2}(x, y)|^{2}dx ≤ Ce^{−2ρr} for all r > 0.

Let p = 1 and p = ∞. Then, for ρ > \frac{ck}{2} the operator P_{2} : φ ↦ ∫_{M} G_{2}(., y)φ(y)dy from L^{p} to L^{p} is bounded.

Proof. We start with p = 1 and estimate for φ ∈ C_{c}^{∞}(M, Σ_{M})

∫_{M} |(P_{2}φ)(x)| dx ≤ ∫_{M} ∫_{M} |G_{2}(x, y)||φ(y)| dy dx = ∫_{M} (∫_{M} |G_{2}(x, y)|dx) |φ(y)| dy

= ∫_{M} (∫_{ℝ_{+}} ∫_{M_{y}(r)} |G_{2}(x, y)| dx̃ dr) |φ(y)|dy

≤ ∫_{M} (∫_{ℝ_{+}} vol(M(r))^{\frac{1}{2}} (∫_{M_{y}(r)} |G_{2}(x, y)|^{2} dx̃)^{\frac{1}{2}} ) |φ(y)| dy

≤ C′ ∫_{r≥r_{0}} sinhc^{\frac{k}{2}} (r)e^{−ρr} dr∥φ∥_{L^{1}}.

where x̃ is the angular part and r the radial part of x. For ρ > \frac{ck}{2} the integral ∫_{r≥r_{0}} sinhc^{\frac{k}{2}} (r)e^{−ρr} dr is bounded. Hence, P_{2} : L^{1} → L^{1} is invertible. Next, we consider the other case p = ∞. Then for φ ∈ L^{∞}(M, Σ_{M})

|(P_{2}φ)(x)| ≤ ∫ ∞_{\frac{R}{2}} ∫_{M_{x}(r)} |G_{2}(x, y)||φ(y)| dỹdr

≤ ∫ ∞_{\frac{R}{2}} sup_{M_{x}(r)} |φ| (∫_{M_{x}(r)} |G_{2}(x, y)| dỹ) dr

≤ ∥φ∥_{L^{∞}} ∫ ∞_{\frac{R}{2}} ∥G_{2}(x, y)∥_{L^{2}(M_{x}(r))}vol(M(r))^{\frac{1}{2}} dr

≤ C∥φ∥_{L^{∞}} ∫ ∞_{\frac{R}{2}} e^{−ρr} sinhc^{\frac{k}{2}} (r) dr ≤ C̃∥φ∥_{∞}.

where for ρ > \frac{ck}{2} the last inequality follows as above. Thus, ∥P_{2}φ∥_{∞} ≤ C̃∥φ∥_{∞}. □

8. σ_{p} contains the L^{p}-spectrum on 𝕄_{c}^{m,k}

In this section we prove one direction of Theorem 1.1.

Proposition 8.1. Let p ∈ [1, ∞]. Let λ_{0}^{2}, λ_{0} ≥ 0, be the lowest eigenvalue of the Dirac square on the closed Riemannian spin manifold N. The L^{p}-spectrum of the Dirac operator on 𝕄_{c}^{m,k} is a subset of

σ_{p} := \{μ ∈ ℂ | μ^{2} = λ_{0}^{2} + κ^{2}, |Im κ| ≤ ck |\frac{1}{p} − \frac{1}{2}|\} .

Proof. We will show that D − μ: H_{1}^{p} ⊂ L^{p} → L^{p} has a bounded inverse for all μ ∈ ℂ \ σ_{p}. Fix μ ∈ ℂ \ σ_{p}, and let κ ∈ ℂ such that μ^{2} = λ_{0}^{2} + κ^{2}. For p = 2, the lemma follows from Subsection 2.6. Let now p ∈ {1, ∞} and μ ∉ σ_{1} = σ_{∞}. Then μ ∉ σ_{2} and (D−μ): H_{1}^{2}(𝕄_{c}^{m,k}) ⊂ L^{2}(𝕄_{c}^{m,k}) → L^{2}(𝕄_{c}^{m,k}) has a bounded inverse given by P_{μ}: φ ↦ ∫_{𝕄_{c}} G_{μ}(x, y)φ(y)dy. By Proposition 7.1,

7.2 and Lemma 6.2 the operator P_{μ}: L^{p} → L^{p} is bounded for |Im κ| > ck |\frac{1}{p} − \frac{1}{2} | = c\frac{k}{2} .

Hence, the L^{1}- and the L^{∞}-spectrum of D on 𝕄_{c}^{m,k} have to be contained in σ_{1} = σ_{∞}. First we deal with the case that Im κ > 0. For p ∈ [1, 2] we use the Stein Interpolation Theorem 2.3: Fix ε > 0 and y_{0} ∈ ℝ. We set h(z) := μ(z)^{2} := λ_{0}^{2} + κ(z)^{2} := λ_{0}^{2} + (y_{0} + \frac{ck}{2} iz + iε)^{2} and A_{z} = (D^{2} − h(z))^{−1}. By Subsection 2.6 the operators

A_{w+iy} = (D^{2} − [λ_{0}^{2} + (y_{0} − \frac{ck}{2} y + i( \underbrace{\frac{ck}{2} w + ε}_{=Im κ(z)>0} ))^{2}])^{−1} ,

for 0 ≤ w ≤ 1 and y ∈ ℝ, are bounded as operators from L^{2} to L^{2}. Furthermore

A_{1+iy} = (D^{2} − (λ_{0}^{2} + (y_{0} − \frac{ck}{2} y + i (\frac{ck}{2} + ε))^{2}))^{−1}

is bounded from L^{1} to L^{1} as seen above. Thus – as required to apply the Stein interpolation theorem – A_{iy} and A_{1+iy} are bounded operators from L^{1} ∩ L^{2} to L^{1} + L^{2}. Let now φ ∈ L^{1}∩L^{2} and ψ ∈ L^{∞}∩L^{2}. Set S := {z ∈ ℂ | 0 ≤ Re z ≤ 1}. We define b_{φ,ψ}(z) = ⟨A_{z}φ, ψ⟩. The map b_{φ,ψ} is analytic in the interior of S, since the resolvent is, see Lemma B.5. Moreover, |b_{φ,ψ}(z)| ≤ ∥A_{z}∥∥φ∥_{L^{2}}∥ψ∥_{L^{2}} ≤ (max_{0≤Re z≤1} ∥A_{z}∥)∥φ∥_{L^{2}}∥ψ∥_{L^{2}} where ∥A_{z}∥ denotes the operator norm for A_{z}: L^{2} → L^{2}. Thus, b_{φ,ψ}(z) is uniformly bounded and continuous on S := {z ∈ ℂ | 0 ≤ Re z ≤ 1}. Thus, we can apply Theorem 2.3 and obtain for t ∈ (0, 1)

and p = \frac{2}{1+t} that A_{t} = (D^{2} − h (\frac{2}{p} − 1))^{−1} = (D^{2} − (λ_{0}^{2} + (y_{0} + cki (\frac{1}{p} − \frac{1}{2}) + iε)^{2}))^{−1}

is bounded from L^{p} to L^{p}. In the case Imκ < 0 we set analogously A_{z} = (D^{2} −g(z))^{−1} for g(z) = λ_{0}^{2} +(y_{0} − \frac{ck}{2} iz − iε)^{2}

and obtain that A_{t} = (D^{2} − g(\frac{2}{p} − 1))^{−1} is bounded from L^{p} to L^{p}. Since y_{0} ∈ ℝ and ε > 0

can be chosen arbitrarily, we get for all μ ∈ ℂ \ σ_{p} that μ^{2} is not in the L^{p}-spectrum of D^{2}. Using Lemma B.8 the claim follows for p ∈ [1, 2] and with Lemma B.3.(i) for p ∈ [2, ∞). □

9. Construction of test spinors on ℍ^{k+1}

In this section we determine the Dirac L^{p}-spectrum of the hyperbolic space. The general case for 𝕄_{c} is given in the next section.

Proposition 9.1. Let p ∈ [1, ∞]. The L^{p}-spectrum of the Dirac operator D on the hyperbolic space ℍ^{k+1} is given by the set

σ_{p}^{ℍ} := \{μ ∈ ℂ | |Im μ| ≤ k |\frac{1}{p} − \frac{1}{2}|\} .

Proof. From Proposition 8.1 we know that the L^{p}-spectrum is contained in σ_{p}^{ℍ}. Thus, it remains to show that each element μ of σ_{p}^{ℍ} is contained in the L^{p}-spectrum of D. For that we start with a similar ansatz as was used in [18, Lemma 7] for the Laplacian. Let the hyperbolic space ℍ^{k+1}, k ≥ 0, be modelled by the space {(y, x_{1}, …, x_{k}) | y > 0} equipped with the metric g = y^{−2}(dx_{1}^{2} + … + dx_{k}^{2} + dy^{2}). We set e_{i} = y \frac{∂}{∂x_{i}} = y∂_{i} for i = 1, …, k and e_{y} = y \frac{∂}{∂y} = y∂_{y}. Then, (e_{y}, e_{1}, …, e_{k}) forms an orthonormal basis, which can assumed to be positively oriented. Further we have [e_{y}, e_{i}] = e_{i} = −[e_{i}, e_{y}]. All other commutators vanish. Then, −Γ_{iy}^{i} = Γ_{ii}^{y} = 1 and all other Christoffel symbols vanish. The orthonormal frame (e_{y}, e_{1}, …, e_{k}) can be lifted to the spin structure ϑ: P_{Spin}(ℍ^{k+1}) → P_{SO}(ℍ^{k+1}), namely we choose a map E: ℍ^{k+1} → P_{Spin}(ℍ^{k+1}) with ϑ(E) = (e_{y}, e_{1}, …, e_{k}). A spinor is by definition a section of the associated bundle Σ_{ℍ^{k+1}} = P_{Spin}(ℍ^{k+1})×_{ρ_{k+1}} Σ_{k+1}, so every spinor can be written as x ↦ [E(x), φ(x)] for a function φ: ℍ^{k+1} → Σ_{k+1}. Hence, identifying (e_{y}, e_{1}, …, e_{k}) with the standard basis of ℝ^{k+1} we obtain [12, (4.8)], [7, Lemma 4.1]

∇_{e_{i}}[E, φ] = [E, ∂_{e_{i}}φ + \frac{1}{2}e_{i} · e_{y} · φ]; ∇_{e_{y}}[E, φ] = [E, ∂_{e_{y}}φ]

and

D[E, φ] = [E, ∑_{i=1}^{k} e_{i} · ∂_{e_{i}}φ + e_{y} · ∂_{e_{y}}φ − \frac{k}{2} e_{y} · φ]

= [E, ∑_{i=1}^{k} ye_{i} · ∂_{i}φ + ye_{y} · ∂_{y}φ − \frac{k}{2} e_{y} · φ].  (10)

Let ψ_{0} ∈ Σ_{k+1} be a unit-length eigenvector of the Clifford multiplication with the vector e_{y} = (1, 0, …, 0)^{t} ∈ ℝ^{k+1} to the eigenvalue ±i, i.e. e_{y} · ψ_{0} = ±iψ_{0}. Set φ_{n}(x, y) = b(x)c_{n}(log y)y^{α}ψ_{0} where α ∈ ℂ, b(x) is any compactly supported function on ℝ^{k}, and where c_{n}: ℝ → ℝ is chosen to be a smooth cut-off function compactly supported on (−4n, −n), c_{n}|_{[−3n,−2n]} ≡ 1 and |c′_{n}| ≤ 2/n. Then for p ∈ [1, ∞) one estimates ∥c′_{n}∥_{p}^{p}/∥c_{n}∥_{p}^{p} ≤ Cn^{−p} → 0 as n → ∞. For p = ∞ we have ∥c′_{n}∥_{∞}/∥c_{n}∥_{∞} ≤ 2/n → 0 as n → ∞. Then we set Φ_{n} := [E, φ_{n}] and obtain

(D − μ)Φ_{n} = [E, yc_{n}(log y) y^{α} ∑_{i=1}^{k}(∂_{i}b) e_{i} · ψ_{0} ± b(x)c′_{n}(log y)y^{α}iψ_{0}

+b(x)c_{n}(log y)(±iα ∓ i\frac{k}{2} − μ)y^{α}ψ_{0}] .  (11)

In the following we will use the notation (X · .) ∈ End(Σ_{k+1}) for the Clifford multiplication by X ∈ ℝ^{k+1}, and obviously its operator norm |(X · .)| equals to the usual norm of X. Let μ = s ± ik (\frac{1}{p} − \frac{1}{2}), s ∈ ℝ. We choose z = log y and α = \frac{k}{2} ∓ iμ = \frac{k}{p} ∓ is. Thus, the last summand in (11) vanishes and pRe α = k. Then, for p ∈ [1, ∞) we have

\frac{∥(D − μ)Φ_{n}∥_{p}}{∥Φ_{n}∥_{p}}

≤ (∫_{ℝ^{k}} | ∑_{i}(∂_{i}b)(e_{i} · .)|^{p} ∫_{0}^{∞} |c_{n}(log y)|^{p}y^{pRe α+p−k−1})^{\frac{1}{p}} ; (∫_{ℝ^{k}} |b(x)|^{p} ∫_{0}^{∞} |c_{n}(log y)|^{p}y^{pRe α−k−1})^{\frac{1}{p}}

+ (∫_{ℝ^{k}} |b(x)|^{p} ∫_{0}^{∞} |c′_{n}(log y)|^{p}y^{pRe α−k−1})^{\frac{1}{p}} ; (∫_{ℝ^{k}} |b(x)|^{p} ∫_{0}^{∞} |c_{n}(log y)|^{p}y^{pRe α−k−1})^{\frac{1}{p}}

≤^{pRe α=k} c (\frac{∫_{ℝ^{k}} ∑_{i} |∂_{i}b|^{p} ∫_{0}^{∞} |c_{n}(log y)|^{p}y^{p−1}}{∫_{ℝ^{k}} |b(x)|^{p} ∫_{0}^{∞} |c_{n}(log y)|^{p}y^{−1}} )^{\frac{1}{p}} + (\frac{∫_{0}^{∞} |c′_{n}(log y)|^{p}y^{−1}}{∫_{0}^{∞} |c_{n}(log y)|^{p}y^{−1}} )^{\frac{1}{p}}

≤^{z=log y} c (\frac{∫_{ℝ^{k}} ∑_{i} |∂_{i}b|^{p} ∫_{−∞}^{0} |c_{n}(z)|^{p}e^{zp}}{∫_{ℝ^{k}} |b(x)|^{q} ∫_{−∞}^{0} |c_{n}(z)|^{p}} )^{\frac{1}{p}} + (\frac{∫_{−∞}^{0} |c′_{n}(z)|^{p}}{∫_{−∞}^{0} |c_{n}(z)|^{p}} )^{\frac{1}{p}}

≤ Ce^{−n} + (\frac{∫_{−∞}^{0} |c′_{n}(z)|^{p}}{∫_{−∞}^{0} |c_{n}(z)|^{p}} )^{\frac{1}{p}} → 0

where the last inequality uses

∫_{−∞}^{0} |c_{n}(z)|^{p}e^{zp} dz = ∫_{−4n}^{−n} |c_{n}(z)|^{p}e^{zp} dz ≤ e^{−np} ∫_{−4n}^{−n} |c_{n}(z)|^{p} dz = e^{−np} ∫_{−∞}^{−0} |c_{n}(z)|^{p} dz.

For p = ∞ we have μ = s ± i\frac{k}{2} , α = ∓s and the estimate above is done analogously. Summarizing, we have shown that ∂σ_{p}^{ℍ}, the boundary of σ_{p}^{ℍ}, is a subset of the Dirac L^{p}- spectrum for ℍ^{k+1} for p ∈ [1, ∞]. Note that σ_{s}^{ℍ} = ⋃_{2≥r≥s} ∂σ_{r}^{ℍ} for s < 2 and σ_{s}^{ℍ} = ⋃_{2≤r≤s} ∂σ_{r}^{ℍ} for s > 2, respectively. Thus, using the Riesz-Thorin interpolation theorem we see that σ_{p}^{ℍ} is a subset of the L^{p}-spectrum of D on ℍ^{k+1} for p ∈ [1, ∞]. □

Remark 9.2. From (10) we obtain

D^{2}[E, φ] =[E, ∑_{i,j} y^{2}e_{i} · e_{j} · ∂_{i}∂_{j}φ + ∑_{i} y^{2}e_{i} · e_{y} · ∂_{i}∂_{y}φ − y \frac{k}{2} ∑_{i} e_{i} · e_{y} · ∂_{i}φ

+ ∑_{i} y^{2}e_{y} · e_{i} · ∂_{y}∂_{i}φ + ∑_{i} ye_{y} · e_{i} · ∂_{i}φ − y^{2}∂_{y}∂_{y}φ − y∂_{y}φ + y \frac{k}{2} ∂_{y}φ

− y \frac{k}{2} ∑_{i} e_{y} · e_{i} · ∂_{i}φ + y \frac{k}{2} ∂_{y}φ − \frac{k^{2}}{4} φ]

=[E, −y^{2} ∑_{i} ∂_{i}^{2}φ − y^{2}∂_{y}^{2}φ + y(k − 1)∂_{y}φ + ∑_{i} ye_{y} · e_{i} · ∂_{i}φ − \frac{k^{2}}{4} φ].

We use μ and φ_{n} = b(x)c_{n}(log y)y^{α}ψ_{0} of the last proposition with b, α, c_{n} and ψ_{0} as therein. For c_{n} we require additionally |c′′_{n}| ≤ 8n^{−2}. Hence, ∥c′′_{n}∥_{p}/∥c_{n}∥_{p} → 0 as n → ∞ for p ∈ [1, ∞]. Then we have

(D^{2} − μ^{2})[E, φ_{n}] = [E, (−y^{2}c_{n}(log y)y^{α} ∑_{i} ∂_{i}^{2}b − y^{2}b∂_{y}^{2}(c_{n}(log y)y^{α}) + y(k − 1)b∂_{y}(c_{n}(log y)y^{α})

−(\frac{k^{2}}{4} + μ^{2})bc_{n}(log y)y^{α}) ψ_{0} − ic_{n}(log y)y^{α} ∑_{i} y(∂_{i}b)e_{i} · ψ_{0}]

= [E, −y^{2}c_{n}(log y)y^{α} ∑_{i} ∂_{i}^{2}bψ_{0} − ic_{n}(log y)y^{α} ∑_{i} y(∂_{i}b)e_{i} · ψ_{0}

−y^{α}b (c′′_{n} + (2α + k − 2)c′_{n} + c_{n} (α(α − 1) − (k − 1)α + \frac{k^{2}}{4} + μ^{2})) ψ_{0}]

=[E, −c_{n}(log y)y^{α+2} ∑_{i} ∂_{i}^{2}b ψ_{0} − ic_{n}(log y)y^{α+1} ∑_{i}(∂_{i}b)e_{i} · ψ_{0}

−y^{α}b (c′′_{n} + (2α + k − 2)c′_{n}) ψ_{0}] ,

and by analogous estimates as in Proposition 9.1 we have ∥(D^{2} −μ^{2})[E, φ_{n}]∥_{p}/∥[E, φ_{n}]∥_{p} → 0 as n → ∞.

Remark 9.3. Note that while the L^{2}-spectrum of the hyperbolic space only consists of continuous spectrum, this is no longer true for the L^{p}-spectrum for p ≠ 2 as can be seen by considering 0 ∈ σ_{p}^{H}: We view the hyperbolic space (ℍ^{k+1}, g_{ℍ}) modelled on the unit ball B_{1}(0) ⊂ ℝ^{k+1} of the Euclidean space and equipped with the metric g_{ℍ} = f^{2}g_{E} where f(x) = \frac{2}{1−|x|^{2}} and |.| denotes the Euclidean norm. Take a constant spinor ψ on B_{1}(0) normalized such that ∥ψ∥_{L^{p}(B_{1}(0),g_{E})} = 1. Then D^{g_{E}}ψ = 0. Using the identification of spinors of conformal metrics set φ := f^{− \frac{k}{2}} ψ. Then D^{g_{ℍ}}φ = 0 and ∥φ∥_{L^{p}(g_{ℍ})}^{p} = ∫_{B_{1}(0)} f^{k+1− \frac{k}{2} p}|ψ|^{p}dvol_{g_{E}}. Thus,

φ is an L^{p}-harmonic spinor if and only if ∫_{B_{1}(0)}(1 − |x|^{2})^{−k−1+ \frac{k}{2} p}dvol_{g_{E}} < ∞, i.e., if and

only if ∫_{0}^{1}(1 − r^{2})^{−1+ \frac{k}{2} (p−2)}r^{k}dr < ∞. This is true precisely if p > 2 and k > 0. Thus, for all p > 2 and k > 0 the L^{p}-kernel of the Dirac operator on (ℍ^{k+1}, g_{ℍ}) is nontrivial.

10. The L^{p}-spectrum on 𝕄_{c}^{m,k} contains σ_{p}

In this section we complete the proof of Theorem 1.1. In Proposition 8.1 it was shown that the L^{p}-spectrum on 𝕄_{c}^{m,k} is contained in σ_{p}. Thus, the converse remains to be shown. The case N = {y} was solved in Proposition 9.1. Recall that by Lemma B.11 and Example B.12 the Dirac L^{p}-spectrum on 𝕄_{c}^{m,k} is point symmetric, i.e., it is symmetric with respect to the reflection λ ↦ −λ.

Let now μ ∈ ∂σ_{p} with μ^{2} = λ_{0}^{2} + κ^{2}, |Im κ| = ck |\frac{1}{p} − \frac{1}{2} | be given. By Proposition 9.1 and

scaling, we see that κ is in the spectrum of the Dirac operator of ℍ_{c}^{k+1}. Then, by Lemma B.8 κ^{2} is in the L^{p}-spectrum of (D^{ℍ_{c}^{k+1}})^{2}, and by Remark 9.2 there is a sequence ψ_{i} ∈ Γ(Σ_{ℍ_{c}^{k+1}}) with ∥((D^{ℍ_{c}^{k+1}})^{2} − κ^{2})ψ_{i}∥_{L^{p}(ℍ_{c}^{k+1})} → 0 while ∥ψ_{i}∥_{L^{q}(ℍ_{c}^{k+1})} = 1. Moreover, by Remark B.7 there is a ψ ∈ Γ(Σ_{N}) with ∥ψ∥_{L^{q}(N)} = 1 and (D^{N})^{2}ψ = λ_{0}^{2}ψ. Assume that at least one of the dimensions of N and ℍ_{c}^{k+1} is even. Then Σ_{𝕄_{c}} = Σ_{ℍ_{c}^{k+1}} ⊗Σ_{N} and by (1) we have D^{2} = (D^{ℍ_{c}^{k+1}})^{2} + (D^{N})^{2}. We set φ_{i} = ψ_{i} ⊗ ψ. Then

∥(D^{2} − μ^{2})φ_{i}∥_{p} =∥ψ_{i} ⊗ ((D^{N})^{2} − λ_{0}^{2})ψ + ((D^{ℍ_{c}^{k+1}})^{2} − κ^{2})ψ_{i} ⊗ ψ∥_{p}
=∥((D^{ℍ_{c}^{k+1}})^{2} − κ^{2})ψ_{i} ⊗ ψ∥_{p} → 0.

Thus, μ^{2} is in the L^{p}-spectrum of D^{2}. By the point symmetry of the spectrum and by Lemma B.8 both μ and −μ are in the L^{p}-spectrum of D. Similarly we obtain the result if both the dimensions of N and ℍ_{c}^{k+1} are odd by setting φ_{i} := ψ_{i} ⊗ (ψ, ψ) in notation of Section 2.5. Up to now we have shown that all μ ∈ ∂σ_{p} are in the L^{p}-spectrum of the Dirac operator on 𝕄_{c}. Following the same arguments as in the last lines of the proof of Proposition 9.1 the proof of Theorem 1.1 is completed.

Remark 10.1. From Theorem 1.1 and Lemma B.8, we can immediately read off the L^{p}- spectrum of D^{2} on 𝕄_{c}^{m,k}. This consists of the closed parabolic region bounded by

s ∈ ℝ ↦ λ_{0}^{2} − c^{2}k^{2} (\frac{1}{p} − \frac{1}{2})^{2} + s^{2} + 2isck |\frac{1}{p} − \frac{1}{2}| .

Let us compare the L^{p}-spectrum for D^{2} on 𝕄_{c}^{k+1,k} = ℍ^{k+1} (c = 1 and λ_{0} = 0)

s ∈ ℝ ↦ −k^{2} (\frac{1}{p} − \frac{1}{2})^{2} + s^{2} + 2isk (\frac{1}{2} − \frac{1}{p}) ,

with the one of the Laplacian on functions whose L^{p}-spectrum is given by the closed parabolic region bounded by [18, (1.5)]

s ∈ ℝ ↦ k^{2} \frac{1}{p} (1 − \frac{1}{p}) + s^{2} + 2isk (\frac{1}{2} − \frac{1}{p}) .

Up to a shift in the real direction this is the same spectrum. However the qualitative difference is that for p ≠ 2 the spectrum of D^{2} contains negative real numbers, in contrast to the Laplacian.

Appendix A. Function spaces

We want to recall some analytical facts which are helpful to define spinorial function spaces on manifolds. Let (M^{n}, g) be an n-dimensional Riemannian spin manifold with the classical Dirac operator D: H_{1}^{2}(M, Σ_{M}) = dom D ⊂ L^{2}(M, Σ_{M}) → L^{2}(M, Σ_{M}). The set of compactly supported spinors C_{c}^{∞}(M, Σ_{M}) is a core of D, i.e., D is the closure of D|_{C_{c}^{∞}(M,Σ_{M})} w.r.t. the graph norm H_{1}^{2}. A distributional spinor (or distribution with spinor values) is a linear map C_{c}^{∞}(M, Σ_{M}) → ℂ with the usual continuity properties of distributions. Any spinor with regularity L_{loc}^{1} defines a distributional spinor by using the standard L^{2}-scalar product on spinors. Then Dφ can be defined in the sense of distributions. Let H_{1}^{s}(M, Σ_{M}) be the set of distributional spinors φ, such that φ und Dφ are in L^{s}, s ∈ [1, ∞]. Equipped with the norm ∥φ∥_{H_{1}^{s}} := ∥φ∥_{s} + ∥Dφ∥_{s} this is a Banach space. This norm is the graph norm of D viewed as an operator in L^{s} to L^{s}. If we consider the operator D: H_{1}^{s} ⊂ L^{s} → L^{s}, we will shortly write D_{s} in the following.

Lemma A.1. Let 1 ≤ s < ∞. C_{c}^{∞}(M, Σ_{M}) is dense in H_{1}^{s}(M, Σ_{M}).

Proof. Assume that φ ∈ H_{1}^{s}(M, Σ_{M}), s < ∞, is given. For a given point p ∈ M and for any R > 0 one can find a compactly supported smooth function η_{R}: M → [0, 1] such that η_{R} ≡ 1 on B_{R}(p) and such that |∇η_{R}| ≤ R^{−1}. Then one easily sees lim_{R→∞} ∥φ−η_{R}φ∥_{s} = 0. Further we calculate

∥D(φ − η_{R}φ)∥_{s} ≤ ∥∇η_{R} · φ∥_{s} + ∥(1 − η_{R})Dφ∥_{s} → 0 as R → ∞.

Thus the elements with compact support are dense in H_{1}^{s}(M, Σ_{M}). Now if ψ ∈ H_{1}^{s}(M, Σ_{M}) has compact support, it follows from standard results that it can be approximated by smooth compactly supported spinors. □

Thus, for s < ∞, H_{1}^{s}(M, Σ_{M}) is equal to the completion of C_{c}^{∞}(M, Σ_{M}) with respect to the graph norm of D: L^{s} → L^{s}. In particular this implies that for s < ∞ the operator D_{s} is a closed extension of D|_{C_{c}^{∞}(M,Σ_{M})} with core C_{c}^{∞}(M, Σ_{M}). Note that D_{∞} is a closed extension of D|_{C_{c}^{∞}(M,Σ_{M})} as well but C_{c}^{∞}(M, Σ_{M}) is in general no longer a core for this operator. Moreover, in the standard literature for L^{p}-theory of the Laplacian, e.g. [18], the operator for s = ∞ is directly defined to be the adjoint operator for s = 1.

Lemma A.2. Let 1 < s < ∞. On manifolds with bounded geometry, the H_{1}^{s}-norm is equivalent to the norm ∥φ∥_{s} + ∥∇φ∥_{s}.

The proof of the lemma relies on local elliptic estimates which follow from the Calderon-Zygmund inequality, e.g. [20, Theorem 9.9], see also [2, Lemma 3.2.2] for the geometric adaptation.

Appendix B. General notes on the L^{p}-spectrum

In this section we collect general facts on the L^{p}-spectrum of the Dirac operator. Unless stated otherwise, we only assume that (M, g) is complete. We examine the adjoint of the operator D_{s}: L^{s} → L^{s} with respect to the duality pairing ⟨., .⟩: L^{s} × (L^{s})^{∗} → ℂ whose restriction to compactly supported spinors coincides with the hermitian L^{2}-product. We use the convention that this pairing is antilinear in the second component. The adjoint D_{s}^{∗} is an operator in (L^{s})^{∗}. For 1 ≤ s < ∞ and s^{−1} + (s^{∗})^{−1} = 1, (L^{s})^{∗} = L^{s^{∗}} whereas (L^{∞})^{∗} is larger than L^{1}. From the formal self-adjointness of D we see, that D_{s^{∗}}|_{C_{c}^{∞}(M,Σ_{M})} = D_{s}^{∗}|_{C_{c}^{∞}(M,Σ_{M})}. Moreover, we have

Lemma B.1. For all φ ∈ H_{1}^{s} and ψ ∈ H_{1}^{s^{∗}}, 1 ≤ s ≤ ∞, we have

(D_{s}φ, ψ) = (φ, D_{s^{∗}}ψ).

Proof. For 1 < s < ∞, let φ_{i}, ψ_{j} ∈ C_{c}^{∞}(M, Σ_{M}) with φ_{i} → φ in H_{1}^{s} and ψ_{j} → ψ in H_{1}^{s^{∗}}. Then,

∫_{M}⟨D_{s}φ, ψ⟩dvol_{g} ← ∫_{M}⟨D_{s}φ_{i}, ψ_{j}⟩dvol_{g} = ∫_{M}⟨φ_{i}, D_{s^{∗}}ψ_{j}⟩dvol_{g} → ∫_{M}⟨φ, D_{s^{∗}}ψ⟩dvol_{g}

as i, j → ∞. Let now s = 1. For φ ∈ C_{c}^{∞}(M, Σ_{M}) the equality follows from the distributional definition of D_{∞}. The rest follows since C_{c}^{∞}(M, Σ_{M}) is dense in H_{1}^{1}. The remaining case s = ∞ just follows from the last one by interchanging s and s^{∗}. □

Lemma B.2. For all 1 ≤ s < ∞ the operators D_{s^{∗}} and D_{s}^{∗} coincide.

Proof. For ψ ∈ H_{1}^{s^{∗}} Lemma B.1 yields (D_{s}φ, ψ) = (φ, D_{s^{∗}}ψ) for all φ ∈ H_{1}^{s} = dom D_{s}. This implies ψ ∈ dom D_{s}^{∗} and D_{s}^{∗}ψ = D_{s^{∗}}ψ. Hence, H_{1}^{s^{∗}} ⊂ dom D_{s}^{∗} and D_{s}^{∗}|_{H_{1}^{s^{∗}}} = D_{s^{∗}}: H_{1}^{s^{∗}} ⊂ L^{s^{∗}} → L^{s^{∗}}. It remains to show that dom D_{s}^{∗} ⊂ H_{1}^{s^{∗}}: Let ψ ∈ dom D_{s}^{∗} ⊂ (L^{s})^{∗} = L^{s^{∗}}. Then there is a ρ ∈ L^{s^{∗}} such that for all φ ∈ dom D_{s} it holds (D_{s}φ, ψ) = (φ, ρ). In particular, this is true for all φ ∈ C_{c}^{∞}(M, Σ_{M}). In other words D_{s}^{∗}ψ = ρ in the sense of distributions. Thus, ψ ∈ H_{1}^{s^{∗}}. □

Since φ ∈ H_{1}^{s} ∩ H_{1}^{r} implies D_{s}φ = D_{r}φ we often denote all those Dirac operators in the following just by D. Moreover, a closed operator P: dom P ⊂ V_{1} → V_{2} between Banach spaces V_{i}, and with dense domain dom P, will be called invertible if there exists a bounded inverse P^{−1} : V_{2} → V_{1}. We will use the phrase “P has a bounded inverse” synonymously.

Lemma B.3. Let 1 ≤ s < ∞.

(i) If \overline{μ} is in the L^{s^{∗}}−spectrum of the Dirac operator where (s^{∗})^{−1} + s^{−1} = 1, then μ is in its L^{s}−spectrum.

(ii) Let D_{s} − μ be invertible. Then, (D_{s^{∗}} − μ̄)^{−1} = ((D_{s} − μ)^{−1})^{∗} and ∥(D_{s^{∗}} − μ̄)^{−1}∥ = ∥(D_{s} − μ)^{−1}∥.

Proof. We prove this for μ = 0. For arbitrary μ this is done analogously. Assume that 0 is not in the L^{s}-spectrum of D, i.e., it has a bounded inverse E = D^{−1} : L^{s} → L^{s} with range ran E = H_{1}^{s}. Let φ ∈ L^{s^{∗}}. Since E is bounded, f: L^{s} → ℂ, ρ ↦ (Eρ, φ) is a bounded functional and, thus, f is in the dual space of L^{s}, i.e., there is ψ ∈ L^{s^{∗}} with (ρ, ψ) = f(ρ) = (Eρ, φ) for all ρ ∈ L^{s}. Hence, φ ∈ dom E^{∗}, i.e., dom E^{∗} = L^{s^{∗}}. Now we can estimate for all φ ∈ H_{1}^{s} and all ψ ∈ L^{s^{∗}} that (Dφ, E^{∗}ψ) = (EDφ, ψ) = (φ, ψ) which implies E^{∗}ψ ∈ dom D^{∗} and D^{∗}E^{∗}ψ = ψ. Thus, ran D^{∗} = L^{s^{∗}} and D^{∗}E^{∗} = Id: L^{s^{∗}} → L^{s^{∗}}. If ρ ∈ L^{s} and φ ∈ dom D^{∗}, we get (ρ, E^{∗}D^{∗}φ) = (Eρ, D^{∗}φ) = (DEρ, φ) = (ρ, φ). Hence, E^{∗}D^{∗} = Id: dom D^{∗} → dom D^{∗}. Together with the corresponding statement from above this gives that (D^{−1})^{∗} = (D^{∗})^{−1}. Thus, 0 is not in the L^{s^{∗}}-spectrum of D. This proves (i) and the first claim of (ii). The operator norm of an operator and its adjoint coincide, see [31, Thm VI.2]. Thus, the equality of the operator norms follows. □

Corollary B.4. If D: H_{1}^{q} → L^{q} has a bounded inverse for some q ∈ (1, ∞). Then as an operator from H_{1}^{s} → L^{s} it has a bounded inverse for all s ∈ [q_{1}, q_{2}] where q_{1} = min{q, q^{∗}}, q_{2} = max{q, q^{∗}}, and (q^{∗})^{−1} + q^{−1} = 1. In particular, the L^{2}-spectrum of D is a subset of the L^{q}-spectrum.

Proof. This Lemma follows directly from the Riesz-Thorin Interpolation Theorem 2.2 (using D = C_{c}^{∞}(M, Σ_{M})) and Lemma B.3. □

Lemma B.5. Let 1 ≤ s ≤ ∞. Let R_{s} = ℂ \ Spec_{L^{s}}(D) be the resolvent set of D: L^{s} → L^{s}. Then, the resolvent

μ ∈ R_{s} ↦ (D − μ)^{−1} ∈ B(L^{s})

is analytic, i.e., the map is locally given by a convergent power series with coefficients in B(L^{s}). Here, B(L^{s}) denotes the set of bounded operators from L^{s} to itself.

See [42, VIII.2 Theorem 1] for a proof. For rounding up our presentation we will next add a lemma not needed in our context but maybe helpful to other applications.

Lemma B.6.

(1) The operator D: H_{1}^{s} ⊂ L^{s} → L^{s}, s ∈ [1, ∞], is an invertible map onto its image if and only if there is a constant C > 0 with ∥Dφ∥_{s} ≥ C∥φ∥_{s} for all φ ∈ H_{1}^{s}.

(2) Under the above conditions the image D(H_{1}^{s}) is closed.

(3) Let s^{−1} + (s^{∗})^{−1} = 1, s < ∞, and assume the conditions from above. Then D is surjective if and only if there is a C > 0 with ∥Dφ∥_{s^{∗}} ≥ C∥φ∥_{s^{∗}} for all φ ∈ H_{1}^{s^{∗}}.

Proof. (1) The proof is straightforward.

(2) The operator D: H_{1}^{s} → D(H_{1}^{s}), where the latter space is equipped with the L^{s}-norm, is a bijective bounded linear map. Hence, D(H_{1}^{s}) is a complete subspace of L^{s} and thus closed.

(3) Suppose that D(H_{1}^{s}) is a proper subspace of L^{s}. Due to Hahn-Banach there is a non-zero continuous functional ψ: L^{s} → ℂ vanishing on D(H_{1}^{s}). We interpret ψ as an element in L^{s^{∗}} using the Riesz representation theorem, i.e. ψ ∈ L^{s^{∗}} is orthogonal on D(H_{1}^{s}). Then, ψ ∈ dom (D_{s})^{∗}, and we even have D_{s}^{∗}ψ = 0. Hence, by Lemma B.2 ψ ∈ H_{1}^{s∗}. This contradicts the estimate. Now assume that D is surjective. Then there is a bounded operator D^{−1} : L^{s} → L^{s}, inverse to D. Thus (D^{−1})^{∗}: L^{s^{∗}} → L^{s^{∗}} is bounded as well, and (D^{−1})^{∗} is the inverse of D^{∗}: H_{1}^{s^{∗}} → L^{s^{∗}}. The fact that the latter map has a bounded inverse is equivalent to the existence of a constant C > 0 with ∥Dφ∥_{s^{∗}} ≥ C∥φ∥_{s^{∗}}.

□

Remark B.7. The L^{s}-spectrum of the Dirac operator D on a closed manifold (M^{m}, g) is independent of s. We sketch the proof: Let φ be an L^{2}-eigenspinor of D. Then regularity theory implies that φ ∈ C^{∞}(M, Σ_{M}) and, hence, φ ∈ L^{s} for all 1 ≤ s ≤ ∞. In particular, Spec_{L^{2}}^{M}(D) ⊂ Spec_{L^{s}}^{M}(D). Let now μ ∉ Spec_{L^{2}}^{M}(D), i.e., (D−μ)^{−1} : L^{2} → L^{2} is bounded. Let G(x, y) be the unique Green function of D − μ, see Proposition 3.2. Then, ∫_{M} |G(., y)|^{2}dy is bounded uniformly in y. Hölder’s inequality implies that also ∫_{M} |G(., y)|dy is bounded uniformly in y. Hence, (D −μ)^{−1} : L^{1} → L^{1} is a bounded operator. Then interpolation gives that (D − μ)^{−1} : L^{s} → L^{s} is bounded for all 1 ≤ s < 2. Because of Spec_{L^{2}}(D) ⊂ ℝ the same is true for (D − μ̄)^{−1} : L^{s} → L^{s}, and by using Lemma B.3 we get that (D − μ)^{−1} : L^{s} → L^{s} is bounded for all 2 < s < ∞. It remains s = ∞: Let r > m. Then by the Sobolev Embedding Theorem H_{1}^{r} ↪ L^{∞} is bounded. Moreover, by the discussion above and using the fact that H_{1}^{r} carries the graph norm of D we know that (D − μ)^{−1} : L^{r} → H_{1}^{r} is bounded for μ ∉ Spec_{L^{2}}^{M}(D) the Hölder inequality gives that

(D − μ)^{−1} : L^{∞} → L^{r} → H_{1}^{r} → L^{∞}

is bounded.

Lemma B.8. Let 1 ≤ s ≤ ∞, and let Spec_{L^{s}}^{M}(D) ≠ ℂ. Then the complex number μ^{2} is in the L^{s}-spectrum of D^{2} if and only if μ or −μ is in the L^{s}-spectrum of D.

Proof. We start with the “only if” part. So assume that both μ and −μ are not in the L^{s}- spectrum of D. Then we have bounded operators (D−μ)^{−1} : L^{s} → L^{s} and (D+μ)^{−1} : L^{s} → L^{s}. It is then easy to verify that (D − μ)^{−1} ∘ (D + μ)^{−1} : L^{s} → L^{s} is a bounded inverse of D^{2} − μ^{2} = (D + μ) ∘ (D − μ). Thus μ^{2} is not in the L^{s}-spectrum of D^{2}. In order to prove the “if” statement, we assume that μ^{2} is not in the spectrum of D^{2}. Then D^{2} − μ^{2} has a bounded inverse P := (D^{2} − μ^{2})^{−1} : L^{s} → L^{s}. Let ψ ∈ P (L^{s}). Then ψ ∈ L^{s} and D^{2}ψ ∈ L^{s}. Next we will show that this implies Dψ ∈ L^{s}. For that we choose λ ∉ Spec_{L^{s}}^{M}(D). Then Dψ = (D − λ)^{−1}(D^{2} − λ^{2})ψ − λψ, and hence Dψ ∈ L^{s}. Thus, P (L^{s}) ⊂ H_{1}^{s}. Hence Q_{1} := (D ± μ) ∘ P is a bounded operator with dom Q_{1} = L^{s}, and one easily checks that this a right inverse to (D ∓ μ). Similarly, one shows that Q_{2} := P ∘ (D ± μ) is a left inverse of (D ∓ μ). A priori Q_{2} is only defined on H_{1}^{s}, but using Q_{1} = Q_{1} ∘ (D ∓ μ) ∘ Q_{2} = Q_{2} it is clear that Q_{2} and Q_{1} coincide on H_{1}^{s}. So the integral kernels of Q_{1} and Q_{2} have to coincide, so Q_{1} is a left and right inverse of (D ∓ μ) and thus ±μ is not in the spectrum of D. □

Remark B.9. In the case 1 < s < ∞ and M of bounded geometry, one can also prove that Spec_{L^{s}}^{M}(D) = ℂ implies Spec_{L^{s}}^{M}(D^{2}) = ℂ: As in the proof of the “if” statement from above one has to show that Dψ ∈ L^{s}. This can be proven using regularity theory on manifolds of bounded geometry.

Lemma B.10 (Pointwise symmetries). Let 1 ≤ s ≤ ∞. Let (M, g) be an m-dimensional Riemannian spin manifold.

(i) m ≡ 0 mod 2: The number μ is in the L^{s}-spectrum of D if and only if −μ is in the L^{s}-spectrum of D if and only if μ̄ is in the L^{s}-spectrum of D.

(ii) m ≡ 1 mod 4: The number μ is in the L^{s}-spectrum of D if and only if −μ̄ is in the L^{s}-spectrum of D.

(iii) m ≡ 3 mod 4: The number μ is in the L^{s}-spectrum of D if and only if μ̄ is in the L^{s}-spectrum of D.

Proof. By [19, Prop. p. 31] we have a map α: Σ_{m} → Σ_{m} that is

• a Spin(m)-equivariant real structure that anticommutes with Clifford multiplication if m ≡ 0, 1 mod 8.

• a Spin(m)-equivariant quaternionic structure that commutes with Clifford multiplication if m ≡ 2, 3 mod 8.

• a Spin(m)-equivariant quaternionic structure that anticommutes with Clifford multiplication if m ≡ 4, 5 mod 8.

• a Spin(m)-equivariant real structure that commutes with Clifford multiplication if

m ≡ 6, 7 mod 8. Note that by definition real structure means that α^{2} = Id and α(iv) = −iα(v). Moreover, quaternionic structure means that α^{2} = − Id and α(iv) = −iα(v). Due to the Spin(m)-equivariance α induces a fiber preserving map α̃ on the spinor bundle with the same properties as above. Thus,

(D − μ) ∘ α̃(φ) = \begin{cases}α̃ ∘ (−D − μ̄)(φ) & m ≡ 0, 1 mod 4 \\ α̃ ∘ (D − μ̄)(φ) & m ≡ 2, 3 mod 4. \end{cases}

Thus, if μ is in the L^{s}-spectrum of D then −μ̄ (resp. μ̄) in the L^{s}-spectrum of D for m ≡ 0, 1 (resp. 2,3) mod 4. This gives (ii) and (iii). If m is even, then D(ω_{M} · φ) = −ω_{M} · Dφ. Thus, the spectrum is symmetric when reflected on the imaginary axis. Together with the symmetries from above, (i) follows. □

Lemma B.11 (Orientation reversing isometry). Let 1 ≤ s ≤ ∞. Assume there is an orientation reversing isometry f: M^{m} → M^{m} that “lifts” to the spin structure as described in the proof. Then μ is in the L^{s}-spectrum of D if and only if −μ is in the L^{s}-spectrum of D.

Proof. The proof follows the lines of [4, Appendix A]. In this reference, f is required to be a reflection at a hyperplane of M. But this doesn’t change the part we need: We lift f to the bundle P_{SO(m)}M of oriented orthonormal frames by mapping the frame E = (e_{1}, …, e_{m}) to f_{∗}E = (−df(e_{1}), df(e_{2}), …, df(e_{m})), so f_{∗}: P_{SO(m)}M → P_{SO(m)}M. Since f is an orientation reserving isometry,

f_{∗}(EA) = f_{∗}(E)JAJ for all A ∈ SO(m)

where J = diag(−1, 1, 1, …, 1). The map f is assumed to lift to the spin structure, i.e., there is a lift f̃_{∗}: P_{Spin(m)}(M) → P_{Spin(m)}(M) with ϑ ∘ f̃_{∗} = f_{∗} ∘ ϑ where ϑ denotes the double covering ϑ: P_{Spin(m)}(M) → P_{SO(m)}(M). By [4, Lemma A.1 and Lemma A.4], f then lifts to a map f_{♯}: Σ_{M} → Σ_{M} on the spinor bundle which fulfils f_{♯}(Dφ) = −D(f_{♯}φ). □

Example B.12.

(i) Let M^{n+1} be a Riemannian spin manifold with a spin structure ϑ as above. Assume that up to isomorphism this is the unique spin structure on M. Let f: M → M be an orientation reversing isometry. By pulling back the double covering P_{Spin}M → P_{SO}M by f_{∗} we obtain the double covering f^{∗}ϑ : f^{∗}P_{Spin}M → P_{SO}M. We then turn f^{∗}P_{Spin}M into a Spin(n + 1)-principal bundle by conjugating the action of Spin(n + 1) on P_{Spin}M with Clifford multiplication with e_{0}. Then f^{∗}ϑ is a spin structure on M. Thus an isomorphism from ϑ to f^{∗}ϑ yields a map f_{♯} as above. (ii) Consider the map f = f_{1} × id: 𝕄_{c}^{m} = ℍ_{c}^{k+1} × N^{n} → 𝕄_{c}^{m,k} where f_{1} is an orientation reversing isometry as in (i). Then, f is again an orientation reversing isometry. Using P_{SO}(ℍ_{c} × N) = (P_{SO}(ℍ_{c}^{k+1}) × P_{SO}(N)) ×ξ̄ SO(m) where ξ̄: SO(k + 1) × SO(n) → SO(m) is the standard embedding and using the analogous describtion for P_{Spin}(ℍ_{c} × N), see Section 2.5, one see that also f lifts to the spin structure.

Appendix C. Dirac eigenvalues of generic metrics

Proposition C.1. Let (M, g) be a closed, connected Riemannian spin manifold, let μ ∈ ℝ. Let U ⊂ M be a nonempty open subset. In case that μ = 0, assume additionally that the α-genus of M is zero. Then, there is a metric g̃ on M with g̃ = g on M \ U and ker (Dg̃ − μ) = {0}.

Proof. For μ = 0, the proposition follows from [4, Theorem 1.1]. For μ ≠ 0, the proof is a direct consequence of the following lemma. □

Lemma C.2. Let (M, g) be a closed, connected Riemannian spin manifold, let μ ∈ ℝ \ {0}, and let U ⊂ M be a nonempty open subset. Then there is a function f ∈ C^{∞}(M, ℝ^{+}) with f|_{M\U} ≡ 1 such that ker (D^{fg} − μ) = {0}.

Proof. Choose f ∈ C^{∞}(M, ℝ^{+}) with f|_{M\U} ≡ 1 such that d = dim(E_{f,μ} := ker (D^{fg} − μ)) is minimal. Assume d > 0, and set g_{0} = fg. For α ∈ C^{∞}(M) with supp α ⊂ U and t close to 0 we define g_{t} := (1 + tα)fg. Then by [10] there are real analytic functions μ_{1}, …, μ_{d}: (−ε, ε) → ℝ with μ_{i}(0) = μ such that Spec_{L^{2}}^{M}(D^{g_{t}}) ∩ (μ − δ, μ + δ) = {μ_{1}(t), …, μ_{d}(t)} including multiplicities. It is shown in [10] that there is an orthonormal basis (ψ^{(1)}, …, ψ^{(d)}) of E_{f,μ} depending on the choice of α such that

\frac{d}{dt}|_{t=0}μ_{i}(t) = −\frac{1}{2} ∫_{M}⟨αg_{0}, Q_{ψ^{(i)}}⟩dvol_{g_{0}}

where Q_{ψ}(X, Y) = \frac{1}{2}Re ⟨X · ∇_{Y} ψ + Y · ∇_{X}ψ, ψ⟩. Thus,

⟨g_{0}, Q_{ψ^{(i)}}⟩ = ∑_{r}⟨e_{r} · ∇_{e_{r}}ψ^{(i)}, ψ^{(i)}⟩ = μ|ψ^{(i)}|^{2}.

As d is minimal, we see that \frac{d}{dt}|_{t=0}μ_{i}(t) = 0, and thus for all α as above

−\frac{1}{2} ∫_{M} αμ ∑_{i=1}^{d} |ψ^{(i)}|^{2}dvol_{g_{0}} = 0.

Note that φ := ∑_{i=1}^{d} |ψ^{(i)}|^{2} ∈ C^{∞}(M) does not depend on the choice of α. This can be seen by direct calculation with base change matrices or alternatively by observing that φ is the pointwise trace of the integral kernel of the projection to E_{f,μ}. With μ ≠ 0 this implies that φ and thus all ψ^{(i)} vanish on U. The unique continuation principle implies then ψ^{(i)} ≡ 0 which gives a contradiction. □

References

[1] Abels, H. Pseudodifferential and singular integral operators. De Gruyter Graduate Lectures. De Gruyter, Berlin, 2012. An introduction with applications. 3

[2] Ammann, B. A variational problem in conformal spin geometry. Habilitation (Hamburg, 2003). A

[3] Ammann, B., Dahl, M., and Humbert, E. Surgery and the spinorial τ-invariant. Comm. Partial Differential Equations 34, 10-12 (2009), 1147–1179. 1

[4] Ammann, B., Dahl, M., and Humbert, E. Harmonic spinors and local deformations of the metric. Math. Res. Lett. 18, 5 (2011), 927–936. B, C

[5] Ammann, B., and Große, N. Relations between threshold constants for Yamabe type bordism invariants. The Journal of Geometric Analysis (Online first) (2015). http://arxiv.org/abs/1502.05232

    and http://dx.doi.org/10.1007/s12220-015-9651-1. 1, 1

[6] Ammann, B., Humbert, E., and Morel, B. Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds. Comm. Anal. Geom. 14, 1 (2006), 163–182. 3.1

[7] Bär, C. Das Spektrum von Dirac-Operatoren. Bonner Mathematische Schriften [Bonn Mathematical Publications], 217. Universität Bonn, Mathematisches Institut, Bonn, 1991. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1990. 9

[8] Bär, C. Extrinsic bounds for eigenvalues of the Dirac operator. Ann. Global Anal. Geom. 16, 6 (1998), 573–596. 2.5, 4.1, 4.2, 4.1

[9] Bär, C., Gauduchon, P., and Moroianu, A. Generalized cylinders in semi-Riemannian and Spin geometry. Math. Z. 249, 3 (2005), 545–580. 4.2 [10] Bourguignon, J.-P., and Gauduchon, P. Spineurs, opérateurs de Dirac et variations de métriques.

Comm. Math. Phys. 144, 3 (1992), 581–599. C

[11] Bunke, U. The spectrum of the Dirac operator on the hyperbolic space. Math. Nachr. 153 (1991), 179–190. 2.6

[12] Cahen, M., Gutt, S., Lemaire, L., and Spindel, P. Killing spinors. Bull. Soc. Math. Belg. Sér. A 38 (1986), 75–102 (1987). 9

[13] Camporesi, R., and Higuchi, A. On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces. J. Geom. Phys. 20, 1 (1996), 1–18. 2.6

[14] Charalambous, N. On the L^{p} independence of the spectrum of the Hodge Laplacian on non-compact manifolds. J. Funct. Anal. 224, 1 (2005), 22–48. 1

[15] Coddington, E. A., and Levinson, N. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. 6

[16] Davies, E. B. Heat kernels and spectral theory, vol. 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990. 1, 2.3

[17] Davies, E. B. L^{p} spectral theory of higher-order elliptic differential operators. Bull. London Math. Soc. 29, 5 (1997), 513–546. 1

[18] Davies, E. B., Simon, B., and Taylor, M. L^{p} spectral theory of Kleinian groups. J. Funct. Anal. 78, 1 (1988), 116–136. 1, 9, 10.1, A

[19] Friedrich, T. Dirac Operators in Riemannian Geometry. Graduate Studies in Mathematics 25. AMS, Providence, Rhode Island, 2000. 2.3, B

[20] Gilbarg, D., and Trudinger, N. S. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. A

[21] Große, N. Solutions of the equation of a spinorial Yamabe-type problem on manifolds of bounded geometry. Comm.Part. Diff. Eq. 37 (2012), 58–76. 1

[22] Hempel, R., and Voigt, J. The spectrum of a Schrödinger operator in L_{p}(R^{ν}) is p-independent. Comm. Math. Phys. 104, 2 (1986), 243–250. 1

[23] Hempel, R., and Voigt, J. On the L_{p}-spectrum of Schrödinger operators. J. Math. Anal. Appl. 121, 1 (1987), 138–159. 1

[24] Ji, L., and Weber, A. Dynamics of the heat semigroup on symmetric spaces. Ergodic Theory Dynam. Systems 30, 2 (2010), 457–468. 1

[25] Ji, L., and Weber, A. L^{p} spectral theory and heat dynamics of locally symmetric spaces. J. Funct. Anal. 258, 4 (2010), 1121–1139. 1

[26] Ji, L., and Weber, A. The L^{p} spectrum and heat dynamics of locally symmetric spaces of higher rank. Ergodic Theory Dynam. Systems 35, 5 (2015), 1524–1545. 1

[27] Kordyukov, Y. A. L^{p}-theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23, 3 (1991), 223–260. 1

[28] Lawson, H. B., and Michelsohn, M.-L. Spin Geometry. Princeton University Press, Princeton, 1989. 3.3, 3

[29] Lohoué, N., and Rychener, T. Die Resolvente von Δ auf symmetrischen Räumen vom nichtkompakten Typ. Comment. Math. Helv. 57, 3 (1982), 445–468. 1

[30] Reed, M., and Simon, B. Methods of modern mathematical physics II. Fourier analysis, selfadjointness. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1975. 2.3

[31] Reed, M., and Simon, B. Methods of modern mathematical physics I. Functional analysis, second ed. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. B

[32] Saloff-Coste, L. Uniformly elliptic operators on Riemannian manifolds. J. Differential Geom. 36, 2 (1992), 417–450. 1

[33] Sepanski, M. R. Compact Lie groups, vol. 235 of Graduate Texts in Mathematics. Springer, New York, 2007. 5

[34] Sturm, K.-T. On the L^{p}-spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118, 2 (1993), 442–453. 1

[35] Taylor, M. E. Pseudodifferential operators. Princeton University Press, Princeton, N.J., 1981. 3

[36] Taylor, M. E. L^{p}-estimates on functions of the Laplace operator. Duke Math. J. 58, 3 (1989), 773–793. 1

[37] Wang, J. The spectrum of the Laplacian on a manifold of nonnegative Ricci curvature. Math. Res. Lett. 4, 4 (1997), 473–479. 1

[38] Warner, F. W. Foundations of differentiable manifolds and Lie groups. Reprint. Graduate Texts in Mathematics, 94. New York etc.: Springer-Verlag. IX, 272 p., 57 figs. DM 58.00; $ 22.50 , 1983. 4.1

[39] Weber, A. Heat kernel estimates and L^{p}-spectral theory of locally symmetric spaces. Dissertation,

    2007. http://dx.doi.org/10.5445/KSP/1000005774. 1

[40] Werner, D. Funktionalanalysis, extended ed. Springer-Verlag, Berlin, 2000. 2.2

[41] Wolf, J. A. Essential self-adjointness for the Dirac operator and its square. Indiana Univ. Math. J. 22 (1972/73), 611–640. 2.3 [42] Yosida, K. Functional analysis, sixth ed., vol. 123 of Grundlehren der Mathematischen Wissenschaften

[Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin-New York, 1980. B

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Email address: bernd.ammann@mathematik.uni-regensburg.de

Mathematisches Institut, Universität Freiburg, 79104 Freiburg, Germany

Email address: nadine.grosse@math.uni-freiburg.de