New York Journal of Mathematics

New York J. Math. 4 (1998) 97–125.

The Normal Symbol on Riemannian Manifolds

Markus J. Pflaum

Abstract. For an arbitrary Riemannian manifold X and Hermitian vector bundles E and F over X we define the notion of the normal symbol of a pseudodifferential operator P from E to F. The normal symbol of P is a certain smooth function from the cotangent bundle T^{∗}X to the homomorphism bundle Hom(E, F) and depends on the metric structures, resp. the corresponding connections on X, E and F. It is shown that by a natural integral formula the pseudodifferential operator P can be recovered from its symbol. Thus, modulo smoothing operators, resp. smoothing symbols, we receive a linear bijective correspondence between the space of symbols and the space of pseudodifferential operators on X. This correspondence comprises a natural transformation between appropriate functors. A formula for the asymptotic expansion of the product symbol of two pseudodifferential operators in terms of the symbols of its factors is given. Furthermore an expression for the symbol of the adjoint is derived. Finally the question of invertibility of pseudodifferential operators is considered. For that we use the normal symbol to establish a new and general notion of elliptic pseudodifferential operators on manifolds.

Contents

1. Introduction 97 2. Symbols 99 3. Pseudodifferential operators on manifolds 101 4. The symbol map 105 5. The symbol of the adjoint 112 6. Product expansions 116 7. Ellipticity and normal symbol calculus 122 References 125

1. Introduction

It is a well-known fact that on Euclidean space one can construct a canonical linear isomorphisms between symbol spaces and corresponding spaces of pseudodifferential operators. Furthermore one has natural formulas which represent a

Received October 14, 1997. Mathematics Subject Classification. 35S05, 58G15. Key words and phrases. pseudodifferential operators on manifolds, asymptotic expansions,

symbol calculus .

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York

ISSN 1076-9803/98

pseudodifferential operator in terms of its symbol, resp. which give an expression for the symbol of a pseudodifferential operator. By using symbols one gets much insight in the structure of pseudodifferential operators on Euclidean space. In particular they give the means to construct a (pseudo) inverse of an elliptic differential operator on ℝ^{d}.

Compared to the symbol calculus for pseudodifferential operators on ℝ^{n} it seems that the symbol calculus for pseudodifferential operators on manifolds is not that well-established. But as the pure consideration of symbols and operators in local coordinates does not reveal the geometry and topology of the manifold one is working on, it is very desirable to build up a general theory of symbols for pseudodifferential operators on manifolds. In his articles [13, 14] Widom gave a proposal for a symbol calculus on manifolds. By using a rather general notion of a phasefunction, Widom constructs a map from the space of pseudodifferential operators on manifolds to the space of symbols and shows by an abstract argument for the case of scalar symbols that this map is bijective modulo smoothing operators, resp. symbols. In our paper we introduce a symbol calculus for pseudodifferential operators between vector bundles having the feature that both the symbol map and its inverse have a concrete representation. In particular we thus succeed in giving an integral representation for the inverse of our symbol map or in other words for the operator map. Essential for our approach is an appropriate notion of a phasefunction. Because of our special choice of a phasefunction the resulting symbol calculus is natural in a category theoretical sense.

Using the integral representation for the operator map, it is possible to write down a formula for the symbol of the adjoint of a pseudodifferential operator and for the symbol of the product of two pseudodifferential operators.

The normal symbol calculus will furthermore give us the means to build up a natural notion of elliptic symbols, respectively elliptic pseudodifferential operators on manifolds. It generalizes the classical notion of ellipticity as defined for example in Hörmander [7] and also the concept of ellipticity introduced by Douglis, Nirenberg [2]. Our framework of ellipticity allows the construction of parametrices of nonclassical, respectively nonhomogeneous, elliptic pseudodifferential operators. Moreover we do not need principal symbols for defining ellipticity. Instead we define elliptic operators in terms of their normal symbol, as the globally defined normal symbol of a pseudodifferential operator carries more information than its principal symbol. Moreover, the normal symbol of a pseudodifferential operator shows immediately whether the corresponding operator is invertible modulo smoothing operators or not.

Let us also mention that another application of the normal symbol calculus on manifolds lies in quantization theory. There one is interested in a quantization map associating pseudodifferential operators to certain functions on a cotangent bundle in a way that Dirac’s quantization condition is fulfilled. But the inverse of a symbol map does exactly this, so it is a quantization map. See Pflaum [8, 9] for details.

Note that our work is also related to the recent papers of Yu. Safarov [11, 10].

2. Symbols

First we will define the notion of a symbol on a Riemannian vector bundle. Let us assume once and for all in this article that μ, ρ, δ ∈ ℝ are real numbers such that 0 ≤ δ < ρ ≤ 1 and 1 ≤ ρ + δ. The same shall hold for triples μ̃, ρ̃, δ̃ ∈ ℝ.

We sometimes use properties of symbols on ℝ^{d} ×ℝ^{N}. As these are well-described in the mathematics literature we only refer the interested reader to Hörmander [6, 7], Shubin [12] or Grigis, Sjöstrand [5] for a general introduction to symbol theory on ℝ^{d} × ℝ^{N} and for proofs.

Definition 2.1. Let E → X be a Riemannian or Hermitian vector bundle over the smooth manifold X, ϱ : T X → X its tangent bundle and π : T^{∗}X → X its cotangent bundle. Then an element a ∈ C^{∞}(π^{∗}(E|_{U})) with U ⊂ X open is called a symbol over U ⊂ X with values in E of order μ and type (ρ, δ) if for every trivialization (x, ζ) : T^{∗}X|_{V} → ℝ^{2d} and every trivialization Ψ : E|V → V × ℝ^{N} with V ⊂ U open the following condition is satisfied:

(Sy) For every α, β ∈ ℕ^{d} and every K ⊂ V compact there exists C = C_{K,α,β} > 0 such that for every ξ ∈ T^{∗}X|_{K} the inequality

||\frac{∂^{|α|}}{∂x^{α}} \frac{∂^{|β|}}{∂ζ^{β}} Ψ(a(ξ))|| ≤ C (1 + ||ξ||)^{μ+δ|α|−ρ|β|} (1)

is valid.

The space of these symbols is denoted by S_{ρ,δ}^{μ}(U, T^{∗}X, E) or shortly S_{ρ,δ}^{μ}(U, E). It gives rise to the sheaf S_{ρ,δ}^{μ}(·, T^{∗}X, E) of symbols on X with values in E of order μ and type (ρ, δ).

By defining S_{ρ,δ}^{−∞}(U, E) = ⋂_{μ∈ℕ} S_{ρ,δ}^{μ}(U, E) and S_{ρ,δ}^{∞}(U, E) = ⋃_{μ∈ℕ} S_{ρ,δ}^{μ}(U, E) we get the sheaf S_{ρ,δ}^{−∞}(·, E) of smoothing symbols resp. the sheaf S_{ρ,δ}^{∞}(·, E) of symbols of type (ρ, δ).

In case E is the trivial bundle X × ℂ of a Riemannian manifold X we write S_{ρ,δ}^{μ} with μ ∈ ℝ ∪ {∞, −∞} for the corresponding symbol sheaves S_{ρ,δ}^{μ}(·, X × ℂ).

The space of symbols S_{ρ,δ}^{μ}(U, T X, E) consists of functions a ∈ C^{∞}(ϱ^{∗}(E|_{U})) such that a fulfills condition (Sy). It gives rise to sheaves S_{ρ,δ}^{μ}(·, T X, E), S_{ρ,δ}^{∞}(·, T X, E) and S_{ρ,δ}^{−∞}(·, T X, E).

It is possible to extend this definition to one of symbols over conic manifolds but we do not need a definition in this generality and refer the reader to Duistermaat [3] and Pflaum [8].

We want to give the symbol spaces S_{ρ,δ}^{μ}(U, E) a topological structure. So first choose a compact set K ⊂ U, a (not necessarily disjunct) partition K = ⋃_{ι∈J} K_{ι} of K

into compact subsets together with local trivializations (x_{ι}, ζ_{ι}) : T^{∗}X|_{V_{ι}} → ℝ^{2d} and trivializations Ψ_{ι} : E|V_{ι} → V_{ι} × ℝ^{N} such that K_{ι} ⊂ V_{ι} ⊂ U and V_{ι} open. Then we can attach for any α, β ∈ ℝ^{d} a seminorm p = p_{(K_{ι},(x_{ι},ζ_{ι})),α,β} : S_{ρ,δ}^{μ}(U, E) → ℝ^{+}∪{0} to the symbol spaces S_{ρ,δ}^{μ}(U, E) by

p(a) = sup_{ι∈J} \{\frac{||\frac{∂^{|α|}}{∂x_{ι}^{α}} \frac{∂^{|β|}}{∂ζ_{ι}^{β}} Ψ_{ι}(a(ξ))||}{(1 + |ξ|)^{μ+|α|δ−|β|ρ}} : ξ ∈ T^{∗}X|_{K_{ι}} \} . (2)

The system of these seminorms gives S_{ρ,δ}^{μ}(U, E) the structure of a Fréchet space such that the restriction morphisms S_{ρ,δ}^{μ}(U, E) → S_{ρ,δ}^{μ}(V, E) for V ⊂ U open are continuous. Additionally we have natural and continuous inclusions S_{ρ,δ}^{μ}(U, E) ⊂ Sμ̃ ; ρ̃,δ̃(U, E) for μ̃ ≥ μ, ρ̃ ≤ ρ and δ̃ ≥ δ. Like in the case of symbols on ℝ^{d} × ℝ^{N} one can show that pointwise multiplication of symbols is continuous with respect to the Fréchet topology on S_{ρ,δ}^{μ}(U, E).

Proposition 2.2. Let a ∈ C^{∞}(π^{∗}(E|_{U})). If there exists an open covering of U by patches V_{ι} with local trivializations (x_{ι}, ζ_{ι}) : T^{∗}X|_{V_{ι}} → ℝ^{2d} and trivializations Ψ_{ι} : E|V_{ι} → V_{ι} × ℝ^{N} such that for every (x_{ι}, ζ_{ι}) the above condition (Sy) holds, then a is a symbol of order μ and type (ρ, δ).

Proof. Let (x, ζ) : T^{∗}X|_{V} → ℝ^{2d}, V ⊂ U open be a trivialization. Then we can write on V_{ι} ∩ V

\frac{∂^{|α|}}{∂x^{α}} \frac{∂^{|β|}}{∂ζ^{β}} = ∑_{{α̃+α′≤α \atop ι}} (f_{α̃,α′,β} ∘ π) ζ_{ι}^{α′} \frac{∂^{|α̃|}}{∂x_{ι}α̃} \frac{∂^{|β+α′|}}{∂ζ_{ι}^{β+α′}} , (3)

where f_{α̃,α′,β} ∈ C^{∞}(V_{ι} ∩ V). As ρ + δ ≥ 1 holds and (Sy) is true for (x_{ι}, ζ_{ι}), the claim now follows. □

Example 2.3. (i) Let X be a Riemannian manifold. Then every smooth function a : T^{∗}U → ℂ which is a polynomial function of order ≤ μ on every fiber lies in S_{1,0}^{μ}(U). In particular we have an embedding D_{0} → S_{ρ,δ}^{∞}, where D_{0} is the sheaf of polynomial symbols on X, i.e., for every U ⊂ X open D_{0}(U) = {f : T^{∗}U → ℂ : f|_{T_{x}^{∗}X} is a polynomial for every x ∈ U}.

(ii) Again let X be Riemannian. Then the mapping l : T^{∗}X → ℂ, ξ ↦ ||ξ||^{2} is a symbol of order 2 and type (1, 0) on X, but not one of order μ < 2. Next regard the function a : T^{∗}X → ℂ, ξ ↦ \frac{1}{1+||ξ||^{2}} . This function is a symbol of order −2 and type (1, 0) on X, but not one of order μ < −2.

(iii) Assume φ : X → ℝ to be smooth and bounded. Then a_{φ} : T^{∗}X → ℝ, ξ ↦ (1 + ||ξ||^{2})^{φ(π(ξ))} comprises a symbol of order μ = sup_{x∈X} φ(x) and type (1, δ), where 0 < δ < 1 is arbitrary.

The following theorem is an essential tool for the use of symbols in the theory of partial differential equations and extends a well-known result for the case of E = ℝ^{d} × ℝ^{N} to arbitrary Riemannian vector bundles.

Theorem 2.4. Let a_{j} ∈ S_{ρ,δ}^{μ_{j}}(U, E), j ∈ ℕ be symbols such that (μ_{j})_{j∈ℕ} is a decreasing sequence with lim_{j→∞} μ_{j} = −∞. Then there exists a symbol a ∈ S_{ρ,δ}^{μ_{0}}(U, E)

unique up to smoothing symbols, such that a − ∑_{j=0}^{k} a_{j} ∈ S_{ρ,δ}^{μ_{k}}(U, E) for every k ∈ ℕ. This induces a locally convex Hausdorff topology on the quotient vector space S_{ρ,δ}^{∞}/S^{−∞}(U, E), which is called the topology of asymptotic convergence.

Proof. It is a well-known fact that the claim holds for U ⊂ ℝ^{d} and E = ℝ^{d} × ℝ^{N} (see [7, 12, 5]). Covering U by trivializations (z_{ι}, ζ_{ι}) : T^{∗}X|_{V_{ι}} → ℝ^{2d} and Ψ_{ι} : E|V_{ι} → V_{ι}×ℝ^{N} we can find a partition (φ_{ι}) of unity subordinate to the covering (V_{ι})

and for every index ι a symbol b_{ι,k} ∈ S_{ρ,δ}^{μ}(V_{ι}, ℝ^{N}) such that b_{ι,k} = ∑_{j=0}^{k} Ψ_{ι} ∘ a_{j}|_{V_{ι}} ∈ S_{ρ,δ}^{μ_{k}}(V_{ι}, ℝ^{N}). Now define a = ∑_{ι}(φ_{ι} ∘ π) Ψ_{ι}^{−1} ∘ b_{ι,k} and check that this a satisfies

the claim. Uniqueness of a up to smoothing symbols is clear again from a local consideration. □

Polynomial symbols are not affected by smoothing symbols. The following proposition gives the precise statement.

Proposition 2.5. Let X be a smooth manifold, and D_{0} the sheaf of polynomial symbols on X. Then by composition with the projection S_{ρ,δ}^{∞} → S_{ρ,δ}^{∞}/S^{−∞} the canonical embedding

\begin{matrix}D_{0} & → & S_{ρ,δ}^{∞} \\ D_{0}(U) ∋ f & ↦ & f ∈ S_{ρ,δ}^{∞}(U), & U ⊂ X open \end{matrix} (4)

gives rise to a monomorphism δ : D_{0} → S_{ρ,δ}^{∞}/S^{−∞} of sheaves of algebras.

Proof. If f, g ∈ D_{0}(U) are polynomial symbols such that f − g ∈ S^{−∞}(U), then f − g is a bounded polynomial function on each fiber of T^{∗}U. Therefore f − g is constant on each fiber, hence f − g = 0 by f − g ∈ S^{−∞}(U). □

3. Pseudodifferential operators on manifolds

Let a be a polynomial symbol defined on the (trivial) cotangent bundle T^{∗}U = U × ℝ^{d} of an open set U ⊂ ℝ^{d} of Euclidean space. Then one can write a = ∑_{α} (a_{α} ∘ π)ζ^{α}, where ζ : T^{∗}U → ℝ^{d} is the projection on the “cotangent vectors” and the a_{α} are smooth functions on U. According to standard results of partial differential equations one knows that the symbol a defines the differential operator A = Op (a):

C^{∞}(U) ∋ f ↦ ∑_{α} a_{α} (−i)^{|α|} \frac{∂^{|α|}}{∂z^{α}} f ∈ C^{∞}(U). (5)

Sometimes A is called the quantization of a. In case f ∈ D(U), the Fourier transform of f is well-defined and provides the following integral representation for Af:

Af = Op (a) (f) = ∫_{ℝ^{d}} e^{i<ξ,·>} a(ξ) f̂(ξ) dξ. (6)

It would be very helpful for structural and calculational considerations to extend this formula to Riemannian manifolds. To achieve this it is necessary to have an appropriate notion of Fourier transform on manifolds. In the following we are going to define such a Fourier transform and will later get back to the problem of an integral representation for the “quantization map” Op on Riemannian manifolds.

Assume X to be Riemannian of dimension d and consider the exponential function exp with respect to the Levi-Civita connection on X. Furthermore let E → X be a Riemannian or Hermitian vector bundle. Choose an open neighborhood W ⊂ T X of the zero section in T X such that (ϱ, exp ) : W → X × X maps W diffeomorphically onto an open neighborhood of the diagonal Δ of X ×X. Then there exists a smooth function ψ : T X → [0, 1] called a cut-off function such that ψ|W̃ = 1 and supp ψ ⊂ W for an open neighborhood W̃ ⊂ W of the zero section in T X. Next let us consider the unique torsionfree metric connection on E. It defines a parallel transport τ_{γ} : E_{γ(0)} → E_{γ_{(}1)} for every smooth path γ : [0, 1] → X. For every v ∈ T X denote by τ_{exp v} or just τ_{y,x} with y = exp v and x = ϱ(v) the parallel transport along [0, 1] ∋ t ↦ exp tv ∈ X. Now the following microlocal lift M_{ψ} is well-defined:

M_{ψ} : C^{∞}(U, E) → S^{−∞}(U, T X, E),
f ↦ f^{ψ}, = \{ \begin{matrix}ψ(v) τ_{exp v}^{−1}(f(exp v)) & for v ∈ W \\ 0 & else. \end{matrix} (7)

M_{ψ} is a linear but not multiplicative map between function spaces.

Over the tangent bundle T X we can define the Fourier transform as the following sheaf morphism:

\begin{matrix}F : S^{−∞}(U, T X, E) & → S^{−∞}(U, T^{∗}X, E), \\ a & ↦ â(ξ) = \frac{1}{(2π)^{n/2}} ∫_{T_{π(ξ)}X}e^{−i<ξ,v>} a(v) dv. \end{matrix} (8)

We also have a reverse Fourier transform:

\begin{matrix}F^{−1} : S^{−∞}(U, T^{∗}X, E) & → S^{−∞}(U, T X, E), \\ b & ↦ b̌(v) = \frac{1}{(2π)^{n/2}} ∫_{T_{π(v)}^{∗}X}e^{i<ξ,v>} b(ξ) dξ. \end{matrix} (9)

It is easy to check that F^{−1} and F are well-defined and inverse to each other.

Composing M_{ψ} and F gives rise to the Fourier transform F_{ψ} = F ∘ M_{ψ} on the Riemannian manifold X. F_{ψ} has the left inverse

S^{−∞}(U, T^{∗}X, E) → C^{∞}(U, E), a ↦ F^{−1}(a)|_{0_{U}} , (10)

where |_{0_{U}} means the restriction to the natural embedding of U into T^{∗}X as zero section.

By definition M_{ψ} and F_{ψ} do depend on the smooth cut-off function ψ, but this arbitrariness will only have minor effects on the study of pseudodifferential operators defined through F_{ψ}. We will get back to this point later on in this section.

The exponential function exp gives rise to normal coordinates z_{x} : V_{x} → ℝ^{d}, where x ∈ V_{x} ⊂ X, V_{x} open and

exp_{x}^{−1}(y) = ∑_{k=1}^{d} z_{x}^{k}(y) · e_{k}(y) (11)

for all y ∈ V_{x} and an orthonormal frame (e_{1}, ..., e_{d}) of T V_{x}. Furthermore we receive bundle coordinates (z_{x}, ζ_{x}) : T^{∗}V_{x} → ℝ^{2d} and (z_{x}, v_{x}) : T V_{x} → ℝ^{2d}. Note that for fixed x ∈ X the map

V_{x} × T^{∗}V_{x} ∋ (y, ξ) ↦ (z_{y}(π(ξ)), ζ_{y}(ξ)) ∈ ℝ^{2d} (12)

is smooth and that by the Gauß Lemma

z_{x}^{k}(y) = −z_{y}^{k}(x) (13)

for every y ∈ V_{x}.

Now let a be a smooth function defined on T^{∗}U and polynomial in the fibers. Then locally a = ∑_{α} (a_{x,α} ∘ π) ζ_{x} with respect to a normal coordinate system at x ∈ U and functions a_{x,α} ∈ C^{∞}(U). Define the operator A : D(U) → C^{∞}(U) by

f ↦ Af = Op_{ψ}(a) (f) = (U ∋ x ↦ \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} a(ξ) f̂^{ψ}(ξ) dξ ∈ ℂ) (14)

and check that

(15)  Af (x) = ∑_{α} a_{x,α}(x) (−i)^{|α|} \frac{∂^{|α|}}{∂v_{x}^{α}} [ψ (f ∘ exp )] (0_{x})

= ∑_{α} a_{x,α}(x) (−i)^{|α|} \frac{∂^{|α|}}{∂z_{x}^{α}} f (x)

for every x ∈ U. Therefore Af is a differential operator independent of the choice of ψ.

However, if a is an arbitrary element of the symbol space S_{ρ,δ}^{∞}(U, Hom(E, F)), Eq. (14) still defines a continuous operator A_{ψ} = Op_{ψ}(a) : D(U, E) → C^{∞}(U, F), which is a pseudodifferential operator but independent of the cut-off function ψ only up to smoothing operators. Let us show this in more detail for the scalar case, i.e., where E = F = X × ℂ. The general case is proven analogously.

First choose μ ∈ ℝ such that a ∈ S_{ρ,δ}^{μ}(U), and consider the kernel distribution D(U × U) → ℂ, (f ⊗ g) ↦ < A_{ψ}f, g > = ∫_{X} g(x) A_{ψ}f(x) dx of A_{ψ}. It can be written in the following way:

< g, A_{ψ}f > = \frac{1}{(2π)^{n}} ∫_{X} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} ψ(v) g(x) f(exp v) e^{−i<ξ,v>} a(ξ) dv dξ dx (16)

= \frac{1}{(2π)^{n}} ∫_{X} ∫_{T_{x}X} ∫_{T_{x}^{∗}X} ψ(v) g(x) f(exp v) e^{−i<ξ,v>} a(ξ) dξ dv dx;

where the first integral is an iterated one, the second one an oscillatory integral. To check that Eq. (16) is true use a density argument or in other words approximate the symbol a by elements a_{k} ∈ S^{−∞}(U) (in the topology of Sμ̃ ; ρ,δ(U) with μ̃ > μ) and prove the statement for the a_{k}. The claim then follows from continuity of both sides of Eq. (16) with respect to a. Let ψ̃ : T X → [0, 1] be another cut-off function such that ψ̃|Õ = 1 on a neighborhood W′ ⊂ W of the zero section of T^{∗}X. We can assume W′ = W̃ and claim the operator A_{ψ} − Aψ̃ to be smoothing. For the proof it suffices to show that the oscillatory integral

K_{A_{ψ}−Aψ̃}(v) = \frac{1}{(2π)^{n}} ∫_{T_{π(v)}^{∗}X} (ψ(v) − ψ̃(v)) e^{−i<ξ,v>} a(ξ) dξ, v ∈ T U, (17)

which gives an integral representation for the kernel of A_{ψ} − Aψ̃ defines a smooth function K_{A_{ψ}−Aψ̃} ∈ C^{∞}(T U). The phase function T_{π(v)}^{∗}X ∋ ξ ↦ −ξ(v) ∈ ℝ has only critical points for v = 0. Hence for v ≠ 0 there exists a (−1)-homogeneous vertical vector field L on T^{∗}X such that L e^{−i<·,v>} = e^{−i<·,v>}. The adjoint L^{†} of L satisfies (L^{†})^{k} a ∈ S_{ρ,δ}^{μ−ρk}(U), where k ∈ ℕ. As the amplitude

(v, ξ) ↦ (ψ(v) − ψ̃(v)) a(ξ)

vanishes on W′ ×_{U} T^{∗}U, the equation

K_{A_{ψ}−Aψ̃}(v) = \frac{1}{(2π)^{n}} ∫_{T_{π(v)}^{∗}X}e^{−i<ξ,v>} (L^{†})^{k} a(ξ) dξ (18)

holds for any integer k fulfilling μ + ρk < −dimX. Note that the integral in Eq. (18) unlike the one in Eq. (17) is to be understood in the sense of Lebesgue. Hence K_{A_{ψ}−Aψ̃} ∈ C^{∞}(T U) and A_{ψ} − Aψ̃ is smoothing.

Let us now prove that A_{ψ} lies in the space Ψ_{ρ,δ}^{μ}(U) of pseudodifferential operators of order μ and type (ρ, δ) over U. We have to check that A_{ψ} is pseudolocal and with respect to some local coordinates looks like a pseudodifferential operator on an open set of ℝ^{d}.

(i) A_{ψ} is pseudolocal: Let u, v ∈ D(U) and suppu∩suppv = ∅. Then the integral kernel K of C^{∞}(U) ∋ f ↦ v A_{ψ}(uf) ∈ C^{∞}(U) has the form

K(v) = \frac{1}{(2π)^{n}} ∫_{T_{π(v)}^{∗}X} ψ(v) a(ξ) v(π(v)) u(exp v) e^{−i<ξ,v>} dξ. (19)

There exists an open neighborhood W′ ⊂ W of the zero section in T X such that the amplitude ψ (v ∘ π) (u ∘ exp ) a vanishes on W′ ×_{U} T^{∗}X. As for all nonzero v ∈ T X the phase function ξ ↦ − < ξ, v > is noncritical, an argument similar to the one above for K_{A_{ψ}−Aψ̃} shows that K is a smooth function on T U, so C^{∞}(U) ∋ f ↦ v A_{ψ}(uf) ∈ C^{∞}(U) is smoothing.

(ii) A_{ψ} is a pseudodifferential operator in appropriate coordinates: Choose for x ∈ U an open neighborhood U_{x} ⊂ U so small that any two points in U_{x} can be connected by a unique geodesic. In particular we assume that U_{x} × U_{x} lies in the range of (π, exp ) : W → X × X. Furthermore let us suppose that ψ ∘ z_{y}(ỹ) = 1 for all y, ỹ ∈ U_{x}. Then we have for f ∈ D(U_{x}):

A_{ψ}f (y) =

= \frac{1}{(2π)^{n/2}} ∫_{T_{y}^{∗}X} a(ξ) f̂^{ψ}(ξ) dξ

= \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} a( \underbrace{T_{z_{x}(y)}^{∗} exp_{x}^{−1}}_{= d_{x}(y)} (ξ)) | det d_{x}(y)| F(f ∘ exp )(d_{x}(y) (ξ)) dξ

= \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{y}X} a(d_{x}(y) (ξ)) | det d_{x}(y)| (f ∘ exp_{y})(v) e^{−i<d_{x}(y) (ξ),v>} dv dξ

= \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} a(d_{x}(y) (ξ)) | det d_{x}(y)| | \underbrace{det (T (z_{y} ∘ z_{x}^{−1})}_{= Z_{y,x}}(v)|

(f ∘ z_{x}^{−1})(v) e^{−i<d_{x}(y) (ξ),z_{y}∘z_{x}^{−1}(v)>} dv dξ

(20)

Now the phase function (v, ξ) ↦ − < d_{x}(y)(ξ), z_{y} ∘ z_{x}^{−1}(v) > vanishes for ξ ≠ 0 if and only if v = z_{x}(y). Hence after possibly shrinking U_{x} the Kuranishi trick gives a smooth function G_{x} : U_{x} × T_{x}X → Iso (T_{x}^{∗}X, T_{y}^{∗}X) ⊂ Lin (T_{x}^{∗}X, T_{y}^{∗}X) such that < d_{x}(y)(ξ), z_{y} ∘ z_{x}^{−1}(v) >=< G_{x}(y, v)(ξ), v − z_{x}(y) > for all (y, v) ∈ U_{x} ×T_{x}X and ξ ∈ T_{x}^{∗}X. A change of variables in Eq. (20) then implies:

(21) A_{ψ}f(y) = \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} a([d_{x}(y) G_{x}^{−1}(y, v)] (ξ)) | det d_{x}(y)| |Z_{y,x}(v)|
| det G_{x}^{−1}(y, v)| (f ∘ z_{x}^{−1})(v) e^{−i<ξ,v−z_{x}(y)>} dv dξ

= \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} ã_{x}(y, v, ξ) (f ∘ z_{x}^{−1})(v) e^{−i<ξ,v−z_{x}(y)>} dv dξ,

where ã_{x}(y, v, ξ) = a([d_{x}(y) G_{x}^{−1}(y, v)] (ξ)) | det d_{x}(y)| |Z_{y,x}(v)| | det G_{x}^{−1}(y, v)| is a symbol lying in S_{ρ,δ}^{μ}(U_{x} × U_{x} × T_{x}X), as ρ + δ ≥ 1 and ρ > δ. Hence A is a pseudodifferential operator with respect to normal coordinates over U_{x} of order μ and type (ρ, δ).

Our considerations now lead us to

Theorem 3.1. Let X be Riemannian, E, F Riemannian or Hermitian vector bundles over X and exp the exponential function corresponding to the Levi-Civita connection on X. Then any smooth cut-off function ψ : T X → [0, 1] gives rise to a linear sheaf morphism Op_{ψ} : S_{ρ,δ}^{∞}(·, Hom(E, F)) → Ψ_{ρ,δ}^{∞}(·, E, F), a ↦ A_{ψ} defined by

A_{ψ}f (x) = \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} a(ξ) f̂^{ψ}(ξ) dξ, (22)

where a ∈ S_{ρ,δ}^{∞}(U, Hom(E, F)), f ∈ D(U, E), x ∈ U and U ⊂ X open. This morphism preserves the natural filtrations of S_{ρ,δ}^{∞}(·, Hom(E, F)) and Ψ_{ρ,δ}^{∞}(·, E, F). In particular it maps the subsheaf S^{−∞}(·, Hom(E, F)) of smoothing symbols to the subsheaf Ψ^{−∞}(·, Hom(E, F)) ⊂ Ψ_{ρ,δ}^{∞}(·, Hom(E, F)) of smoothing pseudodifferential operators.

The quotient morphism \overline{Op} : (S_{ρ,δ}^{∞}/S^{−∞})(·, Hom(E, F)) → (Ψ_{ρ,δ}^{∞}/Ψ^{−∞})(·, E, F) is an isomorphism and independent of the choice of the cut-off function ψ.

Proof. The first part of the theorem has been shown above. So it remains to prove that \overline{Op} : (S_{ρ,δ}^{∞}/S^{−∞})(U, Hom(E, F)) → (Ψ_{ρ,δ}^{∞}/Ψ^{−∞})(U, E, F) is bijective for all open U ⊂ X. Let us postpone this till Theorem 4.2, where we will show that an explicit inverse of \overline{Op} is given by the complete symbol map introduced in the following section. □

By the above Theorem the effect of ψ for the operator Op_{ψ} is only a minor one, so from now on we will write Op instead of Op_{ψ}.

4. The symbol map

In the sequel denote by φ : X × T^{∗}X|_{O} → ℂ the smooth phase function defined by

X × T^{∗}X|_{O} ∋ (x, ξ) ↦ < ξ, exp _{π(ξ)}^{−1}(x) > = < ξ, z_{π(ξ)}(x) > ∈ ℂ, (23)

where O is the range of (ϱ, exp) : W → X × X.

Theorem and Definition 4.1. Let A ∈ Ψ_{ρ,δ}^{μ}(U, E, F) be a pseudodifferential operator on a Riemannian manifold X and ψ : T^{∗}X → [0, 1] a cut-off function. Then the ψ-cut symbol of A with respect to the Levi-Civita connection on X is the section

\begin{matrix}σ_{ψ}(A) = σ_{ψ,A} : T^{∗}U & → π^{∗}(Hom(E, F)), \\ ξ & ↦ [Ξ ↦ A (ψ_{π(ξ)}(·) e^{iφ(·,ξ)} τ_{(·),π(ξ)}Ξ)] (π(ξ)), \end{matrix} (24)

where for every x ∈ X ψ_{x} = ψ ∘exp _{x}^{−1} = ψ ∘z_{x}. σ_{ψ}(A) is an element of S_{ρ,δ}^{μ}(U, X). The corresponding element \overline{σ}(A) in the quotient (S_{ρ,δ}^{μ}/S^{−∞})(U, Hom(E, F)) is called the normal symbol of A. It is independent of the choice of ψ.

Proof. Let us first check that σ_{ψ,A} lies in S_{ρ,δ}^{μ}(U, Hom(E, F)). It suffices to assume that E and F are trivial bundles, hence that A is a scalar pseudodifferential operator. We can find a sequence (x_{ι})_{ι∈ℕ} of points in U and coordinate patches U_{ι} ⊂ U with x_{ι} ∈ U_{ι} such that there exist normal coordinates z_{ι} = z_{x_{ι}} : U_{ι} → T_{x_{ι}}X on the U_{ι}. We can even assume that the operator A_{ι} : D(U_{ι}) → C^{∞}(U_{ι}), u ↦ Au|_{U_{ι}} induces a pseudodifferential operator on T_{x_{ι}}X. In other words there exist symbols a_{ι} ∈ S_{ρ,δ}^{μ}(O_{ι} × O_{ι}, T_{x_{ι}}^{∗}X) with O_{ι} = z_{ι}(U_{ι}) such that A_{ι} is given by the following oscillatory integral:

A_{ι}u(y) = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{xι} X} a_{ι}(z_{ι}(y), v, ξ) (u ∘ z_{ι}^{−1})(v) e^{−i<ξ,v−z_{ι}(y)>} dvdξ. (25)

Now let (ϕ_{ι}) be a partition of unity subordinate to the covering (U_{ι}) of U and for every index ι let ϕ_{ι,1}, ϕ_{ι,2} ∈ C^{∞}(U) such that suppϕ_{ι,1} ⊂ U_{ι}, suppϕ_{ι} ∩suppϕ_{ι,2} = ∅ and ϕ_{ι,1} + ϕ_{ι,2} = 1. Then the symbol σ_{ψ,A} can be written in the following form:

\begin{aligned}σ_{ψ,A}(ζ) = & [A(ψ_{π(ζ)}(·) e^{iφ(·,ζ)})] (π(ζ)) \\ = & \frac{1}{(2π)^{n}} ∑_{ι} [ϕ_{ι,1} A_{ι}(ϕ_{ι}(·) ψ_{π(ζ)}(·) e^{iφ(·,ζ)})] (π(ζ)) \\ & + K( ψ_{π(ζ)}(·) e^{iφ(·,ζ)})(π(ζ)), \end{aligned} (26)

where

K : D(U) → C^{∞}(U) f ↦ ∑_{ι} ϕ_{ι,2} A_{ι}(ϕ_{ι} f), ζ ∈ T^{∗}X_{ι} (27)

is a smoothing pseudodifferential operator. By the Kuranishi trick we can assume that there exists a smooth function G : O_{ι} × O_{ι} → Lin (T_{x_{ι}}^{∗}X, T_{y}^{∗}X) such that G(v, w) is invertible for every v, w ∈ O_{ι} and

< G (z_{ι}(y), T (z_{ι} ∘ z_{y}^{−1})(v)) (ξ), v > = < ξ, z_{ι} ∘ z_{y}^{−1}(v) − z_{ι}(y) > (28) for y ∈ U_{ι}, v ∈ T_{y}X appropriate and ξ ∈ T_{x_{ι}}^{∗}X. Several changes of variables then give the following chain of oscillatory integrals for ζ ∈ T^{∗}U_{ι}:

\begin{matrix}[A_{ι}(ϕ_{ι}(·) ψ_{π(ζ)}(·) e^{iφ(·,ζ)})] (y) \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{xι} X} ϕ_{ι,1}(y) a_{ι}(z_{ι}^{−1}(y), v, ξ) ϕ_{ι}(z_{ι}^{−1}(v)) ψ(z_{y} ∘ z_{x}^{−1}(v)) \\ e^{i<ζ,z_{y}∘z_{ι}^{−1}(v)>} e^{−i<ξ,v−z_{ι}(y)>} dv dξ \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{y}X} ϕ_{ι,1}(y) a_{ι}(z_{ι}^{−1}(y), z_{ι} ∘ z_{y}^{−1}(v), ξ) ϕ_{ι}(z_{y}^{−1}(v)) ψ(v) \\ e^{i<ζ,v>} e^{−i<ξ,z_{ι}∘z_{y}^{−1}(v)−z_{ι}(y)>} dv dξ \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{y}X} ϕ_{ι,1}(y) a_{ι}(z_{ι}^{−1}(y), z_{ι} ∘ z_{y}^{−1}(v), ξ) ϕ_{ι}(z_{y}^{−1}(v)) ψ(v) \\ e^{i<ζ,v>} e^{−i<G(z_{ι}(y),T (z_{ι}∘z_{y}^{−1})(v))(ξ),v>} dv dξ \\ = \frac{1}{(2π)^{n}} ∫_{T_{y}^{∗}X} ∫_{T_{y}X} ϕ_{ι,1}(y) a_{ι}(z_{ι}^{−1}(y), z_{ι} ∘ z_{y}^{−1}(v), G^{−1} (z_{ι}(y), T (z_{ι} ∘ z_{y}^{−1})(v)) (ξ)) \\ ϕ_{ι}(z_{y}^{−1}(v)) ψ(v) | det G^{−1} (z_{ι}(y), T (z_{ι} ∘ z_{y}^{−1})(v))| e^{i<ζ,v>} e^{−i<ξ,v>} dv dξ \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{xι} X} b_{ι}(z_{ι}(y), v, ξ) e^{−i<ξ−T^{∗}(z_{y}∘z_{ι}^{−1})(ζ),v−z_{ι}(y)>} dv dξ, \end{matrix} (29)

where the smooth function b_{ι} on O_{ι} × O_{ι} × T_{x_{ι}}^{∗}X is defined by

\begin{matrix}b_{ι}(z_{ι}(y), v, ξ) = \\ = ϕ_{ι,1}(y) a_{ι} (z_{ι}^{−1}(y), z_{ι} ∘ z_{y}^{−1} ∘ T (z_{y} ∘ z_{ι}^{−1}) (v), G^{−1}(z_{ι}(y), v) ∘ (T^{∗}(z_{ι} ∘ z_{y}^{−1}) (ξ)) \\ ϕ_{ι}(z_{y}^{−1}(T (z_{y} ∘ z_{ι}^{−1}) (v))) ψ(T (z_{y} ∘ z_{ι}^{−1}) (v)) | det G^{−1} (z_{ι}(y), v)| \end{matrix} (30)

and lies in S_{ρ,δ}^{μ}(O_{ι} × O_{ι}, T_{x_{ι}}^{∗}X). Let B_{ι} ∈ Ψ_{ρ,δ}^{μ}(O_{ι}, T_{x_{ι}}X) be the pseudodifferential operator on O_{ι} corresponding to the symbol b_{ι}. The symbol σ_{ι} defined by σ_{ι}(v, ξ) = e^{−i<ξ,v>} B_{ι} (e^{i<ξ,·>}), where v ∈ O_{ι} and ξ ∈ T_{x_{ι}}^{∗}X, now is an element of S_{ρ,δ}^{μ}(O_{ι}, T_{x_{ι}}^{∗}X) as one knows by the general theory of pseudodifferential operators on real vector spaces (see [7, 12, 5]). But we have

[ϕ_{ι,1} A_{ι}(ϕ_{ι}(·) ψ_{π(ζ)}(·) e^{iφ(·,ζ)})] (π(ζ)) = σ_{ι} (z_{ι}(π(ζ)), T^{∗} (z_{π(ζ)} ∘ z_{ι}^{−1}) (ζ)) , (31)

hence Eq. (26) shows σ_{ψ,A} ∈ S_{ρ,δ}^{μ}(U, X) if we can prove that σ_{ψ,K} ∈ S^{−∞}(U, X) for

σ_{ψ,K}(ζ) = K( ψ_{π(ζ)}(·) e^{iφ(·,ζ)})(π(ζ)). So let k : U × U → ℂ be the kernel function of K and write σ_{ψ,K} as an integral:

σ_{ψ,K}(ζ) = ∫_{U} k(π(ζ), y) ψ_{π(ζ)}(y) e^{−iφ(y,ζ)} dy. (32)

Let further Ξ_{1}, ..., Ξ_{k} be differential forms over U and V_{1}, ..., V_{k} the vertical vector fields on T^{∗}U corresponding to Ξ_{1}, ..., Ξ_{k}. Differentiating Eq. (32) under the integral sign and using an adjointness relation then gives

\begin{matrix}|ζ|^{2m} V_{1} · ... · V_{k} σ_{ψ,K} (ζ) = \\ = ∫_{U} k(π(ζ), y) ψ_{π(ζ)}(y) < Ξ_{1}, z_{π(ζ)}(y) > ... < Ξ_{k}, z_{π(ζ)}(y) > \\ i^{2mn+k} \frac{∂^{2mn}}{∂z_{π(ζ)}^{(2m,...,2m)}} e^{−i<ζ,z_{π(ζ)}(·)>}(y) dy \\ = ∫_{U} (\frac{∂^{2mn}}{∂z_{π(ζ)}^{(2m,...,2m)}} )^{†} (k(π(ζ), ·) ψ_{π(ζ)}(·) < Ξ_{1}, z_{π(ζ)}(·) > ... < Ξ_{k}, z_{π(ζ)}(·) >) (y) \\ i^{2mn+k}e^{−i<ζ,z_{π(ζ)}(y)>} dy. \end{matrix} (33)

Therefore |ζ|^{2m} V_{1} · ... · V_{k} σ_{ψ,K} (ζ) is uniformly bounded as long as π(ζ) varies in a compact subset of U. A similar argument for horizontal derivatives of σ_{ψ,K} finally proves σ_{ψ,K} ∈ S^{−∞}(U, X).

Next we have to show that σ_{ψ,A} − σ_{ψ̃,A} is a smoothing symbol for any second cut-off function ψ̃ : T^{∗}X → [0, 1]. It suffices to prove that σ̃_{ι} with

σ̃_{ι}(ζ) = A_{ι} (ϕ_{ι} (ψ − ψ̃) ∘ z_{π(ζ)} e^{−iφ(·,ζ)}) (34)

is smoothing for every ϕ_{ι} ∈ D(U_{ι}). We can find a (−1)-homogeneous first order differential operator L on the vector bundle T_{x_{ι}}^{∗}X × T_{x_{ι}}X → T_{x_{ι}}X such that for all v ≠ z_{ι}(π(ζ)) and ξ ∈ T_{x_{ι}}^{∗}X the equation

Le^{−i<·,v−z_{ι}(π(ζ))>} (ξ) = e^{−i<ξ,v−z_{ι}(π(ζ))>} (35)

holds (see for example Grigis, Sjöstrand [5] Lemma 1.12 for a proof). Because (ψ−ψ̃)∘z_{π(ζ)} ∘z_{ι}^{−1} vanishes on a neighborhood of z_{ι}(π(ζ)), the following equational chain of iterated integrals is also true:

\begin{matrix}σ̃_{ι}(ζ) = \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{xι} X} a_{ι}(z_{ι}^{−1}(y), v, ξ) ϕ_{ι}(z_{ι}^{−1}(v)) (ψ − ψ̃) (z_{π(ζ)} ∘ z_{x}^{−1}(v)) \\ e^{i<ζ,z_{π(ζ)}∘z_{ι}^{−1}(v)>} e^{−i<ξ,v−z_{ι}(π(ζ))>} dv dξ \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{xι} X} a_{ι}(z_{ι}^{−1}(y), v, ξ) ϕ_{ι}(z_{ι}^{−1}(v)) (ψ − ψ̃) (z_{π(ζ)} ∘ z_{x}^{−1}(v)) \\ e^{i<ζ,z_{π(ζ)}∘z_{ι}^{−1}(v)>} L^{k} e^{−i<ξ,v−z_{ι}(π(ζ))>} dv dξ \\ = \frac{1}{(2π)^{n}} ∫_{T_{xι}^{∗} X} ∫_{T_{xι} X} ((L^{†})^{k} a_{ι})(z_{ι}^{−1}(y), v, ξ) ϕ_{ι}(z_{ι}^{−1}(v)) (ψ − ψ̃) (z_{π(ζ)} ∘ z_{x}^{−1}(v)) \\ e^{i<ζ,z_{π(ζ)}∘z_{ι}^{−1}(v)>} e^{−i<ξ,v−z_{ι}(π(ζ))>} dv dξ. \end{matrix} (36)

Note that the last integral of this chain is to be understood in the sense of Lebesgue if k ∈ ℕ is large enough. The same argument which was used for proving that the symbol σ_{ψ,K} from Eq. (32) is smoothing now shows σ_{ι} ∈ S^{−∞}(U_{ι}, X). □

After having defined the notion of a symbol, we will now give its essential properties in the following theorem.

Theorem 4.2. Let A ∈ Ψ_{ρ,δ}^{μ}(U, E, F) be a pseudodifferential operator on a Riemannian manifold X and a = σ_{ψ,A} ∈ S_{ρ,δ}^{μ}(U, Hom(E, F)) be its ψ-cut symbol with respect to the Levi-Civita connection. Then A and the pseudodifferential operator Op_{ψ}(a) defined by Eq. (22) coincide modulo smoothing operators. Moreover the sheaf morphisms \overline{σ} : (Ψ_{ρ,δ}^{∞}/Ψ^{−∞})(·, E, F) → (S_{ρ,δ}^{∞}/S^{−∞})(·, Hom(E, F)) and \overline{Op} : (S_{ρ,δ}^{∞}/S^{−∞})(·, Hom(E, F)) → (Ψ_{ρ,δ}^{∞}/Ψ^{−∞})(·, E, F) are inverse to each other.

Before proving the theorem let us first state a lemma.

Lemma 4.3. The operator

K : D(U, E) → C^{∞}(U, F), f ↦ (U ∋ x ↦ A[(1 − ψ_{x})f] (x) ∈ F ) (37)

is smoothing for any cut-off function ψ : T X → ℂ.

Proof. For simplicity we assume that E = F = X × ℂ. The general case follows analogously. Consider the operators K_{a,ι} : D(U) → C^{∞}(U) where K_{a,ι}f (x) = ϕ_{ι}(x)A[(1 − ψ_{a})f](x) for x ∈ U, a ∈ U_{ι}. The open covering (U_{ι}) of U and a subordinate partition of unity (ϕ_{ι}) are chosen such that for every point a ∈ U_{ι} there exist normal coordinates z_{a} on U_{ι}. We can even assume that ψ(z_{a}(y)) = 1 for all a, y ∈ U_{ι}. Hence supp (1 − ψ_{a}) ∩ supp ϕ_{ι} = ∅, and K_{a,ι} is smoothing, because A is pseudolocal. This yields a family of smooth functions k_{ι} : U × U → ℂ such that

K_{a,ι}f (x) = ∫_{U} k_{ι}(x, y) (1 − ψ_{a}(y)) f(y) dy. (38)

Now the smooth function k ∈ C^{∞}(U × U) with k(x, y) = ∑_{ι} k_{ι}(x, y) (1 − ψ_{x}(y)) is the kernel function of K. □

Proof of Theorem 4.2. Check that for an arbitrary cut-off function ψ̃ : T X → ℂ one can find a cut-off function ψ : T X → ℂ such that the equation

ψ_{x}(y) f(y) = \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} ψ_{x}(y) e^{i<ξ,z_{x}(y)>} f̂ ψ̃(ξ) dξ (39)

holds for x, y ∈ U. Approximating the integral in Eq. (39) by Riemann sums, we can find a sequence (i_{k})_{k∈ℕ} of natural numbers, a family (ξ_{k,ι})_{k∈ℕ,1≤ι≤ι_{k}} of elements ξ_{k,ι} ∈ T_{x}^{∗}X and a sequence (ϵ_{k})_{k∈ℕ} of positive numbers such that

\begin{matrix}lim_{k→∞} ϵ_{k} = 0, \\ [(ψ_{x})f] (y) = lim_{k→∞} \frac{ϵ_{k}^{n}}{(2π)^{n/2}} ∑_{1≤i≤i_{k}} ψ_{x}(y) e^{i<ξ_{k,ι} , z_{x}(y)>} f̂ ψ̃(ξ_{k,ι}). \end{matrix} (40)

By an appropriate choice of the ξ_{k,ι} one can even assume that the series on the right hand side of the last equation converges as a function of y ∈ U in the topology of D(U). As A is continuous on D(U), we now have

(41) A[(ψ_{x})f](x) = lim_{k→∞} \frac{ϵ_{k}^{n}}{(2π)^{n/2}} ∑_{1≤i≤i_{k}} A [ψ_{x}(·) e^{i<ξ_{k,ι} , z_{x}(·)>}](x) f̂ ψ̃(ξ_{k,ι})

= lim_{k→∞} \frac{ϵ_{k}^{n}}{(2π)^{n/2}} ∑_{1≤i≤i_{k}} σ_{A}(ξ_{k,ι}) f̂ ψ̃(ξ_{k,ι})

= \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} σ_{A}(ξ) f̂ ψ̃(ξ) dξ

= Aψ̃f.

On the other hand Lemma 4.3 provides a smoothing operator K such that for f ∈ D(U) and x ∈ U:

Af (x) = A[(ψ_{x})f] (x) + Kf (x) = Aψ̃f (x) + Kf (x). (42)

Thus the first part of the theorem follows.

Now let a ∈ S_{ρ,δ}^{μ}(U, Hom(E, F)) be a symbol, and consider the corresponding pseudodifferential operator A = A_{ψ} ∈ Ψ_{ρ,δ}^{μ}(U, E, F). After possibly altering ψ we can assume that there exists a second cut-off function ψ̃ : T^{∗}X → ℂ such that ψ̃|_{supp ψ} = 1. Then we have for ζ ∈ T_{x}^{∗}X and Ξ ∈ E_{x}

(43)  σ_{A,ψ}(ζ)Ξ = \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} a(ξ) F ((ψ_{x}(·)e^{iφ(·,ζ)}τ_{x,(·)}Ξ)^{ψ̃} ) (ξ) dξ

= \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} a(ξ) F (ψ_{x}(·)e^{iφ(·,ζ)}Ξ) (ξ) dξ

= \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} a(ξ)Ξ ψ(v) e^{−i<ξ,v>} e^{i<ζ,v>} dv dξ,

where the last integral is an iterated one. Next choose a smooth function ϕ : ℝ → [0, 1] such that suppϕ ⊂ [−2, 2] and ϕ|_{[−1,1]} = 1 and define a_{k} ∈ S^{−∞}(U, Hom(E, F)) by a_{k}(ξ) = ϕ (\frac{|ξ|}{k}). Then obviously a_{k} → a in Sμ̃ ; ρ,δ(UHom(E, F),) for μ̃ > μ. We can now write:

σ_{A,ψ}(ζ)Ξ = lim_{k→∞} \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} a_{k}(ξ)Ξ ψ(v) e^{−i<ξ−ζ,v>} dv dξ (44)

= lim_{k→∞} a_{k}(ζ)Ξ − \frac{1}{(2π)^{n/2}} ∫_{T_{x}X} (1 − ψ(v)) [F^{−1}a_{k}](v) e^{i<ζ,v>} dv

= a(ζ) − lim_{k→∞} \frac{1}{(2π)^{n/2}} ∫_{T_{x}X} (1 − ψ(v)) [F^{−1} ((L^{†})^{m} a_{k})](v) e^{i<ζ,v>} dv

= a(ζ) + lim_{k→∞} K_{m}(ζ, a_{k}),

where m ∈ ℕ, L is a smooth (−1)-homogeneous first order differential operator on the vector bundle T^{∗}X ×_{U} Ṫ U → Ṫ U such that Le^{i<·,v>} (ξ) = e^{i<ξ,v>} and

K_{m}(ζ, a_{k}) = −\frac{1}{(2π)^{n/2}} ∫_{T_{x}X} (1 − ψ(v)) [F^{−1} ((L^{†})^{m} a_{k})](−v) e^{i<ζ,v>} dv. (45)

Note that Ṫ U consists of all nonzero vectors v ∈ T U and that (L^{†})^{m} a ∈ S_{ρ,δ}^{μ̃−ρm} for every μ̃ ≥ μ. Now fixing μ̃ ≥ μ and choosing m large enough that ρm > μ̃ + 2(dimX − 1) yields

K_{m}(ζ, a) =: lim_{k→∞} K_{m}(ζ, a_{k}) (46)

= lim_{k→∞} −\frac{1}{(2π)^{n}} ∫_{T_{x}X} ∫_{T_{x}^{∗}X} ((L^{†})^{m} a_{k}) (ξ) (1 − ψ(v)) e^{−i<ξ−ζ,v>} dξ dv

= −\frac{1}{(2π)^{n}} ∫_{T_{x}X} ∫_{T_{x}^{∗}X} ((L^{†})^{m} a) (ξ) (1 − ψ(v)) e^{i<ζ,v>} e^{−i<ξ,v>} dξ dv

in the sense of Lebesgue integrals. As the phase function ξ ↦ − < ξ, v > is noncritical for every nonzero v and 1 − ψ(v) vanishes in an open neighborhood of the zero section of T U, a standard consideration already carried out in preceeding proofs entails that K_{m}(·, a) is a smoothing symbol. Henceforth σ_{A,ψ̃} and a differ by the smoothing symbol K_{m}(·, a). This gives the second part of the theorem. □

In the following let us show that the normal symbol calculus gives rise to a natural transformation between the functor of pseudodifferential operators and the functor of symbols on the category of Riemannian manifolds and isometric embeddings.

First we have to define these two functors. Associate to any Riemannian manifold X the algebra A(X) = Ψ_{ρ,δ}^{∞}/Ψ^{−∞}(X) of pseudodifferential operators on X modulo smoothing ones, and the space S(X) = S_{ρ,δ}^{∞}/S^{−∞}(X) of symbols on X modulo smoothing ones. As Y is Riemannian, we have for any isometric embedding f : X → Y a natural embedding f_{∗} : T^{∗}X → T^{∗}Y. So we can define S(f) : S(Y) → S(X) to be the pull-back f^{∗} of smooth functions on T^{∗}Y via f_{∗}. The morphism A(f) : A(Y) → A(X) is constructed by the following procedure. If f is submersive, i.e., a diffeomorphism onto its image, A(f) is just the pull-back of a pseudodifferential operator on Y to X via f. So we now assume that f is not submersive. Choose an open tubular neighborhood U of f(X) in Y and a smooth cut-off function ϕ : U → [0, 1] with compact support and ϕ|_{V} = 1 on a neighborhood V ⊂ U of f(X). As U is tubular there is a canonical projection Π : U → X such that Π ∘ f = id_{X}. Now let the pseudodifferential operator P represent an element of A(Y). We then define A(f)(P) to be the equivalence class of the pseudodifferential operator

E′ ∋ u ↦ f^{∗} [P (ϕ(Π^{∗}u))] ∈ E′, (47)

where E′ denotes distributions with compact support. Note that modulo smoothing operators f^{∗} [P (ϕ(Π^{∗}u))] does not depend on the choice of ϕ. With these definitions A and S obviously form contravariant functors from the category of Riemannian manifolds with isometric embeddings as morphisms to the category of complex vector spaces and linear mappings.

The following proposition now holds.

Proposition 4.4. The symbol map σ forms a natural transformation from the functor A to the functor S.

Proof. As f is isometric, the relation

f ∘ exp = exp ∘T f (48)

is true. But then the phasefunctions on X and Y are related by

φ_{Y} (y, f_{∗}(ξ)) = φ_{X}(Π(y), ξ), (49)

where y ∈ Y and ξ ∈ T^{∗}X with y and f(π(ξ)) sufficiently close. Hence for a pseudodifferential operator P on Y

(f^{∗}σ_{Y} (P)) (ξ) = [P (ψ_{f(π(ξ))}(·) e^{iφ_{Y} (·,f_{∗}ξ)})] (f(π(ξ))) (50)

and

(σ_{X}f^{∗}(P)) (ξ) = [P (ψ_{f(π(ξ))}(·) ϕ(·) e^{iφ_{X}(Π(·),ξ)})] (π(ξ)). (51)

Eq. (49) now implies

f^{∗}σ_{Y} (P) = σ_{X}f^{∗}(P) (52)

which gives the claim. □

5. The symbol of the adjoint

In the sequel we want to derive an expression for the normal symbol σ_{A^{∗}} of the adjoint of a pseudodifferential operator A in terms of the symbol σ_{A}.

Let E, F be Hermitian (Riemannian) vector bundles over the Riemannian manifold X, and let < , >_{E}, resp. < , >, be the metric on E, resp. F. Denote by D^{E}, resp. D^{F} the unique torsionfree metric connection on E, resp. F, and by D the induced metric connection on Hom(E, F), resp. π^{∗}(Hom(E, F)). Next define the function ρ : O → ℝ^{+} by

(53) ρ(x, y) · ν_{x} = ν(y) ∘ (T_{v} exp_{x} × ⋯ × T_{v} exp_{x}), x, y ∈ O, v = exp_{x}^{−1}(y), where ν is the canonical volume density on T_{x}X and ν_{x} its restriction to a volume density on T_{x}X. Now we claim that the operator Op^{∗}(a) : D(U, F) → C^{∞}(U, E) defined by the iterated integral

Op^{∗}(a)g(x) =\frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} e^{−i<ξ,v>} τ_{exp v}^{−1} [a^{∗} (T_{v}^{∗} exp_{x}^{−1}(ξ)) g(exp v)] ψ̃(v)
ρ^{−1}(exp_{x}(v), x) dv dξ

(54)

is the (formal) adjoint of Op (a), where a ∈ S_{ρ,δ}^{μ}(U, Hom(E, F)), U ⊂ X open and ψ̃

is the cut-off function T X ∋ v ↦ ψ (exp_{exp_{x} v}^{−1}(x)) ∈ [0, 1]. Assuming f ∈ D(V, E), g ∈ D(U, F) with V ⊂ U open, V × V ⊂ O and μ to be sufficiently small the following chain of (proper) integrals holds:

∫_{X} < f(x), Op^{∗}(a)g (x) >_{E} dν(x) =

=^{1} \frac{1}{(2π)^{n}} ∫_{X} ∫_{T_{x}X×T_{x}^{∗}X} e^{i<ξ,v>} < f(x), τ_{exp v} [a^{∗} (T_{v}^{∗} exp_{x}^{−1}(ξ)) g(exp v)] >

ψ̃(v) ρ^{−1}(exp_{x}(v), x) dω_{x}^{n}(v, ξ)dν(x)

=^{2} \frac{1}{(2π)^{n}} ∫_{X} ∫_{T^{∗}V} e^{i<T^{∗} exp_{x}(ξ),exp_{x}^{−1}(π(ξ))>} < τ_{π(ξ),x}f(x), a^{∗}(ξ) g(π(ξ)) >

ψ(exp_{π(ξ)}^{−1}(x)) ρ^{−1}(π(ξ), x) dω^{n}(ξ) dν(x)

=^{3} \frac{1}{(2π)^{n}} ∫_{T^{∗}V} ∫_{X} e^{−i<ξ,exp_{π(ξ)}^{−1}(x)>} < τ_{π(ξ),x}f(x), a^{∗}(ξ) g(π(ξ)) >

ψ(exp_{π(ξ)}^{−1}(x)) ρ^{−1}(π(ξ), x) dν(x) dω^{n}(ξ)

=^{4} \frac{1}{(2π)^{n}} ∫_{T^{∗}V} ∫_{T_{π(ξ)}X} e^{−i<ξ,w>} < a(ξ) τ_{exp w}^{−1}f(exp w), g(x) > ψ(w) dw dω^{n}(ξ)

= \frac{1}{(2π)^{n}} ∫_{X} ∫_{T_{x}X×T_{x}^{∗}X} e^{−i<ξ,w>} < a(ξ) τ_{exp w}^{−1}f(exp w), g(x) > ψ(w) dw dξ dν(x)

= ∫_{X} < Op (a)f(x), g(x) >_{F} dν(x)

(55)

In explanation of the above equalities, we have 1. ω_{x}, ω canonical symplectic forms on T_{x}X × T_{x}^{∗}X, resp. T^{∗}X

2. by T^{∗} exp_{x} : T^{∗}V → T_{x}X × T_{x}^{∗}X symplectic, x ∈ V

3. Gauß Lemma and Fubini’s Theorem

4. coordinate transformation V ∋ x ↦ w = exp_{π(ξ)}^{−1}(x) ∈ T_{π(ξ)}X.

Recall that S^{−∞}(U, Hom(E, F)) is dense in any S_{ρ,δ}^{μ}(U, Hom(E, F)), μ ∈ ℝ with respect to the Fréchet topology of Sμ̃ ; ρ,δ(U, Hom(E, F)) if μ < μ̃. By using a partition of unity and the continuity of Sμ̃ ; ρ,δ(U, Hom(E, F)) ∋ a ↦ Op (a)f ∈ C^{∞}(U, F), resp. Sμ̃ ; ρ,δ(U, Hom(E, F)) ∋ a ↦ Op^{∗}(a)g ∈ C^{∞}(U, E), for any f ∈ D(U, E) and g ∈ D(U, F), Eq. (55) now implies that

∫_{X} < f(x), Op^{∗}(a)g(x) >^{E} dν(x) = ∫_{X} < Op (a)f(x), g(x) >^{F} dν(x) (56)

holds for every f ∈ D(U, E), g ∈ D(U, F) and a ∈ S_{ρ,δ}^{μ}(U) with μ ∈ ℝ. Thus Op^{∗}(a) is the formal adjoint of Op (a) indeed.

The normal symbol σ_{A^{∗}} of A^{∗} = Op^{∗}(a) now is given by

σ_{A^{∗}}(ξ) Ξ ∼ [A^{∗}g(·, ξ, Ξ)] (π(ξ)) = \frac{1}{(2π)^{n}} ∫_{T_{x}X} ∫_{T_{x}^{∗}X} e^{−i<ζ−ξ,v>}

τ_{exp v}^{−1} [a^{∗} (T_{v}^{∗} exp_{x}^{−1}(ζ)) τ_{exp v}Ξ] ψ̃(v) ψ(v) ρ^{−1}(exp_{x}(v), x) dv dζ,

(57)

where ξ ∈ T^{∗}X, x = π(ξ), Ξ ∈ F_{x} and

(58) g(y, ξ, Ξ) = ψ (exp_{ψ(ξ)}^{−1}(y)) e^{i<ξ,exp_{π(ξ)}^{−1}(y)>} τ_{y,π(ξ)}Ξ

with y ∈ X. Thus by Theorem 3.4 of Grigis, Sjöstrand [5] the following asymptotic expansion holds:

σ_{A^{∗}} (ξ) ∼ ∑_{α∈ℕ^{d}} \frac{(−i)^{|α|}}{α!} \frac{∂^{|α|}}{∂v^{α}} |_{v=0} \frac{∂^{|α|}}{∂ζ^{α}} |_{ζ=ξ}

( [τ_{exp v}^{−1} ∘ a^{∗} (T_{v}^{∗} exp_{x}^{−1}(ζ)) ∘ τ_{exp v}] ρ^{−1}(exp_{x}(v), x))

= ∑_{α,β∈ℕ^{d}} \frac{(−i)^{|α+β|}}{α! β!} [ \frac{D^{|α|+|β|}}{∂z_{x}^{β} ∂ζ_{x}^{α}} |_{ξ} a^{∗}] [ \frac{∂^{|α|}}{∂z_{x}^{α}} |_{x} ρ^{−1}(·, x)] ,

(59)

where

\frac{D^{|α|+|β|}}{∂z_{x}^{β} ∂ζ_{x}^{α}} |_{ξ} a^{∗} = \frac{∂^{|β|}}{∂v^{β}} a|_{v=0} \frac{∂^{|α|}}{∂ζ^{α}} a|_{ξ} [τ_{exp v}^{−1} ∘ a (T_{v}^{∗} exp_{x}^{−1}(ζ)) ∘ τ_{exp v}] , (60)

are symmetrized covariant derivatives of a ∈ S^{∞}(X, Hom(E, F)) with respect to the apriorily chosen normal coordinates (z_{x}, ζ_{x}).

Next we will search for an expansion of ρ and its derivatives. For that first write

\frac{∂}{∂z_{x}^{k}} = ∑_{k,l} θ_{k}^{l}(x, ·) e_{l}, (61)

where (e_{1}, ..., e_{d}) is the orthonormal frame of Eq. (11), and the θ_{k}^{l} are smooth functions on an open neighborhood of the diagonal of X × X. Then we have the following relation (see for example Berline, Getzler, Vergne [1], p. 36):

θk l(x, y) = δk l − \frac{1}{6} ∑_{m,n} R^{k}^{mln}(y) zm x(y) zn x(y) + O(|z^{x}(y)|^{3}), (62)

where R^{k}_{mln}(y), resp. their “lowered” versions R_{kmln}(y), are given by the coefficients of the curvature tensor with respect to normal coordinates at y ∈ X,

i.e., R (\frac{∂}{∂z_{y}^{m}} ) = ∑_{k,l,n} R^{k}_{mln}(y)\frac{∂}{∂z_{y}^{k}} ⊗ dz_{y}^{l} ⊗ dz_{y}^{n}. But this implies

\begin{aligned}ρ(x, y) & = |det (θ_{k}^{l}(x, y))| \\ & = 1 − \frac{1}{6} ∑_{k} ∑_{m,n} R^{k}_{mkn}(x) z_{x}^{m}(y) z_{x}^{n}(y) + O(|z_{x}(y)|^{3} \\ & = 1 − \frac{1}{6} ∑_{m,n} Ric_{mn}(x) z_{x}^{m}(y) z_{x}^{n}(y) + O (|z_{x}(y)|^{3}) . \end{aligned} (63)

Using Eq. (13) z_{x}^{k}(y) = −z_{y}^{k}(x) for x and y close enough we now get

\begin{aligned}\frac{∂}{∂z_{x}^{k}} |_{y} ρ^{−1}(·, x) & = \frac{1}{3} ∑_{l} Ric_{kl}(x) z_{x}^{l}(y) + O (|z_{x}(y)|^{2}) , & (64) \\ \frac{∂^{2}}{∂z_{x}^{k}∂z_{x}^{l}} |_{y} ρ^{−1}(·, x) & = \frac{1}{3}Ric_{kl}(x) + O (|z_{x}(y)|) . & (65) \end{aligned}

The above results are summarized by the following theorem.

Theorem 5.1. The formal adjoint A^{∗} ∈ Ψ_{ρ,δ}^{μ}(U, F, E) of a pseudodifferential operator A ∈ Ψ_{ρ,δ}^{μ}(U, E, F) between Hermitian (Riemannian) vector bundles E, F over an open subset U of a Riemannian manifold X is given by

A^{∗}g (x) = Op^{∗}(a)g (x) = \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} e^{−i<ξ,v>}
τ_{exp v}^{−1} [a^{∗} (T_{v}^{∗} exp_{x}^{−1}(ξ)) g(exp v)] ψ̃(v) ρ^{−1}(exp_{x}(v), x) dv dξ

where x ∈ X, g ∈ D(U, F) and a = σ_{A} is the symbol of A. The symbol σ_{A^{∗}} of A^{∗} has asymptotic expansion

\begin{matrix}σ_{A^{∗}}(ξ) ∼ a^{∗}(ξ) + ∑_{{α∈ℕ^{d} \atop |α|≥1}} \frac{D^{2|α|}}{∂z_{π(ξ)}^{α} ∂ζ_{π(ξ)}^{α}} |_{ξ} a^{∗} − \\ − \frac{1}{6} ∑_{α∈ℕ^{d}} ∑_{k,l} (−i)^{|α|} [ \frac{D^{2|α|+2}}{∂z_{π(ξ)}^{α} ∂ζ_{π(ξ)}^{α} ∂ζ_{π(ξ)}^{k} ∂ζ_{π(ξ)}^{l}} |_{ξ} a^{∗}] Ric_{kl}(π(ξ)) + \\ + ∑_{{α,β∈ℕ^{d} \atop |β|≥3}} \frac{(−i)^{|α + β|}}{α! β!} [ \frac{D^{2|α|+|β|}}{∂z_{π(ξ)}^{α} ∂ζ_{π(ξ)}^{α+β}} |_{ξ} a^{∗}] [ \frac{∂^{|α|}}{∂z_{π(ξ)}^{α}} |_{π(ξ)} ρ^{−1}(·, x)] \\ = a^{∗}(ξ) − i ∑_{k} \frac{D^{2}}{∂z_{π(ξ)}^{k} ∂ζ_{π(ξ)}^{k}} |_{ξ} a^{∗} − \frac{1}{2} ∑_{k,l} \frac{D^{4}}{∂z_{π(ξ)}^{k} ∂z_{π(ξ)}^{l} ∂ζ_{π(ξ)}^{k} ∂ζ_{π(ξ)}^{l}} |_{ξ} a^{∗} − \\ − \frac{1}{6} ∑_{k,l} [ \frac{D^{2}}{∂ζ_{π(ξ)}^{k}∂ζ_{π(ξ)}^{l}} |_{ξ} a^{∗}] Ric_{kl}(π(ξ)) + r(ξ) \end{matrix} (66)

with ξ ∈ T^{∗}X and r ∈ S_{ρ,δ}^{μ−2(ρ−δ)}(U, Hom(E, F)).

6. Product expansions

Most considerations of elliptic partial differential operators as well as applications of our normal symbol to quantization theory (see Pflaum [8]) require an expression for the product of two pseudodifferential operators A, B in terms of the symbols of its components. In the flat case on ℝ^{d} it is a well known fact that σ_{AB} has an asymptotic expansion of the form

σ_{AB}(x, ξ) ∼ ∑_{α} \frac{(−i)^{|α|}}{α!} \frac{∂^{|α|}}{∂ζ^{α}} σ_{A}(x, ξ) \frac{∂^{|α|}}{∂x^{α}} σ_{B}(x, ξ). (67)

Note that strictly speaking, the product AB of pseudodifferential operators is only well-defined, if at least one of them is properly supported. But this is only a minor set-back, as any pseudodifferential operator is properly supported modulo a smoothing operator and the above asymptotic expansion also describes a symbol only up to smoothing symbols.

In the following we want to derive a similar but more complicated formula for the case of pseudodifferential operators on manifolds.

Proposition 6.1. Let A ∈ Ψ_{ρ,δ}^{μ}(U, E, F) be a pseudodifferential operator between Hermitian (Riemannian) vector bundles E, F over a Riemannian manifold X and a = σ(A) its symbol. Then for any f ∈ C^{∞}(U, E) the function

σ_{A,f} : T^{∗}U → F, ξ ↦ [A (ψ_{π(ξ)}f e^{iφ(·,ξ)})] (π(ξ)) (68)

is a symbol of order μ and type (ρ, δ) and has an asymptotic expansion of the form

σ_{A,f}(ξ) ∼ ∑_{α} \frac{1}{i^{|α|}α!} [ \frac{D^{|α|}}{∂ζ_{π(ξ)}^{α}} |_{ξ} a] [ \frac{D^{|α|}}{∂z_{π(ξ)}^{α}} |_{π(ξ)} f] . (69)

Proof. Let x = π(ξ), N ∈ ℕ and ψ : T^{∗}X → ℂ a cut-off function. Then the following chain of equations holds:

σ_{A,f}(ξ) = [A (ψ_{π(ξ)}f e^{iφ(·,ξ)})] (x) (70)

= \frac{1}{(2π)^{n/2}} ∫_{T_{x}^{∗}X} a(ζ) [F (f^{ψ} e^{i<ξ,·>})](ζ) dζ

= \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} e^{i<ξ−ζ,v>} a(ζ) τ_{exp v}^{−1}f(exp v) ψ(v) dv dζ

= \frac{1}{(2π)^{n}} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} e^{−i<ζ,v>} a(ξ + ζ) τ_{exp v}^{−1}f(exp v) ψ(v) dv dζ

= \frac{1}{(2π)^{n}} ∑_{|α|≤N} \frac{1}{α!} ∫_{T_{x}^{∗}X} ∫_{T_{x}X} e^{−i<ζ,v>} \frac{D^{|α|}}{∂ζ_{x}^{α}} a(ξ) ζ^{α} τ_{exp v}^{−1}f(exp v) ψ(v) dv dζ

+ r_{N}(ξ)

= ∑_{|α|≤N} \frac{1}{i^{|α|} α!} [ \frac{D^{|α|}}{∂ζ_{x}^{α}} |_{ξ} a]· [ \frac{D^{|α|}}{∂z_{x}^{α}} |_{x} f] + r_{N}(ξ).

Note that in this equation all integrals are iterated ones and check that with Taylor’s formula r_{N} is given by

r_{N}(ξ) = \frac{1}{(2π)^{n}} ∑_{|β|=N+1} \frac{N + 1}{β!} ∫_{T_{x}^{∗}X} ∫_{0}^{1}(1 − t)^{N} \frac{D^{|β|}}{∂ζ_{x}^{β}} a(ξ + tζ) dt ζ^{β} [F (f^{ψ})](ζ) dζ. (71)

Choosing N large enough, the Schwarz inequality and Plancherel’s formula now entail

\begin{aligned}||r_{N}(ξ)|| ≤ & \frac{1}{(2π)^{n/2}} ∑_{|β|=N+1} \frac{N + 1}{β!} ||∫_{0}^{1}(1 − t)^{N} \frac{D^{|β|}}{∂ζ_{x}^{β}} a(ξ + t(·))dt||_{L^{2}(T_{x}^{∗}X)} \\ & ||\frac{D^{|β|}}{∂v_{x}^{β}} f^{ψ}|_{T_{x}X} ||_{L^{2}(T_{x}X)} . \end{aligned} (72)

But for K ⊂ U compact and t ∈ [0, 1] there exist constants D_{N,K}, D̃_{N,K} > 0 such that

||\frac{D^{|β|}}{∂ζ_{x}^{β}} a(ξ + tζ)|| ≤ D_{N,K}(1 + ||ξ|| + ||ζ||)^{μ−ρ(N+1)} (73)

and

||∫_{0}^{1}(1 − t)^{k} \frac{D^{|β|}}{∂ζ_{x}^{β}} a(ξ + t(·))dt||_{L^{2}(T_{x}^{∗}X)} ≤ D_{N,K} ||(1 + ||ξ|| + || · ||)^{μ−ρ(N+1)}||_{L^{2}(T_{x}^{∗}X)} (74)
= D̃_{N,K} (1 + ||ξ||)^{n+μ−ρ(N+1)}

for all x ∈ K, ξ, ζ ∈ T_{x}^{∗}X and all indices β ∈ ℕ^{d} with |β| = N + 1. Inserting this in (72) we can find a C_{N,K} > 0 such that for all x ∈ K and ξ ∈ T_{x}^{∗}X

(75) ||r_{N}(ξ)|| ≤

C_{N,K} (1 + ||ξ||)^{n+μ−ρ(N+1)} ∑_{|γ|≤N+1} sup \{||\frac{D^{|γ|}}{∂z_{x}^{γ}} f(y)|| : y ∈ supp ψ_{π(ξ)}\} .

Because of lim_{N→∞} dimX + μ + ρ(N + 1) = −∞ and a similar consideration for the derivatives of σ_{A,f} the claim now follows. □

Next let us consider two pseudodifferential operators A ∈ Ψ_{ρ,δ}^{∞}(U, F, G) and B ∈ Ψ_{ρ,δ}^{∞}(U, E, F) between vector bundles E, F, G over X. Modulo smoothing operators we can assume one of them to be properly supported, so that AB ∈ Ψ_{ρ,δ}^{∞}(U, E, G) exists. Defining the extended symbol b^{ext} = σ_{B}^{ext} : U × T^{∗}U → Hom(E, F) of B by

\begin{matrix}b^{ext}(x, ξ)Ξ = \\ ψ_{π(ξ)}(x) e^{−iφ(x,ξ)} [B (ψ_{π(ξ)}(·) e^{iφ(·,ξ)}τ_{(·),π(ξ)}Ξ)] (x), x ∈ U, (ξ, Ξ) ∈ π^{∗}(E|_{U}) \end{matrix} (76)

the symbol σ_{AB} of the composition AB is now given modulo S^{−∞}(U, Hom(E, G)) by

\begin{aligned}σ_{AB}(ξ)Ξ & = [AB (ψ_{π(ξ)}e^{iφ(·,ξ)}τ_{(·),π(ξ)}Ξ)] (π(ξ)) \\ & = [A (ψ_{π(ξ)} e^{iφ(·,ξ)} σ_{B}^{ext}(·, ξ)Ξ)] (π(ξ)). \end{aligned} (77)

By Proposition 6.1 this entails that σ_{AB} has an asymptotic expansion of the form

σ_{AB}(ξ) ∼ ∑_{α∈ℕ^{n}} \frac{1}{i^{|α|}α!} [ \frac{D^{|α|}}{∂ζ_{π(ξ)}^{α}} |_{ξ} σ_{A}] [ \frac{D^{|α|}}{∂z_{π(ξ)}^{α}} |_{π(ξ)} σ_{B}^{ext}(·, ξ)] . (78)

As we want to find an even more detailed expansion for σ_{AB}, let us calculate the derivatives \frac{D^{|α|}}{∂z_{π(ξ)}^{α}} |_{π(ξ)} σ_{B}^{ext}(·, ξ) in the following proposition.

Proposition 6.2. Let B ∈ Ψ_{ρ,δ}^{μ}(U, E, F) be a pseudodifferential operator between Hermitian (Riemannian) vector bundles over the Riemannian manifold X and b = σ(B) its symbol. Then for any f ∈ C^{∞}(U, E) the smooth function

σ_{B,f}^{ext} : U × T^{∗}U → F, (x, ξ) ↦ ψ_{π(ξ)}(x) e^{−iφ(x,ξ)} [B (ψ_{π(ξ)}f e^{iφ(·,ξ)})] (x) (79)

is a symbol of order μ and type (ρ, δ) on U × T^{∗}U and has an asymptotic expansion of the form

\begin{aligned}σ_{B,f}^{ext}(x, ξ) ∼ & ∑_{k∈ℕ} ∑_{β,β_{0},...,β_{k}∈ℕ^{n} ; {β_{0}+...+β_{k}=β \atop |β_{1}|,...,|β_{k}|≥2}} \frac{i^{k−|β|}}{k!β_{0}! · ... · β_{k}!} ψ_{π(ξ)}(x) [ \frac{D^{|β|}}{∂ζ_{x}^{β}} |_{d_{x}φ( ,ξ)} b] \\ & [ \frac{D^{|β_{0}|}}{∂z_{x}^{β_{0}}} |_{x} ψ_{π(ξ)}f] [ \frac{∂^{|β_{1}|}}{∂z_{x}^{β_{1}}} |_{x} φ(·, ξ)] · ... · [ \frac{∂^{|β_{k}|}}{∂z_{x}^{β_{k}}} |_{x} φ(·, ξ)] . \end{aligned} (80)

Furthermore the derivative T^{∗}U ∋ ξ ↦ [\frac{∂^{|α|}}{∂z_{π(ξ)}^{α}} σ_{B,f}^{ext}(·, ξ)] (π(ξ)) ∈ F with α ∈ ℕ^{n} is a symbol with values in F of order μ + δ|α| and type (ρ, δ) on T^{∗}U and has an asymptotic expansion of the form

[\frac{D^{|α|}}{∂z_{π(ξ)}^{α}} σ_{B,f}^{ext}(·, ξ)] (π(ξ)) ∼

∼ ∑_{k∈ℕ} ∑_{{α̃,α_{0},...,α_{k}∈ℕ^{n} \atop α̃+α_{0}+...+α_{k}=α}} ∑_{β,β_{0},...,β_{k}∈ℕ^{n} ; {β_{0}+...+β_{k}=β \atop |β_{1}|,...,|β_{k}|≥2}} \frac{i^{k−|β|}α!}{k!α̃!α_{0}! · ... · α_{k}!β_{0}! · ... · β_{k}!}

\{ \frac{D^{|α̃|}}{∂z_{π(ξ)}α̃} |_{π(ξ)} [ \frac{D^{|β|}}{∂ζ_{(−)}^{β}} |_{d_{(−)}φ( ,ξ)} b]\} \{ \frac{D^{|α_{0}|}}{∂z_{π(ξ)}^{α_{0}}} |_{π(ξ)} [ \frac{D^{|β_{0}|}}{∂z_{(−)}^{β_{0}}} |_{(−)} f]\} ·

\{ \frac{∂^{|α_{1}|}}{∂z_{π(ξ)}^{α_{1}}} |_{π(ξ)} [ \frac{∂^{|β_{1}|}}{∂z_{(−)}^{β_{1}}} |_{(−)}φ(·, ξ)]\} · ... · \{ \frac{∂^{|α_{k}|}}{∂z_{π(ξ)}^{α_{k}}} |_{π(ξ)} [ \frac{∂^{|β_{k}|}}{∂z_{(−)}^{β_{k}}} |_{(−)}φ(·, ξ)]\} .

Note 6.3. The differential operators of the form \frac{D^{|α|}}{∂z_{π(ξ)}^{α}} act on the variables de-

noted by (−), the differential operators of the form \frac{D^{|β|}}{∂z_{(−)}^{β}} on the variables (·).

Proof of Proposition 6.2. Let η ∈ C^{∞}(U ×U ×T^{∗}U) be a smooth function such that

η(x, y, ξ) = φ(y, ξ) − φ(x, ξ)− < d_{x}φ( , ξ), z_{x}(y) > (81)

for x, y ∈ supp ψ_{π(ξ)} and such that T^{∗}U ∋ ξ ↦ η(x, y, ξ) ∈ ℂ is linear for all x, y ∈ U. Furthermore denote by F ∈ C^{∞}(U × U × T^{∗}U, E) the function (x, y, ξ) ↦ ψ_{π(ξ)}(y)f(y)e^{iη(x,y,ξ)}. By the Leibniz rule and the definition of η we then have

\begin{aligned}[ \frac{D^{|β|}}{∂z_{x}^{β}} |_{y} F (x, ·, ξ)] & = ∑_{0≤k≤ \frac{|β|}{2}} ∑_{β_{0},...,β_{k}∈ℕ^{n} ; {β_{0}+...+β_{k}=β \atop |β_{1}|,...,|β_{k}|≥2}} \frac{i^{k}}{k!β_{0}! · ... · β_{k}!} [ \frac{D^{|β_{0}|}}{∂z_{x}^{β_{0}}} |_{y} ψ_{π(ξ)}f] · \\ & [ \frac{∂^{|β_{1}|}}{∂z_{x}^{β_{1}}} |_{y} η(x, ·, ξ)] · ... · [ \frac{∂^{|β_{k}|}}{∂z_{x}^{β_{k}}} |_{y} η(x, ·, ξ)] e^{iη(x,y,ξ)}. \end{aligned} (82)

As (x, y, ξ) ↦ [ \frac{∂^{|β_{j} |}}{∂z_{x}^{β_{j}}} |_{y} η(x, ·, ξ)], 0 ≤ j ≤ k is smooth and linear with respect to

ξ, the preceeding equation entails that (x, y, ξ) ↦ [ \frac{D^{|β|}}{∂z_{x}^{β}} |_{y} F (x, ·, ξ)] is absolutely

bounded by a function (x, y, ξ) ↦ C_{β}(x, y) |ξ|^{\frac{|β|}{2}} , where C_{β} ∈ C^{∞}(U × U) and C_{β} ≥ 0. On the other hand we have

σ_{B,f}^{ext}(x, ξ) = ψ_{π(ξ)}(x) [B (ψ_{π(ξ)}(·)f(·)e^{iη(·,x,ξ)} e^{i<d_{x}φ( ,ξ),z_{x}(·)>})] (x) (83)

and by the proof of Proposition 6.1

σ_{B,f}^{ext}(x, ξ) = ψ_{π(ξ)}(x) ∑_{|β|≤N} \frac{i^{|β|}}{β!} [ \frac{D^{|β|}}{∂ξ_{x}^{β}} |_{d_{x}φ( ,ζ)} b] [ \frac{D^{|β|}}{∂z_{x}^{β}} |_{x} F (x, ·, ξ)] + r_{N}(x, ξ) (84)

= ψ_{π(ξ)}(x) ∑_{|β|≤N} ∑_{0≤k≤ \frac{|β|}{2}} ∑_{β_{0},...,β_{k}∈ℕ^{n} ; {β_{0}+...+β_{k}=β \atop |β_{1}|,...,|β_{k}|≥2}} \frac{i^{k−|β|}}{k!β_{0}! · ... · β_{k}!} [ \frac{D^{|β|}}{∂ξ_{x}^{β}} |_{d_{x}φ( ,ζ)} b] ·

[ \frac{D^{|β_{0}|}}{∂z_{x}^{β_{0}}} |_{x} ψ_{π(ξ)}f] · [ \frac{∂^{|β_{1}|}}{∂z_{x}^{β_{1}}} |_{x} φ(·, ξ)] · ... · [ \frac{∂^{|β_{k}|}}{∂z_{x}^{β_{k}}} |_{x} φ(·, ξ)] + r_{N}(x, ξ),

where

\begin{aligned}||r_{n}(x, ξ)|| & ≤ C_{N,K}(1 + ||d_{x}φ( , ξ)||)^{n+μ−ρ(N+1)} \\ ∑_{|γ|≤N+1} & sup & \{||[ \frac{∂^{|γ|}}{∂z_{x}^{γ}} |_{y} F (x, ·, ξ)]|| : y ∈ supp ψ_{x}\} \\ & ≤ D_{N,K}(1 + ||ξ||)^{n+μ−(ρ−1/2)(N+1)} \end{aligned} (85)

holds for compact K ⊂ U, x ∈ K, ξ ∈ T^{∗}X|_{K} and constants C_{N,K}, D_{N,K} > 0. Because ρ > \frac{1}{2}, we have lim_{N→∞} dimX + μ − (ρ − \frac{1}{2})(N + 1) = −∞. By Eqs. (84) and (85) and similar relations for the derivatives of σ_{A,f}^{ext} we therefore conclude that Eq. (80) is true. The rest of the claim easily follows from Leibniz rule. □

The last proposition and Eq. (78) imply the next theorem.

Theorem 6.4. Let A, B ∈ Ψ_{ρ,δ}^{∞}(U, E, F) be two pseudodifferential operators between Hermitian (Riemannian) vector bundles E, F over a Riemannian manifold X. Assume that one of the operators A and B is properly supported. Then the symbol σ_{AB} of the product AB has the following asymptotic expansion.

\begin{matrix}σ_{AB} (ξ) ∼ ∑_{α∈ℕ^{d}} \frac{i^{−|α|}}{α!} [ \frac{D^{|α|}}{∂ζ_{π(ξ)}^{α}} |_{ξ} σ_{A}] [ \frac{D^{|α|}}{∂z_{π(ξ)}^{α}} |_{π(ξ)} σ_{B}] + \\ + ∑_{k≥1} ∑_{{α,α̃,α_{1},...,α_{k}∈ℕ^{n} \atop α̃+α_{1}+...+α_{k}=α}} ∑_{β,β_{1},...,β_{k}∈ℕ^{n} ; {β_{1}+...+β_{k}=β \atop |β_{1}|,...,|β_{k}|≥2}} \frac{i^{k−|α|−|β|}}{k! · α̃! · α_{1}! · ... · α_{k}!β_{1}! · ... · β_{k}!} \\ [ \frac{D^{|α|}}{∂ζ_{π(ξ)}^{α}} |_{ξ} σ_{A}] \{ \frac{D^{|α̃|}}{∂z_{π(ξ)}α̃} |_{π(ξ)} [ \frac{D^{|β|}}{∂ζ_{(−)}^{β}} |_{d_{(−)}φ( ,ξ)} σ_{B}]\} · \\ \{ \frac{∂^{|α_{1}|}}{∂z_{π(ξ)}^{α_{1}}} |_{π(ξ)} [ \frac{∂^{|β_{1}|}}{∂z_{(−)}^{β_{1}}} |_{(−)}φ(·, ξ)]\} · ... · \{ \frac{∂^{|α_{k}|}}{∂z_{π(ξ)}^{α_{k}}} |_{π(ξ)} [ \frac{∂^{|β_{k}|}}{∂z_{(−)}^{β_{k}}} |_{(−)}φ(·, ξ)]\} . \end{matrix} (86)

A somewhat more explicit expansion up to second order is given by

Corollary 6.5. If σ_{A} is a symbol of order μ and σ_{B} a symbol of order μ̃ the coefficients in the asymptotic expansion of σ_{AB} are given up to second order by the following formula:

σ_{AB}(ξ) =σ_{A}(ξ) · σ_{B}(ξ) −

− i ∑_{l} \frac{Dσ_{A}}{∂ζ_{π(ξ),l}} (ξ) \frac{Dσ_{B}}{∂z_{π(ξ)}^{l}} (ξ) − \frac{1}{2} ∑_{k,l} \frac{D^{2}σ_{A}}{∂ζ_{π(ξ),k}∂ζ_{π(ξ),l}} (ξ) \frac{D^{2}σ_{B}}{∂z_{π(ξ)}^{k}∂z_{π(ξ)}^{l}} (ξ) (87)

− \frac{1}{12} ∑_{k,l,m,n} \frac{Dσ_{A}}{∂ζ_{π(ξ),n}} (ξ) \frac{D^{2}σ_{B}}{∂ζ_{π(ξ),l}∂ζ_{π(ξ),m}} (ξ) R^{k}_{mln}(π(ξ)) ζ_{π(ξ),k}(ξ) + r(ξ)

where ξ ∈ T^{∗}U, x = π(ξ), r ∈ S_{ρ,δ}^{μ+μ̃−3}(U) and the R^{m}_{nkl}(y) are the coefficients of

the curvature tensor with respect to normal coordinates at y ∈ X, i.e., R (\frac{∂}{∂z_{y}^{m}} ) =

∑_{k,l,n} R^{k}_{mln}(y)\frac{∂}{∂z_{y}^{k}} ⊗ dz_{y}^{l} ⊗ dz_{y}^{n}.

Proof. First define the order of a summand s = s_{α,α̃,α_{1},...,α_{k}}^{β,β_{1},...,β_{k}} in Eq. (86) by ord(s) =

|α −α̃ +β −k|. Then it is obvious that s ∈ S_{ρ,δ}^{μ+μ̃+ord(s) (ρ−δ)}(U). Therefore we have to calculate only the summands s with ord(s) ≤ 2. To achieve this let us calculate some coordinate changes and their derivatives. Recall the matrix-valued function (θ_{l}^{k}) of Eq. (61) and construct its inverse \binom{}{θ̄_{l}^{k}}. In other words

∑_{k} θ_{k}^{l}(x, y) θ̄_{m}^{k}(x, y) = δ_{m}^{l} (88)

holds for every l, m = 1, ..., d. We can now write

\frac{∂}{∂z_{y}^{k}} |_{z} = ∑_{l,m} θ_{k}^{l}(y, z) θ̄_{l}^{m}(x, z) \frac{∂}{∂z_{x}^{m}} |_{z} (89)

and receive the following formulas:

\begin{matrix}\frac{∂z_{x}^{k}}{∂z_{y}^{l}} (z) = & δ_{l}^{k} − \frac{1}{6} ∑_{m,n} R^{k}_{mln}(y) z_{x}^{m}(z) z_{x}^{n}(z) \\ + \frac{1}{6} ∑_{m,n} R^{k}_{mln}(x)z_{x}^{m}(z) z_{x}^{n}(z) + O (|z_{x}(z) + z_{y}(z)|^{3}) , \end{matrix} (90)

\begin{aligned}\frac{∂^{2}z_{x}^{k}}{∂z_{y}^{l}∂z_{y}^{m}} (z) = & − \frac{1}{6} ∑_{n} (R^{k}_{mln}(y) + R^{k}_{nlm}(y)) z_{x}^{n}(z) \\ & + \frac{1}{6} ∑_{n} (R^{k}_{mln}(x) + R^{k}_{nlm}(x)) z_{x}^{n}(z) \\ & + O (|z_{x}(z) + z_{y}(z)|^{2}) , \end{aligned} (91)

\frac{∂}{∂z_{x}^{n}} |_{y=z} \frac{∂^{2}z_{x}^{k}}{∂z_{y}^{l}∂z_{y}^{m}} (y) = \frac{1}{6} (R^{k}_{mln}(x) + R^{k}_{nlm}(x)) + O (|z_{x}(z)|) . (92)

Next consider the functions

φ_{αβ} : T^{∗}X → ℝ,  ξ ↦ \frac{∂^{|α|}}{∂z_{π(ξ)}^{α}} |_{y=π(ξ)} \frac{∂^{|β|}φ(·, ξ)}{∂^{β}z_{y}^{β}}(y)

which appear in the summands s of Eq. (86). One can write the φ_{αβ} in the form

φ_{αβ}(ξ) = ∑_{k} ζ_{π(ξ),k}(ξ) \frac{∂^{|α|}}{∂z_{π(ξ)}^{α}} |_{y=π(ξ)} \frac{∂^{|β|}z_{π(ξ)}^{k}}{∂z_{y}^{β}} (y) (93)

Thus by Eq. (91)

φ_{0β}(ξ) = 0 (94)

holds for every β ∈ ℕ^{d} with |β| ≥ 2. Furthermore check that

d^{y}φ(·, ξ) = ∑_{k} ζ^{π(ξ),k}(ξ) dz{k \atop π(ξ)}|_{y} (95)

is true.

Now we are ready to calculate the summands s of order ord(s) ≤ 2 by considering the following cases.

(i) ord(s) = 0:

s(ξ) = σ_{A}(ξ) σ_{B}(ξ). (96)

(ii) ord(s) = 1, |α| = 1, k = 0, α̃ = α:

s(ξ) = − i ∑_{l} \frac{Dσ_{A}}{∂ζ_{π(ξ),l}} (ξ) \frac{Dσ_{B}}{∂z_{π(ξ)}^{l}} (ξ). (97)

(iii) ord(s) = 1, |α| = 0, k = 1, |β| = 2: In this case s = 0 because of Eq. (94). (iv) ord(s) = 2, |α| = 2, k = 0, α̃ = α:

s(ξ) = − \frac{1}{2} ∑_{k,l} \frac{D^{2}σ_{A}}{∂ζ_{π(ξ),k}∂ζ_{π(ξ),l}} (ξ) \frac{D^{2}σ_{B}}{∂z_{π(ξ)}^{k}∂z_{π(ξ)}^{l}} (ξ). (98)

(v) ord(s) = 2, |α| = 1, k = 1, |β| = 2: In this case α̃ = 0 by Eq. (94). Hence by Eq. (92)

s(ξ) = − \frac{1}{12} ∑_{k,l,m,n} \frac{Dσ_{A}}{∂ζ_{π(ξ),n}} (ξ) \frac{D^{2}σ_{B}}{∂ζ_{π(ξ),l}∂ζ_{π(ξ),m}} (ξ) R^{k}_{mln}(π(ξ)) ζ_{π(ξ),k}(ξ). (99)

Summing up these summands we receive the expansion of σ_{AB} up to second order. This proves the claim. □

The product expansion of Theorem 6.4 gives rise to a a bilinear map # on the sheaf S_{ρ,δ}^{∞}/S^{−∞}(·, Hom(F, G)) × S_{ρ,δ}^{∞}/S^{−∞}(·, Hom(E, F)) by

S_{ρ,δ}^{∞}(U, Hom(F, G)) × S_{ρ,δ}^{∞}(U, Hom(E, F)) → S_{ρ,δ}^{∞}/S^{−∞}(U, Hom(E, G)),
(a, b) ↦ a#b = σ_{Op(a) Op(b)} (100)

for all U ⊂ X open and vector bundles E, F, G over X. The #-product is an important tool in a deformation theoretical approach to quantization (cf. Pflaum [8]). In the sequel it will be used to study ellipticity of pseudodifferential operators on manifolds.

7. Ellipticity and normal symbol calculus

The global symbol calculus introduced in the preceding paragraphs enables us to investigate the invertibility of pseudodifferential operators on Riemannian manifolds. In particular we are now able define a notion of elliptic pseudodifferential operators which is more general than the usual notion by Hörmander [7] or Douglis, Nirenberg [2].

Definition 7.1. A symbol a ∈ S_{ρ,δ}^{∞}(X, Hom(E, F)) on a Riemannian manifold X is called elliptic if there exists b_{0} ∈ S_{ρ,δ}^{∞}(X, Hom(F, E))) such that

a#b_{0} − 1 ∈ S_{ρ,δ}^{−ε}(X) and b_{0}#a − 1 ∈ S_{ρ,δ}^{−ε}(X) (101)

for an ε > 0. A symbol a ∈ S_{ρ,δ}^{m}(X, Hom(E, F)) is called elliptic of order m if it is elliptic and one can find a symbol b_{0} ∈ S_{ρ,δ}^{m}(X, Hom(F, E)) fulfilling Eq.(101).

An operator A ∈ Ψ_{ρ,δ}^{∞}(X, E, F) is called elliptic resp. elliptic of order m, if its symbol σ_{A} is elliptic resp. elliptic of order m.

Let us give some examples of elliptic symbols resp. operators.

Example 7.2. (i) Let a ∈ S_{1,0}^{m}(X, Hom(E, F)) be a classical symbol, i.e., assume that a has an expansion of the form a ∼ ∑_{j∈ℕ} a_{m−j} with a_{m−j} ∈ S_{1,0}^{m−j}(X) homogeneous of order m − j. Further assume that a is elliptic in the classical sense, i.e., that the principal symbol a_{m} is invertible outside the zero section. By the homogeneity of a_{m} this just means that a_{m} is elliptic of order m. But then a and Op(a) must be elliptic of order m as well.

(ii) Consider the symbols l : T^{∗}X → ℂ, ξ ↦ ||ξ||^{2} and a : T^{∗}X → ℂ, ξ ↦ \frac{1}{1+||ξ||^{2}} of Example 2.3 (ii). Then the pseudodifferential operator corresponding to l is minus the Laplacian: Op(l) = −Δ. Furthermore the symbols l and 1+l are elliptic of order 2, and the relation a#(1 + l) − 1, (1 + l)#a − 1 ∈ S_{1,0}^{−1}(X) is satisfied. In case X is a flat Riemannian manifold, one can calculate directly that modulo smoothing symbols a#l = l#a = 1, hence Op(a) is a parametrix for the differential operator Op(1 + l) = 1 − Δ.

(iii) Let a_{φ}(ξ) = (1 + ||ξ||^{2})^{φ(π(ξ))} be the symbol defined in Example 2.3 (iii), where φ : X → ℝ is supposed to be smooth and bounded. Then a is elliptic and b(ξ) = (1 + ||ξ||^{2})^{−φ(π(ξ))} fulfills the relations (101). In case φ is not locally constant, we thus receive an example of a symbol which is not elliptic in the sense of Hörmander [7] but elliptic in the sense of the above definition.

Let us now show that for any elliptic symbol a ∈ S_{ρ,δ}^{∞}(X, Hom(E, F)) one can find a symbol b ∈ S_{ρ,δ}^{∞}(X, Hom(F, E)) such that even

(102) a#b − 1 ∈ S^{−∞}(X, Hom(F, F)) and b#a − 1 ∈ S^{−∞}(X, Hom(E, E)).

Choose b_{0} according to Definition 7.1 and let r = 1−a#b_{0}, l = 1−b_{0}#a ∈ S^{−∞}(X). Then the symbols

b_{r} = b_{0} (1 + r + r#r + r#r#r + …) and b_{l} = (1 + l + l#l + l#l#l + …) b_{0}

are well-defined and fulfill a#b_{r} = 1 and b_{l}#a = 1 modulo smoothing symbols. Hence b_{r} − b_{l} ∈ S^{−∞}(X, Hom(F, E)), and b = b_{r} is the symbol we were looking for. Thus we have shown the essential part for the proof of the following theorem.

Theorem 7.3. Let A ∈ Ψ_{ρ,δ}^{∞}(X, E, F) be an elliptic pseudodifferential operator. Then there exists a parametrix B ∈ Ψ_{ρ,δ}^{∞}(X, F, E) for A, i.e., the relations

A B − 1 ∈ Ψ^{−∞}(X, F, F) and B A − 1 ∈ Ψ^{−∞}(X, E, E) (103)

hold. If A is elliptic of order m, B can be chosen of order −m. On the other hand, if A is an operator invertible in Ψ_{ρ,δ}^{∞}(X) modulo smoothing operators, then A is elliptic. In case X is compact an elliptic operator A ∈ Ψ_{ρ,δ}^{∞}(X) is Fredholm.

Proof. Let a = σ_{A}. As a is elliptic one can choose b such that (102) holds. If a is elliptic of order m, the above consideration shows that b is of order −m. The operator B = Op(b) then is the parametrix for A. If on the other hand A = Op(a) has a parametrix B = Op(b), then a#b − 1 and b#a − 1 are smoothing, hence A is elliptic. Now recall that a pseudodifferential operator induces continuous mappings between appropriate Sobolev-completions of C^{∞}(X, E), resp. C^{∞}(X, F), and that with respect to these Sobolev-completions any smoothing pseudodifferential operator is compact. Hence the claim that for compact X an elliptic A is Fredholm follows from (103). □

Let us give in the following propositions a rather simple criterion for ellipticity of order m in the case of scalar symbols.

Proposition 7.4. A scalar symbol a ∈ S_{ρ,δ}^{m}(X) is elliptic of order m, if and only if for every compact set K ⊂ X there exists C_{K} > 0 such that

||a(ξ)|| ≥ \frac{1}{C_{K}} ||ξ||^{m} (104)

for all ξ ∈ T^{∗}X with π(ξ) ∈ K and ||ξ|| ≥ C_{K}.

Proof. Let us first show that the condition is sufficient. By assumption there exists a function b ∈ C^{∞}(T^{∗}X) such that for every compact K ⊂ X there is C_{K} > 0 with

a(ξ) · b (ξ) = 1 and ||b(ξ)|| ≤ C_{K}||ξ||^{−m} (105) for ξ ∈ T^{∗}X with π(ξ) ∈ K and ||ξ|| ≥ C_{K}. Hence a · b − 1 ∈ S^{−∞}(X). After differentiating the relation r = a · b − 1 in local coordinates (x, ξ) of T^{∗}X we receive

(106) sup_{ξ∈T^{∗}X|_{K}} |(1 + ||ξ||^{2})^{m/2} \frac{∂^{|α|}}{∂x^{α}} \frac{∂^{|β|}}{∂ξ^{β}} b(ξ)| ≤ C sup_{ξ∈T^{∗}X|_{K}} |a(ξ) \frac{∂^{|α|}}{∂x^{α}} \frac{∂^{|β|}}{∂ξ^{β}} b(ξ) | ≤

≤ C sup_{ξ∈T^{∗}X|_{K}} \{|r(ξ)| + ∑_{{α_{1}+α_{2}=α \atop β_{1}+β_{2}=β} ; |α_{1}+β_{1}|>0} |\frac{∂^{|α_{1}|}}{∂x^{α_{1}}} \frac{∂^{|β_{1}|}}{∂ξ^{β_{1}}} a(ξ) \frac{∂^{|α_{2}|}}{∂x^{α_{2}}} \frac{∂^{|β_{2}|}}{∂ξ^{β_{2}}} b(ξ)|\} ,

hence by induction b ∈ S_{ρ,δ}^{−m}(X) follows. By the product expansion Eq. (87) this implies

a#b − 1 = r + t_{1} and b#a − 1 = r + t_{2}, (107)

where t_{1/2} ∈ S_{ρ,δ}^{−ε}(X) with ε = min{(ρ − δ), 2ρ − 1} > 0. Hence a is an elliptic symbol.

Now we will show the converse and assume the symbol a to be elliptic of order m. According to Theorem 7.3 there exists b ∈ S_{ρ,δ}^{−m}(X) such that Op(b) is a parametrix for Op(a). But this implies by Eq. (87) that ab − 1 ∈ S_{ρ,δ}^{−ε} for some ε > 0, hence |a(ξ)b(ξ) − 1| < \frac{1}{2} for all ξ ∈ T^{∗}X|_{K} with ||ξ|| > C_{K} , where K ⊂ X compact and C_{K} > 0. But then

\frac{1}{2} < |a(ξ)b(ξ)| < C′_{K}|a(ξ)|||ξ||^{−m}, ξ ∈ T^{∗}X|_{K}, ||ξ|| > C_{K}, (108)

which gives the claim. □

In the work of Douglis, Nirenberg [2] a concept of elliptic systems of pseudodifferential operators has been introduced. We want to show in the following that this concept fits well into our framework of ellipticity. Let us first recall the definition by Douglis and Nirenberg. Let E = E_{1} ⊕ ... ⊕ E_{K} and F = F_{1} ⊕ ... ⊕ F_{L} be direct sums of Riemannian (Hermitian) vector bundles over X, and A_{kl} ∈ Ψ_{1,0}^{m_{kl}}(X, E_{k}, F_{l}) with m_{kl} ∈ ℝ be classical pseudodifferential operators. Denote the principal symbol of A_{kl} by p_{kl} ∈ S_{1,0}^{m_{kl}}(X, E_{k}, F_{l}). Douglis, Nirenberg [2] now call the system (A_{kl}) elliptic, if there exist homogeneous symbols b_{kl} ∈ S_{1,0}^{−m_{kl}}(X, E_{k}, F_{l}) such that

∑_{l} p_{kl} · b_{lk′} − δ_{kk′} ∈ S_{1,0}^{−1}(X, E_{k}, F_{l}) and ∑_{k} b_{lk} · p_{kl′} − δ_{ll′} ∈ S_{1,0}^{−1}(X, E_{k}, F_{l}) (109)

for all k, k′, l, l′. The following proposition shows that A = (A_{kl}) is an elliptic pseudodifferential operator, hence according to Theorem 7.3 possesses a parametrix.

Proposition 7.5. Let E = E_{1} ⊕ ... ⊕ E_{K} and F = F_{1} ⊕ ... ⊕ F_{L} be direct sums of Riemannian (Hermitian) vector bundles over X. Assume that A = (A_{kl}) with A_{kl} ∈ S_{1,0}^{m_{kl}}(X, E, F) comprises an elliptic system of pseudodifferential operators in the sense of Douglis, Nirenberg [2]. Then the pseudodifferential operator A is elliptic.

The Normal Symbol on Riemannian Manifolds 125

Proof. Write a_{kl} = σ_{A_{kl}} = p_{kl} + r_{kl} with a_{kl} ∈ S_{1,0}^{m_{kl}}(X, Hom(E_{k}, F_{l}) and r_{kl} ∈ S_{1,0}^{m_{kl}−1}(X, Hom(E_{k}, F_{l}). Furthermore let a = (a_{kl}) ∈ S_{1,0}^{∞}(X, Hom(E, F)), b = (b_{kl}) ∈ S_{1,0}^{∞}(X, Hom(E, F)) and r = (r_{kl}) ∈ S_{1,0}^{∞}(X, Hom(E, F)). By Eq. (109) and the expansion Eq. (87) it now follows

a#b − 1 = (p#b − 1) + r#b ∈ S_{1,0}^{−1}(X, Hom(E, F)),
b#a − 1 = (b#p − 1) + b#r ∈ S_{1,0}^{−1}(X, Hom(E, F)). (110)

But this proves the ellipticity of A. □

So with the help of the calculus of normal symbols one can directly construct inverses to elliptic operators in the algebra of pseudodifferential operators on a Riemannian manifold. In particular after having once established a global symbol calculus it is possible to avoid lengthy considerations in local coordinates. Thus one receives a practical and natural geometric tool for handling pseudodifferential operators on manifolds.

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Humboldt Universität zu Berlin, Mathematisches Institut, Unter den Linden 6, 10099 Berlin

pflaum@mathematik.hu-berlin.de http://spectrum.mathematik.hu-berlin.de/˜ pflaum

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-8.html.