Parametric pseudodifferential operators with point-singularity in the covariable

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb–Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential operators originally proved by Grubb–Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents.


Introduction
In the present paper, we develop a calculus of parameter-dependent pseudodifferential operators (ψdo), both for operators in Euclidean space R n and for operators on sections of vector-bundles over closed Riemannian manifolds, which is closely related to Grubb's calculus of operators with finite regularity number [3] (for a recent application to fractional heat equations see [5]) and to the Grubb-Seeley calculus introduced in [6]. The calculus allows to obtain the classical resolvent-trace expansion for elliptic ψdo due to [6] and a systematic treatment of ψdo of Toeplitz type in the sense of [14,15].
In Sect. 4 we show that a(D; μ) is an operator of order d and with regularity number ν ∈ R in the sense of [3] if a admits a decomposition a = a + p with p ∈ S d and where a admits an expansion of form (1.1), with components satisfying (1.2) but with singular functions a j : introducing polar coordinates (r , φ) on S n + , centered in the "north-pole" (ξ, μ) = (0, 1), they belong to the weighted space r ν− j C ∞ B ( S n + ), where S n + = S n + \{(0, 1)} and C ∞ B means smooth functions which remain bounded on S n + after arbitrary applications of totally characteristic derivatives r ∂ r and usual derivatives in φ.
This observation leads us to consider symbols a = a + p with p ∈ S d but where the homogeneous components of a originate from the weighted spaces is the space of all functions on S n + that, in coordinates (r , φ), extend smoothly up to and including r = 0 (the subscript T stands for Taylor expansion). Symbols of this kind do not only have an expansion (1.1) but intrinsically a further expansion of the form where [ξ, μ] denotes a smooth function that coincides with the usual modulus away from the origin and S m (R n ) is the standard poly-homogeneous Hörmander class of order m without parameter. See Sect. 5 for details. Evidently, expansion (1.3) resembles the one employed by Grubb-Seeley in [6]. While Grubb-Seeley's expansion is in powers of μ and has its origin in a meromorphic (at infinity) dependence on the parameter μ, (1.3) directly originates from the Taylor expansion of the homogeneous components and makes no use of a holomorphic dependence on the parameter. However, expanding [ξ, μ] m in powers of μ allows us to obtain a Grubb-Seeley expansion and ultimately we can recover the resolvent-trace expansion of ψdo shown in [6]. This is discussed in detail in Sects. 6 and 7.6. Ellipticity in our class is most simple for a positive regularity number ν > 0. In this case, the homogeneous principal symbol extends by continuity to the north-pole, and its non-vanishing yields the existence of a parametrix which is the inverse of a(D; μ) for large values of the parameter μ. For ν = 0, ellipticity is more involved and two additional symbolic levels come into play: (a) the principal angular symbol which originates from the leading term of the Taylor expansion of the homogeneous principal symbol, (b) the principal limit-symbol, i.e., the symbol a ∞ [0] from (1.3). Non-vanishing of the homogeneous principal symbol, of the principal angular symbol, and invertibility of the operator a ∞ [0] (D) guarantee the existence of a parametrix in the class which is the inverse for large values of μ. Concerning (a), our calculus appears to be related with Savin-Sternin [8] where a similar structure occurs.
We show that our calculus of operators on R n is invariant under changes of coordinates, see Sect. 7.1. Thus, we can define corresponding classes of ψdo on closed manifolds M, acting on sections of finite-dimensional vector-bundles. While the homogeneous principal symbol and the principal angular symbol have a global meaning as bundle morphisms on (T * M × R + ) \ 0 and T * M \ 0, respectively, expansion (1.3) is shown to have a global analog too, namely (1.4) where α (μ) ∈ L α (E 0 , E 0 ), α ∈ R, denote elliptic elements in Hörmander's class with (scalar) homogeneous principal symbol (|ξ | 2 x + μ 2 ) α/2 , where | · | refers to some fixed Riemannian metric on M. This is shown in Sect. 7.2. The so-called limit-operator A ∞ [ν] takes the place of the above used limit-symbol. In Sect. 7.7 we discuss an application to parameterdependent ψdo of Toeplitz type, here on closed manifolds; originally the concept of Toeplitz type operators emerged in the study of boundary value problems with Atiyah-Patodi-Singer type boundary conditions, see [12,13].
In the present paper, we limit ourselves to ψdo on R n or closed manifolds. However, it is a natural question whether the established calculus allows to build up a corresponding calculus for boundary value problems, in the spirit of [3,4] and [9], leading to a parameterdependent version of the classical Boutet de Monvel algebra [1]. Similarly, one could address the analogous question for manifolds with singularities (conical singularities, in the simplest case), following and extending the approach of Schulze [10,11]. We plan to address these questions in future work.
Hoping to help the reader in reading this paper, we finish this introduction by listing the most important spaces of pseudodifferential symbols used in the sequel: Let f (ω, y) be defined on a set of the form × (R m \ {0}). With slight abuse of language, we shall call f homogeneous of degree d in y if f (ω, t y) = t d f (ω, y) ∀ (ω, y) ∀ t > 0; it would be correct to use the terminology positively homogeneous, but for brevity we shall not do so. Suppose y = (u, v) with u ∈ R k and v ∈ R m−k (k may be equal to m, i.e., y = u).
We shall say that f is homogeneous of degree d in (u, v) for large u if there exists a constant R ≥ 0 (frequently assumed to be equal to 1) such that

Hörmander's class
The uniform Hörmander class S d 1,0 (R n ) consists of those symbols a(x, ξ) : R n × R n → C satisfying the uniform estimates for every multi-indices α, β ∈ N n 0 . This is a Fréchet space with the system of norms for every N , where χ(ξ) is an arbitrary fixed zero-excision function (note that r a,0 = a). Denote by S d (R n ) the space of all such symbols. It is a Fréchet space with the system of norms a j,N := r a,N j , j, N ∈ N 0 , and The ψdo associated with a(x, ξ), denoted by a(x, D), is acting on the Schwartz space S (R n ) of rapidly decreasing functions; here, d¯ξ = (2π) −n dξ . The composition of operators a 0 (x, D) and a 1 (x, D) corresponds to the so-called Leibniz product of symbols, (integration in the sense of oscillatory integrals), cf. for example [7]. If the a j have order d j , then a 1 #a 0 has order d 0 + d 1 . The adjoint symbol gives the formal adjoint operator of a(x, D), i.e., If a(x, ξ; μ) is a symbol that depends on an additional parameter μ, we shall write a(x, D; μ), Leibniz product and adjoint are applied point-wise in μ. Throughout the paper we consider a parameter μ ∈ R + := [0, +∞).

Grubb's calculus
We briefly review a pseudodifferential calculus introduced by Grubb. For further details, we refer the reader to Chapter 2.1 of [3].
The space of smoothing or regularizing symbols, defined as consists of those symbols satisfying, for every N and all orders of derivatives,

Proposition 3.2 The Leibniz product induces maps
Asymptotic summations can be performed within the class, in the following sense: Given a sequence of symbols a ∈ S d− ,ν− 1,0 , there exists an a ∈ S d,ν 1,0 such that If a ∈ S d,ν , its homogeneous principal symbol is defined as Definition 3.4 S d,ν hom denotes the space of all smooth functions a(x, ξ; μ) defined for ξ = 0, which are homogeneous of degree d in (ξ, μ) and satisfy (3.5) for arbitrary orders of derivatives.
If a ∈ S d,ν hom and ν > 0, then a extends by continuity to a function defined for all x and (ξ, μ) = 0; the larger ν is, the more regular (i.e., differentiable) is this extension. This is the justification for the terminology "regularity number." In this case we shall identify a with its extension.
In particular, if a is elliptic then a(x, D; μ) is invertible for large μ.
(3.1) and (3.5) suggest to introduce two subspaces of S d,ν 1,0 and S d,ν hom , respectively, with estimates corresponding to the first and second term on the right-hand side, respectively. These will be discussed in the next two subsections.

Strong parameter-dependence (symbols of infinite regularity)
In this section, we consider the space S d 1,0 and the poly-homogeneous subclass. For clarity we prefer to present it in an independent way. Definition 3.7 S d 1,0 consists of all symbols a(x, ξ; μ) satisfying, for all orders of derivatives, We shall call such symbols also strongly parameter-dependent, since differentiation with respect to ξ or μ improves the decay in (ξ, μ).
The space of regularizing symbols S −∞ = ∩ d∈R S d consists of those symbols which are rapidly decreasing in (ξ, μ) and C ∞ b in x.
where χ(ξ, μ) is an arbitrary zero-excision function. The space of such symbols will be denoted by S d .
We call a 0 the homogeneous principal symbol of a ∈ S d , and Ellipticity of a is defined as in Definition 3.5 and the obvious analog of Theorem 3.6 is valid.

Remark 3.11
In the literature, the space S d is frequently denoted by S d cl and the symbols are called classical rather than poly-homogeneous.

Weakly parameter-dependent symbols
Let us describe the second natural subspace of S d,ν 1,0 .
Definition 3.12 Let S d,ν 1,0 denote the space of all symbols a(x, ξ; μ) which satisfy, for every order of derivatives, Note that S d,ν 1,0 = S d,ν 1,0 whenever ν ≤ 0. In particular, S d−∞,ν−∞ Proposition 3. 13 The Leibniz product induces maps Definition 3.14 Let S d,ν hom denote the space of all functions a(x, ξ; μ) which are defined for ξ = 0, are homogeneous in (ξ, μ) of degree d and satisfy, for every order of derivatives, where χ(ξ) is an arbitrary zero-excision function. The space of such symbols will be denoted by S d,ν .

Theorem 3.17 A symbol a ∈ S d,ν is elliptic if and only if there exists a b ∈ S
Note that ellipticity of a ∈ S d,ν is not equivalent to the point-wise invertibility of the homogeneous principal symbol a 0 on its domain, even not in case of independence of the x-variable (see Theorem 4.4 and the subsequent comment). Moreover, a remainder r ∈ S 0−∞,0−∞ is, in general, only bounded but not decaying as μ → +∞. Therefore, a(x, D; μ) need not be invertible for large μ.

Regularity number and weighted spaces
In any of the so far introduced symbol spaces, the involved variable x enters as a C ∞ b -variable, while the spaces differ by the structures in the variables (ξ, μ). For this reason, and also to keep notation more lean, in this section we ignore the x-dependence and focus on symbols depending only on (ξ, μ).
Let us denote by S n + the unit semi-sphere, Every homogeneous symbol a ∈ S d hom is of the form a(ξ ; μ) = |ξ, μ| d a (ξ, μ) |ξ, μ| , a = a| S n + ∈ C ∞ (S n + ), (4.2) and the map a → a establishes an isomorphism between S d hom and C ∞ (S n + ). A symbol a ∈ S d,ν hom is defined for ξ = 0 only, hence its restriction is defined only on the punctured unit semi-sphere We shall now investigate, which subspace of C ∞ ( S n + ) is in 1-1-correspondence with S d,ν hom . To this end, we shall identify S n + with (0, 1] × S n−1 , using the (polar-)coordinates If E is an arbitrary Fréchet space, we shall denote by C ∞ B ((0, ε), E) the space of all smooth bounded functions u : (0, ε) → E such that (r ∂ r ) u is bounded on (0, ε) for every order of derivatives.
In other words, the index γ indicates the rate of (non-)vanishing in the point (ξ, μ) = (0, 1); we shall also speak of spaces with weight γ . Note that |ξ | = r .
Let a and a be as in the previous definition. Identifying a(ξ, μ) with its local representation a(r , φ), we have the relations In particular, the d-homogeneous extension of a(r , φ) = r ν is a(ξ ; μ) = |ξ | ν |ξ, μ| d−ν .
The following theorem shows that, for weakly parameter-dependent homogeneous components, regularity number and weight are the same thing.

Theorem 4.4 S (d,ν) = S d,ν
hom for every d, ν ∈ R. In particular, the map a → a| S n + establishes an isomorphism between S d,ν hom and r ν C ∞ B ( S n + ).
Proof Let us first prove the inclusion "⊆." Let a(ξ ; μ) be as in Definition 4.2. By multiplication with |ξ, μ| −d , we may assume without loss of generality that d = 0. In view of Lemma 4.3 we may assume that a is supported in a small neighborhood of the point (0, 1). Hence, in representation (4.4) we may assume that a(r , φ) ∈ r ν C ∞ B ((0, 1), C ∞ (S n−1 )). We also may assume ν = 0, since the homogeneous extension of degree d = 0 of r ν is just |ξ | ν |ξ, μ| −ν . Thus we can write Therefore, By induction, we thus find that D α ξ D j μ a(ξ ; μ) is a finite linear combination of terms of the form with a m (r , ξ) ∈ C ∞ B ((0, 1), S −m hom (R n )) and p j+ ∈ S −( j+ ) hom . This yields Next we shall show the inclusion "⊇." Let a ∈ S d,ν hom be given. It is enough to consider the case d = ν = 0, since a ∈ S d,ν hom if and only if |ξ | −ν |ξ, μ| ν−d a ∈ S 0,0 hom and |ξ | −ν |ξ, μ| ν−d = r −ν in polar-coordinates. Again, a can be assumed do have support in a small conical neighborhood containing (0, 1). Thus Using that r v (r )/v(r ) = 1/(r 2 − 1), it is straightforward to see that (r ∂ r ) ∂ α φ a(r , φ) is a linear combination of terms of the form where q is smooth and homogeneous of degree −|α| in φ = 0 and g ∈ C ∞ ([0, 1)). Thus (r ∂ r ) ∂ α φ a(r , φ) is bounded for r ∈ (0, δ] and φ belonging to a small neighborhood of the unit-sphere S n . This shows the claim. In particular, we see that S d,ν hom does not behave well under inversion: if a ∈ r ν C ∞ ( S n + ) is point-wise invertible, the inverse will, in general, not belong to such a weighted space. To guarantee this, an additional control at the singularity of a is needed. This will be addressed in the sequel. Proof The first identity is true in case ν ≤ 0, since then S d hom ⊆ S d,ν hom = S d,ν hom by definition of the involved spaces.
It remains to consider ν > 0. The inclusion ⊇ is clear. By multiplication with |ξ, μ| −d we may assume without loss of generality that d = 0.
Let a ∈ S 0,ν hom be given. We use Theorem 4.4 and show that the restriction of a to S n + is the sum of a smooth function and a function belonging to r ν C ∞ B ( S n + ). By Lemma 4.3 it suffices to find a decomposition for (1 − χ)a.
Let N be the largest natural number with N < ν. It can be shown (see Lemma 2.1.10 and Proposition 2.1.11 in [3]) that a extends as an N -times continuously differentiable function to R n × R + \ {(0, 0)} and if p N (ξ ; μ) denotes the Taylor-polynomial of a in ξ around ξ = 0, then p N is smooth in μ > 0 and Since (1 − χ) p N is smooth on S n + , it remains to verify that the restriction of (1 − χ)r N belongs to r ν C ∞ B ( S n + ). To this end let Then, in polar-coordinates, Derivatives of r α are treated similarly, proceeding as in the proof of Theorem 4.4.
This decomposition also shows how to associate with a symbol a ∈ S d,ν hom a symbol p ∈ S d,ν with homogeneous principal symbol equal to a. In fact, writing a = a + a smooth with a ∈ S d,ν hom and a smooth ∈ S d hom , choose with arbitrary zero-excision functions χ(ξ, μ) and χ(ξ). Changing the cut-off functions induces remainders in S d−∞,ν−∞ ; hence we may assume that χ(ξ)χ(ξ, μ) = χ(ξ) and p = χa + (1 − χ)χa smooth . Then taking another representation a = a + a smooth with associated symbol p , we find Noting that (after restriction to the unit-sphere) In combination with Lemma 4.3 we obtain the following: where S d,ν V ⊂ S d,ν is the subspace of those symbols whose homogeneous components have support in V .

Expansion at infinity
One of the motivations for this paper is to extend the concept of ellipticity in the spaces S d,ν with positive regularity number ν to the case ν = 0. Ellipticity should still be characterized by the invertibility of one or more principal symbols (plus some uniformity assumptions for preserving the C ∞ b structure in x) and should imply invertibility of a(x, D; μ) for large values of the parameter μ. Recall that S d,0 = S d,0 ; for systematic reasons we address this question in S d,ν for arbitrary ν.
In a first step, in Sect. 5.1, we introduce a subclass S d,ν 1,0 of S d,ν 1,0 in which elliptic elements are invertible for large values of μ. The ellipticity involves an estimate of the full symbol and the invertibility of a so-called limit-symbol; the latter plays the role of a new principal symbol. In a second step, we pass to the subclass of poly-homogeneous symbols S d,ν where the full symbol can be replaced by the homogeneous principal symbol.

Symbols with expansion at infinity
The definition does not depend on the choice of the function [ξ, μ], since the difference of two such functions belongs to C ∞ comp (R + × R n ); for further discussion see Sect. 7.2. The coefficients a ∞ [ν+ j] (x, ξ) are uniquely determined by a. S d,ν 1,0 is a Fréchet space when equipped with the projective topology with respect to the mappings a → r a,N : Obviously, the maps a → ξ e a : S d,ν 1,0 −→ S d+e,ν+e i.e., the homogeneous principal symbol of a evaluated in (ξ, μ) Note that the proof of Proposition 5.3 is constructive, i.e., for given a all symbols a ∞ [ j] (x, ξ) can be calculated explicitly.
Step 2: Assume that a | S n + vanishes to order N in (ξ, μ) = (0, 1). Then and therefore Since (a − p )| S n + vanishes to order N in (0, 1), we conclude by Step 2 that a − p ∈ S d− ,N

Lemma 5.4
The following holds true: Proof i) is straight-forward, as is ii) using induction on |α|.
By induction, it is enough to show iii) for j = 1. Observe that Now use the expansion of μ ∈ S 1,0 1,0 , cf. Proposition 5.3, to find a resulting expansion of ∂ μ a.
Proof Let χ(ξ) be a zero-excision function and denote by χ c , c > 0, the operator of mul- Moreover, the following statements are checked by straight-forward calculations: In other words, given a ∈ S d−1,ν−1 1,0 . Now the existence of a follows from Proposition 1.1.17 of [11] The remaining statements are clear.

Ellipticity and parametrix construction
For the following considerations it is convenient to introduce the spaces S d 1,0 (R + ; E) consisting of all smooth functions a(μ) with values in a Frèchet space E, such that for every j and every continuous semi-norm ||| · ||| of E.
whenever T belongs to a unital algebra and 1 − T is invertible.
Step 1: Let us assume that a ∞ [0] = 0. In particular, a ∈ S −1 , we find that 1 − a is invertible with respect to the Leibniz product for large μ and that Using the expansions of a # j ∈ S 0−∞,0−∞ 1,0 and noting that (a # j ) ∞ [0] = 0 for every j due to the multiplicativity of the principal limit-symbol, we find a sequence of symbols for large μ. Thus, for a suitable zero-excision function κ(μ), Step 2: In the general case, again by spectral invariance, we find a has vanishing principal limit-symbol. Apply Step 1 to 1 − a to find a corresponding parametrix 1−b . Then the claim follows by choosing Definition 5. 9 We call a ∈ S d,ν 1,0 elliptic if there exist an R ≥ 0 such that (1) a(x, ξ; μ) is invertible whenever |ξ | ≥ R and Note that condition (2) is equivalent to the existence of an inverse of a ∞ [ν] (x, ξ) with respect to the Leibniz product, with inverse belonging to S −ν 1,0 (R n ).
Proof By order reduction we may assume without loss of generality that d = ν = 0. Step is also elliptic. Thus we can choose a zeroexcision function χ(ξ) such that and The first factor on the right-hand side equals a − r a,N with r a,N ∈ S 0,N . It follows that has vanishing limit-symbol.
Using Proposition 5.8 we thus find a right-parametrix b R ∈ S 0,0 Analogously, we construct a left-parametrix b L . Then the claim follows by choosing

Poly-homogeneous symbols with expansion at infinity
As already mentioned S d,ν does not behave well under inversion because there is no sufficient control at the singularity. We pass to a subclass which also is compatible with the previously introduced expansion at infinity.
In the following definition, we consider the north-pole (0, 1) as a singularity of the semisphere and consider functions having a particular asymptotic structure near this singularity. Asymptotics of this form are well-known in the context of manifolds with conical singularities, cf. for instance [11,Section 2.3].
) is a cut-off function, i.e., ω has compact support in [0, 1) and ω ≡ 1 near the origin. The space of all such functions a(ξ ; μ) will be denoted by r ν C ∞ T ( S n + ).

Definition 5.12 S d,ν
hom consists of all functions of the form Define the principal angular symbol a ν (x, ξ) ∈ S ν hom (R n ) (cf. Section 2.2) of a as Note that, by construction, S d,ν hom ⊆ S d,ν hom . The following proposition shows that such homogeneous components intrinsically admit an expansion at infinity in the sense of Definition 5.1.
Due to Proposition 5.15, condition (2) implies the invertibility of the principal angular symbol of a. Moreover, if the homogeneous principal symbol of a(x, ξ; μ) does not depend on x for large |x|, then condition (1) in Definition 5.16 can be substituted by (1 ) The homogeneous principal symbol a 0 (x, ξ; μ) is invertible whenever ξ = 0.

Refined calculus for symbols of finite regularity
As proved in Theorem 4.6, Grubb's class S d,ν coincides with the non-direct sum S d,ν + S d . In light of the above considerations it is now natural to introduce the following class: The limitation to integer values of ν is needed to ensure compatibility between the spaces S d,ν hom and S d hom in the sense that the Taylor expansions (cf. Definition 5.11) associated with elements of either space only contain integer exponents; in particular, we have S d,ν = S d,ν whenever ν ≤ 0, and S d,ν ⊂ S d,0 whenever ν > 0. The choice of integer ν is also important in view of Lemma 5.19. By Proposition 5.3 and Theorem 5.6, the Leibniz product induces maps By Theorem 5.7 the class is closed under taking the (formal) adjoint. Since in both spaces involved in Definition 5.18 asymptotic summation is possible (cf. Sect. 3.2 and Theorem 5.5), a sequence of symbols a j ∈ S d− j,ν− j can be summed asymptotically in S d,ν .

Lemma 5.19
Let ν be a positive integer 1 . Then the space Proof Write a = a + a 0 with a ∈ r ν C ∞ T ( S n + ) and a 0 ∈ C ∞ (S n + ). Clearly a is smooth on S n + . We proceed in two steps: Step 1: Let us assume that a 0 ≡ 1 in some neighborhood of the point (0, 1). Choose ψ 1 , ψ ∈ C ∞ (S n + ) having their support contained in this neighborhood and such that ψ, ψ 1 ≡ 1 near (0, 1) as well as ψ 1 ≡ 1 on the support of ψ. Let b = −ψ 1 a. Then b ∈ r ν C ∞ T ( S n + ) and, choosing ψ 1 with sufficiently small support, we have that | b| ≤ 1/2 on S n + , since lim (ξ,μ)→(0,1) a(ξ ; μ) = 0. Then By chain rule it is straight-forward to see that Since ν is integer, b j ∈ r ν C ∞ T ( S n + ) for every j. Inserting the Taylor expansion for each b j and noting that c L ∈ r ν+N C ∞ Step 2: Consider the general case. Since a 0 (0; 1) = a(0; 1) is invertible, there exists a b 0 ∈ C ∞ (S n + ) everywhere invertible and such that b 0 = a 0 in a neighborhood of (0, 1).
we conclude that the same is true for a −1 .
In case of x-dependence we need to pose, as usual, an additional uniform bound on the inverse. Since symbols of S d,ν hom are just the homogeneous extensions of degree d of functions from r ν C ∞ T ( S n + ) + C ∞ (S n + ), we immediately have the following corollary.
If a ∈ S d,ν with positive integer ν, then also a ∈ S d,0 . Due to Propositions 5.13 and 5.3, its principal limit-symbol is where a 0 is the homogeneous principal symbol of a (defined on S n + by continuous extension). Recalling Definition 5.16, we find the following result which unifies the notions of ellipticity for symbols of regularity number ν = 0 and ν ∈ N, respectively.

Proposition 5.22 Let ν be a nonnegative integer and a ∈ S d,ν . Then a is elliptic if, and only
if, a is elliptic as an element of S d,0 .

Resolvent-kernel expansions
We shall discuss how our calculus allows to recover the well-known resolvent trace expansion for elliptic ψdo due to Grubb-Seeley, cf. [6].

Preparation
The following Lemma is a slight modification of [6, Lemma 1.3].
Proof For convenience of notation assume independence of x. Obviously it suffices to consider μ ≥ 1. Then a(ξ ; μ) = μ m a(ξ/μ; 1). Let u(t, ξ) = a(tξ ; 1), with certain universal constants c α . Thus the Taylor expansion of u in t centered in t = 0 is of the form with polynomials ζ j (ξ ) as described. Then using the fact that for 0 ≤ t, τ ≤ 1, the above integral belongs to S m + +N 1,0 (R n ) uniformly in 0 ≤ t ≤ 1. Substituting t = 1/μ yields the claim.

A case of particular interest below is that
where, with notation of (6.1), Proof First note that for r (x, ξ; μ) ∈ S d,ν+N , N , S ν+N ). Inserting the expansions the result follows immediately. and write and χ is a zero-excision function such that 1 − χ is supported in the unit-ball centered in the origin. By Corollary 6.2 (with d, ν replaced by d − J , ν − J ) we have with q (x, ξ) ∈ S ν−J + 1,0 (R n ). Recalling (6.2), the associated kernel k r (x, y; μ) satisfies

Now let k j (x, y; μ) denote the kernel associated with χ(ξ)a j (x, ξ; μ). Decompose
Then, for every μ ≥ 1, using the homogeneity of a j , note that the integrand is bounded by ξ d− j , hence integrable since d < −n.

By (6.3) we obtain immediately that
By homogeneity for |ξ | ≥ 1 of the q j, and by using polar-coordinates, 1≤|ξ |≤μ By the second line in (6.3) and the homogeneity of s j,L , If s h j,L denotes the extension by homogeneity of s j,L from |ξ | ≥ 1 to all ξ = 0 (defined by the second term in (6.4)), then j (x, x; μ) and completes the proof.

Application to the resolvent of a Ãdo
Assume we are given two ψdo, p(x, ξ) ∈ S m (R n ) of positive integer order m ∈ N and q(x, ξ) ∈ S ω (R n ) with ω ∈ R. Moreover, let be a sector in the complex plane. Then, for every θ , Note that e iθ a θ (x, ξ; r 1/m ) = re iθ − p(x, ξ). Now assume that a θ is elliptic, uniformly with respect to θ , i.e., uniformly in x ∈ R n and 0 ≤ |θ | ≤ . Using Theorem 5.21, there exists a b θ (x, ξ; μ) ∈ S −m,m , depending uniformly on θ , such that a θ (x, D; μ) is invertible for large μ with a θ (x, D; μ) −1 = b θ (x, D; μ). We then find, for every positive integer , Note that the -fold Leibniz product of b θ belongs to with uniform dependence on θ . If is so large that ω − m < −n, we can apply Theorem 6.3 to both c (1) θ and c (2) θ . This is the key to obtain the following: Theorem 6.4 With the above notation and assumptions, let k(x, y; λ) be the distributional uniformly for λ ∈ with |λ| −→ +∞. Moreover, c j = c j ≡ 0 whenever j is not an integer multiple of m.
To see that the coefficients c j and c j vanish whenever j is not an integer multiple of , one needs to repeat the considerations from [6, Section 2.2] concerning the construction of the parametrix of μ m − p(x, ξ).

Operators on manifolds
We shall show that the various symbol classes introduced so far lead to corresponding operator-classes on smooth compact manifolds. In particular, we shall show that the expansion at infinity and the concept of principal limit-symbol extend to the global setting.

Invariance under change of coordinates
Let κ : R n → R n be a smooth change of coordinates and assume that ∂ j κ k ∈ C ∞ b (R n ) for all 1 ≤ j, k ≤ n, and that |det κ | is uniformly bounded from above and below by positive constants; here, κ denotes the first derivative (Jacobian matrix) of κ. For an operator Its pull-back is κ * A := (κ −1 ) * A. If A(μ) is depending on a parameter μ, pull-back and pushforward are defined in the same way, resulting in families κ * A(μ) and κ * A(μ), respectively. It is then well-known that the classes S d 1,0 and S d are invariant under the change of coordinates x = κ(y).
where κ (y) t denotes the adjoint of the first derivative κ (y).
Next let a(x, ξ; μ) ∈ S d,ν 1,0 be as in Definition 5.1. The invariance follows from the observation that the classes S ν+ j 1,0 (R n ) and S d,ν+N has a complete expansion due to Proposition 5.3. This allows to find the complete expansion of κ * a(x, ξ; μ). Using the formula for the asymptotic expansion of κ * a, one sees that poly-homogeneous symbols remain poly-homogeneous.
Let us have a closer look to the homogeneous principal symbol of a ∈ S d,ν . For convenience of notation let us set p(x, ξ; μ) = (κ * a) 0 (x, ξ; μ) and K(x) = κ (κ −1 (x)) t . To see that p belongs to S d,ν hom we write where, in polar-coordinates, Noting that n is smooth in r up to r = 0 and using the weighted Taylor-expansion of a, one finds that p admits a weighted Taylor-expansion with principal angular symbol This results in the following observation: The principal angular symbols of a and κ * a satisfy the relation In other words, the principal angular symbol transforms as a function on the cotangentbundle of R n .

Remark 7.3
In the above discussion we have focused on changes of coordinates defined on R n , satisfying certain growth conditions at infinity. This is the natural setting for symbols which are globally defined on R n . Alternatively, we could consider arbitrary diffeomorphisms κ : U → V with arbitrary open subsets U , V of R n and the push-forward of ψdo of the form φ a(x, D; μ)ψ with φ, ψ ∈ C ∞ comp (U ). We would obtain a corresponding invariance property; the details are left to the reader.
The invariance under changes of coordinates permits to define corresponding classes for manifolds.  For an alternative description of the operator-classes, let us choose a system of charts κ i : i → U i , i = 1, . . . , m, such that the i cover M; moreover let φ i , ψ i ∈ C ∞ ( i ) such that the φ i are a partition of unity and ψ i ≡ 1 in a neighborhood of the support of φ i . Then L d,ν 1,0 consists of all operators of the form with a i ∈ S d,ν 1,0 . The analogous statement holds for the other classes.

Complete expansion and limit operator
The extension of the concept of complete expansion and principal limit-symbol to manifolds requires some additional analysis. The key is to show that the symbol [ξ, μ] α involved in the definition of S d,ν can be replaced by other ones. It is convenient to use the notation λ α (ξ, μ) = [ξ, μ] α , α ∈ R. Then the expansion of a symbol a(x, ξ; μ) ∈ S d,ν 1,0 takes the form note that here the Leibniz product actually coincides with the point-wise product of the involved symbols.
Note that any λ α in such a family is parameter-elliptic in S α and thus has a parametrix in S −α ; this parametrix coincides with λ −α modulo S −α−1 . Proof of Theorem 7. 7 First we argue that we may assume without loss of generality that ν = 0.
To this end let p s (ξ ) := ξ s , s ∈ R. Then p −ν #a ∈ S d ,0 1,0 for d = d − ν. Given hypothesis a), then p −ν #a ∈ S d ,0 1,0 and we show the existence of an expansion Multiplying from the left with p ν we find the desired expansion for a with a ,∞ [ν+ j] := p ν #b ,∞ [ j] . We argue similarly when starting out from hypothesis b). Now let ν = 0; we show that b) implies a). By Proposition 5.3, λ d− j ∈ S d− j,0 1,0 has an expansion , 0; 1). Next we show that a) implies b) (again with ν = 0). We start out from the expansion the additional super-script 0 is introduced for systematic reasons, since we will now establish an iterative procedure to transform this expansion in an expansion using the family . Write with r 0 ∈ S d−1 . By Proposition 5.3 we have expansions Inserting this in expansion (7.1) and using for j ≥ 1 expansions The second term on the right-hand side equals We conclude that with a ,∞ and resulting symbols a ∞,1 [ j+1] ∈ S j+1 1,0 (R n ). This finishes the first step of the procedure. In the second step we write with r 1 ∈ S d−2 and proceed as above to finally obtain with resulting a ,∞ [1] and a ∞,2 We iterate this procedure until the N -th step which consists in writing as claimed in b). The proof is complete.
The following lemma will be useful in discussing localizations of operator-families.
with respect to some family of order-reducing symbols . Let K ⊂ R n be a compact set and assume that a#φ = a for every function φ ∈ C ∞ comp (R n ) with φ ≡ 1 in an open neighborhood of K . Then, for every such function φ and every j ≥ 0, The analogous result for left-multiplication with φ holds also true (and follows trivially from the uniqueness of the coefficient-symbols in the expansion).
Proof We proceed by induction. Since 1,0 , multiplication from the right with φ yields where [·, ·] is the commutator (with respect to #). Now the third term belongs to The uniqueness of the coefficients in the expansion then implies a ,∞ . Now suppose that (7.2) holds for j = 0, . . . , N − 1. Given a function φ choose φ 0 ∈ C ∞ 0 (R n ) such that φ 0 ≡ 1 near K and φ ≡ 1 near the support of φ 0 . Then, by induction assumption, we have As above, the last term is shown to be in S d,ν+N +1 Using again the induction hypotheses we derive

The leading coefficient A ∞ [ν] is called the limit-operator of A(μ).
Proof The proof of the uniqueness is analogous to the one given after Theorem 7.7. Therefore, we shall focus on the existence of the expansion. Let 1 , . . . , m be a covering of M such that any union i ∪ j is contained in a chart(domain) of M. Let φ i ∈ C ∞ comp ( i ), i = 1, . . . , M, be a sub-ordinate partition of unity. Then A(μ) = i, j φ i A(μ)φ j . It suffices to show the existence of an expansion for each summand.
Thus we may assume from the beginning that there exist a chart κ : → U and two functions φ, ψ ∈ C ∞ comp ( ) such that A(μ) = φ A(μ)ψ. Let a(x, ξ; μ) ∈ S d,ν 1,0 be the symbol of κ * A(μ) and let K be the union of the supports of φ • κ −1 and ψ • κ −1 , respectively. K is a compact subset of U .
Let V be an open neighborhood of K with compact closure contained in U . Take θ ∈ where χ is a zero-excision function and r α ∈ S α−1 . Now define Since θ 0 λ d−ν− j = θ 0 λ d−ν− j by construction, and applying the pull-back under κ, we find

Extension to vector-bundles
Given smooth vector-bundles E j , j = 0, 1, on M of dimension n 0 and n 1 , respectively, the above definitions and results extend in a straight-forward way to operator-families acting as maps (M, E 0 ) → (M, E 1 ) between the spaces of smooth sections of E 0 and E 1 , respectively. The definition of the spaces L d,ν , and L d,ν (E 0 , E 1 ) uses local trivializations of the vector-bundles and (n 1 × n 0 )-matrices a(x, ξ; μ) = a jk (x, ξ; μ) where the symbols a jk are from the corresponding symbolclasses S d,ν 1,0 , etc. We leave the details to the reader. As above, given a bundle E, a family of order-reducing operators is a set E of operators α E (μ) ∈ L α (E, E), α ∈ R, which have (scalar-valued) principal symbol λ α 0 (x, ξ; μ) = (|ξ | 2 x + μ 2 ) α/2 and such that 0 E (μ) is the identity operator. Then we obtain: The leading coefficient A ∞ [ν] is called the limit-operator of A(μ); it behaves multiplicatively under composition.

Recall the compatibility relation
i.e., the principal angular symbol coincides with the homogeneous principal symbol of the limit-operator.
Proposition 7.14 Let A(μ) ∈ L d,ν (E 0 , E 1 ) and assume that both homogeneous principal symbol and principal angular symbol are invertible on their domains. Then there exists a (rough) parametrix B(μ) ∈ L −d,−ν (E 1 , E 0 ), i.e., This result follows from the fact that the invertibility of a homogeneous principal symbol belonging to S d,ν hom ((T * M \ 0) × R + ; E 0 , E 1 ) together with the invertibility of its angular symbol implies that its inverse belongs to the class S −d,−ν hom ((T * M \ 0) × R + ; E 1 , E 0 ), cf. the local situation mentioned after Definition 5.11. Definition 7. 15 We call A(μ) ∈ L d,ν (E 0 , E 1 ) elliptic if its homogeneous principal symbol is invertible on its domain and its principal limit-operator is invertible as a map H s (M, E 0 ) → H s−ν (M, E 1 ) for some s 3 .
Since we can construct in the same way a left-inverse of A(μ) for large μ, the claim follows.

Operators with finite regularity number
In analogy to Sect. 5.4 we introduce the class If ν is positive, the homogeneous principal symbol extends to a bundle homomorphism on (T * M × R + ) \ 0 and ellipticity means invertibility of this extended symbol. Then Theorem 5.21 generalizes in the obvious way to the global setting. Theorem 7.17 With the above notation and assumptions and ∈ N such that ω − m < −dim M, there exist numbers c j , c j , c j , j ∈ N 0 , such that

Resolvent trace expansion
uniformly for λ ∈ with |λ| −→ +∞. Moreover, c j = c j = 0 whenever j is not an integer multiple of m.

Pseudodifferential operators of Toeplitz type
Let us conclude with an application to so-called ψdo of Toeplitz type, cf. [14,15]. To this end, for j = 0, 1, let E j be a vector-bundle over M and P j ∈ L 0 (M; E j , E j ) be idempotent, i.e., P 2 j = P j . The P j define closed subspaces We are interested in the invertibility of Consider P j as an element in L 0,0 (E j , E j ). Since P j is idempotent, so is the homogeneous principal symbol σ (P j ) as morphism of the pull-back of E j to (T * M \ 0) × R + , hence defines a sub-bundle denoted by E j (P j ).

Theorem 7.18
Let notations be as above. Assume that Then there exists a B(μ) ∈ L −d,0 (E 1 , E 0 ) such that, for B(μ) := P 0 B(μ)P 1 , for sufficiently large values of μ. In particular, map (7.8) is an isomorphism for every choice of s and μ large.
As a particular case we can take A(μ) = P 1 (μ d − A)P 0 with a ψdo A ∈ L d (M; E, E), d ∈ N, and two idempotents P 0 , P 1 ∈ L 0 (M; E, E). Note that A(μ) = μ d − A considered as an element of L d,0 has limit-operator A [0] ≡ 1, hence condition ii) in Theorem 7.18 reduces to the requirement that P 1 : H s (M, E; P 0 ) → H s (M, E; P 1 ) isomorphically for some s.

Data Availability Statement
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
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Appendix: Calculus for symbols with expansion at infinity
Let us provide the detailed proofs of Theorems 5.6 and 5.7. They are based on the concept of oscillatory integrals in the spirit of [7], but extended to Frèchet space valued amplitude functions. For an account on this concept see [2].
Let E be a Fréchet space whose topology is described by a system of semi-norms p n , n ∈ N. A smooth function q = q(y, η) : R m × R m → E is called an amplitude function with values in E, provided there exist sequences (m n ) and (τ n ) such that p n D γ η D δ y q(y, η) y τ n η m n for all n and for all orders of derivatives. The space of such amplitude functions is denoted by A(R m × R m , E). We shall frequently make use of the following simple observation:  1 (y, η), q 0 (y, η))) is an amplitude function with values in E.
If χ(y, η) denotes a cut-off function with χ(0, 0) = 1, the so-called oscillatory integral Os − e −iyη q(y, η) dyd¯η := lim ε→0 R n ×R n e −iyη χ(εy, εη)q(y, η) dyd¯η exists and is independent of the choice of χ. Note that for a continuous, E-valued function f with compact support, f (y, η) dydη is the unique element e ∈ E such that e , e = e , f (y, η) dydη for every functional e ∈ E . For simplicity of notation we shall simply write rather than Os − .