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 oerators 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 order to study resolvents of pseudodifferential operators and boundary value problems, Grubb in [2] introduced the concept of regularity for pseudodifferential operators. A symbol 1 a(x, ξ; µ) : R n × R n × R + → C is said to have order d ∈ R and regularity ν ∈ R if (1.1) |D α ξ D β x D j µ a(x, ξ; µ)| ≤ C αβj ξ ν−|α| ξ, µ d−ν−j + ξ, µ d−|α|−j for all orders of derivatives; we shall denote the space of all such symbols by S d,ν 1,0 = S d,ν 1,0 (R n × R n ; R + ). Here, x and ξ have the role of variable and co-variable, respectively, while µ is a parameter. The associated pseudodifferential operator is denoted by a(x, D; µ) or op(a)(µ). In case ν > 0 there is a concept of ellipticity in this class and elliptic elements a(x, ξ; µ) posses a parametrix of same regularity which yields the inverse a(x, D; µ) −1 for large values of the parameter. For a recent application to fractional heat equations see Grubb [3].
If a(x, ξ; µ) is as in (1.1), it satisfies the estimates whenever |α| ≤ ν and, for |α| ≥ ν, the estimates Therefore it is natural to introduce the symbol classes S d 1,0 = S d 1,0 (R n ×R n ; R + ) and S d,ν 1,0 = S d,ν 1,0 (R n × R n ; R + ) of all symbols that satisfy the estimates (1.2) or (1.3), respectively, for every order of derivatives α, β, and j. We shall call the symbols from S d 1,0 strongly parameter-dependent, since differentiation with respect to ξ also improves decay in the parameter µ. In contrast, symbols from S d,ν 1,0 shall be called weakly parameter-dependent. Note that S d,ν 1,0 = S d,ν 1,0 in case ν ≤ 0. In all previuosly mentioned classes one can introduce the subclass of poly-homogeneous symbols that have expansions into homogeneous components of decreasing degree of homogeneity (see Sections 3.1-3.3 for details). The resulting classes are denoted by S d,ν , S d and S d,ν , respectively. In Theorem 4.6 we show that weakly and strongly parameter-dependent symbols generate the whole space of symbols of finite regularity, i.e. (1.4) S d,ν (R n × R n ; R + ) = S d,ν V (R n × R n ; R + ) + S d (R n × R n ; R + ); here V is a conical neighborhood of the form {(ξ, µ) ∈ R n × R + | µ ≥ c|ξ|} with an arbitrary constant c > 0 and the subscript V indicates that all homogeneous components have support in this set. Though in this paper we shall not enter in details concerning boundary value problems, let us mention that the decomposition (1.4) 1 (for simplicity of exposition, we shall focus on scalar-valued symbols though all results easily extend to a matrix-valued setting) PSEUDODIFFERENTIAL OPERATORS OF FINITE REGULARITY 3 remains valid in the respective symbol classes satisfying the transmission condition with respect to (the boundary) x = 0.
In the class S d,ν the regularity parameter ν can be characterized nicely by the structure of the homogeneous components. While the homogeneous components of strongly parameter-dependent symbols restrict to smooth functions on the unit semi-sphere S n + = {(ξ, µ) | |(ξ, µ)| = 1, µ ≥ 0}, this is not the case for weakly parameter-dependent symbols. Generally, they are only smooth on the punctured sphere S n + := S n + \ (0, 1). In fact, introducing polar-coordinates (r, φ) centered in (0, 1), we show in Theorem 4.4 that homogeneous components of regularity ν are in one-to-one correspondence with those functions on S n + that after multiplication with r −ν and after application of arbitrarily many totally characteristic derivatives r∂ r and derivatives with respect to φ remain bounded near (0, 1); let us denote this space by r ν C ∞ B ( S n + ). In this sense, regularity corresponds to a degree of (non-)vanishing in (0, 1), respectively to a (power-)weight. This also reflects in the behaviour of operators under composition, which on symbolic level corresponds to the so-called Leibniz-product: it maps S d0,ν0 × S d1,ν1 into S d0+d1,ν0+ν1 .
In Section 5 we develop a concept of ellipticity in the scale of weakly parameterdependent symbols, which we seek to be as similar as possible to that of strongly parameter-dependent symbols, where ellipticity is encoded by the invertibility of the associated homogeneous principal symbol (in the present set-up of symbols on the non-compact space R n , additional estimates to control the behaviour in x at infinity are required; they disappear when assuming independence of x near infinity). We immediately encounter two problems: The smoothing remainders in the class of weakly parameter-dependent symbols corresponding to weight ν = 0 are characterized by the estimates |D α ξ D β x D j µ r(x, ξ; µ)| ≤ C αβjN ξ −N ξ, µ −j for arbitrary N . In particular, for j = 0, there is no decay in µ, hence 1 + r(x, D; µ) will not be invertible for large µ. Thus, in Section 5.1, an additional structure will be implemented to take care of this decay. We require that symbols from S d,ν have an additional complete expansion of the form [ν+j] (x, ξ) ∈ S ν+j 1,0 (R n × R n ). (1.5) Such an expansion resembles the one employed by Grubb-Seeley in [4], though the motivation for its appearance is different (see below). The leading coefficient, the so-called limit-symbol a ∞ [ν] (x, ξ), will play the role of an additional principal symbol relevant for ellipticity. We shall denote the corresponding (sub-)scale of symbols by S d,ν 1,0 and shall discuss the resulting calculus in datail, including Leibniz-product, adjoint, parametrix construction and invariance under co-ordinate changes.
The second problem concerns the invertibility of homogeneous principal symbols: the space r ν C ∞ B ( S n + ) is not closed under taking the inverse without imposing a control for r → 0. We substitute this space by r ν C ∞ T ( S n + ), where subscript T indicates the existence of a Taylor-expansion in r at r = 0, and implement the leading order coefficient as a further principal symbol, termed principal angular symbol. As it turns out such homogeneous components with Taylor-expansion intrinsically posses a complete expansion (1.5) and the principal angular symbol is the homogeneous principal symbol of the limit-symbol. Therefore, the principal angular symbol only has the role of a subordinate principal symbol. The resulting class of symbols, denoted by S d,ν , is introduced and discussed in Section 5.4. Already in the author's work [12] such a structure has been employed; however, the calculus presented here is more systematic and refined. Our calculus appears to be related to the one developed by Savin and Sternin [7], where also symbols admitting a Taylor expansion in polar coordinates are employed.
In view of the decomposition (1.3), it is natural to introduce which is sub-class of Grubb's class S d,ν . The respective calculus is discussed in Section 6; note that ellipticity in this class also covers the case of vanishing regularity ν = 0. We present two applications. We show that resolvents of suitably elliptic pseudodifferential operators fit into this framework as well as parameter-dependent pseudodifferential operators of Toeplitz type in the spirit of [11], [12].
The main scope of [2] is to construct a parameter-dependent Boutet de Monvel calculus for boundary value problems starting out from the symbol class S d,ν (with transmission condition). It appears to be a natural aim to perform a similar programme starting out from the subclass S d,ν . Ideally one would expect a calculus which is made up of strongly parameter-dependent elements (corresponding to elements of infinite regularity in the sense of [2] or to the parameter-dependent calculus of Schrohe, Schulze [8]) and of suitably weakly parameter-dependent ones. Also it seems interesting to investigate whether this class is suited for building up a calculus of operators on manifolds with conical singularities, with the perspective of treating resolvents of cone pseudodifferential operators.

Notations, symbols, and Leibniz-product
Let us first introduce some notations of which we will make use throughout the whole paper.
2.1. Basic notations. Let y = (1 + |y| 2 ) 1/2 for y ∈ R m with arbitrary m. Let y → [y] : R m → R denote a smooth, strictly positive function that coincides 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 y for large u if

2.2.
Pseudodifferential symbols from Hörmander's class. The (uniform) Hörmander class S d 1,0 (R n × R n ) consists of those smooth functions 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 A symbol a(x, ξ) ∈ S d 1,0 (R n × R n ) is called poly-homogeneous if there exist smooth functions a (d−ℓ) (x, ξ) defined on R n × (R n \ {0}) and homogeneous of degree d − ℓ in ξ such that is an arbitrary fixed zero-excision function (for N = 0 the sum is obsolete, i.e., r a,0 = a). Denote by S d (R n × 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 One associates with a the pseudodifferential operator acting on the Schwarz space S (R n ). If d is the order of a, then op(a) extends by continuity to an operator H s (R n ) → H s−d (R n ) for every s ∈ R, where H s (R n ) is the usual L 2 -Sobolev space of smoothness s. The composition of operators corresponds to the so-called Leibniz-product of symbols, defined as (integration in the sense of oscillatory integrals), cf. [5]. If a j ∈ S dj 1,0 (R n × R n ) then a 1 #a 0 ∈ S d0+d1 1,0 (R n × R n ) and the map (a 1 , a 0 ) → a 1 #a 0 is continuous.
If a(x, ξ; µ) is a symbol that depends on an additional parameter µ such that a(x, ξ; µ) ∈ S d 1,0 (R n × R n ) for every µ and some d, then we shall write op(a)(µ) for the associated operators.

Symbols of finite regularity
We introduce here some of the symbol spaces we shall need in the sequel. For simplicity of presentation we shall assume n ≥ 2; the case n = 1 requires some technical modifications, originating from the non-connectedness of R \ {0}).

3.1.
Grubb's calculus. We briefly review the calculus of Grubb, recalling the main definitions and properties. For proofs and further details we refer the reader to Chapter 2.1 of [2].
In the previous definition, d is the order of the symbol, while ν its regularity.
The space of regularizing symbols of regularity ν ′ ∈ R is defined as it consists of those symbols satisfying for every N and all orders of derivatives (note the minus sign in front of ν ′ on the right-hand side).
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 ℓ ∈ S d−ℓ,ν−ℓ 1,0 , there exists an a ∈ S d,ν 1,0 such that For the Leibniz-product, one has the asymptotic expansion The space of polyhomogeneous symbols of order d and regularity ν is denoted by If a ∈ S d,ν is poly-homogeneous as in the previous definition, its homogeneous principal symbol is defined as Note that a h (x, ξ; µ) = lim t→+∞ t −d a(x, tξ; tµ).
Definition 3.4. S d,ν hom = S d,ν hom (R n × R n ; R + ) 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, for every order of derivatives, Claerly, the homogeneous principal symbol of a(x, ξ; µ) ∈ S d,ν belongs to S d,ν hom . In case ν > 0, a h (x, ξ; µ) extends by continuity to a function defined for all x and (ξ, µ) = 0; the larger ν the more regular (i.e., differentiable) is this extension. Definition 3.5. Let ν > 0. A symbol a(x, ξ; µ) ∈ S d,ν is called elliptic if a h (x, ξ; µ) = 0 for all x and all (ξ, µ) = 0 and Note that if a h (x, ξ; µ) is constant in x for large x, then a is elliptic if and only if a h (x, ξ, µ) is invertible for all x and all (ξ, µ) with |ξ, µ| = 1.
Theorem 3.6. Let a ∈ S d,ν with ν > 0. Then the following two statements are equivalent: Note that if r ∈ S −∞,ν−∞ with ν > 0, then r(µ) µ→+∞ − −−−− → 0 in S −∞ (R n × R n ). In particular, if a is elliptic then op(a)(µ) ∈ L (H s (R n ), H s−d (R n )) is invertible for large µ. 3.2. Symbols of infinite regularity -strong parameter-dependence. In this ection we consider symbols of infinite regularity. Though this is a particular case of the previous section we prefer to present it in a kind of independent way. Definition 3.7. The space S d 1,0 = S d 1,0 (R n × R n ; R + ) consists of all symbols satisfying, for every α, β ∈ N n 0 and every j ∈ N 0 , We shall call such symbols also strongly parameter-dependent ; strong refers to the property that differentiation with respect to ξ or µ improves the decay simultaneously in (ξ, µ). As already stated, in fact S d i.e., symbols that are rapidly decreasing in the variables (ξ, µ).
Proposition 3.8. The Leibniz-product induces maps S d1 Asymptotic summations can be performed, i.e., given a sequence of a ℓ ∈ S d−ℓ Definition 3.9. Let S d hom = S d hom (R n × R n ; R + ) consist of all smooth functions a(x, ξ; µ) defined on R n × (R n × R + \ {(0, 0)}) that are homogeneous of degree d in (ξ, µ) and satisfy, for every order of derivatives, where S n + denotes the unit semi-sphere, Clearly, if χ(ξ, µ) is a zero-excision function then χa ∈ S d 1,0 .
Remark 3.10. Initially one could also ask that the homogeneous components from Definition 3.9 are defined only for ξ = 0. However, since the restriction to any of the domains Q n = {(ξ, µ) | 0 < |ξ| < 1, 1/n < µ < n}, n ∈ N, is smooth and has bounded derivatives of any order, the function extends to a smooth function on the closure of Q n . Thus the homogeneous component has a smooth extension to A symbol a ∈ S d 1,0 is called poly-homogeneous if and only if there exists a sequence of homogeneous symbols a ℓ ∈ S d−ℓ hom such that where χ(ξ, µ) is an aribtrary zero-excision function. The space of such symbols will be denoted by S d = S d (R n × R n ; R + ). The homogeneous principal symbol is 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 = S d cl (R n × R n ; R + ) and the symbols are called "classical" rather than poly-homogeneous. The space of homogeneous symbols S d hom is also denoted by S (d) .

3.3.
Weakly parameter-dependent symbols. Strongly parameter-dependent symbols correspond to the second term on the right-hand side of estimate (3.1).
In this section we introduce a class of symbols correponding to the first term. We shall also call them weakly parameter-dependent, since differentiation with respect to ξ does not improve the decay in µ.
Definition 3.12. S d,ν 1,0 = S d,ν 1,0 (R n × R n ; R + ) denotes the space of all symbols that satisfy, for every order of derivatives, Note that S d 1,0 ⊂ S d,0 1,0 and S d,ν 1,0 = S d,ν 1,0 whenever ν ≤ 0. Also this class is closed under composition: Given a sequence of symbols a ℓ ∈ S d−ℓ,ν−ℓ Definition 3.14. S d,ν hom = S d,ν hom (R n ×R + ) denotes the space of all smooth functions on R n x × (R n ξ \ {0}) × R + which are homogeneous of degree d and satisfy, for every order of derivatives, A symbol a ∈ S d,ν 1,0 is called poly-homogeneous if and only if there exists a sequence of homogeneous symbols a ℓ ∈ S d−ℓ,ν−ℓ hom such that where χ(ξ) is an aribtrary zero-excision function. The space of such symbols a will be denoted by S d,ν = S d,ν (R n × R + ). The homogeneous principal symbol is for all x and all ξ = 0 and Theorem 3.16. For a ∈ S d,ν the following two statements are equivalent:

Decomposition of poly-homogeneous symbols
The homogeneous components of strongly parametrer-dependent symbols are homogeneous extensions (in ξ, µ)) of smooth functions on the unit semi-sphere. This is not the case for symbols of negative regularity and for weakly parameter-dependent symbols. In fact, in the point (0, 1) the components are not smooth or even not defined. In this section we give a precise description of the singular structure near (0, 1), which will be one of the keys for extending ellipticity to such classes. Also we shall derive a useful decomposition for symbols of positive regularity.
For simplicity of exposition, we shall focus on symbols with "constant coefficients", i.e., symbols not depending on the x-variable. The general case is obtained by including x as a smooth variable and requiring uniform bounds of the symbols and all their derivatives with respect to x.

Let us define
We shall identify the interior of 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, 1), E) the space of all smooth bounded functions u : (0, 1) → E such that (r∂ r ) ℓ u is bounded on (0, 1) for every order of derivatives.
Obviously the elements of S (d,ν) (R n ; R + ) are smooth and (positively) homogeneous of degree d on (R n \ {0}) × 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 The following theorem shows that, for weakly parameter-dependent homogeneous components, regularity and weight are the same thing.
Proof. Let us first prove the inclusion "⊆". Let a(ξ; µ) be as in Definition 4.2.
By induction, it is then straightforward to verify that D α ξ D j µ a(ξ; µ) is a linear combination of terms of the form with p j ∈ S −j hom and q |α| being smooth and homogeneous of degree −|α| in ξ = 0. This gives immediately the estimate Next we shall show the inclusion "⊇". Let a ∈ S d,ν hom be given. Again 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 = r −ν in polar-coordinates. Again, a can be assumed do have support in a small conical neighborhood containing (0, 1). Thus Using that rv ′ (r)/v(r) = 1/(r 2 − 1), it is straightforward to see that (r∂ r ) ℓ ∂ α φ a(r, φ) is a linear combination of terms of the form where q |β| is smooth and homogeneous of degree −|β| in φ = 0 and g ∈ C ∞ ([0, 1)). Thus (r∂ r ) ℓ ∂ α φ a(r, φ) is bounded for r ∈ (0, 1 − ε] and φ belonging to a small neighborhood of the unit-sphere S n . This shows the claim.
Proof. The second identity is valid by Theorem 4.4. 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 verify the first identity for 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 will 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 we may assume that a = (1 − χ)a is supported in a small conical neighborhood of the north-pole (0, 1).
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 [2]) that a extends as an N -times continuously differentiable function to R n × R + \ {(0, 0)} and if p N (ξ; µ) denotes the Taylorpolynomial of a in ξ around ξ = 0, then p N is smooth in µ > 0 and To this end it suffices to study the functions with |α| = N and some fixed ε > 0. In polar-coordinates Thus Hence we find that r −ν |r α (r, φ)| is bounded. 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 0-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 .

Weakly parameter-dependent symbols with expansion at infinity
We introduce the two scales S d,ν 1,0 and S d,ν as a refinement of S d,ν and establish a concept of ellipticity in these scales.
Note that the proof of Proposition 5.3 is constructive and allows to determine the coefficients a ∞ [j] (ξ) explicitely. In the following simple example, the coefficients are found by a simple calculation.
It follows that Note that Proposition 5.5. Pointwise multiplication of symbols induces a continuous map In particular, the principal limit-symbol behaves multiplicatively.
Proof. Obviously, a := a 1 a 0 ∈ S d0+d1,ν0+ν1 . By a straightforward calculation, Thus we find, modulo S d0+d1,ν0+ν1+N Both summands on the right-hand side belong to S d0+d1,ν0+ν1+N The continuity follows from the closed graph theorem, since multiplication is a continuous map S d1,ν1 Lemma 5.6. Differentiation induces the following continuous maps: Proof. i) By induction, it suffices to consider the case |α| = 1. Now, using the notation of Definition 5.1,

Now observe that
It follows that for j ≥ 2. Continuity follows again with the closed graph theorem.
ii) One also proceeds by induction. Observe that Now use the expansion of µ ∈ S 1,0 1,0 , cf. Proposition 5.3, to find a resulting expansion of D µ a. We leave the details to the reader.

5.2.
Calculus for symbols with expansion at infinity. The calculus is based on the concept of oscillatory integrals in the spirit of [5], but extended to Frèchet space valued amplitude functions. For a detailed account on this concept see [1], [10].
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 τn η mn 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: Lemma 5.7. Let E 0 , E 1 and E be Fréchet spaces and let ((·, ·)) be a bilinear continuous map from E 1 × E 0 to E. If q j (y, η) are amplitude functions with values in E j , j = 0, 1, then q(y, η) := ((q 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) dydη := lim ε→0 R n ×R n e −iyη χ(εy, εη)q(y, η) dydη 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 − .

Proof.
Step 1: Suppose first that a ∈ S d,ν 1,0 only. We show that q is an amplitude function with values in S d,ν 1,0 . Recall that the topology of S d,ν 1,0 is defined by the semi-norms p N (a) := max If |α| + |β| + j ≤ N and γ, δ ∈ N n 0 are arbitrary, then Step 2: Suppose a ∈ S d,ν 1,0 . Then where a ∞ [ν+j] (y, η; x, ξ) and R a,N (y, η; x, ξ, µ) define amplitude functions with values in S d−ν+j Altogether, this shows the claims for q. q 1 and q 2 are handled in the same way.
Lemma 5.9. Let a(ξ; µ) ∈ S d . Then defines an amplitude function with values in S d .
Proof. By Taylor expansion, Denoting by r N,σ (η, ξ; µ) the integral term, we have We conclude that r N (η, ξ; µ) defines an amplitude function with values in S d−N 1,0 . Writing with a zero-excision function χ and s α,N ∈ S d−N 1,0 , we find that with an amplitude function R N (η, ξ; µ) taking values in S d−N 1,0 . Therefore which are obviously amplitude functions with values in S (d−k) . According to the definition of the topology of S d this shows the claim.
By Proposition 5.8, p is an amplitude function with values in S d0+d1,ν0+ν1 1,0 , hence the oscillatory integral converges in this space.
Since the map a → a ∞ [ν] : S d,ν 1,0 → S ν 1,0 is linear and continuous, we find that Concerning the expansion (5.1) recall that the difference of a 1 #a 0 and the sum in (5.1) is given by x a 0 (y + x, ξ; µ) dydη dθ.
Theorem 5.12 (Asymptotic summation). Let a j ∈ S d−j,ν−j Proof. Let χ(ξ) be a zero-excision function and denote by χ c , c > 0, the operator of multiplication by χ(ξ/c). Then Moreover, the following statements are checked by straight-forward calculations: In particular, Proof. Let us first observe that S 0−∞,0−∞ 1,0 consists of those symbols a for which exists a sequence of symbols a ∞ [j] ∈ S −∞ (R n × R n ) such that, for every N ∈ N 0 , Moreover, note that For the following recall that (1 − T ) −1 = L−1 j=0 T j + T L (1 − T ) −1 whenever T belongs to a unital algebra and 1 − T is invertible.
Proof. By order reduction we may assume without loss of generality that d = ν = 0.
Step 2: Let c as constructed in Step 1. Then, by Theorem 5.10, a#c ≡ ac = χ(ξ) modulo S −1,−1 1,0 . Thus a#c − 1 ∈ S −1,−1 1,0 and the usual Neuman series argument, which is possible in view of Theorem 5.12, allows to construct a symbol c ′ ∈ S 0,0 has vanishing limitsymbol. Using Proposition 5.13 we thus find a right-parametrix b R ∈ S 0,0 1,0 such that a#b R − 1 ∈ C ∞ comp (R + , S −∞ (R n × R n )). Analogously, we construct a left-parametrix b L . Then the claim follows by choosing b = b L or b = b R . 24 JÖRG SEILER 5.4. Poly-homogeneous symbols with expansion at infinity. Ellipticity in the class S d,ν 1,0 is described by the invertibility of the limit-operator and an estimate on the inverse of the complete symbol of the operator. In this section we introduce a subclass of symbols, whose homogeneous components are "half-way" between those of weakly parameter-dependent symbols from S d,ν and those of strongly parameterdependent symbols from S d : Still their restrictions to the (ξ, µ)-unit semi-sphere may not be smooth in (0, 1), but, in polar-coordinates, they have a (weighted) Taylor-expansion there. In this class, ellipticity will be formulated in terms of the homogeneous principal symbol and the limit-operator.
is said to have a weighted Taylorexpansion in (0, 1), if there exist a ν+j ∈ C ∞ (S n−1 ), j ∈ N 0 , such that the representation a(r, φ) = a(rφ; The space of all such functions will be denoted by ρ ν C ∞ T ( S n + ).
Note that, by construction, S d,ν hom ⊆ S d,ν hom .
The previous definition is meaningful according to Proposition 5.18 and Theorem 5.12. If a is as in (5.5) then the principal limit-symbol a ∞ [ν] (x, ξ; µ) belongs to S ν (R n × R n ) and has asymptotic expansion In particular, we have the following: Proposition 5.20. Let a(x, ξ; µ) ∈ S d,ν . Then the homogeneous principal symbol of the principal limit-symbol a ∞ [ν] (x, ξ) coincides with the principal angular symbol of a(x, ξ; µ). Now let us turn to ellipticity and parametrix.
Definition 5.21. A symbol a(x, ξ; µ) ∈ S d,ν is called elliptic if (1) The homogeneous principal symbol a 0 (x, ξ; µ) is invertible whenever ξ = 0 and Note that if the homogeneous principal symbol of a(x, ξ; µ) does not depend on x for large |x|, then condition (1) in Definition 5.21 can be substituted by The homogeneous principal symbol a 0 (x, ξ; µ) is invertible whenever ξ = 0.
Proof. By ellipticity assumption (2), there exists a b(x, ξ) ∈ S −ν (R n × R n ) which is the inverse of a ∞ [ν] (x, ξ) with respect to the Leibniz-product. By Proposition 5.20 it follows that the principal angular symbol of a (i.e., that of a 0 ) is invertible and the inverse is just the homogeneous principal symbol of b. Together with (1) we conclude that the homogeneous principal symbol a 0 (x, ξ; µ) is invertible with inverse belonging to S −d,−ν hom . Thus there exists a c(x, ξ; µ) ∈ S −d,−ν which is a parametrix of a(x, ξ; µ) modulo S −1,−1 . Then proceed as in Step 2 of the proof of Theorem 5.15.

5.5.
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 ′ | are uniformly bounded from above and below by positive constants. Then, given a ∈ S d 1,0 (R n × R n ) there exists a unique symbol a κ in the same class such that [op(a κ )u](y) = [op(a)(u • κ −1 )](κ(y)), y ∈ R n , for every u ∈ S (R n ). In other words, the class S d 1,0 (R n × R n ) is invariant under the change of coordinates x = κ(y).
For parameter-dependent operators we apply this construction for each value of the parameter. 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). It has been shown in Theorem 2.1.21 of [2] that the same is true for the classes S d,ν 1,0 and S d,ν . This includes the invariance of the classes S d,ν 1,0 and S d,ν for ν ≤ 0. If ν > 0, we choose some symbol p(x, ξ) ∈ S −ν (R n × R n ) which is invertible with respect to the Leibniz-product; let q(x, ξ) ∈ S ν (R n × R n ) be the inverse. Given a(x, ξ; µ) ∈ S d,ν 1,0 , we find a κ = (a#p) κ #q κ ∈ S d,ν 1,0 , since a#p ∈ S d−ν,0 . Analogously we argue for poly-homogeneous symbols.
Proposition 5.23. The classes S d,ν 1,0 and S d,ν are invariant under the change of coordinates x = κ(y).
Proof. Let a(x, ξ; µ) be as in Definition 5.1. The result follows from the observation that the classes S ν+j 1,0 (R n × R n ) and S d,ν+N has a complete expansion, cf. Proposition 5.3. This allows to find the complete expansion of a κ (x, ξ; µ). Using the usual formula for the asymptotic expansion of a κ , one sees that poly-homogeneous symbols remain polyhomogeneous.

Refined calculus for symbols of non-negative regularity
We shall introduce a subclass S d,ν of Grubb's class S d,ν in which we have a notion of ellipticity also for the case of regularity ν = 0. We show that resolvents of poly-homogeneous pseudodifferential operators of non-negative order fit into this smaller class, and extend this result even to parameter-dependent pseudodifferential operators of Toeplitz-type.
6.1. Symbols of finite regularity. Motivated by the decomposition result Theorem 4.6, it is natural to make the following definition: The respective class of homogeneous symbols is S d,ν hom := S d,ν hom + S d hom .
By Theorem 5.11 the class is closed under taking the adjoint. Since in both spaces involved in Definition 6.1 asymptotic summation is possible (cf. Section 3.2 and Theorem 5.12), a sequence of symbols a j ∈ S d−j,ν−j can be summed asymptotically in S d,ν . The homogeneous principal symbol is the same as that in the (larger) class S ν,d ; in particular, in case ν > 0, the homogeneous principal symbol extends to a continuous function on the whole unit semi-sphere S n + . For ν > 0, ellipticity is defined as in the class S d,ν .
In the following we shall limit us to the case of ν ∈ N 0 . Note that then S d,ν ⊆ S d,0 . Moreover, the following result is only true for integer ν: Lemma 6.2. Let ν ∈ N and a(ξ; µ) ∈ r ν C ∞ T ( S n + ) + C ∞ (S n + ) be invertible on S n + . Then a(ξ; µ) −1 ∈ r ν C ∞ T ( S n + ) + C ∞ (S n + ).
In case of x-dependence we need to pose an additional uniform bound on the inverse. Since symbols of S d,ν hom are just the homogeneous extensions (in (ξ, µ)) of degree d of functions from r ν C ∞ T ( S n + ) + C ∞ (S n + ), we immediately have the following corollary.
After this observation it is clear that we can construct a parametrix in the class: Theorem 6.4. Let ν ∈ N and a(x, ξ; µ) ∈ S d,ν be elliptic. Then there exists a parametrix b(x, ξ; µ) ∈ S −d,ν such that both a#b − 1 and b#a − 1 belong to C ∞ comp (R + , S −∞ (R n × R n )).
If a ∈ S d,ν with ν ∈ N, then also a ∈ S d,0 = S d,0 . Due to Propositions 5.18 and 5.3, a ∞ [0] (x, ξ) = a 0 (x, 0; 1), where a 0 is the homogeneous principal symbol of a and a ∞ [0] its principal limitoperator. Recalling Definition 5.21, we find the following result which unifies the notions of ellipticity for symbols of regularity ν = 0 and ν ∈ N, respectively. Proposition 6.5. Let ν ∈ N and a ∈ S d,ν . Then a is elliptic if, and only if, a is elliptic as an element of S d,0 = S d,0 .
In particular, Γ is a ray of minimal growth for p(x, D), considered as unbounded operator in H s (R n ) with domain H s+d (R n ).
In the previous theorem, the uniform estimate is satisfied automatically if p 0 (x, ξ) does not depend on x for large |x| and p(x, ξ) does not assume values in Γ for all ξ = 0.
6.3. Pseudodifferential operators of Toeplitz-type. Though we have discussed sofar only operators with scalar-valued symbols (for convenience of exposition) in this section we shall work with matrix-valued symbols, using the obvious generalizations of the previous results to this setting.
Let p j (x, ξ) ∈ S 0 (R n × R n , C k×k ), j = 0, 1, be such that P j := p j (x, D) is a projection, i.e., P 2 j = P j . These projections define closed subspaces of the Sobolev spaces H s (R n , C k ), H s (R n , P j ) := P j (H s (R n , C k )), s ∈ R.
Let us consider the limit-symbol. Define Since (6.3) is an isomorphism, we find that T is injective with image L 2 (R n , P 1 ) ⊕ L 2 (R n , 1 − P 0 ). Hence T is upper semi-Fredholm. It follows that T * T is an injective, self-adjoint Fredholm operator, hence an isomorphism. However, by construction, , where a ∞ [0] (x, ξ) is the limit-symbol of a(x, ξ; µ) (note also that the limit-symbol of [ξ, µ] is constant 1).
In a similar way (working in the finite-dimensional spaces C k ) one shows that the homogeneous principal symbol a 0 (x, ξ; µ) of a(x, ξ; µ) is invertible whenever ξ = 0.
Let P be a continuous projection in a Hilbert space. Hence P (H) is a closed subspace and its dual space can be identified with P * (H). Having this in mind, the adjoint A(µ) * can be identified with the operator P * 0 a ( * ) (x, D; µ)P * 1 and we have the analogue of conditions (1) and (2). Again we can conclude that A(µ) * is invertible from the left for large µ and thus find that A(µ) is invertible for large µ. This shows the claim.