Asymptotic Expansions for Bounded Solutions

to Semilinear Fuchsian Equations

Xiaochun Liu and Ingo Witt

Received: Jun 12, 2002 Revised: July 15, 2004

Communicated by Bernold Fielder

Abstract. It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptotic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schulze’s notion of asymptotic type for conormal asymptotic expansions near a conical point is refined. This in turn allows to perform explicit computations on asymptotic types — modulo the resolution of the spectral problem for determining the singular exponents in the asymptotic expansions.

2000 Mathematics Subject Classification: Primary: 35J70; Secondary: 35B40, 35J60 Keywords and Phrases: Calculus of conormal symbols, conormal asymptotic expansions, discrete asymptotic types, weighted Sobolev spaces with discrete asymptotics, semilinear Fuchsian equations

Contents

1 Introduction 208

2 Asymptotic types 212 2.1 Fuchsian differential operators . . . . . . . . . . . . . . . . . . . 212 2.2 Definition of asymptotic types . . . . . . . . . . . . . . . . . . . 217 2.3 Pseudodifferential theory . . . . . . . . . . . . . . . . . . . . . . 230 2.4 Function spaces with asymptotics . . . . . . . . . . . . . . . . . 231 3 Applications to semilinear equations 238

3.1 Multiplicatively closed asymptotic types . . . . . . . . . . . . . 238 3.2 The bootstrapping argument . . . . . . . . . . . . . . . . . . . 244 3.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . 245

3.4 Example: The equation Δu = Au^{2} + B(x)u in three space dimensions 247

1 Introduction

In this paper, we study solutions u = u(x) to semilinear elliptic equations of the form

Au = F (x, B_{1}u, …, B_{K}u) on X^{∘} = X \ ∂X.  (1.1)

Here, X is a smooth compact manifold with boundary, ∂X, and of dimension n + 1, A, B_{1}, …, B_{K} are Fuchsian differential operators on X^{∘}, see Definition 2.1, with real-valued coefficients and of orders μ, μ_{1}, …, μ_{K}, respectively, where μ_{J} < μ for 1 ≤ J ≤ K, and F = F (x, ν): X^{∘} × ℝ^{K} → ℝ is a smooth function subject to further conditions as x → ∂X. In case A is elliptic in the sense of Definition 2.2 (a) we shall prove that bounded solutions u: X^{∘} → ℝ to Eq. (1.1) possess complete conormal asymptotic expansion of the form

u(t, y) ∼ ∑_{j=0}^{∞} ∑_{k=0}^{m_{j}} t^{−p_{j}} log^{k} t c_{jk}(y) as t → +0.  (1.2)

Here, (t, y) ∈ [0, 1) × Y are normal coordinates in a neighborhood U of ∂X, Y is diffeomorphic to ∂X, and the exponents p_{j} ∈ ℂ appear in conjugated pairs, Re p_{j} → −∞ as j → ∞, m_{j} ∈ ℕ, and c_{jk}(y) ∈ C^{∞}(Y). Note that such conormal asymptotic expansions are typical of solutions u to linear equations of the form (1.1), i.e., in case F (x) = F (x, ν) is independent of ν ∈ ℝ^{K}. The general form (1.2) of asymptotics was first thoroughly investigated by Kondrat’ev in his nowadays classical paper [9]. After that to assign asymptotic types to conormal asymptotic expansions of the form (1.2) has proved to be very fruitful. In its consequence, it provides a functional-analytic framework for treating singular problems, both linear and non-linear ones, of the kind (1.1). Function spaces with asymptotics will be discussed in Sections 2.4, 3.1. In its standard setting, going back to Rempel–Schulze [14] in case n = 0 (when Y is always assumed be a point) and Schulze [15] in the general case, an asymptotic type P for conormal asymptotic expansions of the form (1.2) is given by a sequence {(p_{j}, m_{j}, L_{j})}_{j=0}^{∞}, where p_{j} ∈ ℂ, m_{j} ∈ ℕ are as in (1.2), and L_{j} is a finite-dimensional linear subspace of C^{∞}(Y) to which the coefficients c_{jk}(y) for 0 ≤ k ≤ m_{j} are required to belong. (In case n = 0, the spaces L_{j} = ℂ disappear.) A function u(x) is said to have conormal asymptotics of type P as x → ∂X if u(x) obeys a conormal asymptotic expansion of the form (1.2), with the data given by P.

When treating semilinear equations we shall encounter asymptotic types belonging to bounded functions u(x), i.e., asymptotic types P for which

\{ \begin{matrix}p_{0} = 0, m_{0} = 0, L_{0} = span\{1\}, \\ Re p_{j} < 0 for all j ≥ 1, \end{matrix}  (1.3)

where 1 ∈ L_{0} is the function on Y being constant 1. It turns out that this notion of asymptotic type resolves asymptotics not fine enough to suit a treatment of semilinear problems. The difficulty with it is that only the aspect of the production of asymptotics is emphasized — via the finite-dimensionality of the spaces L_{j} — but not the aspect of their annihilation. For semilinear problems, however, the latter affair becomes crucial. Therefore, in Section 2, we shall introduce a refined notion of asymptotic type, where additionally linear relations between the various coefficients c_{jk}(y) ∈ L_{j}, even for different j, are taken into account. Let \underline{As}(Y) be the set of all these refined asymptotic types, while \underline{As}^{♯}(Y) ⊂ \underline{As}(Y) denotes the set of asymptotic types belonging to bounded functions according to (1.3). For R ∈ \underline{As}(Y), let C_{R}^{∞}(X) be the space of smooth functions u ∈ C^{∞}(X^{∘}) having conormal asymptotic expansions of type R, and C_{R}^{∞}(X × ℝ^{K}) = C^{∞}(ℝ^{K}; C_{R}^{∞}(X)), where C_{R}^{∞}(X) is equipped with its natural (nuclear) Fréchet topology. In the formulation of Theorem 1.1, below, we will assume that F ∈ C_{R}^{∞}(X × ℝ^{K}), where

ω(t)t^{μ−μ̄−ε}C_{R}^{∞}(X) ⊂ L^{∞}(X)  (1.4)

for some ε > 0. Here, μ̄ = max_{1≤J≤K} μ_{J} < μ and ω = ω(t) is a cut-off function supported in U, i.e., ω ∈ C^{∞}(X), supp ω ⋐ U. Here and in the sequel, we always assume that ω = ω(t) depends only on t for 0 < t < 1 and ω(t) = 1 for 0 < t ≤ 1/2. Condition (1.4) means that, given the operator A and then compared to the operators B_{1}, …, B_{K}, functions in C_{R}^{∞}(X) cannot be too singular as t → +0. There is a small difference between the set \underline{As}^{b}(Y) of all bounded asymptotic types and the set \underline{As}^{♯}(Y) of asymptotic types as described by (1.3); \underline{As}^{♯}(Y) ⊊ \underline{As}^{b}(Y). The set \underline{As}^{♯}(Y) actually appears as the set of multiplicatively closable asymptotic types, see Lemma 3.4. This shows up in the fact that when only boundedness is presumed asymptotic types belonging to \underline{As}^{b}(Y) — but not to \underline{As}^{♯}(Y) — need to be excluded from the considerations by the following non-resonance type condition (1.5), below: Let H^{−∞,δ}(X) = ⋃_{s∈ℝ} H^{s,δ}(X) for δ ∈ ℝ be the space of distributions u = u(x) on X^{∘} having conormal order at least δ. (The weighted Sobolev space H^{s,δ}(X), where s ∈ ℝ is Sobolev regularity, is introduced in (2.31).) Note that ⋃_{δ∈ℝ} H^{−∞,δ}(X) is the space of all extendable distributions on X^{∘} that in turn is dual to the space C_{O}^{∞}(X) of all smooth functions on X vanishing to infinite order at ∂X. Note also that the conormal order δ for δ → ∞ is the parameter in which the asymptotics (1.2) are understood.

Now, fix δ ∈ ℝ and suppose that a real-valued u ∈ H^{−∞,δ}(X) satisfying Au ∈ C_{O}^{∞}(X) has an asymptotic expansion of the form

u(x) ∼ Re (∑_{j=0}^{∞} ∑_{k=0}^{m_{j}} t^{l+j+iβ} log^{k} t c_{jk}(y)) as t → +0,

where l ∈ ℤ, β ∈ ℝ, β ≠ 0 (and l > δ − 1/2 provided that c_{0m_{0}}(y) ≢ 0 due to the assumption u ∈ H^{−∞,δ}(X)). Then, for each 1 ≤ J ≤ K, it is additional required that

B_{J}u = O(1) as t → +0 implies B_{J}u = o(1) as t → +0,  (1.5)

where O and o are Landau’s symbols. Condition (1.5) means that there is no real-valued u ∈ H^{−∞,δ}(X) with Au ∈ C_{O}^{∞}(X) such that B_{J}u admits an asymptotic series starting with the term Re(t^{iβ}d(y)) for some β ∈ ℝ \ {0}, d(y) ∈ C^{∞}(Y). This condition is void if δ ≥ 1/2 + μ̄. Our main theorem states:

Theorem 1.1. Let δ ∈ ℝ and A ∈ Diff_{Fuchs}^{μ}(X) be elliptic in the sense of Definition 2.2 (a), B_{J} ∈ Diff_{Fuchs}^{μ_{J}}(X) for 1 ≤ J ≤ K, where μ_{J} < μ, and F ∈ C_{R}^{∞}(X × ℝ^{k}) for some asymptotic type R ∈ \underline{As}(Y) satisfying (1.4). Further, let the non-resonance type condition (1.5) be satisfied. Then there exists an asymptotic type P ∈ \underline{As}(Y) expressible in terms of A, B_{1}, …, B_{K}, R, and δ such that each solution u ∈ H^{−∞,δ}(X) to Eq. (1.1) satisfying B_{J}u ∈ L^{∞}(X) for 1 ≤ J ≤ K belongs to the space C_{P}^{∞}(X).

Under the conditions of Theorem 1.1, interior elliptic regularity already implies u ∈ C^{∞}(X^{∘}). Thus, the statement concerns the fact that u possesses a complete conormal asymptotic expansion of type P near ∂X. Furthermore, the asymptotic type P can at least in principle be calculated once A, B_{1}, …, B_{K}, R, and δ are known. Some remarks about Theorem 1.1 are in order: First, the solution u is asked to belong to the space H^{−∞,δ}(X). Thus, if the non-resonance type condition (1.5) is satisfied for all δ ∈ ℝ — which is generically true — then the foregoing requirement can be replaced by the requirement for u being an extendable distribution. In this case, P_{δ} ≼ P_{δ′} for δ ≥ δ′ in the natural ordering of asymptotic types, where P_{δ} denotes the asymptotic type associated with the conormal order δ. Moreover, jumps in this relation occur only for a discrete set of values of δ ∈ ℝ and, generically, P_{δ} eventually stabilizes as δ → −∞. Secondly, for a solution u ∈ C_{P}^{∞}(X) to Eq. (1.1), neither u nor the righthand side F (x, B_{1}u(x), …, B_{K}u(x)) need be bounded. Unboundedness of u, however, requires that, up to a certain extent, asymptotics governed by the elliptic operator A are canceled jointly by the operators B_{1}, …, B_{K}. Again, this is a non-generic situation. Furthermore, in applications one often has that one of the operators B_{J}, say B_{1}, is the identity — belonging to Diff_{Fuchs}^{0}(X) — i.e., B_{1}u = u for all u. Then this leads to u ∈ L^{∞}(X) and explains the term “bounded solutions” in the paper’s title.

Remark 1.2. Theorem 1.1 continues to hold for sectional solutions in vector bundles over X. Let E_{0}, E_{1}, E_{2} be smooth vector bundles over X, A ∈ Diff_{Fuchs}^{μ}(X; E_{0}, E_{1}) be elliptic in the above sense, B ∈ Diff_{Fuchs}^{μ−1}(X; E_{0}, E_{2}), and F ∈ C_{R}^{∞}(X, E_{2}; E_{1}). Then, under the same technical assumptions as above, each solution u to Au = F (x, Bu) in the class of extendable distributions with Bu ∈ L^{∞}(X; E_{2}) belongs to the space C_{P}^{∞}(X; E_{0}) for some resulting asymptotic type P.

Theorem 1.1 has actually been stated as one, though basic example for a more general method for deriving — and then justifying — conormal asymptotic expansions for solutions to semilinear elliptic Fuchsian equations. This method always works if one has boundedness assumptions as made above, but boundedness can often successfully be replaced by structural assumptions on the nonlinearity. An example is provided in Section 3.4. The proposed method works indeed not only for elliptic Fuchsian equations, but for other Fuchsian equations as well. In technical terms, what counts is the invertible of the complete sequence of conormal symbols in the algebra of complete Mellin symbols under the Mellin translation product, and this is equivalent to the ellipticity of the principal conormal symbol (which, in fact, is a substitute for the non-characteristic boundary in boundary problems). For elliptic Fuchsian differential operator, this latter condition is always fulfilled. The derivation of conormal asymptotic expansions for solutions to semilinear Fuchsian equations is a purely algebraic business once the singular exponents and their multiplicities for the linear part are known. However, a strict justification of these conormal asymptotic expansions — in the generality supplied in this paper — requires the introduction of the refined notion of asymptotic type and corresponding function spaces with asymptotics. For this reason, from a technical point of view the main result of this paper is Theorem 2.42 which states the existence of a complete sequence of holomorphic Mellin symbols realizing a given proper asymptotic type in the sense of exactly annihilating asymptotics of that given type. (The term “proper” is introduced in Definition 2.22.) The construction of such Mellin symbols relies on the factorization result of Witt [21].

Remark 1.3. Behind part of the linear theory, there is Schulze’s cone pseudodifferential calculus. The interested reader should consult Schulze [15, 16]. We do not go much into the details, since for most of the arguments this is not needed. Indeed, the algebra of complete Mellin symbols controls the production and annihilation of asymptotics, and it is this algebra that is detailed discussed. The relation with conical points is as follows: A conical point leads — via blowup, i.e., the introduction of polar coordinates — to a manifold with boundary. Vice versa, each manifold with boundary gives rise to a space with a conical point — via shrinking the boundary to a point. Since in both situations the analysis is taken place over the interior of the underlying configuration, i.e., away from the conical point and the boundary, respectively, there is no essential difference between these two situations. Thus, the geometric situation is given by the kind of degeneracy admitted for, say, differential operators. In the case considered in this paper, this degeneracy is of Fuchsian type. The first part of this paper, Section 2, is devoted to the linear theory and the introduction of the refined notion of asymptotic type. Then, in a second part, Theorem 1.1 is proved in Section 3.

2 Asymptotic types

In this section, we introduce the notion of discrete asymptotic type. A comparison of this notion with the formerly known notions of weakly discrete asymptotic type and strongly discrete asymptotic type, respectively, can be found in Figure 1. The definition of discrete asymptotic type is modeled on part of the Gohberg-Sigal theory of the inversion of finitely meromorphic, operator-valued functions at a point, see Gohberg-Sigal [4]. See also Witt [18] for the corresponding notion of local asymptotic type, i.e., asymptotic types at one singular exponent p ∈ ℂ in (1.2) only. Finally, in Section 2.4, function spaces with asymptotics are introduced. The definition of these function spaces relies on the existence of complete (holomorphic) Mellin symbols realizing a prescribed proper asymptotic type. The existence of such complete Mellin symbols is stated and proved in Theorem 2.42.

Added in proof. To keep this article of reasonable length, following the referee’s advice, proofs of Theorems 2.6, 2.30, and 2.42 and Propositions 2.28 (b), 2.31, 2.32, 2.35, 2.36, 2.40, 2.44, 2.46, 2.47, 2.48, 2.49, and 2.52 are only sketchy or missing at all. They are available from the second author’s homepage^{1}.

2.1 Fuchsian differential operators

Let X be a compact C^{∞} manifold with boundary, ∂X. Throughout, we fix a collar neighborhood U of ∂X and a diffeomorphism χ: U → [0, 1) × Y, with Y being a closed C^{∞} manifold diffeomorphic to ∂X. Hence, we work in a fixed splitting of coordinates (t, y) on U, where t ∈ [0, 1) and y ∈ Y. Let (τ, η) be the covariables to (t, y). The compressed covariable tτ to t is denoted by τ̃, i.e., (τ̃, η) is the linear variable in the fiber of the compressed cotangent bundle T̃^{∗}X|_{U}. Finally, let dim X = n + 1.

Definition 2.1. A differential operator A with smooth coefficients of order μ on X^{∘} = X \ ∂X is called Fuchsian if

χ_{∗} (A|_{U}) = t^{−μ} ∑_{k=0}^{μ} a_{k}(t, y, D_{y})(−t∂_{t})^{k},  (2.1)

where a_{k} ∈ C^{∞}([0, 1); Diff^{μ−k}(Y)) for 0 ≤ k ≤ μ. The class of all Fuchsian differential operators of order μ on X^{∘} is denoted by Diff_{Fuchs}^{μ}(X).

1 http://www.ma.imperial.ac.uk/˜ ifw/asymptotics.html

Documenta Mathematica 9 (2004) 207–250

Weakly discrete asymptotic types Singular exponents with multiplicities, (p_{j}, m_{j}), are prescribed, the coefficients c_{jk}(y) ∈ C^{∞}(Y) are arbitrary. The general form of asymptotics is observed, cf., e.g., Kondrat’ev (1967), Melrose (1993), Schulze (1998).

Strongly discrete asymptotic types Singular exponents with multiplicities, (p_{j}, m_{j}), are prescribed, c_{jk}(y) ∈ L_{j} ⊂ C^{∞}(Y), where dim L_{j} < ∞. The production of asymptotics is observed, cf. Rempel–Schulze (1989), Schulze (1991).

Discrete asymptotic types Linear relation between the various coefficients c_{jk}(y) ∈ L_{j}, even for different j, are additionally allowed. Thus the production/annihilation of asymptotics is observed, cf. this article.

Figure 1: Schematic overview of asymptotic types

Henceforth, we shall suppress writing the restriction ·|_{U} and the operator pushforward χ_{∗} in expressions like (2.1). For A ∈ Diff_{Fuchs}^{μ}(X), we denote by

σ_{ψ}^{μ}(A)(t, y, τ, η) = t^{−μ} ∑_{k=0}^{μ} σ_{ψ}^{μ−k}(a_{k}(t))(y, η)(itτ)^{k}

the principal symbol of A, by σ̃_{ψ}^{μ}(A)(t, y, τ̃, η) its compressed principal symbol related to σ_{ψ}^{μ}(A)(t, y, τ, η) via

σ_{ψ}^{μ}(A)(t, y, τ, η) = t^{−μ}σ̃_{ψ}^{μ}(A)(t, y, tτ, η)

in (T̃^{∗}X \ 0)|_{U}, and by σ_{M}^{μ}(A)(z) its principal conormal symbol,

σ_{M}^{μ}(A)(z) = ∑_{k=0}^{μ} a_{k}(0)z^{k}, z ∈ ℂ.

Further, we introduce the jth conormal symbol σ_{M}^{μ−j}(A)(z) for j = 1, 2, … by

σ_{M}^{μ−j}(A)(z) = ∑_{k=0}^{μ} \frac{1}{j!} \frac{∂^{j}a_{k}}{∂t^{j}} (0)z^{k}, z ∈ ℂ.

Note that σ̃_{ψ}^{μ}(A)(t, y, τ̃, η) is smooth up to t = 0 and that σ_{M}^{μ−j}(z) for j = 0, 1, 2, … is a holomorphic function in z taking values in Diff^{μ}(Y). Moreover, if A ∈ Diff_{Fuchs}^{μ}(X), B ∈ Diff_{Fuchs}^{ν}(X), then AB ∈ Diff_{Fuchs}^{μ+ν}(X),

σ_{M}^{μ+ν−l}(AB)(z) = ∑_{j+k=l} σ_{M}^{μ−j}(A)(z + ν − k)σ_{M}^{ν−k}(B)(z)  (2.2)

for all l = 0, 1, 2, … This formula is called the Mellin translation product (due to the shifts of ν − k in the argument of the first factors).

Definition 2.2. (a) The operator A ∈ Diff_{Fuchs}^{μ}(X) is called elliptic if A is an elliptic differential operator on X^{∘} and

σ̃_{ψ}^{μ}(A)(t, y, τ̃, η) ≠ 0, (t, y, τ̃, η) ∈ (T̃^{∗}X \ 0)|_{U}.  (2.3)

(b) The operator A ∈ Diff_{Fuchs}^{μ}(X) is called elliptic with respect to the weight δ ∈ ℝ if A is elliptic in the sense of (a) and, in addition,

σ_{M}^{μ}(A)(z): H^{s}(Y) → H^{s−μ}(Y), z ∈ Γ_{(n+1)/2−δ},  (2.4)

is invertible for some s ∈ ℝ (and then for all s ∈ ℝ). Here, Γ_{β} = {z ∈ ℂ; Re z = β} for β ∈ ℝ.

Under the assumption of interior ellipticity of A, (2.3) can be reformulated as

∑_{k=0}^{μ} σ_{ψ}^{μ−k}(a_{k}(0))(y, η)(iτ̃)^{k} ≠ 0

for all (0, y, τ̃, η) ∈ (T̃^{∗}X\0)|_{∂U}. This relation implies that σ_{M}^{μ}(A)(z)|_{Γ_{(n+1)/2−δ}}

is parameter-dependent elliptic as an element in L_{cl}^{μ}(Y; Γ_{(n+1)/2−δ}), where the latter is the space of classical pseudodifferential operators on Y of order μ with parameter z varying in Γ_{(n+1)/2−δ}, for

σ_{ψ}^{μ}(σ_{M}^{μ}(A))(y, z, η)|_{z=(n+1)/2−δ−τ̃} = σ̃_{ψ}^{μ}(A)(0, y, τ̃, η),

where σ_{ψ}^{μ}(·) on the left-hand side denotes the parameter-dependent principal symbol. Thus, if (a) is fulfilled, then it follows that σ_{M}^{μ}(A)(z) in (2.4) is invertible for z ∈ Γ_{(n+1)/2−δ}, |z| large enough.

Lemma 2.3. If A ∈ Diff_{Fuchs}^{μ}(X) is elliptic, then there exists a discrete set D ⊂ ℂ with D ∩ {z ∈ ℂ; c_{0} ≤ Re z ≤ c_{1}} is finite for all −∞ < c_{0} < c_{1} < ∞ such that (2.4) is invertible for all z ∈ ℂ \ D. In particular, there is a discrete set D ⊂ ℝ such that A is elliptic with respect to the weight δ for all δ ∈ ℝ \ D; D = Re D.

Proof. Since σ_{M}^{μ}(A)(z)|_{Γ_{β}} ∈ L^{μ}(Y; Γ_{β}) is parameter-dependent elliptic for all β ∈ ℝ, for each c > 0 there is a C > 0 such that σ_{M}^{μ}(A)(z) ∈ L^{μ}(Y) is invertible for all z with | Re z| ≤ c, | Im z| ≥ C. Then the assertion follows from results on the invertibility of holomorphic operator-valued functions. See Proposition 2.5, below, or Schulze [16, Theorem 2.4.20].

Next, we introduce the class of meromorphic functions arising in point-wise inverting parameter-dependent elliptic conormal symbols σ_{M}^{μ}(A)(z). The following definition is taken from Schulze [16, Definition 2.3.48]:

Definition 2.4. (a) M_{O}^{μ}(Y) for μ ∈ ℤ ∪ {−∞} is the space of all holomorphic functions f(z) on ℂ taking values in L_{cl}^{μ}(Y) such that f(z)|_{z=β+iτ} ∈ L_{cl}^{μ}(Y; ℝ_{τ}) uniformly in β ∈ [β_{0}, β_{1}] for all −∞ < β_{0} < β_{1} < ∞.

(b) M_{as}^{−∞}(Y) is the space of all meromorphic functions f(z) on ℂ taking values in L^{−∞}(Y) that satisfy the following conditions:

(i) The Laurent expansion around each pole z = p of f(z) has the form

f(z) = \frac{f_{0}}{(z − p)^{ν}} + \frac{f_{1}}{(z − p)^{ν−1}} + ⋯ + \frac{f_{ν−1}}{z − p} + ∑_{j≥0} f_{ν+j}(z − p)^{j},  (2.5)

where f_{0}, f_{1}, …, f_{ν−1} ∈ L^{−∞}(Y) are finite-rank operators. (ii) If the poles of f(z) are numbered someway, p_{1}, p_{2}, …, then | Re p_{j}| → ∞ as j → ∞ if the number of poles is infinite. (iii) For any ⋃_{j}{p_{j}}-excision function χ(z) ∈ C^{∞}(ℂ), i.e., χ(z) = 0 if dist(z, ⋃_{j}{p_{j}}) ≤ 1/2 and χ(z) = 1 if dist(z, ⋃_{j}{p_{j}}) ≥ 1, we have χ(z)f(z)|_{z=β+iτ} ∈ L^{−∞}(Y; ℝ_{τ}) uniformly in β ∈ [β_{0}, β_{1}] for all −∞ < β_{0} < β_{1} < ∞.

(c) Finally, we set M_{as}^{μ}(Y) = M_{O}^{μ}(Y) + M_{as}^{−∞}(Y) for μ ∈ ℤ. (Note that M_{O}^{μ}(Y) ∩ M_{as}^{−∞}(Y) = M_{O}^{−∞}(Y).) Functions f(z) belonging to M_{as}^{μ}(Y) are called Mellin symbols of order μ.

⋃_{μ∈ℤ} M_{as}^{μ}(Y) is a filtered algebra under pointwise multiplication. For f ∈ M_{as}^{μ}(Y) for μ ∈ ℤ and f(z) = f_{0}(z) + f_{1}(z), where f_{0} ∈ M_{O}^{μ}(Y), f_{1} ∈ M_{as}^{−∞}(Y), the parameter-dependent principal symbol σ_{ψ}^{μ}(f_{0}(z)|_{z=β+iτ} ) is independent of the choice of the decomposition of f and also independent of β ∈ ℝ. It is called the principal symbol of f. The Mellin symbol f ∈ M_{as}^{μ}(Y) is called elliptic if its principal symbol is everywhere invertible. For the next result, see Schulze [16, Theorem 2.4.20]:

Proposition 2.5. The Mellin symbol f ∈ M_{as}^{μ}(Y) for μ ∈ ℤ is invertible in the filtered algebra ⋃_{μ∈ℤ} M_{as}^{μ}(Y), i.e., there is a g ∈ M_{as}^{−μ}(Y) such that (fg)(z) = (gf)(z) = 1 on ℂ, if and only if f is elliptic.

For f ∈ M_{as}^{μ}(Y), p ∈ ℂ, and N ∈ ℕ, we denote by [f(z)]_{p}^{N} the Laurent series of f(z) around z = p truncated after the term containing (z − p)^{N}, i.e.,

[f(z)]_{p}^{N} = \frac{f_{−ν}}{(z − p)^{ν}} + ⋯ + \frac{f_{−1}}{z − p} + f_{ν} + f_{1}(z − p) + ⋯ + f_{N}(z − p)^{N}. (2.6)

Furthermore, [f(z)]_{p}^{∗} = [f(z)]_{p}^{−1} denotes the principal part of the Laurent series of f(z) around z = p. In various constructions, it is important to have examples of elliptic Mellin symbols f ∈ M_{as}^{μ}(Y) of controlled singularity structure:

Theorem 2.6. Let μ ∈ ℤ and {p_{j}}_{j=1,2,...} ⊂ ℂ be a sequence obeying the property mentioned in Definition 2.4 (b) (ii). Let, for each j = 1, 2, …, operators f_{−ν_{j}}^{j}, …, f_{N_{j}}^{j} in L_{cl}^{μ}(Y), where ν_{j} ≥ 0, N_{j} + ν_{j} ≥ 0, be given such that

• f_{−ν_{j}}^{j}, …, f_{min{N_{j},0}}^{j} ∈ L^{−∞}(Y) are finite-rank operators,

• there is an elliptic g ∈ M_{O}^{μ}(Y) such that, for all j, 0 ≤ k ≤ N_{j},

f_{k}^{j} − \frac{1}{k!} g^{(k)}(p_{j}) ∈ L^{−∞}(Y)  (2.7)

(in particular, f_{k}^{j} ∈ L_{cl}^{μ−k}(Y) for 0 ≤ k ≤ N_{j} and f_{0}^{j} ∈ L_{cl}^{μ}(Y) is elliptic of index zero).

Then there is an elliptic Mellin symbol f(z) ∈ M_{as}^{μ}(Y) such that, for all j,

[f(z)]_{p_{j}}^{N_{j}} = \frac{f_{−ν_{j}}^{j}}{(z − p_{j})^{ν_{j}}} + ⋯ + \frac{f_{−1}^{j}}{z − p_{j}} + f_{0}^{j} + ⋯ + f_{N_{j}}^{j}(z − p_{j})^{N_{j}},  (2.8)

while f(q) ∈ L_{cl}^{μ}(Y) is invertible for all q ∈ ℂ \ ⋃_{j=1,2,...}{p_{j}}.

If n = 0, condition (2.7) is void. In case n > 0, however, this condition expresses several compatibility conditions among the σ_{ψ}^{μ−l}(f_{k}^{j}), where j = 0, 1, 2, …, 0 ≤ k ≤ N_{j}, and l ≥ k, and also certain topological obstructions that must be fulfilled. For instance, for any f ∈ M_{O}^{μ}(Y),

σ_{ψ}^{μ−j}(f(z))(y, η) = ∑_{k=0}^{j} \frac{(z − p)^{k}}{k!} σ_{ψ}^{μ−j}(f^{(k)}(p))(y, η), j = 0, 1, 2, …

in local coordinates (y, η) — showing, among others, that σ_{ψ}^{μ−j}(f(z)) is polynomial of degree j with respect to z ∈ ℂ. The point is that we do not assume g(q) ∈ L_{cl}^{μ}(Y) be invertible for q ∈ ℂ \ ⋃_{j=1,2,...}{p_{j}}.

Proof of Theorem 2.6. This can be proved using the results of Witt [21]. In particular, the factorization result there gives directly the existence of f(z) if the sequence {p_{j}} ⊂ ℂ is void.

Now, we are going to introduce the basic object of study — the algebra of complete conormal symbols. This algebra will enable us to introduce the refined notion of asymptotic type and to study the behavior of conormal asymptotics under the action of Fuchsian differential operators.

Definition 2.7. (a) For μ ∈ ℤ, the space Symb_{M}^{μ}(Y) consists of all sequences 𝔖^{μ} = {𝔰^{μ−j}(z); j ∈ ℕ} ⊂ M_{as}^{μ}(Y).

(b) An element 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) is called holomorphic if 𝔖^{μ} = {𝔰^{μ−j}(z); j ∈ ℕ} ⊂ M_{O}^{μ}(Y).

(c) ⋃_{μ∈ℤ} Symb_{M}^{μ}(Y) is a filtered algebra under the Mellin translation product, denoted by ♯_{M}. Namely, for 𝔖^{μ} = {𝔰^{μ−j}(z); j ∈ ℕ} ∈ Symb_{M}^{μ}(Y), 𝔗^{ν} = {𝔱^{ν−k}(z); k ∈ ℕ} ∈ Symb_{M}^{ν}(Y), we define 𝔘^{μ+ν} = 𝔖^{μ}♯_{M}𝔗^{ν} ∈ Symb_{M}^{μ+ν}(Y), where 𝔘^{μ+ν} = {𝔲^{μ+ν−l}(z); l ∈ ℕ}, by

𝔲^{μ+ν−l}(z) = ∑_{j+k=l} 𝔰^{μ−j}(z + ν − k)𝔱^{ν−k}(z)  (2.9)

for l = 0, 1, 2, … See also (2.2).

From Proposition 2.5, we immediately get:

Lemma 2.8. 𝔖^{μ} = {𝔰^{μ−j}(z); j ∈ ℕ} ∈ Symb_{M}^{μ}(Y) is invertible in the filtered algebra ⋃_{μ∈ℤ} Symb_{M}^{μ}(Y) if and only if 𝔰^{μ}(z) ∈ M_{as}^{μ}(Y) is elliptic.

In the case of the preceding lemma, 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) is called elliptic. A holomorphic elliptic 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) is called elliptic with respect to the weight δ ∈ ℝ if the line Γ_{(n+1)/2−δ} is free of poles of 𝔰^{μ}(z)^{−1}. Notice that a holomorphic elliptic 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) is elliptic for all, but a discrete set of δ ∈ ℝ. The inverse to 𝔖^{μ} with respect to the Mellin translation product is denoted by (𝔖^{μ})^{−1}. The set of elliptic elements of Symb_{M}^{μ}(Y) is denoted by Ell Symb_{M}^{μ}(Y). There is a homomorphism of filtered algebras,

⋃_{μ∈ℕ} Diff_{Fuchs}^{μ}(X) → ⋃_{μ∈ℤ} Symb_{M}^{μ}(Y), A ↦ {σ_{M}^{μ−j}(A)(z); j ∈ ℕ}.

By the remark preceding Lemma 2.3, {σ_{M}^{μ−j}(A)(z); j ∈ ℕ} ∈ Symb_{M}^{μ}(Y) is elliptic if A ∈ Diff_{Fuchs}(X) is elliptic in the sense of Definition 2.2 (a).

2.2 Definition of asymptotic types

We now start to introduce discrete asymptotic types.

2.2.1 The spaces E^{δ}(Y) and E_{V} (Y)

Here, we construct the “coefficient” space E^{δ}(Y) = ⋃_{V ∈C^{δ}} E_{V} (Y) that admits the non-canonical isomorphism (2.13), below,

C_{as}^{∞,δ}(X)/C_{O}^{∞}(X) ⟶^{≅} E^{δ}(Y),

where C_{as}^{∞,δ}(X) is the space of smooth functions on X^{∘} obeying conormal asymptotic expansions of the form (1.2) of conormal order at least δ, i.e., Re p_{j} < (n + 1)/2 − δ holds for all j (with the condition that the singular exponents p_{j} appear in conjugated pairs dropped), and C_{O}^{∞}(X) is the subspace of all smooth functions on X^{∘} vanishing to infinite order at ∂X.

Definition 2.9. A carrier V of asymptotics for distributions of conormal order δ is a discrete subset of ℂ contained in the half-space {z ∈ ℂ; Re z < (n+1)/2− δ} such that, for all β_{0}, β_{1} ∈ ℝ, β_{0} < β_{1}, the intersection V ∩ {z ∈ ℂ; β_{0} < Re z < β_{1}} is finite. The set of all these carriers is denoted by C^{δ}.

In particular, V_{p} = p − ℕ for p ∈ ℂ is such a carrier of asymptotics. Note that V_{p} ∈ C^{δ} if and only if Re p < (n + 1)/2 − δ. We set T^{ϱ}V = ϱ + V ∈ C^{−ϱ+δ} for ϱ ∈ ℝ and V ∈ C^{δ}. We further set C = ⋃_{δ∈ℝ} C^{δ}. Let [C^{∞}(Y)]^{∞} = ⋃_{m∈ℕ}[C^{∞}(Y)]^{m} be the space of all finite sequences in C^{∞}(Y), where the sequences (ϕ_{0}, …, ϕ_{m−1}) and (\underbrace{0, …, 0}_{h times}, ϕ_{0}, …, ϕ_{m−1}) for

h ∈ ℕ are identified. For V ∈ C^{δ}, we set E_{V} (Y) = ∏_{p∈V} [C^{∞}(Y)]_{p}^{∞}, where [C^{∞}(Y)]_{p}^{∞} is an isomorphic copy of [C^{∞}(Y)]^{∞}, and define E^{δ}(Y) to be the space of all families Φ ∈ E_{V} (Y) for some V ∈ C^{δ} depending on Φ. Thereby, Φ ∈ E_{V} (Y), Φ′ ∈ E_{V′}(Y) for possibly different V, V′ ∈ C^{δ} are identified if Φ(p) = Φ′(p) for p ∈ V ∩ V′, while Φ(p) = 0 for p ∈ V \ V′, Φ′(p) = 0 for p ∈ V′ \ V. Under this identification,

E^{δ}(Y) = ⋃_{V ∈C^{δ}} E_{V} (Y).  (2.10)

Moreover, E_{V} (Y) ∩ E_{V′}(Y) = E_{V ∩V′}(Y). On [C^{∞}(Y)]^{∞}, we define the right shift operator T by

(ϕ_{0}, …, ϕ_{m−2}, ϕ_{m−1}) ↦ (ϕ_{0}, …, ϕ_{m−2}).

On E^{δ}(Y), the right shift operator T acts component-wise, i.e., (T Φ)(p) = T (Φ(p)) for Φ ∈ E_{V} (Y) and all p ∈ V. Remark 2.10. To designate different shift operators with the same symbol T, once T^{ϱ} for ϱ ∈ ℝ for carriers of asymptotics, once T, T^{2}, etc. for vectors in E^{δ}(Y) should not confuse the reader. For Φ ∈ E^{δ}(Y), we define c-ord(Φ) = (n + 1)/2 − max{Re p; Φ(p) ≠ 0}. In particular, c-ord(0) = ∞. Note that c-ord(Φ) > δ if Φ ∈ E^{δ}(Y). For Φ_{i} ∈ E^{δ}(Y), α_{i} ∈ ℂ for i = 1, 2, … satisfying c-ord(Φ_{i}) → ∞ as i → ∞, the sum

Φ = ∑_{i=1}^{∞} α_{i}Φ_{i},  (2.11)

is defined in E^{δ}(Y) in an obvious fashion: Let Φ_{i} ∈ E_{V_{i}}(Y), where V_{i} ∈ C^{δ_{i}}, δ_{i} ≥ δ, and δ_{i} → ∞ as i → ∞. Then V = ⋃_{i} V_{i} ∈ C^{δ}, and Φ ∈ E_{V} (Y) is defined by Φ(p) = ∑_{i=1}^{∞} α_{i}Φ_{i}(p) for p ∈ V, where, for each p ∈ V, the sum on the right-hand side is finite.

Lemma 2.11. Let Φ_{i} ∈ E^{δ}(Y) for i = 1, 2, …, c-ord(Φ_{i}) → ∞ as i → ∞. Then (2.11) holds if and only if

c-ord(Φ − ∑_{i=1}^{N} α_{i}Φ_{i}) → ∞ as N → ∞.  (2.12)

Note that (2.12) already implies that c-ord(α_{i}Φ_{i}) → ∞ as i → ∞.

Definition 2.12. Let Φ_{i}, i = 1, 2, …, be a sequence in E^{δ}(Y) with the property that c-ord(Φ_{i}) → ∞ as i → ∞. Then this sequence is called linearly independent if, for all α_{i} ∈ ℂ,

∑_{i=1}^{∞} α_{i}Φ_{i} = 0

implies that α_{i} = 0 for all i. A linearly independent sequence Φ_{i} for i = 1, 2, … in J for a linear subspace J ⊆ E^{δ}(Y) is called a basis for J if every vector Φ ∈ J can be represented in the form (2.11) with certain (then uniquely determined) coefficients α_{i} ∈ ℂ.

Note that ∑_{i=1}^{∞} α_{i}Φ_{i} = 0 in E^{δ}(Y) if and only if c-ord(∑_{i=1}^{N} α_{i}Φ_{i}) → ∞ as N → ∞ according to Lemma 2.11. We also obtain:

Lemma 2.13. Let Φ_{i}, i = 1, 2, …, be a sequence in E^{δ}(Y) such that c-ord(Φ_{i}) → ∞ as i → ∞. Further, let {δ_{j}}_{j=1}^{∞} be a strictly increasing sequence such that δ_{j} > δ for all j and δ_{j} → ∞ as j → ∞. Assume that the Φ_{i} are numbered in such a way that c-ord(Φ_{i}) ≤ δ_{j} if and only if 1 ≤ i ≤ e_{j}. Then the sequence Φ_{i}, i = 1, 2, …, is linearly independent provided that, for each j = 1, 2, …,

Φ_{1}, …, Φ_{e_{j}} are linearly independent over the space E^{δ_{j}}(Y).

We now introduce the notion of characteristic basis:

Definition 2.14. Let J ⊆ E^{δ}(Y) be a linear subspace, T J ⊆ J, and Φ_{i} for i = 1, 2, … be a sequence in J. Then Φ_{i}, i = 1, 2, …, is called a characteristic basis of J if there are numbers m_{i} ∈ ℕ ∪ {∞} such that T^{m_{i}}Φ_{i} = 0 if m_{i} < ∞, while the sequence {T^{k}Φ_{i}; i = 1, 2, …, 0 ≤ k < m_{i}} forms a basis for J.

Remark 2.15. This notion generalizes a notion of Witt [18]: There, given a finite-dimensional linear space J and a nilpotent operator T: J → J, the sequence Φ_{1}, …, Φ_{e} in J has been called a characteristic basis, of characteristic (m_{1}, …, m_{e}), if

Φ_{1}, T Φ_{1}, …, T^{m_{1}−1}Φ_{1}, …, Φ_{e}, T Φ_{e}, …, T^{m_{e}−1}Φ_{e},

constitutes a Jordan basis of J. The numbers m_{1}, …, m_{e} appear as the sizes of Jordan blocks; dim J = m_{1} +⋯+m_{e}. The tuple (m_{1}, …, m_{e}) is also called the characteristic of J (with respect to T), e is called the length of its characteristic, and Φ_{1}, …, Φ_{e} is sometimes said to be a an (m_{1}, …, m_{e})-characteristic basis of J. The space {0} has empty characteristic of length e = 0.

The question of the existence of a characteristic basis obeying one more special property is taken up in Proposition 2.20. We also need following notion:

Definition 2.16. Φ ∈ E^{δ}(Y) is called a special vector if Φ ∈ E_{V_{p}}^{δ}(Y) for some p ∈ ℂ.

Thus, Φ ∈ E_{V} (Y) is a special vector if there is a p ∈ ℂ, Re p < (n+1)/2−δ such that Φ(p′) = 0 for all p′ ∈ V, p′ ∉ p−ℕ. Obviously, if Φ ≠ 0, then p is uniquely determined by Φ, by the additional requirement that Φ(p) ≠ 0. We denote this complex number p by γ(Φ). In particular, c-ord(Φ) = (n + 1)/2 − Re γ(Φ).

2.2.2 First properties of asymptotic types

In the sequel, we fix a splitting of coordinates U → [0, 1) × Y, x ↦ (t, y), near ∂X. Then we have the non-canonical isomorphism

C_{as}^{∞,δ}(X)/C_{O}^{∞}(X) ⟶^{≅} E^{δ}(Y),  (2.13)

assigning to each formal asymptotic expansion

u(x) ∼ ∑_{p∈V} ∑_{k+l=m_{p}−1} \frac{(−1)^{k}}{k!} t^{−p} log^{k} t ϕ_{l}^{(p)}(y) as t → +0  (2.14)

for some V ∈ C^{δ}, m_{p} ∈ ℕ, the vector Φ ∈ E_{V} (Y) given by

Φ(p) = \begin{cases}(ϕ_{0}^{(p)}, ϕ_{1}^{(p)}, …, ϕ_{m_{p}−1}^{(p)}) & if p ∈ V, \\ 0 & otherwise, \end{cases}

see also (2.30). “Non-canonical” in (2.13) means that the isomorphism depends explicitly on the chosen splitting of coordinates U → [0, 1)×Y, x ↦ (t, y), near ∂X. Coordinate invariance is discussed in Proposition 2.32. Note the shift from m_{p} to m_{p} − 1 that for notational convenience has appeared in formula (2.14) compared to formula (1.2).

Definition 2.17. An asymptotic type, P, for distributions as x → ∂X, of conormal order at least δ, is represented — in the given splitting of coordinates near ∂X — by a linear subspace J ⊂ E_{V} (Y) for some V ∈ C^{δ} such that the following three conditions are met:

(a) T J ⊆ J.

(b) dim J^{δ+j} < ∞ for all j ∈ ℕ, where J^{δ+j} = J/(J ∩ E^{δ+j}(Y)).

(c) There is a sequence {p_{j}}_{j=1}^{M} ⊂ ℂ, where M ∈ ℕ∪{∞}, Re p_{j} < (n+1)/2−δ, and Re p_{j} → −∞ as j → ∞ if M = ∞, such that V ⊆ ⋃_{j=1}^{M} V_{p_{j}} and

J = ⨁_{j=1}^{M} (J ∩ E_{V_{p_{j}}} (Y)) .  (2.15)

The empty asymptotic type, O, is represented by the trivial subspace {0} ⊂ E^{δ}(Y). The set of all asymptotic types of conormal order δ is denoted by \underline{As}^{δ}(Y).

Definition 2.18. Let u ∈ C_{as}^{∞,δ}(X) and P ∈ \underline{As}^{δ}(Y) be represented by J ⊂ E_{V} (Y). Then u is said to have asymptotics of type P if there is a vector Φ ∈ J such that

u(x) ∼ ∑_{p∈V} ∑_{k+l=m_{p}−1} \frac{(−1)^{k}}{k!} log^{k} t ϕ_{l}^{(p)}(y) as t → +0,  (2.16)

where Φ(p) = (ϕ_{0}^{(p)}, ϕ_{1}^{(p)}, …, ϕ_{m_{p}−1}^{(p)}) for p ∈ V. The space of all these u is denoted by C_{P}^{∞}(X).

Thus, by representation of an asymptotic type it is meant that P that — in the philosophy of asymptotic algebras, see Witt [20] — is the same as the linear subspace C_{P}^{∞}(X)/C_{O}^{∞}(X) ⊂ C_{as}^{∞,δ}(X)/C_{O}^{∞}(X), is mapped onto J by the isomorphism (2.13). For P ∈ \underline{As}^{δ} represented by J ⊂ E_{V} (Y), we introduce

δ_{P} = min{c-ord(Φ); Φ ∈ J},  (2.17)

Notice that δ_{P} > δ and δ_{P} = ∞ if and only if P = O. Obviously, \underline{As}^{δ}(Y) ⊆ \underline{As}^{δ′}(Y) if δ ≥ δ′. We likewise set

\underline{As}(Y) = ⋃_{δ∈ℝ} \underline{As}^{δ}(Y).

On asymptotic types P ∈ \underline{As}^{δ}(Y), we have the shift operation T^{ϱ} for ϱ ∈ ℝ, namely T^{ϱ}P is represented by the space

T^{ϱ}J = {Φ ∈ E_{T^{ϱ}V}^{ϱ+δ} (Y); Φ(p) = Φ̄(p − ϱ), p ∈ ℂ, for some Φ̄ ∈ J},

where J ⊂ E_{V} (Y) represents P. Furthermore, for J ⊂ E_{V} (Y) as in Definition 2.17,

J_{p} = {Φ(p); Φ ∈ J} ⊂ [C^{∞}(Y)]^{∞}

for p ∈ ℂ is the localization of J at p. Note that T J_{p} ⊆ J_{p} and dim J_{p} < ∞; thus, J_{p} is a local asymptotic type in the sense of Witt [18].

We now investigate common properties of linear subspaces J ⊂ E_{V} (Y) satisfying (a) to (c) of Definition 2.17. Let Π_{j}: J → J^{δ+j} be the canonical surjection. For j′ > j, there is a natural surjective map Π_{jj′}: J^{δ+j′} → J^{δ+j} such that Π_{jj′′} = Π_{jj′}Π_{j′j′′} for j′′ > j′ > j and

(J, Π_{j}) = proj lim_{j→∞} (J^{δ+j}, Π_{jj′}).  (2.18)

Note that T: J^{δ+j} → J^{δ+j} is nilpotent, where T denotes the map induced by T: J → J. Furthermore, for j′ > j, the diagram

\begin{matrix}J^{δ+j′} & ⟶^{Π_{jj′}} & J^{δ+j} \\ ^{T} ↓ & & ↓^{T} \\ J^{δ+j′} & ⟶^{Π_{jj′}} & J^{δ+j} \end{matrix}  (2.19)

commutes and the action of T on J is that one induced by (2.18), (2.19).

Proposition 2.19. Let J ⊂ E_{V} (Y) be a linear subspace for some V ∈ C^{δ}. Then there is a sequence Φ_{i} for i = 1, 2, … of special vectors with c-ord(Φ_{i}) → ∞ as i → ∞ such that the vectors T^{k}Φ_{i} for i = 1, 2, …, k = 0, 1, 2 … span J if and only if J fulfills conditions (a), (b), and (c).

In the situation just described, we write J = ⟨Φ_{1}, Φ_{2}, … ⟩.

Proof. Let J ⊂ E_{V} (Y) fulfill conditions (a) to (c). Due to (c) we may assume that V = V_{p} for some p ∈ ℂ. Suppose that the special vectors Φ_{1}, …, Φ_{e} ∈ J have already been chosen (where e = 0 is possible). Then we choose the vector Φ_{e+1} among the special vectors Φ ∈ J which do not belong to ⟨Φ_{1}, …, Φ_{e}⟩ such that Re γ(Φ_{e+1}) is minimal. We claim that J = ⟨Φ_{1}, Φ_{2}, … ⟩. In fact, c-ord(Φ_{i}) = (n + 1)/2 − Re γ(Φ_{i}) → ∞ as i → ∞ and, if Φ is a special vector in J, then Φ ∈ ⟨Φ_{1}, …, Φ_{e}⟩, where e is such that Re γ(Φ_{e}) ≤ Re γ(Φ), while Re γ(Φ_{e+1}) > Re γ(Φ). Otherwise, Φ_{e+1} would not have been chosen in the (e + 1)th step. The other direction is obvious.

For j ≥ 1, let (m_{1}^{j}, …, m_{e_{j}}^{j}) denote the characteristic of the space J^{δ+j}, see Remark 2.15

Proposition 2.20. Let J ⊂ E_{V} (Y) be a linear subspace and assume that the special vectors Φ_{i} for i = 1, 2, …, e, where e ∈ ℕ ∪ {∞}, as constructed in Proposition 2.19, form a characteristic basis of J. Then the following conditions are equivalent:

(a) For each j, Π_{j}Φ_{1}, …, Π_{j}Φ_{e_{j}}^{j} is an (m_{1}^{j}, …, m_{e_{j}}^{j})-characteristic basis of J^{δ+j};

(b) For each j, T^{m_{1}^{j}−1}Φ_{1}, …, T^{m_{e_{j}} −1}Φ_{e_{j}} are linearly independent over the space E^{δ+j}(Y), while T^{k}Φ_{i} ∈ E^{δ+j}(Y) if either 1 ≤ i ≤ e_{j}, k ≥ m_{i}^{j} or i > e_{j}. In particular, if (a), (b) are fulfilled, then, for any j′ > j, Π_{jj′}Φ_{1}^{j′}, …, Π_{jj′}Φ_{e_{j}}^{j′}

is a characteristic basis of J^{δ+j}, while Π_{jj′}Φ_{e_{j}+1}^{j′} = ⋯ = Π_{jj′}Φ_{e′_{j}}^{j′} = 0. Here,

Φ_{i}^{j′} = Π_{j′}Φ_{i} for 1 ≤ i ≤ e_{j′}.

Proof. This is a consequence of Lemma 2.13 and Witt [18, Lemma 3.8].

Notice that, for a linear subspace J ⊂ E_{V} (Y) satisfying conditions (a) to (c) of Definition 2.17, a characteristic basis possessing the equivalent properties of Proposition 2.20 need not exist. We provide an example:

Example 2.21. Let the space J = ⟨Φ_{1}, Φ_{2}⟩ ⊂ E_{V_{p}}(Y) for some p ∈ ℂ, Re p < (n+1)/2−δ, be spanned by two vectors Φ_{1}, Φ_{2} in the sense of Proposition 2.19. We further assume that Φ_{1}(p) = (ψ_{0}, ⋆), Φ_{1}(p − 1) = (ψ_{1}, ⋆, ⋆), Φ_{2}(p) = 0, and Φ_{2}(p−1) = (ψ_{1}, ⋆), where ψ_{0}, ψ_{1} ∈ C^{∞}(Y) are not identically zero and ⋆ stands for arbitrary entries, see Figure 2. Then, the asymptotic type represented by J is non-proper. In fact, assume that Re p ≥ (n + 1)/2 − δ + 1. Then

\overbrace{Φ_{1}}  \overbrace{Φ_{2}} p − 1  p  p − 1  p ψ_{1} ⋆ ⋆  ψ_{0} ⋆  ψ_{1} ⋆

Figure 2: Example of a non-proper asymptotic type

Π_{2}Φ_{1}, T Π_{2}Φ_{1} − Π_{2}Φ_{2} is a (3, 1)-characteristic basis of J^{δ+2}, and any other characteristic basis of J^{δ+2} is, up to a non-zero multiplicative constant, of the form

\{ \begin{matrix}Π_{2}Φ_{1} + α_{1}T Π_{2}Φ_{1} + α_{2}T^{2}Π_{2}Φ_{1} + α_{3}Π_{2}Φ_{2}, \\ β_{1}(T Π_{2}Φ_{1} − Π_{2}Φ_{2}) + β_{2}T^{2}Π_{2}Φ_{1}, \end{matrix}  (2.20)

where α_{1}, α_{2}, α_{3}, β_{1}, β_{2} ∈ ℂ and β_{1} ≠ 0. But then the conclusion in Proposition 2.20 is violated, since both vectors in (2.20) have non-zero image under the projection Π_{12}, while Π_{1}Φ_{1} is a (2)-characteristic basis of J^{δ+1}.

Definition 2.22. An asymptotic type P ∈ \underline{As}^{δ}(Y) represented by the linear subspace J ⊂ E_{V} (Y) is called proper if J admits a characteristic basis Φ_{1}, Φ_{2}, … satisfying the equivalent conditions in Proposition 2.20. The set of all proper asymptotic types is denoted by \underline{As}_{prop}^{δ}(Y) ⊊ \underline{As}^{δ}(Y).

For Φ ∈ E^{δ}(Y), p ∈ ℂ, and Φ(p) = (ϕ_{0}^{(p)}, ϕ_{1}^{(p)}, …, ϕ_{m_{p}−1}^{(p)}) we shall use, for any q ∈ ℂ, the notation

Φ(p)[z − q] = \frac{ϕ_{0}^{(p)}}{(z − q)^{m_{p}}} + \frac{ϕ_{1}^{(p)}}{(z − q)^{m_{p}−1}} + ⋯ + \frac{ϕ_{m_{p}−1}^{(p)}}{z − q} ∈ M_{q}(C^{∞}(Y)),

where M_{q}(C^{∞}(Y)) is the space of germs of meromorphic functions at z = q taking values in C^{∞}(Y). Analogously, A_{q}(C^{∞}(Y)) is the space of germs of holomorphic functions at z = p taking values in C^{∞}(Y).

Definition 2.23. For 𝔖^{μ} = {𝔰^{μ−j}(z); j ∈ ℕ} ∈ Symb_{M}^{μ}(Y), the linear space L_{𝔖^{μ}}^{δ} ⊆ C_{as}^{∞,δ}(X)/C_{O}^{∞}(X) is represented by the space of Φ ∈ E^{δ}(Y) for which there are functions ϕ̃^{(p)}(z) ∈ A_{p}(C^{∞}(Y)) for p ∈ ℂ, Re p < (n + 1)/2 − δ, such that

∑_{j=0}^{[(n+1)/2−δ+μ−Re q]^{−}} 𝔰^{μ−j}(z − μ + j)(Φ(q − μ + j)[z − q]

+ ϕ̃^{(q−μ+j)}(z − μ + j)) ∈ A_{q}(C^{∞}(Y)) (2.21) for all q ∈ ℂ, Re q < (n + 1)/2 − δ + μ. Here, [a]^{−} for a ∈ ℝ is the largest integer strictly less than a, i.e., [a]^{−} ∈ ℤ and [a]^{−} < a ≤ [a]^{−} + 1.

Remark 2.24. (a) If Φ ∈ E_{V} (Y) for V ∈ C^{δ}, then condition (2.21) is effective only if

q ∈ ⋃_{j=0}^{[(n+1)/2−δ+μ−Re q]^{−}} T^{μ−j}V.

(b) If Φ ∈ E^{δ}(Y) belongs to the representing space of L_{𝔖^{μ}}^{δ}, and if u ∈ C_{as}^{∞,δ}(X) possesses asymptotics given by the vector Φ according to (2.16), then there is a v ∈ C_{O}^{∞}(X) such that

∑_{j=0}^{∞} ω(c_{j}t)t^{−μ+j} op_{M}^{(n+1)/2−δ} (𝔰^{μ−j}(z)) ω̃(c_{j}t) (u + v) ∈ C_{O}^{∞}(X).

Here, the numbers c_{j} > 0 are chosen so that c_{j} → ∞ as j → ∞ sufficiently fast so that the infinite sum converges. For the notation op_{M}^{(n+1)/2−δ}(…) see (2.35), below.

Definition 2.25. For P ∈ \underline{As}^{δ}(Y) being represented by J ⊂ E_{V} (Y) and 𝔖^{μ} ∈ Symb_{M}^{μ}(Y), the push-forward Q^{δ−μ}(P; 𝔖^{μ}) of P under 𝔖^{μ} is the asymptotic type in \underline{As}^{δ−μ}(Y) represented by the linear subspace K ⊂ E_{T^{−μ}V} (Y) consisting of all vectors Ψ ∈ E_{T^{−μ}V} (Y) such that there is a Φ ∈ J and there are functions ϕ̃^{(p)}(z) ∈ A_{p}(C^{∞}(Y)) for p ∈ V such that

Ψ(q)[z − q] = ∑_{j=0}^{[(n+1)/2−δ+μ−Re q]^{−}}

[𝔰^{μ−j}(z − μ + j) (Φ(q − μ + j)[z − q] + ϕ̃^{(q−μ+j)}(z − μ + j))]_{q}^{∗} , (2.22)

holds for all q ∈ T^{μ}V, see (2.6).

Remark 2.26. For a holomorphic 𝔖^{μ} ∈ Symb_{M}^{μ}(Y), one needs not to refer to the holomorphic functions ϕ̃^{(p)}(z) ∈ A_{p}(C^{∞}(Y)) for p ∈ V in order to define the push-forward Q^{δ−μ}(P; 𝔖^{μ}) in (2.22). We then also write Q(P; 𝔖^{μ}) instead of Q^{δ−μ}(P; 𝔖^{μ}).

Extending the notion of push-forward from asymptotic types to arbitrary linear subspaces of C_{as}^{∞,δ}(X)/C_{O}^{∞}(X), the space L_{𝔖^{μ}}^{δ} ⊆ C_{as}^{∞,δ}(X)/C_{O}^{∞}(X) for 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) appears as the largest subspace of C_{as}^{∞,δ}(X)/C_{O}^{∞}(X) for which

Q^{δ−μ}(L_{𝔖^{μ}}^{δ}; 𝔖^{μ}) = Q^{δ−μ}(O; 𝔖^{μ}).  (2.23)

In this sense, it characterizes the amount of asymptotics of conormal order at least δ annihilated by 𝔖^{μ} ∈ Symb_{M}^{μ}(Y).

Definition 2.27. A partial ordering on \underline{As}^{δ}(Y) is defined by P ≼ P′ for P, P′ ∈ \underline{As}^{δ}(Y) if and only if J ⊆ J′, where J, J′ ⊂ E^{δ}(Y) are the representing spaces for P and P′, respectively.

Proposition 2.28. (a) The p.o. set (\underline{As}^{δ}(Y), ≼) is a lattice in which each non-empty subset S admits a meet, ⋀ S, represented by ⋂_{P ∈S} J_{P}, and each bounded subset T admits a join, ⋁ T , represented by ∑_{Q∈T} J_{Q}, where J_{P} and J_{Q} represent the asymptotic types P and Q, respectively. In particular, ⋀ \underline{As}^{δ}(Y) = O.

(b) For P ∈ \underline{As}^{δ}(Y), 𝔖^{μ} ∈ Symb_{M}^{μ}(Y), we have Q^{δ−μ}(P; 𝔖^{μ}) ∈ \underline{As}^{δ−μ}(Y).

Proof. (a) is immediate from the definition of asymptotic type and (b) can be checked directly on the level of (2.22).

Remark 2.29. Each element 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) induces a natural action C_{as}^{∞,δ}(X) → C_{as}^{∞,δ}(X)/C_{O}^{∞}(X). Its expression in the splitting of coordinates U → [0, 1) × Y, x ↦ (t, y), is given by (2.22). In the language of Witt [20], this means that the quadruple (⋃_{μ∈ℤ} Symb_{M}^{μ}(Y), C_{as}^{∞,δ}(X), C_{O}^{∞}(X), \underline{As}^{δ}(Y)) is an asymptotic algebra that is even reduced; thus providing justification for the above choice of the notion of asymptotic type.

Theorem 2.30. For a holomorphic 𝔖^{μ} ∈ Ell Symb_{M}^{μ}(Y), we have L_{𝔖^{μ}}^{δ} ∈ \underline{As}_{prop}^{δ}(Y).

Proof. Let 𝔖^{μ} = {𝔰^{μ−j}(z); j ∈ ℕ} ⊂ M_{O}^{μ}(Y). Assume that, for some p ∈ ℂ, Re p < (n + 1)/2 − δ, Φ_{0} ∈ L_{𝔰^{μ}(z)} at z = p, with the obvious meaning, for this see Witt [18]. (Notice that L_{𝔰^{μ}(z)} at z = p is contained in the space [C^{∞}(Y)]^{∞}.) We then successively calculate the sequence Φ_{0}, Φ_{1}, Φ_{2}, … from the relations, at z = p,

𝔰^{μ}(z − j)Φ_{j}[z − p] + 𝔰^{μ−1}(z − j + 1)Φ_{j−1}[z − p]

+ ⋯ + 𝔰^{μ−j}(z)Φ_{0}[z − p] ∈ A_{p}(C^{∞}(Y)), j = 0, 1, 2, …, (2.24)

see (2.22) and Remark 2.26. In each step, we find Φ_{j} ∈ [C^{∞}(Y)]^{∞} uniquely determined modulo L_{𝔰^{μ}(z)} at z = p − j such that (2.24) holds. We obtain the vector Φ ∈ E_{V_{p}}(Y) define by Φ(p − j) = Φ_{j} that belongs to the linear subspace J ⊂ E^{δ}(Y) representing L_{𝔖^{μ}}^{δ}. Conversely, each vector in J is a sum like in (2.11) of vectors Φ obtained in that way. Thus, upon choosing in each space L_{𝔰^{μ}(z)} at z = p a characteristic basis and then, for each characteristic basis vector Φ_{0} ∈ [C^{∞}(Y)]^{∞}, exactly one vector Φ ∈ E_{V_{p}}(Y) as just constructed, we obtain a characteristic basis of J in the sense of Definition 2.14 consisting completely of special vectors (since L_{𝔰^{μ}(z)} at z = p equals zero for all p ∈ ℂ, Re p < (n + 1)/2 − δ, but a set of p belonging to C^{δ}). In particular, J ⊂ E_{V} (Y) for some V ∈ C^{δ} and (a) to (c) of Definition 2.17 are satisfied. By its very construction, this characteristic basis fulfills condition (b) of Proposition 2.20. Therefore, the asymptotic type L_{𝔖^{μ}}^{δ} represented by J is proper.

In conclusion, we obtain:

Proposition 2.31. Let 𝔖^{μ} ∈ Ell Symb_{M}^{μ}(Y). Then:

(a) L_{𝔖^{μ}}^{δ} = Q^{δ}(O; (𝔖^{μ})^{−1}) and L_{(𝔖^{μ})^{−1}}^{δ−μ} = Q^{δ−μ}(O; 𝔖^{μ}).

(b) There is an order-preserving bijection

\{P ∈ \underline{As}^{δ}(Y); P ≽ L_{𝔖^{μ}}^{δ}\} → \{Q ∈ \underline{As}^{δ−μ}(Y); Q ≽ L_{(𝔖^{μ})^{−1}}^{δ−μ}\},  (2.25)
P ↦ Q^{δ−μ}(P; 𝔖^{μ}),

with the inverse given by Q ↦ Q^{δ}(Q; (𝔖^{μ})^{−1}).

Proof. Using Proposition 2.28 (b), the proof consists of a word-by-word repetition of the arguments given in the proof of Witt [18, Proposition 2.5].

In its consequence, Proposition 2.31 enables one to perform explicit calculations on asymptotic types. We conclude this section with the following basic observation:

Proposition 2.32. The notion of asymptotic type, as introduced above, is invariant under coordinates changes.

Proof. Let κ: X → X be a C^{∞} diffeomorphism and let κ_{∗}: C^{∞}(X^{∘}) → C^{∞}(X^{∘}) be the corresponding push-forward on the level of functions, i.e., (κ_{∗}u)(x) = u(κ^{−1}(x)) for u ∈ C^{∞}(X^{∘}), where κ^{−1} denotes the inverse C^{∞} diffeomorphism to κ. As is well-known, κ_{∗} restricts to κ_{∗}: C_{as}^{∞,δ}(X) → C_{as}^{∞,δ}(X) for any δ ∈ ℝ, see, e.g., Schulze [15, Theorem 1.2.1.11]. We have to prove that, for each P ∈ \underline{As}^{δ}(Y), there is a κ_{∗}P ∈ \underline{As}^{δ}(Y) so that the push-forward κ_{∗} restricts further to a linear isomorphism κ_{∗}: C_{P}^{∞}(X) → C_{κ_{∗}P}^{∞} (X), i.e., we have to show that there is a κ_{∗}P ∈ \underline{As}^{δ}(Y) so that κ_{∗}(C_{P}^{∞}(X)) = C_{κ_{∗}P}^{∞} (X). Using Proposition 2.19, we eventually have to prove that, for each u ∈ C_{as}^{∞,δ}(X) such that

u(x) ∼ ∑_{j=0}^{∞} ∑_{k+l=m_{j}−1} \frac{(−1)^{k}}{k!} log^{k} t ϕ_{l}^{(j)}(y) as t → +0,  (2.26)

where Φ ∈ E_{V_{p}}(Y) for a certain p ∈ ℂ, Re p < (n + 1)/2 − δ, and Φ(p − j) = (ϕ_{0}^{(j)}, ϕ_{1}^{(j)}, …, ϕ_{m_{j}−1}^{(j)}) for all j ∈ ℕ, see (2.16), the push-forward κ_{∗}u is again of the form (2.26), with some other κ_{∗}Φ ∈ E_{V_{p}}(Y) in place of Φ ∈ E_{V_{p}}(Y). But this is immediate from a direct computation.

2.2.3 Characteristics of proper asymptotic types

We introduce the notion of characteristic of a proper asymptotic type. This will be the main ingredient in the prove of Theorem 2.42. Let P ∈ \underline{As}_{prop}^{δ}(Y) be represented by J ⊂ E_{V} (Y) and let Φ_{1}, Φ_{2}, … by a characteristic basis of J according to Definition 2.22. As before, let (m_{1}^{j}, …, m_{e_{j}}^{j}) be the characteristic of the space J^{δ+j}. From Proposition 2.20, we conclude that e_{1} ≤ e_{2} ≤ … In the next lemma, we find a suitable “path through” the numbers m_{i}^{j} for j ≥ j_{i}, where j_{i} = min{j; e_{j} ≥ i}, i.e., an appropriate re-ordering of the tuples (m_{1}^{j}, …, m_{e_{j}}^{j}).

Lemma 2.33. The numbering within the tuples (m_{1}^{j}, …, m_{e_{j}}^{j}) can be chosen in such a way that, for each j ≥ 1, there is a characteristic (m_{1}^{j}, …, m_{e_{j}}^{j})-basis (Φ_{1}^{j}, …, Φ_{e_{j}}^{j}) of J^{δ+j} such that, for all j′ > j,

Π_{jj′}Φ_{i}^{j′} = \begin{cases}Φ_{i}^{j} & if 1 ≤ i ≤ e_{j}, \\ 0 & if e_{j} + 1 ≤ i ≤ e_{j′} \end{cases}

holds. Furthermore, the scheme

e_{1} rows \{ e_{2} − e_{1} rows \{ e_{3} − e_{2} rows \{ \{ \begin{matrix}m_{1}^{1} & m_{1}^{2} & m_{1}^{3} & m_{1}^{4} & … \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & … \\ m_{e_{1}}^{1} & m_{e_{1}}^{2} & m_{e_{1}}^{3} & m_{e_{1}}^{4} & … \\ m_{e_{1}+1}^{2} & m_{e_{1}+1}^{3} & m_{e_{1}+1}^{4} & … \\ . . . . . . . . . . . . . . . . . . . . . . . & … \\ m_{e_{2}}^{2} & m_{e_{2}}^{3} & m_{e_{2}}^{4} & … \\ m_{e_{2}+1}^{3} & m_{e_{2}+1}^{4} & … \\ . . . . . . . . . . . . . . & … \\ m_{e_{3}}^{3} & m_{e_{3}}^{4} & … \\ ⋮ & ... , \end{matrix}  (2.27)

where in the jth column the characteristic of the space J^{δ+j} appears, is uniquely determined up to permutation of the kth and the k′th row, where e_{j}+1 ≤ k, k′ ≤ e_{j+1} for some j (e_{0} = 0).

Proof. This is a reformulation of Proposition 2.20 in terms of the characteristics of the spaces J^{δ+j}. Notice that one can recover the characteristic basis Φ_{1}, Φ_{2}, … of J, that was initially given, from the property that Π_{j}Φ_{i} = Φ_{i}^{j} holds for all 1 ≤ i ≤ e_{j}, while Π_{j}Φ_{i} = 0 for i > e_{j}.

Performing the constructions of the foregoing lemma for each space J ∩E_{V_{p_{j}}} (Y) in (2.15) separately, one sees that the following notion is correctly defined:

Definition 2.34. Let P ∈ \underline{As}_{prop}^{δ}(Y) and J ⊂ E_{V} (Y) represent P. If Φ_{1}, Φ_{2}, … is a characteristic basis of J according to Definition 2.22 and if the tuples (m_{1}^{j}, …, m_{e_{j}}^{j}) are re-ordered according to Lemma 2.33, then the sequence

char P = {(γ(Φ_{i}) | m_{i}^{j_{i}}, m_{i}^{j_{i}+1}, m_{i}^{j_{i}+2}, …)}_{i=1}^{e}  (2.28)

is called the characteristic of P.

The characteristic char P of an asymptotic type P ∈ \underline{As}_{prop}^{δ}(Y) is unique up to permutation of the kth and the k′th entry, where e_{j} + 1 ≤ k, k′ ≤ e_{j+1} for some j. So far, it is an invariant associated with the representing space J; so it still depends on the splitting of coordinates. However, we have:

Proposition 2.35. The characteristic char P of an asymptotic type P ∈ \underline{As}_{prop}^{δ}(Y) is independent of the chosen splitting of coordinates U → [0, 1) × Y, x ↦ (t, y), near ∂X.

Proof. Follow the proof of Proposition 2.32 to get the assertion.

Now, let {(p_{i} | m_{i}^{j_{i}}, m_{i}^{j_{i}+1}, …)}_{i=1}^{e} ⊂ ℂ × ℕ^{ℕ} be any given sequence, where we additionally assume that Re p_{i} < (n + 1)/2 − δ for all i, Re p_{i} → −∞ as i → ∞ when e = ∞, the p_{i} are ordered so that Re p_{i} ≥ (n + 1)/2 − δ − j holds if and only if i ≤ e_{j} for a certain (then uniquely determined) sequence e_{1} ≤ e_{2} ≤ … satisfying e = sup_{j} e_{j}, and

1 ≤ m_{i}^{j_{i}} ≤ m_{i}^{j_{i}+1} ≤ m_{i}^{j_{i}+2} ≤ …,

where j_{i} = min{j; e_{j} ≥ i} as above.

Proposition 2.36. Let the characteristic {(p_{i} | m_{i}^{j_{i}}, m_{i}^{j_{i}+1}, …)}_{i=1}^{e} satisfy the properties just mentioned. If n = 0, then we assume, in addition, that p_{i} ≠ p_{i′} for i ≠ i′ and, for all i, k > 0,

m_{i}^{j_{i}+k} − m_{i}^{j_{i}+k−1} = a > 0 ⟺ p_{i′} = p_{i} − k for some i′ and m_{i′}^{j_{i′}} = a

(where j_{i′} = j_{i} + k). Then there exists a holomorphic 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) that is elliptic with respect to the weight δ ∈ ℝ such that L_{𝔖^{μ}}^{δ} ∈ \underline{As}_{prop}^{δ}(Y) has exactly this characteristic.

Proof. Multiplying 𝔖^{μ} by an elliptic element 𝔗^{−μ} = {𝔱^{−μ}(z), 0, 0, … } such that 𝔱^{−μ}(z) ∈ M_{O}^{−μ} and 𝔱^{−μ}(z)^{−1} ∈ M_{O}^{μ}, we can assume μ = 0. If n = 0, then we choose an elliptic 𝔰^{0}(z) ∈ M_{O}^{0} that has zeros precisely at z = p_{i} of order m_{i}^{j_{i}} for i = 1, 2, … according to Theorem 2.6. In case dim Y > 0, let {ϕ_{i}}_{i=1}^{e} be an orthonormal set in C^{∞}(Y) with respect to a fixed C^{∞}-density dμ on Y. Let Π_{i} for i = 1, …, e be the orthogonal projection in L^{2}(Y, dμ) onto the subspace spanned by ϕ_{i}. We then choose an elliptic 𝔰^{μ}(z) ∈ M_{O}^{μ}(Y) such that, for every p ∈ V_{p_{i}} and all i,

[𝔰^{μ}(z)]_{p}^{N_{p}} = (1 − ∑_{p_{i′} −k=p} Π_{i′}) + ∑_{p_{i′} −k=p}(z − p)^{m_{i′}^{j_{i′} +k}}Π_{i′}

where the sums are extended over all i′, k such that p_{i′} − k = p, for some N_{p} sufficiently large, while 𝔰^{μ}(q) ∈ L_{cl}^{μ}(Y) is invertible for all q ∈ ℂ \ V, again according to Theorem 2.6. In both cases, we set 𝔖^{μ} = {𝔰^{μ−j}(z)}_{j=0}^{∞} with 𝔰^{μ−j}(z) ≡ 0 for j > 0. Then 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) is elliptic with respect to the weight δ, and the proper asymptotic type L_{𝔖^{μ}}^{δ} has characteristic {(p_{i} | m_{i}^{j_{i}}, m_{i}^{j_{i}+1}, …)}_{i=1}^{e}.

2.2.4 More properties of asymptotic types

Here, we study further properties of asymptotic types. First, asymptotic types are composed of elementary building blocks:

Proposition 2.37. (a) An asymptotic type P ∈ \underline{As}^{δ}(Y) is join-irreducible, i.e., P ≠ O and P = P_{0} ∨ P_{1} for P_{0}, P_{1} ∈ \underline{As}^{δ}(Y) implies P = P_{0} or P = P_{1}, if and only if there is a Φ ∈ E^{δ}(Y), Φ ≠ 0, such that the representing space, J, for P, in the given splitting of coordinates near ∂X, has characteristic basis Φ, i.e., J = ⟨Φ⟩. In particular, every join-irreducible asymptotic type is proper.

(b) The join-irreducible asymptotic types are join-dense in \underline{As}^{δ}(Y).

Proof. (a) Let P ≠ O. Assume that, for some j ≥ 1, J^{δ+j} has characteristic of length larger 1. Then J^{δ+j} = K_{0} + K_{1} for certain linear subspaces K_{i} ⊊ J^{δ+j} satisfying T K_{i} ⊆ K_{i}, for i = 0, 1. Setting J_{i} = {Φ ∈ J; Π_{j}Φ ∈ K_{i}}, we get that J = J_{0} + J_{1}, J_{i} ⊊ J, and T J_{i} ⊆ J_{i} for i = 0, 1. Since this decomposition can be chosen compatible with (2.15), we obtain that a necessary condition for P to be join-irreducible is that each space J^{δ+j} for j ≥ 1 has characteristic of length at most 1, i.e., J = ⟨Φ⟩ for some Φ ≠ 0. Vice versa, if J = ⟨Φ⟩ for some Φ ≠ 0, then P is join-irreducible, since the subspace ⟨T^{k}Φ⟩ ⊆ J for k ∈ ℕ are the only subspaces of J that are invariant under the action of T.

(b) This follows directly from Proposition 2.19.

Note that, by the foregoing proposition, also the proper asymptotic types are join-dense in \underline{As}^{δ}(Y). We will utilize this fact in the definition of cone Sobolev spaces with asymptotics. In constructing asymptotic types P ∈ \underline{As}^{δ}(Y) obeying certain properties, one often encounters a situation in which P is successively constructed on strips {z ∈ ℂ; (n + 1)/2 − δ − β_{h} ≤ Re z < (n + 1)/2 − δ} of finite width, where the sequence {β_{h}}_{h=0}^{∞} ⊂ ℝ_{+} is strictly increasing and β_{h} → ∞ as h → ∞. We will meet an example in Section 3.3. To formulate the result, we need one more definition:

Definition 2.38. Let P, P′ ∈ \underline{As}^{δ}(Y) be represented by J ⊂ E_{V} (Y) and J′ ⊂ E_{V} (Y), respectively. Then, for ϑ ≥ 0, the asymptotic types P and P′ are said to be equal up to the conormal order δ + ϑ if Π_{ϑ}J = Π_{ϑ}J′, where Π_{ϑ}: J → J/(J ∩ E^{δ+ϑ}(Y)) is the canonical projection. Similarly, P and P′ are said to be equal up to the conormal order δ + ϑ − 0 if they are equal up to the conormal order δ + ϑ − ϵ, for any ϵ > 0. (Similarly for the order relation ≼ instead of equality.)

Proposition 2.39. Let \{P_{ι}\}_{ι∈I} ⊂ \underline{As}^{δ}(Y) be an increasing net of asymptotic types. Then the join ⋁_{ι∈I} P_{ι} exists if and only if, for each j ≥ 1, there is an ι_{j} ∈ I such that P_{ι} = P_{ι′} up to the conormal order δ + j for all ι, ι′ ≥ ι_{j}.

Proof. The condition is obviously sufficient. Conversely, suppose that the join ⋁_{ι∈I} P_{ι} exists. Let P_{ι} be represented by the subspace J_{ι} ⊂ E_{V_{ι}}(Y) for V_{ι} ∈ C^{δ}. Since the join ⋁_{ι∈I} P_{ι} exists, the carriers V_{ι} can be chosen in such way that ⋃_{ι∈I} V_{ι} ⊆ V for some V ∈ C^{δ}. Thus J_{ι} ⊂ E_{V} (Y) for all ι. Now, for each j ≥ 1, dim(∑_{ι∈I} J_{ι}^{δ+j}) < ∞, otherwise ⋁_{ι∈I} P_{ι} does not exist. But since the net {J_{ι}^{δ+j}}_{ι∈I} is increasing, this already implies that there is some ι_{j} ∈ I such that J_{ι}^{δ+j} = J_{ι′}^{δ+j} for ι, ι′ ≥ ι_{j}, i.e., P_{ι} = P_{ι′} up to the conormal order δ + j for ι, ι′ ≥ ι_{j}.

An equivalent condition is that the net \{P_{ι}\}_{ι∈I} ⊂ \underline{As}^{δ}(Y) of asymptotic types be bounded on each strip {z ∈ ℂ; (n + 1)/2 − δ − j ≤ Re z < (n + 1)/2 − δ} of finite width.

2.3 Pseudodifferential theory

Here, we establish an analogue of Witt [18, Theorem 1.2]. We need:

Proposition 2.40. Let P, P_{0} ∈ \underline{As}_{prop}^{δ}(Y), Q ∈ \underline{As}_{prop}^{δ−μ}(Y) for μ ∈ ℝ. Assume that P ∧ P_{0} = O. Then there is a holomorphic 𝔖^{μ} ∈ Ell Symb_{M}^{μ}(Y) that is elliptic with respect to the weight δ such that L_{𝔖^{μ}}^{δ} = P_{0} and Q(P; 𝔖^{μ}) = Q if and only if P and Q have the same characteristic shifted by μ, i.e., we have char P = char Q − μ (with the obvious meaning of char Q − μ).

Proof. It is readily seen that P ∈ \underline{As}_{prop}^{δ}(Y), Q ∈ \underline{As}_{prop}^{δ−μ}(Y) have the same characteristic shifted by μ if there is a holomorphic 𝔖^{μ} ∈ Ell Symb_{M}^{μ}(Y) such that Q(P; 𝔖^{μ}) = Q. Suppose that char P = char Q − μ. First, we deal with the case P_{0} = O. Let the asymptotic types P, Q be represented by J ⊂ E_{V} (Y) and K ⊂ E_{T^{μ}V} (Y), respectively. Let {Φ_{i}}_{i=1}^{e} and {Ψ_{i}}_{i=1}^{e} be characteristic bases of J and K corresponding to char P and char Q, respectively. We have to choose the sequence {𝔰^{μ−k}(z); k ∈ ℕ} ⊂ M_{O}^{μ}(Y). By Theorem 2.6, it suffices to construct the finite parts [𝔰^{μ−k}(z)]_{p′}^{N_{p′k}} for p′ ∈ V, k ∈ ℕ, and N_{p′k} sufficiently large appropriately. Thereby, we can assume that V = V_{p} for some p ∈ ℂ, Re p < (n + 1)/2 − δ. Let e_{1} ≤ e_{2} ≤ …, where e = sup_{j∈ℕ} e_{j}, be such that γ(Φ_{i}) = γ(Ψ_{i})−μ = p−j for e_{j−1} + 1 ≤ i ≤ e_{j} (and e_{0} = 0). Then the finite parts [𝔰^{μ−k}(z)]_{p−j}^{m^{j+k}} for all j, k must be chosen so that, for each j ∈ ℕ,

Φ_{i}(p − j)^{[𝔰^{μ}(z)]_{p−j}^{m^{j}}} + Φ_{i}(p − j + 1)^{[𝔰^{μ−1}(z)]_{p−j+1}^{m^{j}}}

+ ⋯ + Φ_{i}(p)^{[𝔰^{μ−j}(z)]_{p}^{m^{j}}} = Ψ_{i}(p + μ − j) (2.29)

for 1 ≤ i ≤ e_{j}, where m^{j} = sup_{1≤i≤e_{j}} m_{i}^{j}, and Φ_{i}(p − k) = 0 if e_{k} + 1 ≤ i ≤ e_{j}. Here, (m_{1}^{j}, …, m_{e_{j}}^{j}) is the characteristic of J^{δ+j} and, for Φ = (ϕ_{0}, …, ϕ_{m−1}), Ψ = (ψ_{0}, …, ψ_{m−1}) ∈ [C^{∞}(Y)]^{∞}, and 𝔰(z) ∈ M_{O}^{μ}(Y), the relation

Φ^{[𝔰(z)]_{p}^{m}} = Ψ

stands for the linear system

𝔰(p)ϕ_{0} = ψ_{0},
𝔰(p)ϕ_{1} + \frac{𝔰′(p)}{1!} ϕ_{0} = ψ_{1},
⋮
𝔰(p)ϕ_{m−1} + \frac{𝔰′(p)}{1!} ϕ_{m−2} + ⋯ + \frac{𝔰^{(m−1)}(p)}{(m − 1)!} ϕ_{0} = ψ_{m−1}.

System (2.29) can successively be solved for [𝔰^{μ}(z − k)]_{p−j+k}^{m^{j}} for j = 0, 1, 2, … and 0 ≤ k ≤ j. In fact, this can be done by choosing [𝔰^{μ−k}(z)]_{p−j+k}^{m^{j}} for k > 0 arbitrarily. In particular, we may choose 𝔰^{μ−k}(z) ≡ 0 for k > 0. The case P_{0} ≠ O can be reduced to the case P_{0} = O as in the proof of Witt [18, Lemma 3.16], since the three rules from Witt [18, Lemma 2.3] applied there continues to hold in the present situation.

Remark 2.41. (a) The proof of Proposition 2.40 shows that the holomorphic 𝔖^{μ} = {𝔰^{μ−j}; j ∈ ℕ} ∈ Ell Symb_{M}^{μ}(Y) satisfying L_{𝔖^{μ}}^{δ} = P_{0} and Q(P; 𝔖^{μ}) = Q can always be chosen so that 𝔰^{μ−j}(z) ≡ 0 for j > 0.

(b) Proposition 2.40 in connection with Theorem 2.30 also shows that \underline{As}_{prop}^{δ}(Y) consists precisely of those asymptotic types that are of the form L_{𝔖^{μ}}^{δ} for some holomorphic 𝔖^{μ} ∈ Ell Symb_{M}^{μ}(Y) that is elliptic with respect to the weight δ. (Choose P = Q = O in Proposition 2.40.) Now, we reach the final aim of this section:

Theorem 2.42. Let P ∈ \underline{As}_{prop}^{δ}(Y) and Q ∈ \underline{As}_{prop}^{δ−μ}(Y). Then there exists a 𝔖^{μ} ∈ Symb_{M}^{μ}(Y) that is elliptic with respect to the weight δ such that L_{𝔖^{μ}}^{δ} = P and L_{(𝔖^{μ})^{−1}}^{δ−μ} = Q always when dim Y > 0 and if and only if P ∧ T^{−μ}Q = O when dim Y = 0.

Proof. The condition P ∧ T^{−μ}Q = O is obviously necessary if dim Y = 0. In the general case, choose P_{1} ∈ \underline{As}_{prop}^{δ}(Y), Q_{1} ∈ \underline{As}_{prop}^{δ−μ}(Y) having the same characteristics as P and Q, respectively, such that P_{1} ∧ T^{−μ}Q_{1} = O. As in the proof of Witt [18, Theorem 1.2], it then suffices to construct holomorphic 𝔖^{0} ∈ Ell Symb_{M}^{μ}(Y), 𝔗^{0} ∈ Ell Symb_{M}^{−μ}(Y) that are elliptic with respect to the weight δ such that

L_{𝔖^{0}}^{δ} = P_{1}, Q^{δ}(Q_{1}; 𝔖^{0}) = Q, L_{𝔗^{0}}^{δ} = Q_{1}, Q^{δ}(P_{1}; 𝔗^{0}) = P.

This is achieved by using Proposition 2.40.

2.4 Function spaces with asymptotics

The definition of cone Sobolev spaces with asymptotics is based on the Mellin transformation. See Schulze [15, Sections 1.2, 2.1] for this idea and also Remark 2.45. For more details on the Mellin transformation, see Jeanquartier [5].

2.4.1 Weighted cone Sobolev spaces

Let Mu(z) = ũ(z) = ∫_{0}^{∞} t^{z−1}u(t) dt, z ∈ ℂ, be the Mellin transformation, first defined for u ∈ C_{0}^{∞}(ℝ_{+}) and then extended to larger distribution classes. In particular, u will be allowed to be vector-valued. Recall the following properties of M:

M_{t→z}{(−t∂_{t} − p)u}(z) = (z − p)ũ(z),
M_{t→z}{t^{−p}u}(z) = ũ(z − p),  p ∈ ℂ,

whenever both sides are defined, M: L^{2}(ℝ_{+}) → L^{2}(Γ_{1/2}; (2πi)^{−1}dz) is an isometry, and

M_{t→z} \{\frac{(−1)^{k}}{k!} t^{−p} log^{k} t χ_{(0,1)}(t)\} (z) = \frac{1}{(z − p)^{k+1}} ,  (2.30)

where χ_{(0,1)} is the characteristic function of the interval (0, 1). We infer that

h(z) = M_{t→z} {(−1)^{k}ω(t)t^{−p} log^{k} t/k!} (z) ∈ M_{as}^{−∞} is a meromorphic function of z having a pole precisely at z = p, and the principal part of the Laurent expansion around this pole is given by the right-hand side of (2.30), i.e., [h(z)]_{p}^{∗} = (z − p)^{−(k+1)}. Here, ω(t) is a cut-off function near t = 0. For s, δ ∈ ℝ, let H^{s,δ}(X) denote the space of u ∈ H_{loc}^{s}(X^{∘}) such that M_{t→z}{ωu}(z) ∈ L_{loc}^{2}(Γ_{(n+1)/2−δ}; H^{s}(Y)) and the expression

∥u∥_{H^{s,δ}(X)} = \{\frac{1}{2πi} ∫_{Γ_{(n+1)/2−δ}} ∥R^{s}(z)M_{t→z}\{ωu\}(z)∥_{L^{2}(Y)}^{2}\}^{1/2}  (2.31)

is finite. Here, R^{s}(z) ∈ L_{cl}^{s}(Y; Γ_{(n+1)/2−δ}) is an order-reducing family, i.e., R^{s}(z) is parameter-dependent elliptic and R^{s}(z): H^{r}(Y) → H^{r−s}(Y) is an isomorphism for some r ∈ ℝ (and then for all r ∈ ℝ) and all z ∈ Γ_{(n+1)/2−δ}. For instance, if f(z) ∈ M_{O}^{s}(Y) is elliptic and the line Γ_{(n+1)/2−δ} is free of poles of f(z)^{−1}, then f(z) is such an order-reduction. We will employ this observation in the next section when defining cone Sobolev spaces with asymptotics.

2.4.2 Cone Sobolev spaces with asymptotics

Let s, δ ∈ ℝ, P ∈ \underline{As}_{prop}^{δ}(Y). By Theorem 2.42, there is an elliptic Mellin symbol h_{P}^{s} (z) ∈ M_{O}^{s}(Y) such that the line Γ_{(n+1)/2−δ} is free of poles of h_{P}^{s} (z)^{−1} and L_{𝔖^{s}}^{δ} = P for 𝔖^{s} = {h_{P}^{s} (z), 0, 0, … } ∈ Symb_{M}^{s}(Y).

Definition 2.43. Let s, δ ∈ ℝ, ϑ ≥ 0, and P ∈ \underline{As}^{δ}(Y).

(a) For P ∈ \underline{As}_{prop}^{δ}(Y), the space H_{P,ϑ}^{s,δ}(X) consists of all functions u ∈ H^{s,δ}(X) such that M_{t→z}{ωu}(z), which is a priori holomorphic in {z ∈ ℂ; Re z > (n + 1)/2 − δ} taking values in H^{s}(Y), possesses a meromorphic continuation to the half-space {z ∈ ℂ; Re z > (n + 1)/2 − δ − ϑ}, moreover,

h_{P}^{s} (z)M_{t→z}{ωu}(z) ∈ A({z ∈ ℂ; Re z > (n + 1)/2 − δ − ϑ}; L^{2}(Y))

and the expression

sup_{δ<δ′<δ+ϑ} \{\frac{1}{2πi} ∫_{Γ_{(n+1)/2−δ′}} ∥h_{P}^{s} (z)M_{t→z}\{ωu\}(z)∥_{L^{2}(Y)}^{2} dz\}^{1/2}  (2.32)

is finite.

(b) For a general P ∈ \underline{As}^{δ}(Y), represented as the join P = ⋁_{ι∈I} P_{ι} for a bounded family \{P_{ι}\}_{ι∈I} ⊂ \underline{As}_{prop}^{δ}(Y), we define H_{P,ϑ}^{s,δ}(X) = ∑_{ι∈I} H_{P_{ι},ϑ}^{s,δ}(X).

It is readily seen that Definition 2.43 (a) is independent of the choice of the Mellin symbol h_{P}^{s} (z). Moreover, under the condition that (2.32) is finite the limit

h_{P}^{s} (z)M_{t→z}{ωu}(z)|_{z=(n+1)/2−δ′+iτ} → w(τ) as δ′ → δ + ϑ − 0

exists in L^{2}(ℝ_{τ}; L^{2}(Y)). Thus, H_{P,ϑ}^{s,δ}(X) is a Hilbert space with the norm

∥u∥_{H_{P,ϑ}^{s,δ}(X)} = {∥w∥_{L^{2}(ℝ_{τ};L^{2}(Y))}^{2} + ∥u∥_{H^{s,δ}(X)}^{2}}^{1/2} .  (2.33)

Definition 2.43 (b) is justified by Proposition 2.37 (b), since we obviously have H_{P,ϑ}^{s,δ}(X) = H^{s,δ+ϑ}(X) for P ∈ \underline{As}_{prop}^{δ}(Y) and δ_{P} > δ+ϑ. Again, this definition is seen to be independent of the choice of the representing family {P_{ι}}_{ι∈I} ⊂ \underline{As}_{prop}^{δ}(Y), and it also yields a Hilbert space structure for H_{P,ϑ}^{s,δ}(X).

Proposition 2.44. Let s, δ ∈ ℝ, ϑ ≥ 0, and P ∈ \underline{As}_{prop}^{δ}(Y). Further, let 𝔖^{s} = {𝔰^{s−j}(z) j = 0, 1, 2, … } ∈ Symb_{M}^{s}(Y) be elliptic with respect to the weight δ and L_{𝔖^{s}}^{δ} = P, L_{(𝔖^{s})^{−1}}^{δ−s} = O. (Condition L_{(𝔖^{s})^{−1}}^{δ−s} = O means that the Mellin symbols 𝔰^{s−j}(z) are holomorphic when Re z > (n + 1)/2 − δ.) Then a function u ∈ H^{s,δ}(X) belongs to the space H_{P,ϑ}^{s,δ}(X) if and only if M_{t→z}{ωu}(z) possesses a meromorphic continuation to the half-space {z ∈ ℂ; Re z > (n+1)/2−δ−ϑ},

∑_{j=0}^{M} 𝔰^{s−j}(z − s + j)M_{t→z}{ωu}(z − s + j)

∈ A({z ∈ ℂ; Re z > (n + 1)/2 − δ + s − ϑ}; L^{2}(Y)),

and the expression

sup_{δ<δ′<δ+ϑ}\{\frac{1}{2πi} ∫_{Γ_{(n+1)/2−δ′+s}}

∥∑_{j=0}^{M} 𝔰^{s−j}(z − s + j)M_{t→z}{ωu}(z − s + j)∥_{L^{2}(Y)}^{2} dz}^{1/2}

is finite. Here, M is any integer larger than ϑ.

Proof. This is an application (of an adapted version) of Witt [18, Proposition 2.6]. Note that 𝔰^{s−j}(z − s + j)M_{t→z}{ωu}(z − s + j) ∈ A({z ∈ ℂ; Re z > (n + 1)/2 − δ + s − j}; L^{2}(Y)) so that the condition is actually independent of the choice of the integer M > ϑ.

For s, δ ∈ ℝ, ϑ > 0, and P ∈ \underline{As}^{δ}(Y), we will also employ the spaces

H_{P,ϑ−0}^{s,δ}(X) = ⋂_{ϵ>0} H_{P,ϑ−ϵ}^{s,δ}(X).  (2.34)

These space H_{P,ϑ−0}^{s,δ}(X) are Fréchet-Hilbert spaces, i.e., Fréchet spaces whose topology is given by a countable family of Hilbert semi-norms. We will also use notations like

H_{P,ϑ}^{∞,δ}(X) = ⋂_{s∈ℝ} H_{P,ϑ}^{s,δ}(X),  H_{P,ϑ}^{−∞,δ}(X) = ⋃_{s∈ℝ} H_{P,ϑ}^{s,δ}(X),
H_{P,ϑ+0}^{s,δ}(X) = ⋃_{ϵ>0} H_{P,ϑ+ϵ}^{s,δ}(X),  etc.

Remark 2.45. In case P is a strongly discrete asymptotic type, the spaces H_{P,ϑ−0}^{s,δ}(X) are the function spaces introduced by Schulze [15, Section 2.1.1]. There, the notation H_{P}^{s,δ}(X)_{Δ} with the half-open interval Δ = (−ϑ, 0] has been used. The definition of the function spaces H_{P}^{s,δ}(X)_{Δ} refers to fixed splitting of coordinates near ∂X and is, in general, not coordinate invariant.

2.4.3 Functional-analytic properties

We list some properties of the function spaces H_{P,ϑ}^{s,δ}(X):

Proposition 2.46. Let s, s′, δ, δ′ ∈ ℝ, ϑ ≥ 0, P ∈ \underline{As}^{δ}(Y), P′ ∈ \underline{As}^{δ′}(Y), and \{P_{ι}\}_{ι∈I} ⊂ \underline{As}^{δ}(Y) be a family of asymptotic types. Then:

(a) H_{P,0}^{s,δ}(X) = H^{s,δ}(X).

(b) H_{P,ϑ}^{s,δ}(X) = H_{P,ϑ+a}^{s,δ−a}(X) for any a > 0.

(c) H_{O,ϑ}^{s,δ}(X) = H^{s,δ+ϑ}(X).

(d) We have

H_{P,ϑ}^{s,δ}(X) = H_{O,ϑ}^{s,δ}(X)

⊕ \{ω(t) ∑_{\begin{matrix}p∈V, \\ Re p>(n+1)/2−δ−ϑ \end{matrix}} ∑_{k+l=m_{p}−1} \frac{(−1)^{k}}{k!} t^{−p} log^{k} t ϕ_{l}^{(p)}(y);

Φ(p) = (ϕ_{0}^{(p)}, …, ϕ_{m_{p}−1}^{(p)}) for some Φ ∈ J},

where J ⊂ E_{V} (Y) is the linear subspace representing the asymptotic type P, provided that Re p ≠ (n + 1)/2 − δ − ϑ holds for all p ∈ V.

(e) We have H_{P,ϑ}^{s,δ}(X) ⊆ H_{P′,ϑ′}^{s′,δ′}(X) if and only if s ≥ s′, δ + ϑ ≥ δ′ + ϑ′, and P ≼ P′ up to the conormal order δ′ + ϑ′.

(f) H_{⋀_{ι∈I} P_{ι},ϑ}^{s,δ}(X) = ⋂_{ι∈I} H_{P_{ι},ϑ}^{s,δ}(X) if the family {P_{ι}}_{ι∈I} is non-empty.

(g) H_{⋁_{ι∈I} P_{ι},ϑ}^{s,δ}(X) = ∑_{ι∈I} H_{P_{ι},ϑ}^{s,δ}(X) if the family {P_{ι}}_{ι∈I} is bounded (where the sum sign stands for the non-direct sum of Hilbert spaces);

(h) C_{P}^{∞}(X) = ⋂_{s∈ℝ, ϑ≥0} H_{P,ϑ}^{s,δ}(X).

(i) C_{P}^{∞}(X) is dense in H_{P,ϑ}^{s,δ}(X).

Proof. The proofs of (a) to (i) are straightforward.

From (e) we get, in particular, H_{P,ϑ}^{s,δ}(X) = H_{P′,ϑ′}^{s′,δ′}(X) if and only if s = s′, δ + ϑ = δ′ + ϑ′, and P = P′ up to the conormal order δ + ϑ. (b) and also (c), in view of (a), are special cases.

Proposition 2.47. For δ ∈ ℝ, P ∈ \underline{As}^{δ}(Y), and any a ∈ ℝ, the family {H_{P,s−a}^{s,δ}(X); s ≥ a} of Hilbert spaces forms an interpolation scale with respect to the complex interpolation method.

Proof. This is immediate from the definition.

Proposition 2.48. The spaces H_{P,ϑ}^{s,δ}(X) are invariant under coordinate changes, where this has to be understood in the sense of Proposition 2.32.

Proof. Basically, this follows from the invariance of the spaces C_{P}^{∞}(X) under coordinate changes, where the latter is just a reformulation of the fact that the asymptotic types in \underline{As}^{δ}(Y) are coordinate invariant.

2.4.4 Mapping properties and elliptic regularity

We finally take the step from the algebra of complete conormal symbols to elliptic Fuchsian differential operators and their parametrices. These parametrices are cone pseudodifferential operators, where for the latter we refer to Schulze [16, Chapter 2]. While for general cone pseudodifferential operators, there might be a difference between the conormal asymptotics produced on the level of complete conormal symbols and operators, respectively — due to the appearance of so-called singular Green operators — for Fuchsian differential operators this does not happen. In cone pseudodifferential calculus, one encounters operators of the form ω(t)t^{−μ} op_{M}^{(n+1)/2−δ}(h) ω̃(t), where h(t, z) ∈ C^{∞}(\overline{ℝ}_{+}; M_{as}^{μ}(Y)). Here,

op_{M}^{(n+1)/2−δ}(h(t, z))u = \frac{1}{2πi} ∫_{Γ_{(n+1)/2−δ}} t^{−z}h(t, z)ũ(z) dz  (2.35)

is a pseudodifferential operator, whose definition is based on the Mellin transformation instead of the Fourier transformation. The mapping properties of these operators in the spaces H_{P,ϑ}^{s,δ}(X) are as follows:

Proposition 2.49. Let h(t, z) ∈ C^{∞}(\overline{ℝ}_{+}; M_{as}^{μ}(Y)) and assume that the line Γ_{(n+1)/2−δ} is free of poles of ∂^{j}h(0, z)/∂t^{j} for all j = 0, 1, 2, … Then, for all P ∈ \underline{As}^{δ}(Y), s ∈ ℝ, ϑ ≥ 0,

ω(t)t^{−μ} op_{M}^{(n+1)/2−δ}(h) ω̃(t): H_{P,ϑ}^{s,δ}(X) → H_{Q,ϑ}^{s−μ,δ−μ}(X),

where ω(t), ω̃(t) are cut-off functions, 𝔖^{μ} = \{\frac{1}{j!} \frac{∂^{j}h}{∂t^{j}} (0, z); j = 0, 1, 2, … \} ∈

Symb_{M}^{μ}(Y), and Q = Q^{δ−μ}(P, 𝔖^{μ}) ∈ \underline{As}^{δ−μ}(Y).

Proof. The previous definitions are made to let this result hold.

Notation. Proposition 2.49 implies that, given a cone pseudodifferential operator A in Schulze’s cone calculus C^{μ}(X, (δ, δ − μ, (−∞, 0])), see Schulze [16, Chapter 2] again, for each P ∈ \underline{As}^{δ}(Y), there is a Q ∈ \underline{As}^{δ−μ}(Y) such that, for all s ∈ ℝ, ϑ ≥ 0,

A: H_{P,ϑ}^{s,δ}(X) → H_{Q,ϑ}^{s−μ,δ−μ}(X).  (2.36)

Given P ∈ \underline{As}^{δ}(Y), the minimal such asymptotic type Q ∈ \underline{As}^{δ−μ}(Y), that exists by virtue of Proposition 2.28 (a) and Proposition 2.46 (f), is denoted by Q^{δ−μ}(P; A). If A is elliptic, given Q ∈ \underline{As}^{δ−μ}(Y), the minimal asymptotic type P ∈ \underline{As}^{δ}(Y) such that, for all s ∈ ℝ, ϑ ≥ 0, u ∈ H^{−∞,δ}(X), Au ∈ H_{Q,ϑ}^{s−μ,δ−μ}(X) implies u ∈ H_{P,ϑ}^{s,δ}(X) is denoted by P^{δ}(Q; A). We shall employ this push-forward notation also if more than one operator A is involved, i.e., Q^{δ−μ}(P; A_{1}, …, A_{m}) denotes the minimal asymptotic type Q for which A_{j}: H_{P,ϑ}^{s,δ}(X) → H_{Q,ϑ}^{s−μ,δ−μ}(X) for 1 ≤ j ≤ m.

Theorem 2.50. For A ∈ Diff_{Fuchs}^{μ}(X), P ∈ \underline{As}^{δ}(Y), Q ∈ \underline{As}^{δ−μ}(Y), we have Q^{δ−μ}(P; A) = Q^{δ−μ}(P; 𝔖^{μ}), where 𝔖^{μ} = {σ_{M}^{μ−j}(A)(z); j = 0, 1, … } ∈ Symb_{M}^{μ}(Y), as well as, in case A is elliptic, P^{δ}(Q; A) = Q^{δ}(Q; (𝔖^{μ})^{−1}).

Proof. In fact, Q^{δ−μ}(P; A) = Q^{δ−μ}(P; 𝔖^{μ}) follows from Proposition 2.49. Furthermore, it is known that formal asymptotic solutions u ∈ C_{as}^{∞}(X) to the equation Au = f for f ∈ C_{R}^{∞}(X) and any R ∈ \underline{As}^{δ−μ}(Y) can be constructed, see, e.g. Melrose [13, Lemma 5.13]. More precisely, it can be shown that there is a right parametrix B to A, B: H^{s−μ,δ−μ}(X) → H^{s,δ}(X) for all s ∈ ℝ, such that

AB = I + R, R: H^{−∞,δ−μ}(X) → C_{O}^{∞}(X),

i.e., R is smoothing over X^{∘} and flattening to infinite order near ∂X. In fact, B ∈ C^{−μ}(X, (δ − μ, δ, (−∞, 0])) and, in particular, B ∈ L_{cl}^{−μ}(X^{∘}). Now let BA = I + R_{0}. Obviously, R_{0} is smoothing over X^{∘} such that R_{0} : H^{s,δ}(X) → H^{∞,δ−μ}(X) for any s ∈ ℝ. Furthermore, A(I + R_{0}) = ABA = (I + R)A so that

AR_{0} = RA.

We conclude that R_{0} : H^{s,δ}(X) → C_{P_{0}}^{∞}(X), where P_{0} = Q^{δ}(O; (𝔖^{μ})^{−1}). Hence, for u ∈ H^{−∞,δ}(X), Au = f ∈ H_{Q,ϑ}^{s−μ,δ−μ}(X), we get

u = Bf − R_{0}u ∈ H_{P,ϑ}^{s,δ}(X),

where P = Q^{δ}(Q; (𝔖^{μ})^{−1}). Thus P^{δ}(Q; A) = Q^{δ}(Q; (𝔖^{μ})^{−1}) as claimed. See also Witt [20, Remark after Proposition 5.5].

Notation. For A ∈ Diff_{Fuchs}^{μ}(X), Q^{δ−μ}(P; A) is even independent of δ ∈ ℝ in view of the holomorphy of the conormal symbols σ_{M}^{μ−j}(A)(z) for j = 0, 1, 2, … In this case, we simply write Q(P; A) = Q^{δ−μ}(P; A).

Proposition 2.51. Let A ∈ Diff_{Fuchs}^{μ}(X) be elliptic. Then there is an orderpreserving bijection

\{P ∈ \underline{As}^{δ}(Y); P ≽ L_{𝔖^{μ}}^{δ}\} → \underline{As}^{δ−μ}(Y), P ↦ Q(P; A),  (2.37)

with its inverse given by Q ↦ P^{δ}(Q; A). In particular, L_{𝔖^{μ}}^{δ} is mapped to the empty asymptotic type, O.

Proof. This is implied by Proposition 2.31 and Theorem 2.50. Note that L_{(𝔖^{μ})^{−1}}^{δ−μ} = O, since the σ_{M}^{μ−j}(A)(z) for j = 0, 1, 2, … are holomorphic.

Eventually, we have the following locality principle:

Proposition 2.52. Let A ∈ Diff_{Fuchs}^{μ}(X) be elliptic, Q_{0}, Q_{1} ∈ \underline{As}^{δ−μ}(Y), and P_{0} = P^{δ}(Q_{0}; A), P_{1} = P^{δ}(Q_{1}; A). Then, for any ϑ > 0, P_{0} = P_{1} up to the conormal order δ + ϑ if Q_{0} = Q_{1} up to the conormal order δ − μ + ϑ.

Proof. This follows from P_{0} = Q^{δ}(Q_{0}; (𝔖^{−μ})^{−1}), P_{1} = Q^{δ}(Q_{1}; (𝔖^{−μ})^{−1}), where 𝔖^{μ} = {σ_{M}^{μ−j}(A)(z); j ∈ ℕ} ∈ Ell Symb_{M}^{μ}(Y).

Remark 2.53. Combined with Theorem 2.30, Theorem 2.50 shows that each solution u ∈ C_{as}^{∞,δ}(X) to the equation Au = f ∈ C_{O}^{∞}(X), where A ∈ Diff_{Fuchs}^{μ}(X) is elliptic, can be written over finite weight intervals as a finite sum of functions of the form (2.16) modulo the corresponding flat class, where the Φ are taken from a characteristic basis of the linear subspace of E^{δ}(Y) representing P^{δ}(O; A). If Φ(p) = (ϕ_{0}, …, ϕ_{m−1}) for such a vector Φ, where p = γ(Φ), then we say that A admits an asymptotic series starting with the term t^{−p} log^{m−1} t ϕ_{0}. Since this is then the most singular term (when γ(Φ) is highest possible), if it coefficient can be shown to vanish, then the whole series must vanish, up to the next appearance of a starting term for another asymptotic series.

3 Applications to semilinear equations

In this section, Theorem 1.1 is proved. To this end, multiplicatively closable and multiplicatively closed asymptotic types are investigated in Section 3.1. This allows the derivation of results concerning the action of nonlinear superposition operators on cone Sobolev spaces with asymptotics. In Section 3.2, the general scheme for establishing results of the type of Theorem 1.1 is established. This scheme is specified to multiplicatively closable asymptotic types in Section 3.3, then completing the proof of Theorem 1.1.

3.1 Multiplicatively closed asymptotic types

Here, we investigate multiplicative properties of asymptotic types and the behavior of cone Sobolev spaces H_{P,ϑ}^{s,δ}(X) under the action of nonlinear superposition.

Notation. In connection with pointwise multiplication, it is useful to employ the following notation:

H_{P,ϑ}^{s}(X) = \begin{cases}H_{P,δ_{P} −δ+ϑ}^{s,δ}(X) & if ϑ ≥ 0, \\ H^{s,δ_{P} +ϑ}(X) & otherwise, \end{cases}

where P ∈ \underline{As}^{δ}(Y), P ≠ O, and δ < δ_{P} in the first line. (Proposition 2.46 (b) shows that this definition is independent of the choice of δ.) Thus, starting from δ_{P}, the conormal order is improved by ϑ upon allowing asymptotics of type P. Similarly for H_{P,ϑ−0}^{s}(X). Furthermore, we write {ϑ} if we mean either ϑ or ϑ − 0. For instance, {ϑ} ≥ 0 means ϑ ≥ 0 if {ϑ} = ϑ and ϑ > 0 if {ϑ} = ϑ − 0.

3.1.1 Multiplication of asymptotic types

The result admitting nonlinear superposition for function spaces with asymptotics is stated first:

Lemma 3.1. Given P ∈ \underline{As}(Y), Q ∈ \underline{As}(Y), there is a minimal asymptotic type, P ∘ Q ∈ \underline{As}(Y), such that

C_{P}^{∞}(X) × C_{Q}^{∞}(X) → C_{P ∘Q}^{∞}(X), (u, v) ↦ uv.  (3.1)

Proof. Suppose that the asymptotic types P, Q are represented by subspaces J ⊂ E_{V} (Y) and K ⊂ E_{W} (Y), respectively, for suitable V, W ∈ C. Then the asymptotic type P ∘ Q is carried by the set V + W, and it is represented by the linear subspace of E_{V +W} (Y) consisting of all Θ ∈ E_{V +W} (Y) for which there are Φ ∈ J, Ψ ∈ K such that Θ(r) = ∑\begin{matrix}p+q=r, \\ p∈V, q∈W \end{matrix} Φ(p)×Ψ(q) holds for all r ∈ V +W.

Here,

Φ × Ψ = (\binom{m + n}{m}ϕ_{0}ψ_{0}, \binom{m + n − 1}{m}ϕ_{0}ψ_{1} + \binom{m + n − 1}{m − 1}ϕ_{1}ψ_{0},

\binom{m + n − 2}{m}ϕ_{0}ψ_{2} + \binom{m + n − 2}{m − 1}ϕ_{1}ψ_{1} + \binom{m + n − 2}{m − 2}ϕ_{0}ψ_{2},

…, \binom{1}{1}ϕ_{m−1}ψ_{n} + \binom{1}{0}ϕ_{m}ψ_{n−1}, \binom{0}{0}ϕ_{m}ψ_{n})

for Φ = (ϕ_{0}, ϕ_{1}, …, ϕ_{m}), Ψ = (ψ_{0}, ψ_{1}, …, ψ_{n}) ∈ [C^{∞}(Y)]^{∞}. For this, see (2.16). Note that T (Φ × Ψ) = (T Φ) × Ψ + Φ × (T Ψ) and, for Φ ∈ E_{V_{p}}(Y), Ψ ∈ E_{V_{q}}(Y), we have Φ × Ψ ∈ E_{V_{p+q}}(Y) showing that the linear subspace of E_{V +W} (Y) described above actually represents an asymptotic type.

The multiplication of asymptotic types possesses a unit, denoted by 1, that is represented by the space span{(1)} ⊂ E_{{0}}(Y), where 1 is the function identically 1 on Y.

Definition 3.2. An asymptotic type Q ∈ \underline{As}(Y) is called multiplicatively closed if Q ∘ Q = Q. An asymptotic type Q ∈ \underline{As}(Y) is called multiplicatively closable if it is dominated by a multiplicatively closed asymptotic type. In this case, the minimal multiplicatively closed asymptotic type dominating Q is called the multiplicative closure of Q and is denoted by Q̃.

From the proof of Lemma 3.1,

δ_{P ∘Q} ≥ δ_{P} + δ_{Q} − (n + 1)/2,  (3.2)

where equality holds if P = Q or if dim Y = 0. Especially, δ_{Q} = (n + 1)/2 if Q is multiplicatively closed and δ_{Q} ≥ (n + 1)/2 if Q is multiplicatively closable. Furthermore, it is also seen Q ≽ 1 for any multiplicatively closed asymptotic type Q, see also Lemma 3.4 below.

3.1.2 The class \underline{As}^{♯}(Y) of multiplicatively closable asymptotic types

We study the class of asymptotic types that belong to bounded functions. It turns out that this class is intimately connected to the multiplication of asymptotic types.

Definition 3.3. (a) The class \underline{As}^{b}(Y) of bounded asymptotic types consists of all asymptotic types Q ∈ \underline{As}(Y) for which δ_{Q} ≥ (n + 1)/2. Equivalently, a bounded asymptotic type Q is represented by a subspace J ⊂ E_{V} (Y) for some V ∈ C, where V ⊂ {z ∈ ℂ; Re z ≤ 0}.

(b) The class \underline{As}^{♯}(Y) consists of all bounded asymptotic types Q represented by a subspace J ⊂ E(Y) such that J_{0} ⊆ span{(1)} and J_{p} = {0} for Re p = 0, p ≠ 0.

Lemma 3.4. For Q ∈ \underline{As}(Y), the following conditions are equivalent:

(a) Q is multiplicatively closable;

(b) the join ⋁_{k≥1} Q^{k} does exist, where Q^{k} = \underbrace{Q ∘ ⋯ ∘ Q}_{k times} is the k-fold product;

(c) Q ∈ \underline{As}^{♯}(Y). In case (a) to (c) are fulfilled, we have Q̃ = 1 ∨ ⋁_{k≥1} Q^{k}.

Proof. (a) and (b) are obviously equivalent. Moreover, (c) implies (b). It remains to show that (a) also implies (c). If Q is multiplicatively closable, then Q̃ exists and δQ̃ = (n + 1)/2. In particular, Q̃ ∈ \underline{As}^{b}(Y). Let Q̃ be represented by J ⊂ E_{V} (Y) for some V ∈ C, V ⊂ {z ∈ ℂ; Re z ≤ 0}. Suppose that ϕ ∈ J_{p} for p ∈ ℂ, Re p = 0, where ϕ ≠ 0. We immediately get ϕ^{l} ∈ J_{lp} for any l ∈ ℕ, l ≥ 1. For p ≠ 0, we obtain the contradiction {lp; l ∈ ℕ} ⊆ V ∈ C. For p = 0 and ϕ not being constant, we obtain a contradiction to the fact that dim J_{0} < ∞. Thus, Q̃ ∈ \underline{As}^{♯}(Y) and, therefore, Q ∈ \underline{As}^{♯}(Y).

Lemma 3.5. For each Q ∈ \underline{As}(Y), there are asymptotic types Q^{b} ∈ \underline{As}^{b}(Y) and Q^{♯} ∈ \underline{As}^{♯}(Y) which are maximal among all asymptotic types possessing the property

Q^{b} ≼ Q and Q^{♯} ≼ Q,  (3.3)

respectively. In particular, Q^{♯} ≼ Q^{b}.

Proof. The proof is straightforward.

3.1.3 Nonlinear superposition

We investigate expressions like F (x, v(x)), where F (x, ν) ∈ C_{R}^{∞}(X×ℝ) for some R ∈ \underline{As}(Y) and v ∈ H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X) with s ≥ 0, ϑ > 0, and Q ∈ \underline{As}^{♯}(Y). For later reference, we recall the following facts:

Proposition 3.6. (a) For s > (n + 1)/2, 0 ≤ s′ ≤ s, γ, δ ∈ ℝ, pointwise multiplication induces a bilinear continuous map

H^{s,γ}(X) × H^{s′,δ}(X) → H^{s′,γ+δ−(n+1)/2}(X).  (3.4)

(b) For s, δ ∈ ℝ, H^{s,δ}(X) ⊂ L^{∞}(X) if and only if s > (n+1)/2, δ ≥ (n+1)/2.

(c) For s ≥ 0, γ, δ ≥ (n + 1)/2, pointwise multiplication induces a bilinear continuous map

(H^{s,γ}(X) ∩ L^{∞}(X)) × (H^{s,δ}(X) ∩ L^{∞}(X)) → H^{s,γ+δ−(n+1)/2}(X) ∩ L^{∞}(X).

(d) For s ≥ 0, δ ∈ ℝ, p ∈ ℂ, c(y) ∈ C^{∞}(Y), the multiplication operator

ω(t)t^{−p}c(y): H^{s,δ}(X) → H^{s,δ−Re p}(X),

where ω(t) is a cut-off function, is continuous.

(e) For s ≥ 0, v_{1}, …, v_{K} ∈ (1 + H^{s,(n+1)/2}(X)) ∩ L^{∞}(X), and F ∈ C^{∞}(ℝ^{K}), we have

F (v_{1}, …, v_{K}) ∈ (1 + H^{s,(n+1)/2}(X)) ∩ L^{∞}(X).  (3.5)

The map ((1 + H^{s,(n+1)/2}(X)) ∩ L^{∞}(X))^{K} → (1 + H^{s,(n+1)/2}(X)) ∩ L^{∞}(X) induced by (3.5) is continuous and sends bounded sets to bounded sets.

Proof. A proof of (3.4) in case s′ = s has been supplied by Witt [19, Lemma 2.7] using a result of Dauge [2, Theorem (AA.3)]. The other proofs are similar.

Remark 3.7. Property (d) fails if logarithms appear and has to be replaced by

ω(t)t^{−p} log^{k} t c(y): H^{s,δ}(X) → H^{s,δ−Re p−0}(X)

is continuous when k ∈ ℕ, k ≥ 1. First, Lemma 3.1 is sharpened:

Proposition 3.8. For s > (n + 1)/2, 0 ≤ s′ ≤ s, ϑ > 0, and P, Q ∈ \underline{As}(Y), pointwise multiplication induces a bilinear continuous map

H_{P,ϑ−0}^{s}(X) × H_{Q,ϑ−0}^{s′}(X) → H_{P ∘Q,ϑ−0}^{s′}(X).  (3.6)

Proof. Let u ∈ H_{P,ϑ−0}^{s}(X), v ∈ H_{Q,ϑ−0}^{s′}(X). Then u = u_{0} + u_{1}, v = v_{0} + v_{1}, where

u_{0} = ∑_{j=0}^{M} ∑_{k=0}^{m_{j}} ω(t)t^{−p_{j}} log^{k} t c_{jk}(y), v_{0} = ∑_{j′=0}^{N} ∑_{k′=0}^{n_{j′}} ω(t)t^{−q_{j′}} log^{k′} t d_{j′k′}(y),

(3.7)

ω(t) is a cut-off function, the sequences {(p_{j}, m_{j}, c_{jk})}, {(q_{j′}, n_{j′}, d_{j′k′})} are given by the asymptotic types P and Q, respectively, according to Definition 2.18, and M, N are chosen so that u_{1} ∈ H^{s,δ_{P} +ϑ−0}(X), v_{1} ∈ H^{s′,δ_{Q}+ϑ−0}(X). Since u_{0} ∈ H^{∞,δ_{P} −0}(X), v_{0} ∈ H^{∞,δ_{Q}−0}(X), we obtain

uv = u_{0}v_{0} + u_{1}v_{0} + u_{0}v_{1} + u_{1}v_{1},

where u_{1}v_{0} + u_{0}v_{1} + u_{1}v_{1} ∈ H^{s′,δ_{P ∘Q}+ϑ−0}(X) by (3.4) and

u_{0}v_{0} = ∑_{j,j′=0}^{M,N} ∑_{k,k′=0}^{m_{j},n_{j′}} ω^{2}(t)t^{−(p_{j}+q_{j′})} log^{k+k′} t c_{jk}(y)d_{j′k′}(y) ∈ H_{P ∘Q,ϑ−0}^{∞}(X),

for ω^{2}(t) is a cut-off function and the sequence

{(r_{j′′}, o_{j′′}, ∑_{p_{j}+q_{j′} =r_{j′′}} ∑_{k+k′=k′′} c_{jk}d_{j′k′})},

where o_{j′′} = max{m_{j} + n_{j′}; p_{j} + q_{j′} = r_{j′′}}, is associated with an asymptotic type that equals P ∘Q up to the conormal order δ_{P ∘Q} +ϑ−0. This immediately gives uv ∈ H_{P ∘Q,ϑ−0}^{s′}(X).

The significance of the class \underline{As}^{b}(Y) is uncovered by the next result:

Proposition 3.9. For s ≥ 0, δ ∈ ℝ, δ + \{ϑ\} ≥ (n + 1)/2, and Q ∈ \underline{As}^{δ}(Y),

H_{Q,{ϑ}}^{s,δ}(X) ∩ L^{∞}(X) = H_{Q^{b},{ϑ}}^{s,δ}(X) ∩ L^{∞}(X).  (3.8)

Proof. Let u ∈ H_{Q,{ϑ}}^{s,δ}(X) ∩ L^{∞}(X) and write

u(x) = ∑_{j=0}^{M} ∑_{k=0}^{m_{j}} ω(t)t^{−p_{j}} log^{k} t c_{jk}(y) + u_{1}(x),

where the sequences {(p_{j}, m_{j}, c_{jk})} is given by the asymptotic type Q and M is chosen so that u_{1} ∈ H^{s,(n+1)/2−0}(X). Since u ∈ L^{∞}(X) ⊂ H^{0,(n+1)/2−0}(X), we get that ∑_{j=0}^{M} ∑_{k=0}^{m_{j}} ω(t)t^{−p_{j}} log^{k} t c_{jk}(y) ∈ H^{0,(n+1)/2−0}(X) which implies c_{jk}(y) = 0 for Re p_{j} > 0. Thus u ∈ H_{Q^{b},ϑ}^{s,δ}(X).

Lemma 3.10. For s ≥ 0, ϑ > 0, and P, Q ∈ \underline{As}^{b}(Y), pointwise multiplication induces a bilinear continuous map

(H_{P,ϑ−0}^{s}(X) ∩ L^{∞}(X)) × (H_{Q,ϑ−0}^{s}(X) ∩ L^{∞}(X)) → H_{P ∘Q,ϑ−0}^{s}(X) ∩ L^{∞}(X).

Proof. Represent u = u_{0} + u_{1} ∈ H_{P,ϑ−0}^{s}(X) ∩ L^{∞}(X), v = v_{0} + v_{1} ∈ H_{Q,ϑ−0}^{s}(X) ∩ L^{∞}(X) as in the proof of Proposition 3.8. Since u_{0}, v_{0} ∈ L^{∞}(X) due to the assumption P, Q ∈ \underline{As}^{b}(Y), we get that u_{1} ∈ H^{s,δ_{P} +ϑ−0}(X) ∩ L^{∞}(X), v_{1} ∈ H^{s,δ_{Q}+ϑ−0}(X) ∩ L^{∞}(X) and, therefore, u_{1}v_{0} + u_{0}v_{1} + u_{1}v_{1} ∈ H^{s,δ_{P ∘Q}+ϑ−0}(X) ∩ L^{∞}(X) in view of Proposition 3.6 (c). The assertion follows.

A more precise statement is possible if P, Q ∈ \underline{As}^{♯}(Y):

Lemma 3.11. For s ≥ 0, ϑ ≥ 0, and P, Q ∈ \underline{As}^{♯}(Y) satisfying P ≽ 1, Q ≽ 1, pointwise multiplication induces a bilinear continuous map

(H_{P,ϑ}^{s}(X) ∩ L^{∞}(X)) × (H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X)) → H_{P ∘Q,ϑ}^{s}(X) ∩ L^{∞}(X). (3.9)

Especially, for s ≥ 0, ϑ ≥ 0, and Q ∈ \underline{As}^{♯}(Y) being multiplicatively closed, H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X) is an algebra under pointwise multiplication.

Proof. We may assume that ϑ > 0. Write u = u_{0} + u_{1} ∈ H_{P,ϑ}^{s}(X) ∩ L^{∞}(X), v = v_{0} + v_{1} ∈ H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X) as in the proof of Proposition 3.8, where u_{0} = u_{00} + u_{01}, v_{0} = v_{00} + v_{01}, u_{00} = ω(t)c_{00}, and v_{00} = ω(t)d_{00} with c_{00}, d_{00} being constants and in the expressions for u_{01}, v_{01} only appear exponents with Re p_{j} < 0 and Re q_{j′} < 0, respectively. Then

u_{1}v_{01} + u_{01}v_{1} + u_{1}v_{1} ∈ H^{s,(n+1)/2+ϑ+0}(X),

u_{00}v ∈ H_{Q,ϑ}^{s}(X) ⊆ H_{P ∘Q,ϑ}^{s}(X), uv_{00} ∈ H_{P,ϑ}^{s}(X) ⊆ H_{P ∘Q,ϑ}^{s}(X), and

u_{01}v_{01} ∈ H_{P ∘Q,ϑ+0}^{∞}(X),

which proves the assertion.

The fact which has actually been used in the last proof is that Proposition 3.6 (d) applies to the function ω(t)1 (p = 0, c(y) ≡ 1). This is also used in part (b) of the next result:

Lemma 3.12. (a) Let s ≥ 0, ϑ > 0, and R, Q ∈ \underline{As}(Y). Then pointwise multiplication induces a continuous map

C_{R}^{∞}(X) × H_{Q,ϑ−0}^{s}(X) → H_{R∘Q,ϑ−0}^{s}(X).  (3.10)

(b) If, in addition, R ∈ \underline{As}(Y) is so that the multiplicities of its highest singular values are one, i.e., J_{r} ⊆ [C^{∞}(Y)]^{1} for each r ∈ V, Re r = (n+1)/2−δ_{R}, where J ⊂ E_{V} (Y) represents R, then pointwise multiplication induces a continuous map

C_{R}^{∞}(X) × H_{Q,ϑ}^{s}(X) → H_{R∘Q,ϑ}^{s}(X).

Proof. (a) is immediate from Proposition 3.8. To get (b), we argue as in the proof of Lemma 3.11.

Proposition 3.13. Let s ≥ 0, ϑ ≥ 0, and Q ∈ \underline{As}^{♯}(Y) be multiplicatively closed. Then v_{1}, …, v_{K} ∈ H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X) and F ∈ C^{∞}(ℝ^{K}) implies that

F (v_{1}, …, v_{K}) ∈ H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X).  (3.11)

Proof. We are allowed to assume that ϑ > 0. Then v ∈ H_{Q,ϑ}^{s}(X) implies that v|_{∂X} is a constant, where v|_{∂X} means the factor in front of t^{0} in the asymptotic expansion (1.2) (with u replaced with v) of v as t → +0. Let β_{J} = v_{J} |_{∂X} for 1 ≤ J ≤ K be these constants. Using Taylor’s formula, we obtain

F (v_{1}, …, v_{K}) = ∑_{|α|<N} \frac{1}{α!} (∂^{α}F)(β_{1}, …, β_{K})(v_{1} − β_{1})^{α_{1}} … (v_{K} − β_{K})^{α_{K}}

+ N ∑_{|α|=N} ∫_{0}^{1} \frac{(1 − σ)^{N−1}}{α!} (∂^{α}F)(β_{1} + σ(v_{1} − β_{1}), …, β_{K} + σ(v_{K} − β_{K})) dσ

× (v_{1} − β_{1})^{α_{1}} … (v_{K} − β_{K})^{α_{K}} . (3.12)

By Lemma 3.11, (v_{1} − β_{1})^{α_{1}} … (v_{K} − β_{K})^{α_{K}} ∈ H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X) for any α ∈ ℕ^{K}, thus the first summand on the right-hand side of (3.12) belongs to H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X). On the other hand, choosing N sufficiently large, we can arrange that (v_{1} − β_{1})^{α_{1}} … (v_{K} − β_{K})^{α_{K}} ∈ H^{s,(n+1)/2+ϑ}(X) ∩ L^{∞}(X) for |α| ≥ N, since v_{J} − β_{J} ∈ H^{s,(n+1)/2+0}(X) ∩ L^{∞}(X) for 1 ≤ J ≤ K. By (3.5), {(∂^{α}F)(β_{1} + σ(v_{1} − β_{1}), …, β_{K} + σ(v_{K} − β_{K})) dσ; 0 ≤ σ ≤ 1} is a bounded set in (1 + H^{s,(n+1)/2}(X)) ∩ L^{∞}(X) for any α ∈ ℕ^{K}. This shows that the second summand on the right-hand side of (3.12) belongs to H^{s,(n+1)/2+ϑ}(X) ∩ L^{∞}(X).

Proposition 3.14. (a) Let s ≥ 0, ϑ > 0. Further let Q ∈ \underline{As}^{♯}(Y) be multiplicatively closed and R ∈ \underline{As}(Y). Then v_{1}, …, v_{K} ∈ H_{Q,ϑ−0}^{s}(X) ∩ L^{∞}(X) and F ∈ C_{R}^{∞}(X × ℝ^{K}) implies that

F (x, v_{1}, …, v_{K}) ∈ H_{R∘Q,ϑ−0}^{s}(X).  (3.13)

(b) If, in addition, R satisfies the assumption of Lemma 3.12 (b), then v_{1}, …, v_{K} ∈ H_{Q,ϑ}^{s}(X) ∩ L^{∞}(X) and F ∈ C_{R}^{∞}(X × ℝ^{K}) implies that

F (x, v_{1}, …, v_{K}) ∈ H_{R∘Q,ϑ}^{s}(X).

Proof. We prove (a), (b) is analogous. Since C_{R}^{∞}(X × ℝ^{K}) = C_{R}^{∞}(X) ˆ⊗_{π} C^{∞}(ℝ^{K}), we can write

F (x, υ) = ∑_{j=0}^{∞} α_{j} φ_{j}(x) F_{j}(υ),

where {α_{j}}_{j=0}^{∞} ∈ l^{1} and {φ_{j}}_{j=0}^{∞} ⊂ C_{R}^{∞}(X) and {F_{j}}_{j=0}^{∞} ⊂ C^{∞}(ℝ^{K}), respectively, are null sequences. By the preceeding proposition,

F_{j}(v_{1}, …, v_{K}) → 0 as j → ∞ in H_{Q,ϑ−0}^{s}(X).

By Lemma 3.12,

φ_{j}(x)F_{j}(v_{1}, …, v_{K}) → 0 as j → ∞ in H_{R∘Q,ϑ−0}^{s}(X).

Thus

F (x, v_{1}, …, v_{K}) = ∑_{j=0}^{∞} α_{j} φ_{j}(x)F_{j}(v_{1}, …, v_{K}) ∈ H_{R∘Q,ϑ−0}^{s}(X),

where the sum on the right-hand side is absolutely convergent.

3.2 The bootstrapping argument

We consider the equation

Au = Π(u),  (3.14)

where A ∈ Diff_{Fuchs}^{μ}(X) is an elliptic Fuchsian differential operator. Properties of the nonlinear operator u ↦ Π(u) are discussed below. The method proposed for deriving elliptic regularity for solutions to (3.14) amounts to balancing two asymptotic types — one for the left-hand and the other one for the right-hand side of (3.14). We assume: There are asymptotic types P̄ ∈ \underline{As}^{δ}(Y), Q̄ ∈ \underline{As}^{δ−μ}(Y), numbers a, b, s_{0}, ϑ_{0} ∈ ℝ with

a < μ, b < δQ̄ − δP̄ + μ, s_{0} ≥ a^{+}, δP̄ + {ϑ_{0}} ≥ δ,

and a subset U ⊆ H_{P̄,{ϑ_{0}}}^{s_{0}}(X) such that the following conditions are met: (A) A is elliptic with respect to the conormal order δ and P̄ ≽ P^{δ}(Q̄; A), i.e.,

u ∈ H^{−∞,δ}(X), Au ∈ C∞ ; Q̄(X) implies u ∈ C∞ ; P̄(X);

(B) For s ≥ s_{0}, ϑ ≥ ϑ_{0}, we have

Π: U ∩ H_{P̄,{ϑ}}^{s}(X) → H_{Q̄,{ϑ}−b}^{s−a}(X).

Note that {ϑ_{0}} − b + δQ̄ ≥ δ − μ.

Proposition 3.15. Under the conditions (A), (B), each solution u ∈ U ⊆ H_{P̄,{ϑ_{0}}}^{s_{0}}(X) to (3.14) belongs to the space C∞ ; P̄(X).

Proof. We prove by induction on j that

u ∈ H_{P̄,{ϑ_{0}}+j(μ−b+δQ̄−δP̄)}^{s_{0}+j(μ−a)}(X)  (3.15)

for all j ∈ ℕ. Since μ − a > 0, μ − b + δQ̄ − δP̄ > 0, this implies u ∈ C∞ ; P̄(X). By assumption, (3.15) holds for j = 0. Now suppose that (3.15) for some j has already been proven. From (B) we conclude that Π(u) ∈ H_{Q̄,{ϑ_{0}}+j(μ−b+δQ̄−δP̄)−b}^{s_{0}+j(μ−a)−a}(X). In view of (A), elliptic regularity gives u ∈

H_{P̄,{ϑ_{0}}+(j+1)(μ−b+δQ̄−δP̄)}^{s_{0}+(j+1)(μ−a)}(X).

Example 3.16. Here, we provide an example for a nonlinearity Π satisfying (B). Let Π(u) = K_{0}(u)/K_{1}(u), where K_{0}, K_{1} are polynomials of degree m_{0} and m_{1}, respectively. Let u ∈ H_{P,ϑ−0}^{s}(X), where s > (n + 1)/2, δ_{P} + ϑ > (n + 1)/2, and ϑ > 0. Further, we assume that the multiplicities of the highest singular values for P are simple and the coefficient functions for these singular values nowhere vanish on Y. Then we have K_{0}(u) ∈ H_{P_{0},ϑ−0}^{s}(X), K_{1}(u) ∈ H_{P_{1},ϑ−0}^{s}(X) for resulting asymptotic types P_{0}, P_{1}. In particular, P_{0} is dominated by 1∨⋁_{k=1}^{m_{0}} P^{k} and P_{1} is dominated by 1 ∨ ⋁_{k=1}^{m_{1}} P^{k}. Furthermore, it is readily seen that v ∈ H_{P_{1},ϑ−0}^{s}(X) and v ≠ 0 everywhere on X^{∘} implies that 1/v ∈ H_{Q_{1},ϑ′−0}^{s}(X) for some resulting asymptotic type Q_{1}. Hence, we are allowed to set P̄ = P, Q̄ = P_{0} ∘ Q_{1}, and

U = {u ∈ H_{P,ϑ−0}^{s}(X); K_{1}(u) ≠ 0 everywhere on X^{∘}}.

The condition s > (n + 1)/2 can be replaced by s ≥ 0. Then we additionally need u ∈ L_{loc}^{∞}(X^{∘}).

3.3 Proof of the main theorem

The main step consists in constructing asymptotic types P̄, Q̄ so that Proposition 3.15 applies. Thereby, upon choosing δ ∈ ℝ even smaller if necessary, we can assume that

δ ≤ μ̄ + (n + 1)/2

and that A ∈ Diff_{Fuchs}^{μ}(X) is elliptic with respect to the conormal order δ. Set Δ = δ_{R} + (μ − μ̄) − (n + 1)/2. By assumption (1.4), Δ > 0.

3.3.1 Construction of asymptotic types P, Q

We construct by induction on h sequences \{P_{h}\}_{h=0}^{∞} ⊂ \underline{As}^{δ}(Y) and \{Q_{h}\}_{h=0}^{∞} ⊂ \underline{As}^{♯}(Y) of asymptotic types as follows: Set P_{0} = P^{δ}(O; A). Suppose that P_{0}, …, P_{h} and Q_{0}, …, Q_{h−1} for some h have already been constructed. Then

Q_{h} = (Q(P_{h}; B_{1}, …, B_{K})^{♯}) ,  (3.16)
P_{h+1} = P^{δ}(R ∘ Q_{h}; A).  (3.17)

Lemma 3.17. For each h ≥ 0,

P_{h} = P_{h+1} up to the conormal order δ_{R} + μ + hΔ − 0,  (3.18)
Q_{h} = Q_{h+1} up to the conormal order δ_{R} + (μ − μ̄) + hΔ − 0.  (3.19)

In particular, the joins P = ⋁_{h=0}^{∞} P_{h} and Q = ⋁_{h=0}^{∞} Q_{h} exist.

Proof. We set Q_{−1} = O and proceed by induction on h. (3.19) holds for h = −1, since Q_{0} ∈ \underline{As}^{♯}(Y) and, therefore, Q_{0} = O up to the conormal order (n + 1)/2 − 0. Suppose that Q_{h−1} = Q_{h} up to the conormal order δ_{R} + (μ − μ̄) + (h − 1)Δ − 0 for some h ≥ 0 has already been proved. Then R ∘ Q_{h−1} = R ∘ Q_{h} up to the conormal order δ_{R} + hΔ − 0 and P_{h} = P_{h+1} up to the conormal order δ_{R} + μ + hΔ − 0, since P_{h} = P^{δ}(R ∘ Q_{h}; A), P_{h+1} = P^{δ}(R ∘ Q_{h+1}; A). Now suppose that P_{h} = P_{h+1} up to the conormal order δ_{R} + μ + hΔ − 0. We obtain Q(P_{h}; B_{1}, …, B_{K}) = Q(P_{h+1}; B_{1}, …, B_{K}) up to the conormal order δ_{R} + (μ − μ̄) + hΔ − 0 and, therefore, Q_{h} = Q_{h+1} up to the conormal order δ_{R} + (μ − μ̄) + hΔ − 0, since Q_{h} = (Q(P_{h}; B_{1}, …, B_{K})^{♯}) , Q_{h+1} = (Q(P_{h+1}; B_{1}, …, B_{K})^{♯}) . This completes the inductive proof.

Lemma 3.18. The asymptotic types P = ⋁_{h=0}^{∞} P_{h} ∈ \underline{As}^{δ}(Y), Q = ⋁_{h=0}^{∞} Q_{h} ∈ \underline{As}^{♯}(Y) satisfy:

(a) Q(P; B_{1}, …, B_{K})^{b} = Q(P; B_{1}, …, B_{K})^{♯} and Q = (Q(P; B_{1}, …, B_{K})^{♯}) ;

(b) P = P^{δ}(R ∘ Q; A);

(c) Q is multiplicatively closed. Furthermore, P, Q are minimal among all asymptotic types in \underline{As}^{δ}(Y) and \underline{As}^{♯}(Y), respectively, satisfying (a) to (c).

Proof. The assertions immediately follow from the description of the asymptotic types P_{h}, Q_{h} given in the previous lemma. Only Q(P; B_{1}, …, B_{K})^{b} = Q(P; B_{1}, …, B_{K})^{♯} needs an argument: But P = P_{0} up to the conormal order δ_{R} + μ − 0, so we get Q(P; B_{1}, …, B_{K}) = Q(P_{0}; B_{1}, …, B_{K}) up to the conormal order δ_{R}+(μ−μ̄)−0 = (n+1)/2+Δ−0 > (n + 1)/2, and Q(P_{0}; B_{1}, …, B_{K})^{b} = Q(P_{0}; B_{1}, …, B_{K})^{♯} is exactly the nonresonance condition (1.5).

Note that, by the non-resonance condition (1.5) and Proposition 3.9,

B_{J}u ∈ H_{Q(P;B_{1},...,B_{K}),ϑ−0}^{s−μ̄,δ−μ̄}(X) ∩ L^{∞}(X)

⊆ H_{Q(P;B_{1},...,B_{K})^{♯},ϑ−0}^{s−μ̄,δ−μ̄}(X) ⊆ H_{Q,ϑ−0}^{s−μ̄,δ−μ̄}(X) (3.20)

if u ∈ H_{P,ϑ−0}^{s,δ}(X), δ − μ̄ + ϑ > (n + 1)/2, and B_{J}u ∈ L^{∞}(X).

3.3.2 End of the proof of Theorem 1.1

Since B_{J}u ∈ L^{∞}(X) ⊂ H^{0,(n+1)/2−0}(X) for all 1 ≤ J ≤ K, we have F (x, B_{1}u, …, B_{K}u) ∈ H^{0,δ_{R}−0}(X) and

u ∈ H_{P_{0},δ_{R}+μ−δ_{P} −0}^{μ}(X) = H_{P,δ_{R}+μ−δ_{P} −0}^{μ}(X)

by elliptic regularity. To conclude the proof of Theorem 1.1, we apply Proposition 3.15 with Πu = F (x, B_{1}u, …, B_{K}u), P̄ = P, Q̄ = R ∘ Q, where P ∈ \underline{As}^{δ}(Y), Q ∈ \underline{As}^{♯}(Y) have been constructed in Lemmas 3.17, 3.18, s_{0} = μ, {ϑ_{0}} = δ_{R} + μ − δ_{P} − 0, a = μ̄, b = (n + 1)/2 − δ_{P} + μ̄, and

U = {u ∈ H_{P,δ_{R}+μ−δ_{P} −0}^{μ}(X); B_{J}u ∈ L^{∞}(X), 1 ≤ J ≤ K}.  (3.21)

Then a < μ, b < δ_{R∘Q} − δ_{P} + μ for δ_{R∘Q} = δ_{R}, Δ > 0, and δ_{P} + ϑ_{0} = δ_{R} + μ > μ̄ + (n + 1)/2 ≥ δ, i.e., δ_{P} + {ϑ_{0}} ≥ δ. Moreover, condition (A) is fulfilled. To check condition (B), note that u ∈ U ∩H_{P,ϑ−0}^{s}(X) for s ≥ μ, ϑ ≥ δ_{R}+μ−δ_{P} implies

F (x, B_{1}u, …, B_{K}u) ∈ H_{R∘Q,δ_{P} −μ̄−(n+1)/2+ϑ−0}^{s−μ̄}(X)

by (3.20) and Proposition 3.14. Thus Proposition 3.15 applies to yield u ∈ C_{P}^{∞}(X). Remark 3.19. From (3.21) it is seen that the asymptotic type P ∈ \underline{As}^{δ}(Y) can be taken smaller, namely instead of P = P^{δ}(R ∘ Q; A) we can choose the asymptotic type

⋁\{P′ ∈ \underline{As}^{δ}(Y); P′ ≼ P^{δ}(R ∘ Q; A), Q(P′; B_{1}, …, B_{K}) ∈ \underline{As}^{♯}(Y)\}.

In concrete problems, the resulting asymptotic type for u can be even smaller, e.g., due to nonlinear interaction caused by the special structure of the nonlinearity.

3.4 Example: The equation Δu = Au^{2} + B(x)u in three space dimensions

Let Ω be a bounded, smooth domain in ℝ^{3} containing 0. We are going to study singular solutions to the equation

Δu = Au^{2} + B(x)u on Ω \ {0},  (3.22)
γ_{0}u = c_{0}, u|_{∂Ω} = ϕ,  (3.23)

where γ_{0}u = lim_{x→0} |x|u(x), A ∈ ℝ, and B ∈ C^{∞}(\overline{Ω}) is real-valued. Since the quadratic polynomial Au^{2} + B(x)u rather than a general nonlinearity F (x, u) enters, we may admit complex-valued solutions u to (3.22). In particular, c_{0} ∈ ℂ. Remark 3.20. By results in Véron [17], one expects the limit lim_{x→0} |x|u(x) exist for the solutions u = u(x) to (3.22). On Ω \ {0}, we introduce polar coordinates (t, y) ∈ ℝ_{+} × S^{2}, t = |x|, y = x/|x|. We further introduce the function spaces

X^{2} = {c_{0}t^{−1} + c_{11} log t + u_{0}(x); c_{0}, c_{11} ∈ ℂ, u_{0} ∈ H^{2}(Ω)},
Y^{0} = {d_{0}t^{−2} + v_{0}(x); d_{0} ∈ ℂ, v_{0} ∈ L^{2}(Ω)},

the definition of which is suggested by formal asymptotic analysis. On the space X^{2}, we have the trace operators γ_{0}, γ_{1}, γ_{11}, where γ_{11}u = lim_{t→+0}(u(x)− (γ_{0}u)t^{−1})/ log t, γ_{1}u = lim_{t→+0}(u(x) − (γ_{0}u)t^{−1} − (γ_{11}u) log t).

Proposition 3.21. Suppose that B(x) ≥ 0 for all x ∈ \overline{Ω}. Then, for all c_{0} ∈ ℂ, ϕ ∈ H^{3/2}(∂Ω) with |c_{0}|+∥ϕ∥_{H^{3/2}(∂Ω)} small enough, the boundary value problem (3.22), (3.23) admits a unique small solution u ∈ X^{2}. This solution u = u(x) obeys a complete conormal asymptotic expansion as x → 0 that can successively be calculated. Especially,

c_{11} = Ac_{0}^{2},  (3.24)

where c_{11} = γ_{11}u.

Proof. Let us consider the nonlinear operator

Ψ: X^{2} → Y^{0} × ℂ × H^{3/2}(∂Ω), u ↦ (Δu − Au^{2} − B(x)u, γ_{0}u, u|_{∂Ω}) .

It is readily seen that the linearization of Ψ about u = 0 is an isomorphism between the indicated spaces. Thus, the existence of a unique small solution u ∈ X^{2} to (3.22), (3.23) is implied by the inverse function theorem. (3.24) likewise follows. Furthermore, writing this solution in the form u(x) = c_{0}t^{−1} + c_{11} log t + u_{0}(x), where u_{0} ∈ H^{2}(Ω), we get that u_{0} fulfills the equation

c_{11}t^{−2} + Δu_{0} = A (c_{0}^{2}t^{−2} + 2c_{0}c_{11}t^{−1} log t + c_{11}^{2} log^{2} t)

+ 2A (c_{0}t^{−1} + c_{11} log t) u_{0} + Au_{0}^{2} + B(x) (c_{0}t^{−1} + c_{11} log t) + B(x)u_{0}. (3.25)

This can be brought into the form (1.1) with A = Δ,

F (x, ν) = (2Ac_{0}c_{11}t^{−1} log t + B(x)c_{0}t^{−1} + Ac_{11}^{2} log^{2} t + B(x)c_{11} log t)

+ (2Ac_{0}t^{−1} + 2Ac_{11} log t + B(x)) ν + Aν^{2},

since Δ = t^{−2} ((−t∂_{t})^{2} − (−t∂_{t}) + Δ_{S^{2}}) ∈ Diff_{Fuchs}^{2}(Ω \ {0}), where 0 ∈ Ω is considered as conical point with cone base S^{2} = {x ∈ ℝ^{3}; |x| = 1}, cf. Remark 1.3, and Δ_{S^{2}} being the Laplace-Beltrami operator on S^{2}. The conditions (1.4), (1.5) are obviously satisfied.

Thus, Theorem 1.1 applies to u_{0} ∈ H^{2}(Ω) ⊂ L^{∞}(Ω) to yield that u_{0} and, therefore, u obey a complete conormal asymptotic expansion.

Remark 3.22. (a) Taking for P the asymptotic type in \underline{As}^{0}(S^{2}) that comes out of the calculation of the conormal asymptotic expansion for u, i.e., we have u ∈ C_{P}^{∞}(Ω \ \{0\}), and for Q the resulting asymptotic type in \underline{As}^{−2}(S^{2}) for the right-hand side of (3.22), we are in a situation in which Proposition 3.15 directly applies without having boundedness assumptions for u.

(b) Allowing more general functions B ∈ C_{R}^{∞}(Ω\\{0\}) for some R ∈ \underline{As}^{−1/2}(S^{2}) (the conormal order −1/2 ensures that the term Ac_{0}^{2}t^{−2} dominates on the righthand side of (3.25)) rather than B ∈ C_{P_{0}}^{∞}(Ω \ {0}), where P_{0} is the asymptotic type for Taylor asymptotics, one can perform the same analysis as before upon replacing the space H^{2}(Ω) in the definition of X^{2} accordingly.

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Xiaochun Liu

School of Mathematics and Statistics Wuhan University Wuhan, 430072 P.R. China

  liuxc72@public.wh.hb.cn

Ingo Witt

Department of Mathematics Imperial College 180 Queen’s Gate London SW7 2AZ United Kingdom

                                   i.witt@imperial.ac.uk