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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


Introduction
In this paper, we study solutions u = u(x) to semilinear elliptic equations of the form 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 t −pj log k t c jk (y) as t → +0. ( 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 .
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 As(Y ) be the set of all these refined asymptotic types, while As ♯ (Y ) ⊂ As(Y ) denotes the set of asymptotic types belonging to bounded functions according to (1.3).For R ∈ As(Y ), let C ∞ R (X) be the space of smooth functions u ∈ C ∞ (X • ) having conormal asymptotic expansions of type R, and , where C ∞ R (X) is equipped with its natural (nuclear) Fréchet topology.In the formulation of Theorem 1.1, below, we will assume that where for some ε > 0. Here, μ = 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 As b (Y ) of all bounded asymptotic types and the set As ♯ (Y ) of asymptotic types as described by (1.3);As ♯ (Y ) As b (Y ).The set 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 As b (Y ) -but not to 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 where l ∈ Z, β ∈ R, β = 0 (and l > δ − 1/2 provided that c 0m0 (y) ≡ 0 due to the assumption u ∈ H −∞,δ (X)).Then, for each 1 ≤ J ≤ K, it is additional required that 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 + μ.Our main theorem states: Fuchs (X) be elliptic in the sense of Definition 2.2 (a), B J ∈ Diff µJ Fuchs (X) for 1 ≤ J ≤ K, where µ J < µ, and F ∈ C ∞ R (X × R k ) for some asymptotic type R ∈ As(Y ) satisfying (1.4).Further, let the non-resonance type condition (1.5) be satisfied.Then there exists an asymptotic type P ∈ 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 0 Fuchs (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 µ Fuchs (X; E 0 , E 1 ) be elliptic in the above sense, B ∈ Diff µ−1 Fuchs (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.

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.

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 τ , i.e., (τ , η) is the linear variable in the fiber of the compressed cotangent bundle T * X U .Finally, let dim X = n + 1.

Discrete asymptotic types
Linear relation between the various coefficients c jk (y) ∈ Lj, even for different j, are additionally allowed.Thus the production/annihilation of asymptotics is observed, cf.this article.
(2.3) (b) The operator A ∈ Diff µ Fuchs (X) is called elliptic with respect to the weight δ ∈ R if A is elliptic in the sense of (a) and, in addition, is invertible for some s ∈ R (and then for all s ∈ R).Here, Under the assumption of interior ellipticity of A, (2.3) can be reformulated as for all (0, y, τ , η) ∈ ( T * X \0) ∂U .This relation implies that σ µ M (A)(z) Γ (n+1)/2−δ is parameter-dependent elliptic as an element in L µ cl 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 where σ µ ψ (•) on the left-hand side denotes the parameter-dependent principal symbol.Thus, if (a) is fulfilled, then it follows that 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 µ M (A)(z).The following definition is taken from Schulze [16, Definition 2.3.48]: 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 where as (Y ) are called Mellin symbols of order µ.
µ∈Z M µ as (Y ) is a filtered algebra under pointwise multiplication.For f ∈ M µ as (Y ) for µ ∈ Z and f (z) = f 0 (z) + f 1 (z), where as (Y ), the parameter-dependent principal symbol σ µ ψ 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 µ as (Y ) is called elliptic if its principal symbol is everywhere invertible.For the next result, see Schulze [16,Theorem 2.4.20]: In various constructions, it is important to have examples of elliptic Mellin symbols f ∈ M µ as (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 , where ν j ≥ 0, N j + ν j ≥ 0, be given such that • there is an elliptic Then there is an elliptic Mellin symbol f (z) ∈ M µ as (Y ) such that, for all j, ) If n = 0, condition (2.7) is void.In case n > 0, however, this condition expresses several compatibility conditions among the σ µ−l ψ (f j k ), 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 µ O (Y ), in local coordinates (y, η) -showing, among others, that σ µ−j ψ (f (z)) is polynomial of degree j with respect to z ∈ C. The point is that we do not assume g(q) ∈ L µ cl (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.
From Proposition 2.5, we immediately get: In the case of the preceding lemma, By the remark preceding Lemma 2.3,

Definition of asymptotic types
We now start to introduce discrete asymptotic types.

The spaces E
where C ∞,δ as (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 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 , where the sequences (φ 0 , . . ., φ m−1 ) and (0, . . ., 0 (2.10) , where, for each p ∈ V , the sum on the right-hand side is finite. (2.12) Note that (2.12) already implies that c-ord Definition 2.12.Let Φ i , i = 1, 2, . . ., be a sequence in E δ (Y ) with the property that c-ord 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) Lemma 2.11.We also obtain: 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 ej are linearly independent over the space E δj (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 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 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: 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).

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 assigning to each formal asymptotic expansion 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 ∂Xby a linear subspace J ⊂ E V (Y ) for some V ∈ C δ such that the following three conditions are met: (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 As δ (Y ).
Documenta Mathematica 9 (2004) 207-250 Definition 2.18.Let u ∈ C ∞,δ as (X) and P ∈ 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 where Φ(p) = (φ mp−1 ) 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 , is mapped onto J by the isomorphism (2.13).For P ∈ As δ represented by J ⊂ E V (Y ), we introduce Notice that δ P > δ and δ P = ∞ if and only if On asymptotic types P ∈ As δ (Y ), we have the shift operation T ̺ for ̺ ∈ R, namely T ̺ P is represented by the space for some Φ ∈ J , where J ⊂ E V (Y ) represents P .Furthermore, for J ⊂ E V (Y ) as in Definition 2.17, 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 (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 commutes and the action of T on J is that one induced by (2.18), (2.19).
In the situation just described, we write 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 , 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 j 1 , . . ., m j ej ) 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 j ej is an (m j 1 , . . ., m j ej )-characteristic basis of J δ+j ; (b) For each j, T m j 1 −1 Φ 1 , . . ., T me j −1 Φ ej are linearly independent over the space E δ+j (Y ), while i or i > e j .In particular, if (a), (b) are fulfilled, then, for any j ′ > j, Here, Proof.This is a consequence of Lemma 2.13 and Witt [18,Lemma 3.8].
Definition 2.23.For S µ = {s µ−j (z); j ∈ N} ∈ Symb µ M (Y ), the linear space 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] as (X) possesses asymptotics given by the vector Φ according to (2.16), then there is 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 Definition 2.25.For P ∈ As δ (Y ) being represented by J ⊂ E V (Y ) and S µ ∈ Symb µ M (Y ), the push-forward Q δ−µ (P ; S µ ) of P under S µ is the asymptotic type in 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 holds for all q ∈ T µ V , see (2.6).
Extending the notion of push-forward from asymptotic types to arbitrary linear subspaces of

.23)
In this sense, it characterizes the amount of asymptotics of conormal order at least δ annihilated by S µ ∈ Symb µ M (Y ).
Definition 2.27.A partial ordering on As δ (Y ) is defined by P P ′ for P, P ′ ∈ 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 (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, is immediate from the definition of asymptotic type and (b) can be checked directly on the level of (2.22).
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 is an asymptotic algebra that is even reduced ; thus providing justification for the above choice of the notion of asymptotic type.
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, ), 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 Vp (Y ) define by Φ(p − j) = Φ j that belongs to the linear subspace J ⊂ E δ (Y ) representing L δ S µ .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 Vp (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 δ S µ represented by J is proper.
In conclusion, we obtain: There is an order-preserving bijection 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.

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 ∈ As δ prop (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 j 1 , . . ., m j ej ) 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 j i for j ≥ j i , where j i = min{j; e j ≥ i}, i.e., an appropriate re-ordering of the tuples (m j 1 , . . ., m j ej ).
Lemma 2.33.The numbering within the tuples (m j 1 , . . ., m j ej ) can be chosen in such a way that, for each j ≥ 1, there is a characteristic (m j 1 , . . ., m j ej )-basis (Φ j 1 , . . ., Φ j ej ) of J δ+j such that, for all j ′ > j, holds.Furthermore, the scheme 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 j i 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 Vp j (Y ) in (2.15) separately, one sees that the following notion is correctly defined: . . is a characteristic basis of J according to Definition 2.22 and if the tuples (m j 1 , . . ., m j ej ) are re-ordered according to Lemma 2.33, then the sequence char (2.28) is called the characteristic of P .
The characteristic char P of an asymptotic type P ∈ As δ prop (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: ⊂ 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 where j i = min{j; e j ≥ i} as above.
Proposition 2.36.Let the characteristic p i m ji i , m ji+1 i , . . .e i=1 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, Then there exists a holomorphic S µ ∈ Symb µ M (Y ) that is elliptic with respect to the weight δ ∈ R such that L δ S µ ∈ As δ prop (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 −µ O and t −µ (z) −1 ∈ M µ O , we can assume µ = 0.If n = 0, then we choose an elliptic s 0 (z) ∈ M 0 O that has zeros precisely at z = p i of order m ji i for i = 1, 2, . . .according to Theorem 2.6.In case dim Y > 0, let {φ i } e i=1 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 µ O (Y ) such that, for every p ∈ V pi and all 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 µ cl (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 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 As δ (Y ).We will utilize this fact in the definition of cone Sobolev spaces with asymptotics.
In constructing asymptotic types P ∈ 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 ′ ∈ 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 − ǫ, for any ǫ > 0. (Similarly for the order relation instead of equality.)Proposition 2.39.Let {P ι } ι∈I ⊂ 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 Vι (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 δ+j ι < ∞, otherwise ι∈I P ι does not exist.But since the net {J δ+j ι } ι∈I is increasing, this already implies that there is some ι j ∈ I such that J δ+j ι = J δ+j ι ′ for ι, ι ′ ≥ ι j , i.e., P ι = P ι ′ up to the conormal order δ + j for ι, ι ′ ≥ ι j .

Pseudodifferential theory
Here, we establish an analogue of Witt [18, Theorem 1.2].We need: that is elliptic with respect to the weight δ such that L δ S µ = 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
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 } e i=1 and {Ψ i } e i=1 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 µ O (Y ).By Theorem 2.6, it suffices to construct the finite parts [s µ−k (z)] 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)] m j+k p−j for all j, k must be chosen so that, for each j ∈ N, for 1 ≤ i ≤ e j , where m j = sup 1≤i≤ej m j i , and Here, (m j 1 , . . ., m j ej ) is the characteristic of J δ+j and, for Φ = (φ 0 , . . ., φ m−1 ), Ψ = (ψ 0 , . . ., stands for the linear system System (2.29) can successively be solved for [s µ (z − k)] m j p−j+k for j = 0, 1, 2, . . .and 0 ≤ k ≤ j.In fact, this can be done by choosing [s µ−k (z)] m j p−j+k 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.
having the same characteristics as P and Q, respectively, such that that are elliptic with respect to the weight δ such that This is achieved by using Proposition 2.40.

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].and the expression sup (2.32) is finite.
(b) For a general P ∈ As δ (Y ), represented as the join P = ι∈I P ι for a bounded family {P ι } ι∈I ⊂ As δ prop (Y ), we define H s,δ P,ϑ (X) = ι∈I H s,δ Pι,ϑ (X).It is readily seen that Definition 2.43 (a) is independent of the choice of the Mellin symbol h s P (z).Moreover, under the condition that (2.32) is finite the limit . Thus, H s,δ P,ϑ (X) is a Hilbert space with the norm Definition 2.43 (b) is justified by Proposition 2.37 (b), since we obviously have H s,δ P,ϑ (X) = H s,δ+ϑ (X) for P ∈ As δ prop (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 ⊂ As δ prop (Y ), and it also yields a Hilbert space structure for H s,δ P,ϑ (X).
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.
Remark 2.45.In case P is a strongly discrete asymptotic type, the spaces H s,δ P,ϑ−0 (X) are the function spaces introduced by Schulze [15, Section 2.1.1].There, the notation H s,δ P (X) ∆ with the half-open interval ∆ = (−ϑ, 0] has been used.The definition of the function spaces H s,δ P (X) ∆ refers to fixed splitting of coordinates near ∂X and is, in general, not coordinate invariant.
Proposition 2.47.For δ ∈ R, P ∈ As δ (Y ), and any a ∈ R, the family H s,δ P,s−a (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 s,δ P,ϑ (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 As δ (Y ) are coordinate invariant.

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 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 s,δ P,ϑ (X) are as follows: We conclude that R 0 : H s,δ (X) → C ∞ P0 (X), where where Notation.For A ∈ Diff µ Fuchs (X), Q δ−µ (P ; A) is even independent of δ ∈ R in view of the holomorphy of the conormal symbols σ µ−j M (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 µ Fuchs (X) be elliptic.Then there is an orderpreserving bijection with its inverse given by Q → P δ (Q; A).In particular, L δ S µ is mapped to the empty asymptotic type, O.
Proof.This follows from Remark 2.53.Combined with Theorem 2.30, Theorem 2.50 shows that each solution u ∈ C ∞,δ as (X) to the equation , where A ∈ Diff µ Fuchs (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.
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.

Multiplicatively closed asymptotic types
Here, we investigate multiplicative properties of asymptotic types and the behavior of cone Sobolev spaces H s,δ P,ϑ (X) under the action of nonlinear superposition.
Notation.In connection with pointwise multiplication, it is useful to employ the following notation: 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 s P,ϑ−0 (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.

Multiplication of asymptotic types
The result admitting nonlinear superposition for function spaces with asymptotics is stated first: 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) = Here, 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 .
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 Q.
From the proof of Lemma 3.1, where equality holds if Furthermore, it is also seen Q 1 for any multiplicatively closed asymptotic type Q, see also Lemma 3.4 below.

The class 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.
In case (a) to (c) are fulfilled, we have Proof.
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, Q ∈ As ♯ (Y ) and, therefore, Q ∈ As ♯ (Y ).
respectively.In particular, Q Proof.The proof is straightforward.

Documenta Mathematica 9 (2004) 207-250
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 ∈ As(Y ).Then pointwise multiplication induces a continuous map Proof.(a) is immediate from Proposition 3.8.To get (b), we argue as in the proof of Lemma 3.11.

Proposition 2 .
35.The characteristic char P of an asymptotic type P ∈ As δ prop (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 ji i , m ji+1 i , . . . ) e i=1

Remark 2 .
41. (a) The proof of Proposition 2.40 shows that the holomorphic S µ = {s µ−j ; j ∈ N} ∈ Ell Symb µ M (Y ) satisfying L δ S µ = 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 As δ prop (Y ) consists precisely of those asymptotic types that are of the form L δ S µ for some holomorphic S µ ∈ Ell Symb µ M (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 ∈ As δ prop (Y ) and Q ∈ As δ−µ prop (Y ).Then there exists a S µ ∈ Symb µ M (Y ) that is elliptic with respect to the weight δ such that L δ S µ = P and L δ−µ (S µ ) −1 = Q always when dim Y > 0 and if and only if

Definition 3 . 3 .Lemma 3 . 4 .
(a) The class As b (Y ) of bounded asymptotic types consists of all asymptotic types Q ∈ 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 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.For Q ∈ As(Y ), the following conditions are equivalent: (a) Q is multiplicatively closable; (b) the join k≥1 Q k does exist, where Q (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 Q exists and δ

Lemma 3 . 5 .
For each Q ∈ As(Y ), there are asymptotic types Q b ∈ As b (Y ) and Q ♯ ∈ As ♯ (Y ) which are maximal among all asymptotic types possessing the property Q

. 10 )
(b) If, in addition, R ∈ 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
The class of all Fuchsian differential operators of order µ on X • is denoted by Diff µ Fuchs (X).
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 An asymptotic type P ∈ As δ (Y ) is join-irreducible, i.e., P = O and P = P 0 ∨ P 1 for P 0 , P 1 ∈ 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 As δ (Y ).