Bulletin T.CXXXIII de l’Académie serbe des sciences et des arts − 2006 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 31

ON DUALITY THEORY AND PSEUDODIFFERENTIAL TECHNIQUES FOR

COLOMBEAU ALGEBRAS: GENERALIZED DELTA FUNCTIONALS, KERNELS AND WAVE FRONT SETS^{1}

CLAUDIA GARETTO, G. HÖRMANN^{2}

(Presented at the 1st Meeting, held on February 24, 2006)

A b s t r a c t. Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functionals and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis.

AMS Mathematics Subject Classification (2000): 46F30, 46A20, 47G30 Key Words: Colombeau generalized functions, duality theory, pseudodif-

ferential operators, microlocal analysis

1. Topological structures in Colombeau algebras and duality theory

Apart from some early and inspiring work by Biagioni, Pilipović, Scarpalézos [1, 16, 18, 19, 20], topological questions have played a marginal role in

1This paper was presented at the Conference GENERALIZED FUNCTIONS 2004, Topics in PDE, Harmonic Analysis and Mathematical Physics, Novi Sad, September 22– 28, 2004

2Supported by FWF grant P16820-N04

the existing Colombeau literature. However, the recent papers on pseudodifferential operators acting on algebras of generalized functions [2, 6, 7] and a preliminary kernel theory introduced in [6] motivate a renewed interest in topological issues concerning a class of spaces sufficiently large and general to model the most common Colombeau algebras. In this section we recall the major points of the theory of topological ℂ̃-modules and locally convex topological ℂ̃-modules developed in [4, 5]. As a topic of particular interest, the foundations of duality theory are provided within this framework, dealing with the ℂ̃-module L(G, ℂ̃) of all ℂ̃-linear and continuous functionals on G. In particular, due to the fact that many algebras of generalized functions can be easily viewed as locally convex topological ℂ̃-modules, all the previous theoretical concepts and results can be applied to the topological dual of a Colombeau algebra. We discuss generalized delta functionals as examples of elements of this type of dual space. Among generalized operator kernels those associated with Fourier integral operators have a rich structure, which is of relevance in applications such as microlocal analysis. Section 2 of the paper is devoted to a review of new methods and examples in this direction.

1.1 Topological and locally convex topological ℂ̃-modules

We begin by recalling some basic notions. The ring ℂ̃ of complex generalized numbers is defined factorizing the algebra E_{M} := {(u_{ε})_{ε} ∈ ℂ^{(0,1]} : ∃N ∈ ℕ |u_{ε}| = O(ε^{−N}) as ε → 0} with respect to the ideal N := {(u_{ε})_{ε} ∈ ℂ^{(0,1]} : ∀q ∈ ℕ |u_{ε}| = O(ε^{q}) as ε → 0} and can be endowed with a structure of a topological ring, making use of the function

v : E_{M} → (−∞, +∞] : (u_{ε})_{ε} → sup{b ∈ ℝ : |u_{ε}| = O(ε^{b}) as ε → 0} (1.1)

on E_{M}. The properties of v enables us to use (1.1) in defining the valuation

v_{ℂ}(u) := v((u_{ε})_{ε})  (1.2)

of the complex generalized number u = [(u_{ε})_{ε}]. Let now

| · |e := ℂ̃ → [0, +∞) : u → |u|e := e^{−v_{ℂ}(u)}.  (1.3)

The coarsest topology on ℂ̃ such that the map |·|e is continuous (the “sharp” topology on ℂ̃ according to [16, 18, 19, 20]) is compatible with the ring structure.

A topology τ on a ℂ̃-module (G, +) is said to be ℂ̃-linear if the addition G × G → G : (u, v) → u + v and the product ℂ̃ × G → G : (λ, u) → λu are continuous. A topological ℂ̃-module G is a ℂ̃-module with a ℂ̃-linear topology. Clearly ℂ̃ is a topological module on itself.

Our investigation of the topological aspects of a ℂ̃-module is based on the following ℂ̃-linear-adaptation of the notions of absorbent, balanced and convex subsets of a vector space.

Definition 1.1.. A subset A of a ℂ̃-module G is ℂ̃-absorbent if for all u ∈ G there exists a ∈ ℝ such that u ∈ [(ε^{b})_{ε}]A for all b ≤ a. A ⊆ G is ℂ̃-balanced if λA ⊆ A for all λ ∈ ℂ̃ with |λ|e ≤ 1. A ⊆ G is ℂ̃-convex if A + A ⊆ A and [(ε^{b})_{ε}]A ⊆ A for all b ≥ 0.

A subset A which is both ℂ̃-balanced and ℂ̃-convex is called absolutely ℂ̃-convex. The ℂ̃-convexity cannot be considered as a generalization of the corresponding concept in vector spaces. In fact the only subset A of ℂ which is ℂ̃-convex is the trivial set {0}. In the sequel, we shall simply talk of absorbent, balanced or convex subsets, omitting the prefix ℂ̃, when we deal with ℂ̃-modules.

As in the classical theory of topological vector spaces every topological ℂ̃-module has a base of absorbent and balanced neighborhoods of the origin. In detail, let G be a topological ℂ̃-module and U be a base of neighborhoods of the origin. Then for each U ∈ U, (i) U is absorbent, (ii) there exists V ∈ U with V + V ⊆ U, (iii) there exists a balanced neighborhood of the origin W such that W ⊆ U ([4, Proposition 1.3]). This fact ensures the following characterization: G is separated if and only if ∩_{U∈U}U = {0}. The notion of ℂ̃-convexity introduced above is employed in the definition of locally convex topological ℂ̃-module.

Definition 1.2.. A locally convex topological ℂ̃-module is a topological ℂ̃-module which has a base of ℂ̃-convex neighborhoods of the origin.

Combining this definition with the properties of bases of neighborhoods of the origin in topological ℂ̃-modules one can easily prove that every locally convex topological ℂ̃-module G has a base of absolutely convex and absorbent neighborhoods of the origin ([4, Proposition 1.7]).

We now want to deduce some more information on the topology of G from the nature of the neighborhoods. Our aim is to find a suitable ℂ̃-version of the Minkowski functional, and therefore a suitable ℂ̃-version of seminorm, which will allow to characterize the topology of a locally convex topological ℂ̃-module G as the topology determined by the “ℂ̃”-Minkowski functionals of all absolutely convex and absorbent subsets of G. This is possible by means of the notions of valuation and ultra-pseudo-seminorm.

Definition 1.3.. Let G be a ℂ̃-module. A valuation on G is a function v : G → (−∞, +∞] such that

(i) v(0) = +∞,

(ii) v(λu) ≥ v_{ℂ}(λ) + v(u) for all λ ∈ ℂ̃, u ∈ G,

(ii)’ v(λu) = v_{ℂ}(λ) + v(u) for all λ = [(cε^{a})_{ε}], c ∈ ℂ, a ∈ ℝ, u ∈ G,

(iii) v(u + v) ≥ min{v(u), v(v)}.

An ultra-pseudo-seminorm on G is a function P : G → [0, +∞) such that

(i) P(0) = 0,

(ii) P(λu) ≤ |λ|eP(u) for all λ ∈ ℂ̃, u ∈ G,

(ii)’ P(λu) = |λ|eP(u) for all λ = [(cε^{a})_{ε}], c ∈ ℂ, a ∈ ℝ, u ∈ G,

(iii) P(u + v) ≤ max{P(u), P(v)}.

P(u) = e^{−v(u)} is a typical example of an ultra-pseudo-seminorm obtained by means of a valuation on G. An ultra-pseudo-norm is an ultra-pseudo-seminorm P such that P(u) = 0 implies u = 0. | · |e introduced in (1.3) is an ultra-pseudo-norm on ℂ̃. In the language of ℂ̃-modules the Minkowski functional of a subset A becomes a “Minkowski valuation” v_{A}.

Proposition 1.4.. ([4]) Let A be an absolutely convex and absorbent subset of a ℂ̃-module G. Then

v_{A}(u) := sup{b ∈ ℝ : u ∈ [(ε^{b})_{ε}]A}  (1.4)

is a valuation on G. Moreover, for P_{A}(u) := e^{−v_{A}(u)} and η > 0 the chain of inclusions {u ∈ G : P_{A}(u) < η} ⊆ [(ε^{− log(η)})_{ε}]A ⊆ {u ∈ G : P_{A}(u) ≤ η} holds.

We usually call P_{A} the gauge of A. The properties which characterize an ultra-pseudo-seminorm together with Proposition 1.4 yield the following assertions:

(i) the topology induced by the family of ultra-pseudo-seminorms {P_{i}}_{i∈I} on G provides G with the structure of a locally convex topological ℂ̃-module.

(ii) In a locally convex topological ℂ̃-module G the original topology is induced by the family of ultra-pseudo-seminorms {P_{U}}_{U∈U}, where U is a base of absolutely convex and absorbent neighborhoods of the origin.

Remark 1.5. An inspection of the neighborhoods of the origin gives some information about “metrizability” and “normability”. More precisely the topology of a separated locally convex topological ℂ̃-module with a countable base of neighborhoods of the origin is induced by a metric invariant under translation and if a separated locally convex topological ℂ̃-module has a bounded neighborhood of the origin then its topology is induced by a ultra-pseudo-norm. By definition a subset A of G is bounded if and only if it is absorbed by every neighborhood U of the origin, i.e., there exists a ∈ ℝ such that A ⊆ [(ε^{b})_{ε}]U for all b ≤ a.

Ultra-pseudo-seminorms are particularly useful in proving the continuity of ℂ̃-linear maps between locally convex topological ℂ̃-modules.

Theorem 1.6 ([4]). Let (G, {P_{i}}_{i∈I}) and (H, {Q_{j}}_{j∈J}) be locally convex topological ℂ̃-modules. A ℂ̃-linear map T : G → H is continuous if and only if it is continuous at the origin if and only if for all j ∈ J there exists a finite subset I_{0} ⊆ I and a constant C > 0 such that for all u ∈ G

Q_{j}(T u) ≤ C max_{i∈I_{0}} P_{i}(u).  (1.5)

We conclude this subsection focusing our attention on examples of topological ℂ̃-modules from Colombeau’s theory of generalized functions, starting with the notion of strict inductive limit.

Example 1.7. Strict inductive limit of locally convex topological ℂ̃-modules

Consider a ℂ̃-module G and a sequence (G_{n})_{n∈ℕ} of submodules of G such that G_{n} ⊆ G_{n+1} for all n and G = ∪_{n∈ℕ}G_{n}. Assume that G_{n} is equipped with a locally convex ℂ̃-linear topology τ_{n} such that the topology induced by τ_{n+1} on G_{n} is τ_{n}. G endowed with the finest ℂ̃-linear locally convex topology τ which makes each embedding ι_{n} : G_{n} → G continuous is called strict inductive limit of the sequence (G_{n})_{n∈ℕ} of locally convex topological ℂ̃-modules. This topology is determined by the ultra-pseudo-seminorms {P_{U}}_{U} where U is any absolutely convex subset of G such that U ∩ G_{n} is a neighborhood of 0 in G_{n} for all n. By Proposition 1.21 in [4] we have that the topology τ on G induces the original topology τ_{n} on each G_{n} and that G is separated if each G_{n} is separated. Finally, under the additional assumption of G_{n} closed in G_{n+1} according to τ_{n+1}, one can prove that G is complete if and only if every G_{n} is complete and that A ⊆ G is bounded if and only if A is contained in some G_{n} and bounded there ([4, Theorems 1.32, 1.26]).

We recall that as in the classical theory of inductive limits a ℂ̃-linear map T : G → ℂ̃ is continuous if and only if T_{|_{Gn}} is continuous for all n ∈ ℕ ([4, Proposition 1.19]).

Example 1.8. The ℂ̃-module G_{E} of generalized functions based on a locally convex topological vector space E

Let E be a locally convex topological vector space topologized through the family of seminorms {p_{i}}_{i∈I}. The elements of

M_{E} := {(u_{ε})_{ε} ∈ E^{(0,1]} : ∀i ∈ I ∃N ∈ ℕ p_{i}(u_{ε}) = O(ε^{−N}) as ε → 0} (1.6)

and

N_{E} := {(u_{ε})_{ε} ∈ E^{(0,1]} : ∀i ∈ I ∀q ∈ ℕ p_{i}(u_{ε}) = O(ε^{q}) as ε → 0}, (1.7)

are called E-moderate and E-negligible, respectively. We define the ℂ̃- modules of generalized functions based on E as the factor space G_{E} := M_{E}/N_{E}. Since the growth in ε of an E-moderate net is estimated in terms of any seminorm p_{i} of E, it is natural to introduce the p_{i}-valuation of (u_{ε})_{ε} ∈ M_{E} as

v_{p_{i}}((u_{ε})_{ε}) := sup{b ∈ ℝ : p_{i}(u_{ε}) = O(ε^{b}) as ε → 0}.  (1.8)

It is immediate to see that (1.8) can be used for defining the p_{i}-valuation v_{p_{i}}(u) = v_{p_{i}}((u_{ε})_{ε}) of a generalized function u = [(u_{ε})_{ε}] ∈ G_{E}. In particular, v_{p_{i}} is a valuation in the sense of Definition 1.3 and thus P_{i}(u) := e^{−v_{p_{i}}(u)} is an ultra-pseudo-seminorm on the ℂ̃-module G_{E}. G_{E} endowed with the topology of the ultra-pseudo-seminorms {P_{i}}_{i∈I} is a locally convex topological ℂ̃-module. Following [16, 18, 19, 20] we use the adjective “sharp” for the topology induced by the ultra-pseudo-seminorms {P_{i}}_{i∈I}. The sharp topology on G_{E}, here denoted by τ_{♯} is independent of the choice of the family of seminorms which determines the original locally convex topology on E. (G_{E}, τ_{♯}) is a separated locally convex topological ℂ̃-module and it is complete when E has a countable base of neighborhoods of the origin ([4, Propositions 3.2, 3.4]).

Example 1.9. Colombeau algebras obtained as ℂ̃-modules G_{E} Particular choices of E give several known Colombeau algebras of gen-

eralized functions as ℂ̃-modules G_{E} and the corresponding sharp topologies. This is of course the case for E = ℂ and G_{E} = ℂ̃ which is an ultra-pseudo-normed ℂ̃-module.

Consider now an open subset Ω of ℝ^{n}. E = E(Ω), i.e. the space C^{∞}(Ω) topologized through the family of seminorms p_{K_{i},j}(f) = sup_{x∈K_{i},|α|≤j} |∂^{α}f(x)|, where K_{0} ⊂ K_{1} ⊂ ....K_{i} ⊂ ... is a countable and exhausting sequence of compact subsets of Ω, provides G_{E} = G(Ω). G(Ω) endowed with the sharp topology determined by {P_{K_{i},j}}_{i∈ℕ,j∈ℕ} is a Fréchet ℂ̃-module (i.e., metrizable and complete). Another example of a Fréchet ℂ̃-module is given by G_{E} when E is 𝒮 (ℝ^{n}). In this way we construct the algebra G_{𝒮} (ℝ^{n}) := G_{𝒮 (ℝ^{n})} (see [2, 6]).

Example 1.10. The Colombeau algebra Gc(Ω) of compactly supported generalized functions

For K ⋐ Ω we denote by G_{K}(Ω) the space of all generalized functions in G(Ω) with support contained in K. Note that G_{K}(Ω) is contained in G_{D_{K′}(Ω)} for all compact subsets K′ of Ω such that K ⊆ int(K′), where D_{K′}(Ω) is the space of all smooth functions f with supp f ⊆ K′. We recall that in G(Ω) the p_{K,n}-valuation where p_{K,n}(f) = sup_{x∈K,|α|≤n} |∂^{α}f(x)| is obtained as the valuation of the complex generalized number sup_{x∈K,|α|≤n} |∂^{α}u(x)| := (sup_{x∈K,|α|≤n} |∂^{α}u_{ε}(x)|)_{ε} + N. Hence for K, K′ ⋐ Ω, K ⊆ int(K′),

v_{K,n}(u) = v_{p_{K′,n}}(u)  (1.9)

is a valuation on G_{K}(Ω). More precisely (1.9) does not depend on K′. One can easily check that G_{K}(Ω), with the topology induced by the ultra-pseudo-seminorms {P_{G_{K}(Ω),n}(u) := e^{−v_{K,n}(u)}}_{n∈ℕ}, is a locally convex topological ℂ̃- module and by construction its topology coincides with the topology induced by any G_{D_{K′}(Ω)} with K ⊆ int(K′). In particular, (G_{K}(Ω), {P_{G_{K}(Ω),n}}_{n∈ℕ}) is a Fréchet ℂ̃-module. Let now (K_{n})_{n∈ℕ} be an exhausting sequence of compact subsets of Ω such that K_{n} ⊆ K_{n+1}. Clearly G_{c}(Ω) = ∪_{n∈ℕ}G_{K_{n}}(Ω) and the assumptions of Example 1.7 are satisfied by G_{n} = G_{K_{n}}(Ω). Therefore G_{c}(Ω) endowed with the strict inductive limit topology of the sequence (G_{K_{n}}(Ω))_{n} is a separated and complete locally convex topological ℂ̃-module.

Example 1.11. Algebras of regular generalized functions For any locally convex topological vector space (E, {p_{i}}_{i∈I}) the set

M_{E}^{∞} := {(u_{ε})_{ε} ∈ E^{(0,1]} : ∃N ∈ ℕ ∀i ∈ I p_{i}(u_{ε}) = O(ε^{−N}) as ε → 0}

(1.10)

is a subspace of the set M_{E} of E-moderate nets. Therefore the corresponding factor space G_{E}^{∞} := M_{E}^{∞}/N_{E} is a subspace of G_{E} whose elements are called regular generalized functions based on E. The moderateness properties of M_{E}^{∞} allow us to define the valuation v_{E}^{∞} : M_{E}^{∞} → (−∞, +∞] as

v_{E}^{∞}((u_{ε})_{ε}) = sup{b ∈ ℝ : ∀i ∈ I p_{i}(u_{ε}) = O(ε^{b}) as ε → 0}

which can be obviously extended to G_{E}^{∞}. This yields the existence of the ultra-pseudo-norm P_{E}^{∞}(u) := e^{−v_{E}^{∞}(u)} on G_{E}^{∞} which determines a topology finer than the one induced by G_{E}. One can prove that when E has a countable base of neighborhoods of the origin then the associated space G_{E}^{∞} of regular generalized functions is a complete and ultra-pseudo-normed ℂ̃-module. A concrete example of G_{E}^{∞} is given by the Colombeau algebra of 𝒮 -regular generalized functions G_{𝒮}^{∞}(ℝ^{n}) (see [2, 6]), whose definition is precisely G_{E}^{∞} with E = 𝒮 (ℝ^{n}).

The Colombeau algebra G^{∞}(Ω) of regular generalized functions can be seen as the intersection ∩_{K⋐Ω}G^{∞}(K), where G^{∞}(K) is the space of all u ∈ G(Ω) such that there exists a representative (u_{ε})_{ε} satisfying the condition:

∃N ∈ ℕ ∀α ∈ ℕ^{n} sup_{x∈K} |∂^{α}u_{ε}(x)| = O(ε^{−N}) as ε → 0.

On G^{∞}(K) we define the ultra-pseudo-seminorm P_{G^{∞}(K)} via the valuation v_{G^{∞}(K)}(u) = sup{b ∈ ℝ : ∀α ∈ ℕ^{n} sup_{x∈K} |∂^{α}u_{ε}(x)| = O(ε^{b})}. It follows that G^{∞}(Ω) is a Frechét ℂ̃-module if topologized trough the family of ultra-pseudo-seminorms P_{G^{∞}(K_{n})}, where {K_{n}}_{n∈ℕ} is an arbitrary exhausting sequence of compact subsets of Ω.

Finally, a strict inductive limit procedure equips Gc^{∞}(Ω) = ∪_{n∈ℕ}G_{K_{n}}^{∞}(Ω) with a complete and separated locally convex topology. Each G_{K_{n}}^{∞}(Ω) := G_{K}(Ω) ∩ G^{∞}(Ω) is a complete ultra-pseudo-normed ℂ̃-module with respect to P_{G_{Kn}^{∞}(Ω)}(u) = e^{− v_{Kn}^{∞}(u)}, where v_{K_{n}}^{∞}(u) = sup{b ∈ ℝ : ∀α ∈ ℕ^{n} sup_{x∈K′_{n}} |∂^{α}u_{ε}(x)| = O(ε^{b})} and K_{n} ⊆ int(K′_{n}) ⊆ K′_{n} ⋐ Ω.

1.2. Duality theory for topological ℂ̃-modules

This subsection is devoted to the dual of a topological ℂ̃-module G i.e., the ℂ̃-module L(G, ℂ̃) of all ℂ̃-linear and continuous maps on G with values in ℂ̃. Since G with L(G, ℂ̃) forms a pairing, the topological dual of G can be endowed with the weak topology σ(L(G, ℂ̃), G) that is the coarsest topology such that each map ⟨u, ·⟩ : L(G, ℂ̃) → ℂ̃ : T → T (u) is continuous for u varying in G. This is a separated locally convex ℂ̃-linear topology determined by the family of ultra-pseudo-seminorms P_{u}(T) = |T (u)|e.

Before introducing other topologies on the dual L(G, ℂ̃) we recall that the polar of a subset A of G is the set A^{∘} of all T ∈ L(G, ℂ̃) such that |T (u)|_{ε} ≤ 1 for all u ∈ A. As proved in [4, Proposition 2.4] A^{∘} is a balanced and convex subset of L(G, ℂ̃) closed with respect to σ(L(G, ℂ̃), G). Moreover A^{∘} is absorbent if and only if A is bounded in (G, σ(G, L(G, ℂ̃))). It follows that for any σ(G, L(G, ℂ̃))-bounded subset of G we can define the gauge P_{A^{∘}} according to Proposition 1.4.

Definition 1.12. Let G be a topological ℂ̃-module. We call strong topology the topology β(L(G, ℂ̃), G) determined by the ultra-pseudo-seminorms {P_{A^{∘}}}_{A∈A} where A is the collection of all σ(G, L(G, ℂ̃))-bounded subsets of G. Restricting A to the family of all bounded subset of G we define the topology β_{b}(L(G, ℂ̃), G) of uniform convergence on bounded subsets of G.

It is clear that σ(L(G, ℂ̃), G) ⪯ β_{b}(L(G, ℂ̃), G) ⪯ β(L(G, ℂ̃), G), where ⪯ stands for “is coarser than”. Proposition 3.26 in [4] proves that β_{b}(L(G, ℂ̃), G) and β(L(G, ℂ̃), G) coincide when G is a ℂ̃-module of generalized functions based on a normed space E. We now focus on some special families of topological ℂ̃-modules. In order to state the following definition we recall that a subset S of G is said to be bornivourus if it absorbs every bounded subset of G. Finally S is a barrel of G if it is absorbent, balanced, convex and closed.

Definition 1.13. A locally convex topological ℂ̃-module G is bornological if every balanced, convex and bornivorous subset of G is a neighborhood of the origin. A locally convex topological ℂ̃-module is barrelled if every barrel is a neighborhood of the origin.

The Frechét ℂ̃-modules are examples of bornological and barrelled ℂ̃- modules as well as the strict inductive limit of Frechét ℂ̃-modules (see [4, Propositions 2.9, 2.14, 2.15]). Moreover by Proposition 2.10 in [4] we have that the dual of a bornological ℂ̃-module is complete with respect to both the topologies β(L(G, ℂ̃), G) and β_{b}(L(G, ℂ̃, G).

1.3. Topological dual of a Colombeau algebra: generalized delta functionals and operator kernels

The theoretical background provided by the previous subsection can be applied to the topological duals of the Colombeau algebras Gc(Ω), G(Ω) and G_{𝒮} (ℝ^{n}). Since G(Ω) and G_{𝒮} (ℝ^{n}) are Frechét ℂ̃-modules and Gc(Ω) is a strict inductive limit of Frechét ℂ̃-modules we know that the duals L(Gc(Ω), ℂ̃), L(G(Ω), ℂ̃) and L(G_{𝒮} (ℝ^{n}), ℂ̃) are complete locally convex topological ℂ̃- modules when equipped with the strong topology or with the topology of uniform convergence on bounded subsets. As an analogy with distribution theory we recall that L(Gc(Ω), ℂ̃) is a sheaf on Ω and that L(G(Ω), ℂ̃) can be identified with the set of all functionals in L(Gc(Ω), ℂ̃) having compact support ([5, Theorem 1.2]).

Point value theory for generalized functions [8, 17] and a useful characterization of the ideals of negligible nets which occur in the definition of some Colombeau algebras as G(Ω), G_{𝒮} (ℝ^{n}), G_{p,p}(ℝ^{n}) := G_{W^{∞,p}(ℝ^{n})} (see [5, 8]) are the tools employed in proving the following embedding theorem. The Colombeau algebras considered in the sequel are endowed with the topologies discussed in Subsection 1.1 and the corresponding duals are equipped with the topology β_{b} of uniform convergence on bounded subsets. It is clear that the identity between representatives defines the ℂ̃-linear continuous embeddings Gc(Ω) ⊆ G(Ω), G^{∞}(Ω) ⊆ G(Ω), Gc^{∞}(Ω) ⊆ Gc(Ω) and G_{𝒮}^{∞}(ℝ^{n}) ⊆ G_{𝒮} (ℝ^{n}).

Theorem 1.14 ([5]). By integration we obtain ℂ̃-linear continuous embeddings, u → (v → ∫_{Ω} u(y)v(y) dy), of

(i) G(Ω) into L(Gc(Ω), ℂ̃),

(ii) Gc(Ω) into L(G(Ω), ℂ̃),

(iii) G_{𝒮} (ℝ^{n}) into L(G_{𝒮} (ℝ^{n}), ℂ̃).

Theorem 1.14 yields the following chains of continuous embeddings G^{∞}(Ω) ⊆ G(Ω) ⊆ L(Gc(Ω), ℂ̃), Gc^{∞}(Ω) ⊆ Gc(Ω) ⊆ L(G(Ω), ℂ̃), G_{𝒮}^{∞}(ℝ^{n}) ⊆ G_{𝒮} (ℝ^{n}) ⊆ L(G_{𝒮} (ℝ^{n}), ℂ̃), which play an important role in regularity theory of differential and pseudodifferential operators with generalized symbols (c.f. [3, 6]). Concluding we discuss some interesting examples of functionals in L(Gc(Ω), ℂ̃), L(G(Ω), ℂ̃) and L(G_{𝒮} (ℝ^{n}), ℂ̃). They are mainly provided by point value theory and kernel theory for Fourier integral operators with generalized symbols.

Example 1.15. The generalized delta functional δx̃

Let x̃ be a generalized point in Ω̃_{c}. We can define a ℂ̃-linear map δx̃ : G(Ω) → ℂ̃ associating with each u ∈ G(Ω) its point value u(x̃) at x̃. It is clear that δx̃ belongs to L(G(Ω), ℂ̃) since |δx̃(u)|e ≤ e^{−v(sup_{x∈K} |u_{ε}(x)|)}, where (x_{ε})_{ε} is a representative of x̃ contained in a compact set K for ε small enough.

Since every u ∈ Gc(Ω) is a generalized function in some G_{D_{K′}(Ω)} it is meaningful to define the point value u(x̃) even when x̃ is in Ω̃ \ Ω̃_{c}. In this case δx̃ is a ℂ̃-linear map from Gc(Ω) into ℂ̃ and by |δx̃(u)|e ≤ P_{K,0}(u) on G_{K}(Ω) it is continuous. Finally, as proved in [5], a point value theory with generalized points in ℝ̃^{n} can be formulated for generalized functions in G_{𝒮} (ℝ^{n}). One easily sees that for x̃ ∈ ℝ̃^{n} the functional δx̃ : G_{𝒮} (ℝ^{n}) → ℂ̃ belongs to L(G_{𝒮} (ℝ^{n}), ℂ̃).

Remark 1.16. Properties of δx̃

Let x̃ = [(x_{ε})_{ε}] ∈ Ω̃. Defining supp x̃, the support supp x̃ of x̃, to be the complement of the set

\{x_{0} ∈ Ω : \begin{matrix}∃V (x_{0}) neighborhood of x_{0} ∃η ∈ (0, 1] ∀ε ∈ (0, η] \\ x_{ε} ∉ V (x_{0}) \end{matrix} \} , (1.11)

in Ω, one can prove that supp δx̃ = supp x̃. Note that by (1.11) the support of x̃ ∈ Ω̃ is the set of all accumulation points of a representing net (x_{ε})_{ε}. As a consequence and differently from the distributional case, δx̃ may have support empty or unbounded. Another difference to distribution theory consists in the existence of elements of L(G(Ω), ℂ̃) having support {0} which are not ℂ̃-linear combinations of δ_{0} and its derivatives. As an example take [(ε)_{ε}] ∈ ℝ̃ and δ_{[(ε)_{ε}]} ∈ L(G(ℝ), ℂ̃). By (1.11) supp δ_{[(ε)_{ε}]} = {0} but for any m ∈ ℕ and any choice of c_{i} ∈ ℂ̃ the equality δ_{[(ε)_{ε}]} = ∑_{i=0}^{m} c_{i}δ_{0}^{(i)} does not hold.

The following theorem proves that the restriction of the generalized delta functional δx̃ to the set of regular generalized functions is an integral operator. This implies an integral representation of regular generalized functions as in [5, Theorems 2.3, Propositions 2.6].

Theorem 1.17.

(i) For all x̃ ∈ Ω̃_{c} there exists v ∈ Gc(Ω) such that for all u ∈ G^{∞}(Ω)

u(x̃) = ∫_{Ω} v(y)u(y) dy.  (1.12)

(ii) (i) holds with Ω̃, G(Ω) and Gc^{∞}(Ω) in place of Ω̃_{c}, Gc(Ω), and G^{∞}(Ω), respectively.

(iii) For all x̃ ∈ ℝ̃^{n} there exists v ∈ G_{𝒮} (ℝ^{n}) such that for all u ∈ G_{𝒮}^{∞}(ℝ^{n})

u(x̃) = ∫_{ℝ^{n}} v(y)u(y) dy.  (1.13)

The convolution with a mollifier in 𝒮 (ℝ^{n}) is the basic idea in proving Theorem 1.17. Indeed, in (iii) we have that v is given by [(φ_{ε}(x_{ε} − ·))_{ε}] ∈ G_{𝒮} (ℝ^{n}), where (x_{ε})_{ε} is any representative of x̃. In general (1.12) and (1.13) do not hold when u is not regular. See [5, Remarks 2.2, 2.5] for details.

Example 1.18. Generalized kernels

The ℂ̃-linear functional on Gc(Ω) defined by the (generalized) integral

∫_{Ω} k(y)u(y)dy

is called integral operator with kernel k ∈ G(Ω). It is an element of L(Gc(Ω), ℂ̃), since with u ∈ G_{K}(Ω), K ⊆ int(K′) ⊆ K′ ⋐ Ω, the estimate

|∫_{Ω} k_{ε}(y)u_{ε}(y) dy| ≤ c sup_{y∈K′} |k_{ε}(y)| sup_{y∈K′} |u_{ε}(y)|

yields

|∫_{Ω} k(y)u(y) dy|e ≤ C P_{G_{K}(Ω),0}(u),

where C = e^{−v(sup_{y∈K′} |k_{ε}(y)|)}. Analogously we prove that u → ∫_{Ω} k(y)u(y) dy belongs to L(G(Ω), ℂ̃) when k ∈ Gc(Ω) and that the functional u → ∫_{ℝ^{n}} k(y)u(y)dy is an element of L(G_{𝒮} (ℝ^{n}), ℂ̃) for k ∈ G_{𝒮} (ℝ^{n}).

Let us now consider a phase function ϕ on Ω × ℝ^{p} and a generalized symbol a = [(a_{ε})_{ε}] ∈ S̃_{ρ,δ}^{m}(Ω × ℝ^{p}) as defined in [6, Definition 3.3] with ρ > 0 and δ < 1. The generalized oscillatory integral

I_{ϕ}(au) := ∫_{Ω×ℝ^{p}} e^{iϕ(y,ξ)}a(y, ξ)u(y) dy dξ := [(∫_{Ω×ℝ^{p}} e^{iϕ(y,ξ)}a_{ε}(y, ξ)u_{ε}(y) dy dξ)_{ε}]

is an example of a functional in L(Gc(Ω), ℂ̃). In fact, denoting the symbol seminorm sup_{y∈K′,ξ∈ℝ^{p}} sup_{|α+β|≤j}⟨ξ⟩^{−m+ρ|α|−δ|β|}|∂_{ξ}^{α}∂_{y}^{β}a_{ε}(y, ξ)| by |a_{ε}|_{K′,j,ρ,δ}^{(m)} and working on I_{ϕ}(au) at the level of representatives we have that

|∫_{Ω×ℝ^{p}} e^{iϕ(y,ξ)}a_{ε}(y, ξ)u_{ε}(y) dy dξ| ≤ c_{j,K}|a_{ε}|_{K′,j,ρ,δ}^{(m)} sup_{y∈K′,|γ|≤j} |∂^{γ}u_{ε}(y)|

is valid for u ∈ G_{K}(Ω), m − js < −p and for some constant c_{j,K} depending on j and K.

When ϕ(x, y, ξ) is a phase function on Ω′ × Ω × ℝ^{p} and, in addition, a phase function in (y, ξ) for all x, the map A : Gc(Ω) → G(Ω′) given by

\begin{matrix}Au(x) = ∫_{Ω×ℝ^{p}} e^{iϕ(x,y,ξ)}a(x, y, ξ)u(y) dy d^{−}ξ \\ := [(∫_{Ω×ℝ^{p}} e^{iϕ(x,y,ξ)}a_{ε}(x, y, ξ)u_{ε}(y) dy d^{−}ξ)_{ε}] \end{matrix}

is called Fourier integral operator with generalized amplitude a ∈ S̃_{ρ,δ}^{m}(Ω′ × Ω × ℝ^{p}) ([6, Proposition 3.10]). The kernel of A is the ℂ̃-linear functional on Gc(Ω′ × Ω) determined by

k_{A}(u) := ∫_{Ω′} A(u(x, ·)) dx.

Since k_{A} can be expressed by the oscillatory integral

∫_{Ω′×Ω×ℝ^{p}} e^{iϕ(x,y,ξ)}a(x, y, ξ)u(x, y) dy dx d^{−}ξ

in dx, dy, dξ, from the previous reasoning it follows that k_{A} belongs to the dual L(Gc(Ω′ × Ω), ℂ̃). We recall that when Ω′ = Ω and ϕ(x, y, ξ) = (x − y)ξ we obtain the kernel of a generalized pseudodifferential operator.

2. Pseudodifferential techniques and microlocal analysis

2.1. The wave front set

Let u be a distribution or generalized function on Ω. In both cases, the basic idea of the wave front set of u, WF(u), can be sketched as follows: it consists of pairs (x; ξ) in Ω × ℝ^{d} ≅ T^{∗}Ω, where x_{0} marks the location of singularities in Ω, whereas ξ_{0} ≠ 0 in ℝ^{d} represents the directions of highfrequency contributions in their Fourier spectrum. The focusing procedure for this “localization with attached directional spectral analysis” can be based on spatial cut-off functions φ ∈ D(Ω) and cones Γ ⊆ ℝ^{d}\0, where rapid decrease of the corresponding Fourier transform F(φu) is to be tested. To be more precise, the microlocal regularity test for u = [(u_{ε})_{ε}] at (x_{0}; ξ_{0}) ∈ T^{∗}(Ω)\0 (cotangent space deprived of its zero section) at scale 0 < γ_{ε} ↗ ∞ (ε → 0) requires the following: find an open neighborhood U of x_{0} and an open cone ξ_{0} ∈ Γ ⊆ ℝ^{d}\0 such that for all φ ∈ D(U) the generalized Fourier transform F(φu) is G-rapidly decreasing with scale γ in Γ, i.e., ∃N ∀l ∃ε_{0} such that

|F(φu_{ε})(ξ)| ≤ γ_{ε}^{N}(1 + |ξ|)^{−l} ∀ξ ∈ Γ, ∀ε ∈ (0, ε_{0}).

The complement of the set of regular points, denoted by WF_{g}^{γ}(u), is the generalized wave front set of u (at scale γ). If v ∈ D′(Ω) then we have full consistency with the distribution theoretic notion in the sense that WF_{g}^{γ}(ι(v)) = WF(v). In particular, the G^{∞}-regular embedded distributions are precisely the smooth functions.

The distributional wave front set can be described (in fact, was originally defined in [9]) in terms of characteristic sets of pseudodifferential operators: If m ∈ ℝ arbitrary then for any v ∈ D′

WF(v) = ⋂ ; \begin{matrix}A ∈ Ψ^{m} \\ Av ∈ C^{∞} \end{matrix} Char(A).  (2.14)

Here, Char(A) is the characteristic set of A = a(x, D) (a ∈ S^{m} the space of smooth symbols of order m and type (0, 1)), i.e., the complement of those points (x_{1}, ξ_{1}), where a is (micro-)elliptic in the sense that an estimate

|a(x, ξ)| ≥ C(1 + |ξ|)^{m}

holds for x near x_{1} and ξ with |ξ| ≥ 1 in some conic neighborhood of ξ_{1}. Equation (2.14) is an efficient distribution theoretic lever when applied to questions of noncharacteristic regularity or propagation of singularities for solutions to partial differential equations, or to directly deduce geometric invariance properties of the wave front set.

Slow scale micro-ellipticity: A key property for generalized symbols is that of slow scale growth, which has already been used in several contexts of regularity theory (cf. [12, 13, 6, 7]). The basic measurement is done by estimating with elements of the following set of strongly positive slow scale nets:

Π_{sc} := {(ω_{ε})_{ε} ∈ ℝ^{(0,1]} : ∃c > 0 ∀p ≥ 0 ∃c_{p} > 0 ∀ε : c ≤ ω_{ε} and ω_{ε}^{p} ≤ c_{p} ε^{−1}}.

Definition 2.1. Let m be a real number. The set \underline{S}sc^{m}(Ω × ℝ^{n}) of slow scale nets of symbols of order m consists of all (a_{ε})_{ε} ∈ S^{m}[Ω×ℝ^{n}] such that:

\begin{matrix}∀K ⋐ Ω ∃(ω_{ε})_{ε} ∈ Π_{sc}∀α, β ∈ ℕ^{n} ∃c > 0 ∀ε : \\ sup_{x∈K,ξ∈ℝ^{n}}⟨ξ⟩^{−m+|α|}|∂_{ξ}^{α}∂_{x}^{β}a_{ε}(x, ξ)| ≤ c ω_{ε}. \end{matrix}

The subset \underline{N}^{m}(Ω × ℝ^{n}) of negligible nets of symbols of order m is defined by (a_{ε})_{ε} ∈ S^{m}[Ω × ℝ^{n}] with the property

\begin{matrix}∀K ⋐ Ω ∀α, β ∈ ℕ^{n} ∀q ∈ ℕ, ∃c > 0 ∀ε : \\ sup_{x∈K,ξ∈ℝ^{n}}⟨ξ⟩^{−m+|α|}|∂_{ξ}^{α}∂_{x}^{β}a_{ε}(x, ξ)| ≤ c ε^{q}. \end{matrix}

The elements of the factor space \underline{S̃}sc^{m}(Ω × ℝ^{n}) := \underline{S}sc^{m}(Ω × ℝ^{n})/\underline{N}^{m}(Ω × ℝ^{n}) are the classes of slow scale generalized symbols of order m.

Slow scale generalized symbols enable us to design a simple and useful notion of micro-ellipticity (cf. [7]).

Definition 2.2. Let a ∈ \underline{S̃}sc^{m}(Ω × ℝ^{n}) and (x_{0}, ξ_{0}) ∈ T^{∗}(Ω) \ 0. We say that a is slow scale micro-elliptic at (x_{0}, ξ_{0}) if it has a representative (a_{ε})_{ε} satisfying the following: there is a relatively compact open neighborhood U of x_{0}, a conic neighborhood Γ of ξ_{0}, and (r_{ε})_{ε}, (s_{ε})_{ε} in Π_{sc} such that

|a_{ε}(x, ξ)| ≥ \frac{1}{s_{ε}} ⟨ξ⟩^{m}  (x, ξ) ∈ U × Γ, |ξ| ≥ r_{ε}, ε ∈ (0, 1]. (2.15)

We denote by Ellsc(a) the set of all (x_{0}, ξ_{0}) ∈ T^{∗}(Ω) \ 0 where a is slow scale micro-elliptic. The symbol a is called slow scale elliptic if there exists (a_{ε})_{ε} ∈ a such that (2.15) holds at all points in T^{∗}(Ω) \ 0.

In case of classical symbols the notion of slow scale micro-ellipticity coincides with the classical one, which is defined equivalently as the set of noncharacteristic points. More general definitions of ellipticity have been investigated in [2, 6, 12, 13]. But the above slow scale variant has a stability property, similar to classical ellipticity. In fact, due to the overall slow scale conditions in Definition 2.1, any symbol which is slow scale micro-elliptic at (x_{0}, ξ_{0}) fulfills the stronger hypoellipticity estimates [6, Definition 6.1]; furthermore, 2.15 is stable under lower order (slow scale) perturbations (cf. [7, Proposition 1.3]), and the simple slow scale ellipticity condition in Definition 2.2 already guarantees the existence of a parametrix.

Pseudodifferential characterization of the generalized wave front set: Let _{pr}Ψsc^{m}(Ω) denote the set of all properly supported operators a(x, D) where a belongs to \underline{S̃}sc^{m}(Ω × ℝ^{n}).

Theorem 2.3 ([7]). Let m ∈ ℝ arbitrary. For every u ∈ G(Ω)

WFg(u) = ⋂_{{a(x,D)∈ _{pr}Ψsc^{m}(Ω) \atop a(x,D)u ∈ G^{∞}(Ω)}}Ellsc(a)c = ⋂_{{A∈Ψ^{m}(Ω) \atop Au ∈ G^{∞}(Ω)}}Char(A).

Noncharacteristic G^{∞}-regularity: A first application of Theorem 2.3 is the following extension of classical noncharacteristic regularity for solutions of pseudodifferential equations (cf. [10, Theorem 18.1.28]).

Theorem 2.4 ([7]). If P = p(x, D) is a properly supported pseudodifferential operator with slow scale symbol and u ∈ G(Ω) then

WFg(P u) ⊆ WFg(u) ⊆ WFg(P u) ∪ Ellsc(p)c.  (2.16)

As can be seen from various examples in [12], relation (2.16) does not hold in general for regular symbols p which satisfy estimate (2.15). In this sense, the overall slow scale properties of the symbol are crucial in the above statement and are not just technical convenience. In fact, adapting the reasoning in [12, Example 4.6] to the symbol p_{ε}(x, ξ) = 1 + c_{ε}x^{2}, c_{ε} ≥ 0, we obtain the following: p_{ε}(x, ξ) ≥ 1 whereas the unique solution u of pu = 1 is G^{∞} if and only if (c_{ε})_{ε} is a slow scale net.

2.2. Propagation of G^{∞}-singularities

1. The case of smooth coefficients: In [7] the pseudodifferential approach to wave front sets provides the key tool to extend a distributional result on propagation of singularities. We allow for Colombeau generalized functions as solutions and initial values in first-order strictly hyperbolic partial differential equations with smooth coefficients. Hyperbolicity will be assumed with respect to time direction, corresponding to the variable t.

Theorem 2.5 ([7]). Let P (t, x, D_{x}) be a first-order partial differential operator with real principal symbol p_{1} and coefficients in C^{∞}(ℝ × Ω), which are constant outside some compact subset of Ω. Let u ∈ G(Ω × ℝ) be the (unique) solution to the homogeneous Cauchy problem

D_{t}u + P (t, x, D_{x})u = 0  (2.17)
u(., 0) = g ∈ G(Ω).  (2.18)

1. If Φ_{t} denotes the Hamilton flow corresponding to p_{1}(t, ., .) on T^{∗}(Ω) then we have for all t ∈ ℝ

WFg(u(., t)) = Φ_{t}(WFg(g)).  (2.19)

2. Denote by γ(x_{0}, ξ_{0}) the maximal bicharacteristic curve (of Q := D_{t} + P) passing through (x_{0}, 0; ξ_{0}, −P_{1}(0, x_{0}, ξ_{0})) ∈ Char(Q) ⊆ T^{∗}(Ω × ℝ) \ 0. Then the generalized wave front of u is given by

WFg(u) = ⋃_{(x_{0},ξ_{0})∈WFg(g)} γ(x_{0}, ξ_{0}).  (2.20)

2. Colombeau-type coefficients and generalized (null) bicharacteristics:

We study global problems with Ω = ℝ^{n}. Consider the first-order partial differential operator

P (t, x; D_{x}) = ∑_{j=1}^{n} a_{j}(x, t)D_{x_{j}} + a_{0}(x, t),

where a_{k} ∈ G(ℝ^{n+1}) (k = 0, …, n) and a := (a_{1}, …, a_{n}) ∈ G(ℝ^{n+1})^{n} is real, in the sense that each component has real-valued representatives. In addition, we make the following assumptions on the coefficients:

1. a_{0}, …, a_{n} are equal to real constants for large |x|,

2. ∂_{x_{j}}a_{k} (k = 1, …, n; j = 1, …, n) as well as a_{0} are of log-type (i.e., satisfy asymptotic L^{∞}-estimates with bound O(log(1/ε))).

Then we deduce from [15] that the corresponding Cauchy problem

Qu := D_{t}u + P u = f ∈ G(ℝ^{n+1})  (2.21)
u |_{t=0} = g ∈ G(ℝ^{n})  (2.22)

has a unique solution u ∈ G(ℝ^{n+1}). The, now, generalized principal symbols p_{1}(t, ., .) = a(., t) · ξ ∈ G(T^{∗}(Ω)) (∀t ∈ ℝ) and q_{1} = τ + p_{1} ∈ G(T^{∗}(ℝ^{n+1})) define the generalized Hamilton vector fields

H_{p_{1}(t)} = (a(., t); −d_{x}a(., t) · ξ) and H_{q_{1}} = (a, 1; −d_{x}a · ξ, −∂_{t}a · ξ)

on T^{∗}(ℝ^{n}) and T^{∗}(ℝ^{n+1}) respectively. Let (x_{0}, t_{0}; ξ_{0}, 0) ∈ T^{∗}(ℝ^{n+1}) satisfy q_{1}(x_{0}, 0, ξ_{0}, τ_{0}) = 0 (in ℂ̃) and consider the bicharacteristic system of generalized ordinary differential equations. Note that, due to the simple symbol structure with respect to τ, the t-component may be used as parameter along the curves; thus, the equations read

ẋ(t) = a(x(t), t)  x(0) = x_{0}  (2.23)
ξ̇(t) = −d_{x}a(x(t), t) · ξ(t)  ξ(0) = ξ_{0}  (2.24)
τ̇(t) = −∂_{t}a(x(t), t) · ξ(t)  τ(0) = τ_{0}.  (2.25)

Here, (2.23) is independent of the other equations and is uniquely solvable for x ∈ G(ℝ)^{n} by [8, Theorem 1.5.2]. Since d_{x}a is of log-type and Equation (2.24) is linear with respect to ξ, the standard Gronwall estimates (in terms of vector and matrix norms) yield unique existence of the generalized solution ξ ∈ G(ℝ)^{n}. Finally, τ is obtained by simply integrating (2.25). We summarize this in the following statement.

Proposition 2.6. Under Assumptions (i) and (ii) above for the coefficients of the operator Q = D_{t} + P (t, x, D_{x}) there is a unique generalized bicharacteristic curve β ∈ G(ℝ)^{2n+2}, β(t) = (x(t), t; ξ(t), τ(t)), passing through every point (x_{0}, 0; ξ_{0}, τ_{0}) ∈ T^{∗}(ℝ^{n+1}) at which the principal symbol q_{1} vanishes.

Remark 2.7. (i) As a matter of fact, (2.23) establishes a generalized (c-bounded) flow on ℝ^{n} in the sense of [14], whereas (2.23-2.24) defines a (not necessarily c-bounded) flow on T^{∗}(ℝ^{n}) as in [8, Equations (1.33-35)] (and without satisfying the hypothesis of Theorem 1.5.12 there, since the (x, ξ)-gradient of the vector field H_{p_{1}(t)} need not be of log-type). (ii) Observe that, by construction, we have that

q_{1}(x(t), t, ξ(t), τ(t)) = 0 in ℝ̃  ∀t ∈ ℝ,  (2.26)

i.e., the Hamilton function vanishes along the generalized null bicharacteristics.

We may then ask, if the generalized bicharacteristic flow still describes the propagation of microlocal singularities in a similar way as Theorem 2.5 illustrates this for the case of smooth coefficients. In other words: How does the generalized wave front set, WFg(u), of the solution u to (2.21-2.22) relates to the generalized bicharacteristic curves (or the flow) obtained in Proposition 2.6?

On a conceptual level, one immediately realizes that a direct set theoretic relation is not meaningful since WFg(u) ⊆ T^{∗}(ℝ^{n+1}) ≅ ℝ^{2n+2}, whereas the bicharacteristic curves give generalized point values β(t) ∈ ℝ̃^{2n+2} at fixed t ∈ ℝ. Here, we will only touch upon this new research issue by discussing a simple example in space dimension n = 1.

The Hurd-Sattinger Example revisited: Let ρ ∈ Cc^{∞}(ℝ) be symmetric, nonnegative, with supp (ρ) ⊆ [−1, 1] and ∫ ρ = 1. We use the (slow) scale γ_{ε} := log(1/ε), put ρ^{ε}(x) := γ_{ε}ρ(γ_{ε}x), and define (H denoting the Heaviside function)

Θ := [(H(−.) ∗ ρ^{ε})_{ε∈(0,1]}] ≈ H(−x)

considered as element in G(ℝ^{2}) (i.e., independent of the t-variable). Note that Θ as well as ∂_{x}Θ = Θ′ are of log-type and that Θ equals 1 when x < −1 and equals 0 when x > 1.

Let u ∈ G(ℝ^{2}) be the solution to the Cauchy problem

Qu := D_{t}u + Θ∂_{x}u + Θ′u = 0  (2.27)
u |_{t=0} = d_{−s_{0}}  (2.28)

where s_{0} > 0 and d_{−s_{0}} = [(ρ^{ε}(. + s_{0}))_{ε}] ≈ δ_{−s_{0}} (the Dirac measure at −s_{0}). The bicharacteristic differential equations read

ẋ(t) = Θ(x(t))  x(0) = x_{0}  (2.29)
ξ̇(t) = −Θ′(x(t))ξ(t)  ξ(0) = ξ_{0}  (2.30)
τ̇(t) = 0 ⇒ τ(t) = τ_{0} ∀t.  (2.31)

Integrating (2.30) we get

ξ_{ε}(t) = ξ_{0} exp ( −∫_{0}^{t} Θ′_{ε}(x_{ε}(r)) dr) = ξ_{0} exp ( ∫_{0}^{t} ρ_{ε}(−x_{ε}(r)) dr). (2.32)

Note that, by construction, we have the following characteristic relation along the bicharacteristics: for all t and q

(q_{1}(x(t), t, ξ(t), τ(t)))_{ε} = τ_{ε}(t) + Θ_{ε}(x_{ε}(t)) ξ_{ε}(t) = O(ε^{q}) (ε → 0). (2.33)

A detailed description of the solution to (2.29), the characteristic flow, and an analysis of the generalized wave front set of u can be found in [11].

To be precise, the microlocal regularity properties are studied there on the scale γ_{ε}, indicated by the notation WFg^{γ}(u), in order to correctly trace back the influence of the “original distributional” coefficient singularity at x = 0. Accordingly, the initial value singularity at x_{0} = −s_{0} < 0 propagates towards the axis x = 0 with speed 1 as long as x < 0 and “gets stuck” at x = 0 from time t = s_{0} on. The wave front set turns out to be

WFg^{γ}(u) = ({t = x + s_{0}, x < 0} × {ξ + τ = 0, (ξ, τ) ≠ (0, 0)})

∪({(0, s_{0})} × (ℝ^{2} \ {(0, 0)}))

∪({x = 0, t > s_{0}} × {τ = 0, ξ ≠ 0}).

(2.34)

We obtain the following picture for the representatives and the limit(as ε →0) of the (bi)characteristic curve passing through (−s_{0}, 0) (resp. (−s_{0}, 0; ξ_{0}, τ_{0})):

Here, t_{ε} is unique with the property x_{ε}(t_{ε}) = 0 and t_{ε} → s_{0} as ε → 0, where the characteristic limit curve (thick gray line) is non-differentiable. The admissible directions for (ξ_{0}, τ_{0}) are simply described by τ_{0}/ξ_{0} = ±1. Since τ(t) = τ_{0} for all t, it remains to investigate lim_{ε→0} ξ_{ε}(t) for the cases t < s_{0}, t = s_{0}, and t > s_{0}.

As long as t < s_{0} we directly obtain from equation (2.32), the fact that x_{ε} → t−s_{0} < 0 as ε → 0 (cf. [11, Subsection 4.2]), and the support properties of ρ that eventually (as soon as ε is sufficiently small) ξ_{ε}(t) = ξ_{0}, which is in accordance with the wave front set in (2.34).

If t > s_{0} then lim_{ε→0} ẋ_{ε}(t) = 0 since the characteristic limit curve is a vertical line in that region. Thus, noting that ẋ_{ε}(t) ≥ 0, upon inserting (2.29) into (2.33) we conclude that lim_{ε→0} ξ_{ε}(t) = ±∞, the sign corresponding to that of −τ_{0}. This yields the horizontal directions of the wave front set (2.34) as limit directions of (ξ_{ε}(t), τ_{ε}(t)).

Finally, consider t = s_{0}, so we are right at the kink in the limit of the characteristic curves. We note that ẋ_{ε}(t_{ε}) = Θ_{ε}(0) = ∫_{−∞}^{0} ρ = 1/2 and, by monotonicity of Θ_{ε} and x_{ε}, the inequality ẋ_{ε}(s_{0}) ≥ ẋ_{ε}(t_{ε}) ≥ 0 is valid. Hence (2.33) and (2.32) imply |τ_{0}| = |ξ_{ε}(s_{0})|ẋ_{ε}(s_{0}) ≥ |ξ_{ε}(s_{0})|ẋ_{ε}(t_{ε}) = |ξ_{ε}(s_{0})|/2 ≥ |ξ_{0}|/2 > 0, so that

τ_{ε}(s_{0}) = τ_{0}, |ξ_{0}| ≤ |ξ_{ε}(s_{0})| ≤ 2|τ_{0}| ∀ε ∈ (0, 1],

which shows that at the base point (0, s_{0}) the possible accumulation points of the bicharacteristic flow can cover only a true subcone of all the directions actually occurring in the wave front set (2.34) at this singularity. In this sense, the generalized bicharacteristic flow-out fails to generate the full wave front set.

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Dipartimento di Matematica Università di Torino Italia

garettoc@dm.unito.it

Institut für Mathematik

Universität Wien

Austria

                                        guenther.hoermann@univie.ac.at