PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 64 (78), 1998, 98–106

CONVERGENCE STRUCTURES AND S-ASYMPTOTIC BEHAVIOUR OF FOURIER HYPERFUNCTIONS

B. Stanković

Communicated by Stevan Pilipović

Abstract. Some structural theorems for the convergence in the space of Fourier hyperfunctions are proved and applied to the S-asymptotic behaviour of elements in this space.

Preliminaries

We denote by D^{n} the compactification of R^{n}, D^{n} = R^{n} ∪ S_{∞}^{n−1} and supply it with the usual topology. The sheaves Õ and Q on D^{n} + iR^{n} are defined as follows (cf. [4], [5]). For any open set U ⊂ D^{n} + iR^{n}, and δ ≥ 0 Õ^{−δ}(U)(Õ^{0}(U) = Õ(U)) consists of those elements of O(U ∩ C^{n}) which satisfy |F (z)| ≤ C_{V,ε} exp(−(δ − ε) × | Re z|) uniformly for any open set V ⊂ C^{n}, V̄ ⊂ U, and for every ε > 0. Hence, Õ|_{C^{n}} = O. The derived sheaf H_{D^{n}}^{n}(Õ), denoted by Q, is called the sheaf of Fourier hyperfunctions. It is a flabby sheaf on D^{n} [5]. We need only the space of global sections Q(D^{n}).

Let I_{k}, k = 1,… be open intervals, neighbourhoods of 0 ∈ R, let I = I_{1} × ⋯ × I_{n} be a convex neighbourhood of 0 ∈ R^{n} and U_{j} = {(D^{n} + iI) ∩ {Im z_{j} ≠ 0}}, j = 1,…, n. The family {D^{n} +iI, U_{j}; j = 1,…, n} gives a relative Leray covering for the pair {D^{n} + iI, (D^{n} + iI) \ D^{n}} relative to the sheaf Õ. Thus

Q(D^{n}) = Õ((D^{n} + iI)#D^{n})/ ∑_{j=1}^{n} Õ((D^{n} + iI)#_{j}D^{n}),

where (D^{n} + iI)#D^{n} = U_{1} ∩ ⋯ ∩ U_{n} and

(D^{n} + iI)#_{j}D^{n} = U_{1} ∩ ⋯ ∩ U_{j−1} ∩ U_{j+1} ∩ ⋯ ∩ U_{n}.

AMS Subject Classification (1991): Primary 46F10 Supported by Ministry of Science and Technology of Serbia, grant number 04M03/C

Similarly Q^{−δ}, δ > 0, is defined using Õ^{−δ} instead of Õ (cf. Definition 8.2.5. in [4]).

We shall use the notation Λ for the set of n-vectors with entry {−1, 1}; the corresponding open orthants in R^{n} will be denoted by Γ_{σ}, σ ∈ Λ. By Γ we denote a convex cone in R^{n}.

A global section f = [F] ∈ Q(D^{n}) is defined by F ∈ Õ((D^{n} + iI)#D^{n}); F = (F_{σ}), where F_{σ} ∈ Õ(D^{n} + iI_{σ}), D^{n} + iI_{σ} is an infinitesimal wedge of type R^{n} + iΓ_{σ}0, σ ∈ Λ, I_{σ} = I ∩ Γ_{σ}. F is the defining function for f.

Recall the topological structure of Q(D^{n}). Let f = [F] ∈ Q(D^{n}) and F ∈ Õ((D^{n} + iI)#D^{n}). Then, a family of semi-norms is defined by P_{K,ε}(F) = sup_{z∈R^{n}+iK} |F (z) exp(−ε| Re z|)|, ε > 0, K ⊂⊂ I \{0}; Õ((D^{n} +iI)#D^{n}) is a Fréchet

and Montel space, as well as Q(D^{n}).

Let f = [F] ∈ Q(D^{n}). Then we associate to f, f(x) ≅ ∑_{σ∈Λ} sgn σF_{σ}(x +

iΓ_{σ}0), F_{σ} ∈ Õ(D^{n} + iI_{σ}) (cf. [4, Theorem 8.5.3 and Definition 8.3.1]).

The Fourier transform of Q(D^{n}) is defined by the use of functions χ_{σ} = χ_{σ_{1}} … χ_{σ_{n}}, where σ_{k} = ±1, k = 1,…, n, σ = (σ_{1},…, σ_{n}) and χ_{1}(t) = e^{t}/(1 + e^{t}),

χ_{−1}(t) = 1/(1+e^{t}), t ∈ R. Let u(x) ≅ ∑_{σ∈Λ} U_{σ}(x+iΓ_{σ}0) = ∑_{σ∈Λ} ∑_{σ̃∈Λ}(χσ̃U_{σ})(x+iΓ_{σ}0),

where χσ̃U_{σ} ∈ Q(D^{n} + iI_{σ}), σ, σ̃ ∈ Λ and decreases exponentially along the real axis outside the closed σ̃-th orthant.

The Fourier transform of u is defined by

F(u) ≅ ∑_{σ∈Λ} ∑_{σ̃∈Λ} F(χσ̃U_{σ})(ξ − iΓσ̃0)

∑_{σ∈Λ} ∑_{σ̃∈Λ} ∫_{Im z=y^{σ}} e^{izζ}(χσ̃U_{σ})(z)dx, y^{σ} ∈ I_{σ}, ζ = ξ + iη,

where F(χσ̃U_{σ}) ∈ Õ(D^{n} − iIσ̃) and |F(χσ̃U_{σ})(z)| = O(e^{−w|x|}) for a suitable w > 0 along the real axis outside the closed σ-orthant. F is an automorphism of Q(D^{n}).

A continuous function v defined on R^{n} is of infra-exponential growth if for every ε > 0 there exists C_{ε} > 0 such that |v(x)| ≤ C_{ε}e^{ε|x|}, x ∈ R^{n}. By ∥v∥_{ε} = sup_{x∈R^{n}} e^{−ε|x|}|v(x)|, ε > 0 we define a family of seminorms in the space of functions

of infra-exponential growth. By ℓv we denote the Fourier hyperfunction defined by v.

An infinite-order differential operator J(D) = ∑_{|α|≥0}b_{α}D^{α} with lim_{|α|→∞}\sqrt{[}|α|]|b_{α}|α!

= 0 is called a local operator. J(D) acts on Q as a sheaf homomorphism and continuously on Q(D^{n}).

We shall use the following proposition proved in [7] and [8].

Proposition 1. Let I be a convex neighbourhood of 0 ∈ R^{n} and I_{σ} = I ∩ Γ_{σ}, σ ∈ Λ. Let {f_{h}; h ∈ Γ} be a family in Q(D^{n}) such that f_{h} ≅ ∑_{σ∈Λ} G_{h,σ}(x + iΓ_{σ}0),

where G_{h,σ} ∈ Õ(D^{n} + iI_{σ}), h ∈ Γ, σ ∈ Λ.

A necessary and sufficient condition that f_{h} converges in Q(D^{n}) to f ≅ ∑_{σ∈Λ} G_{σ}(x + iΓ_{σ}0) as ∥h∥ → ∞, h ∈ Γ is the existence of families {F_{h,σ}; h ∈ Γ}

⊂ Õ(D^{n} + iI_{σ}), σ ∈ Λ, such that 1) F_{h} = (F_{h,σ}) belongs to the same class as G_{h} = (G_{h,σ}), h ∈ Γ;

2) For every σ ∈ Λ, F_{h,σ} converges to F_{σ} in Õ(D^{n}+iI_{σ}) as ∥h∥ → ∞, h ∈ Γ, where F = (F_{σ}) belongs to the same class as G = (G_{σ}).

Convergence structure for Fourier hyperfunctions

We shall prove some structural theorems for the convergence in the space of Fourier hyperfunctions Q(D^{n}). First we shall give a modified form to Theorem 1.3 in [3]. We give the complete proof of it although we shall use the same method of the proof as in the theorem mentioned above.

Let B[K] be the set of hyperfunctions with supports in the compact set K ⊂ R^{n}.

Proposition 2. Let K′ be a compact set in R^{n} and K the convex hull of K′. Then for every u = [U] ∈ B[K] there exist an elliptic local operator J_{0}(D) and a function v ∈ C^{∞} ∩ L^{2} with the properties:

a) |v(x)| ≤ C_{ε}e^{−(1−ε)|x|}, x ∈ R^{n}, for every ε > 0 and ℓv ∈ Q^{−1}(D^{n})

b) v(x) = (u ∗ ℓg)(x), where the function g has the same property a) as the function v.

c) u = J_{0}(D)v.

Proof. The Fourier transform û of u is an entire function and satisfies the following growth condition

|u(ζ)| ≤ C exp (\frac{|ζ|}{φ(|ζ|)} + H_{K}(Im ζ)),

where φ(r) is a monotone increasing function of r > 0 and satisfies: φ(0) = 1, φ(r) → ∞ when r → ∞; H_{K}(η) = sup_{x∈K} xη. (cf. Lemma 1.1 in [3]). We take

an elliptic local operator J(D) which corresponds to the chosen φ as it is done in Lemma 1.2 in [3]. Now we can consider û as a Fourier hyperfunction with defining function U = (û(ζ), 0,…, 0). The corresponding function to J(D), J(ζ) is an entire function, |J(ζ)| ≤ C_{ε}e^{ε|ζ|} for every ε > 0. Also, for any prescribed positive constants A, C, we have

J(ζ) ≥ C exp(A|ζ|/φ(|ζ|)) for |η| ≤ 1

or

|J^{−2}(ζ)| ≤ C^{−2} exp(−2A|ζ|/φ(|ζ|)), |η| ≤ 1.

We can take without loss of generality that |ζ|/φ(|ζ|) ≥ c|ζ|^{γ} with some c > 0 and 0 < γ < 1. We have now the following estimate:

|û(ζ)/J^{2}(ζ)| ≤ C exp(−c|ζ|^{γ}), |η| ≤ 1.  (1)

Let V = û/J^{2}, then û = J^{2}V and u = J^{2}(D)F^{−1}(V). Since û is also an entire function, û/J^{2} belongs to O(D^{n} + i{|η| < 1}). By Theorem 8.2.6 in [4], F^{−1}(V) ∈ Q^{−1}(D^{n}). Consequently, v = F^{−1}(V) ∈ Q^{−1}(D^{n}).

Now we prove that v is rapidly decreasing. For V we know that V = û/J^{2} = (û(ζ)/J^{2}(ζ), 0,…,). Then

F^{−1}(V) ≃ ∑_{σ∈Λ} F^{−1}(χ_{σ}(ζ)û(ζ)/J^{2}(ζ)) = ∑_{σ∈Λ} F^{−1}(V)_{σ}(x − iΓ_{σ}0)

Consider F^{−1}(V)_{σ}(z) for a σ ∈ Λ and z ∈ R^{n} − iI_{σ} :

|F^{−1}(V)_{σ}(z)| = \frac{1}{(2π)^{n}} | ∫_{R^{n}} e^{−ζzi} \frac{û(ζ)}{J^{2}(ζ)} χ_{σ}(ζ) dξ|

≤ \frac{1}{(2π)^{n}} ∫_{R^{n}} e^{ξy+ηx}| \frac{û(ζ)}{J^{2}(ζ)} χ_{σ}(ζ))| dξ, ζ = ξ + iη

uniformly for |η| ≤ (1 − ε) for every ε > 0. Thus we can write η = −(1 − ε)x/|x| and

|F^{−1}(V)_{σ}(z)| ≤ \frac{1}{(2π)^{n}} e^{−(1−ε)|x|} ∫_{R^{n}} e^{ξy}| \frac{û(ζ)}{J^{2}(ζ)} χ_{σ}(ζ)| dξ,

where the integral is convergent since y ∈ −I_{σ}. It follows that F^{−1}(V)_{σ}(x + iy), y ∈ −I_{σ}, σ ∈ Λ, can be extended to real axis and that v(x) is rapidly decreasing.

Furthermore we can give another analytic form to ℓv. For the holomorphic function J^{−2}(ζ), |η| < 1, we can take

|J^{−2}(ζ)| ≤ C exp(−|ζ|^{γ}), 0 < γ < 1, |η| < 1.  (2)

By Theorem 8.2.6 in [4], J^{−2}(ζ) is the Fourier transform of a g ∈ Q^{−1}(D^{n}). By the same reasons as for v we have |g(x)| ≤ Ce^{−(1−ε)|x|} for every ε > 0, x ∈ R^{n}. By the Paley–Wiener–Ehrenpreis theorem, û(ζ) is an entire function of ζ and for any ε > 0 and any |η| ≤ δ, δ > 0, there exists C_{ε,δ} such that:

|û(ζ)| ≤ C_{ε,δ}e^{ε|ζ|}, |η| ≤ δ.  (3)

Consequently, û(ζ) is a slowly increasing analytic function and (cf. Theorem 8.4.3 in [4]), F(u ∗ g) = û · J^{−2}.

Since (û · J^{−2})(ξ) ∈ L^{2}, then (u ∗ g)(x) ∈ L^{2}, as well. We know that B[K] ⊂ Q^{−1}(D^{n}), then by Theorem 8.2.8 in [4], u ∗ g ∈ Q^{−1}(D^{n}).

Now we have in correspondence to [3]:

1) ℓv ∈ Q^{−1}(D^{n}) is defined by the function v belonging L^{2} ∩ C^{∞} (cf. [3]).

2) v(x) = (u ∗ ℓg)(x), where ℓg ∈ Q^{−1}(D^{n}), |g(x)| ≤ Ce^{−(1−ε)|x|}, for every ε > 0, x ∈ R^{n}.

3) |v(x)| ≤ Ce^{−(1−ε)|x|}, for every ε > 0, where C > 0. □

Corollaries of Proposition 2. 1) If we take in Proposition 2, u = δ, and v = q, then

δ = J_{0}(D)ℓq,  (4)

where J_{0} and q have the following properties:

J_{0}(D) = J(D)J(D), J(D) is an elliptic local operator constructed by any monotone increasing function φ(r), of r > 0 with the properties: φ(0) = 1, φ(r) → ∞ (see Lemma 1.2 in [3]).

ℓq ∈ Q^{−1}(D^{n}), q ∈ L^{2} ∩ C^{∞} and |q(x)| ≤ Ce^{−(1−ε)|x|}, x ∈ R^{n}, for every ε > 0. Also F(q)(ζ) = J^{−2}(ζ),

Proof. In the proof of Proposition 2 we have only to take F(u) = F(δ) = 1. Then (1) is satisfied for every J, we constructed using the function φ mentioned above, because of (2).

2) If f ∈ Q(D^{n}), then there exist an elliptic local operator J_{0}(D) and a function v ∈ C^{∞} which is of “infra exponential” growth such that f = J_{0}(D)v, ℓv ∈ Q(D^{n}).

Assertion 2) was first proved in [3] but without the property of v to be of “infra exponential” growth. The complete assertion 2) was proved in [2]. We shall show how it can be proved using 1).

Proof. By (4), by properties of q and Proposition 8.4.8 in [4]

f = (J_{0}(D)q) ∗ f = J_{0}(D)(ℓq ∗ f).

It remains only to show that (ℓq ∗ f) is a function v with the required properties. Because of 1) we can choose for J the same function as it is done in [2].

Let (ℓq ∗ f) = v. By Proposition 8.4.3 in [4]

F(ℓq ∗ f)(ζ) = v̂(ζ) = J^{−2}(ζ)F(f)(ζ) = ∑_{σ,σ̃∈Λ} J^{−2}F(χσ̃F_{σ})(x − iΓσ̃0).

or F^{−1} F(ℓq ∗ f) = v.

Let η ∈ Iσ̃. We consider

v_{σ,σ̃}(z) = \frac{1}{(2π)^{n}} ∫_{Imζ=η} e^{−izζ} \frac{F(χσ̃F_{σ})(ζ)}{J^{2}(ζ)}dζ, σ, σ̃ ∈ Λ.

Then for every σ, σ̃

|v_{σ,σ̃}^{(p)}(z)| ≤ Ce^{xη} ∫_{R^{n}} e^{yζ}e^{−|ζ|^{γ}} |ζ|^{p}dζ, z ∈ R^{n} + iI_{σ}, p = 0, 1,…

This implies that v_{σ,σ̃}^{(p)}(z), z ∈ R^{n} + iI_{σ}, p = 0, 1,… and σ, σ̃ ∈ Λ, is

continuable to a continuous function v_{σ,σ̃}^{(p)}(x) up to real axis and |v_{σ,σ̃}(x)| ≤ C_{ε}e^{ε|x|} for every ε > 0. The function v(x) = ∑_{σ,σ̃∈Λ} v_{σ,σ̃}(x) has the required properties. □

Theorem 1. Let I be a convex neighbourhood of 0 ∈ R^{n} and I_{σ} = I ∩ Γ_{σ}, σ ∈ Λ. Let {f_{h}; h ∈ Γ} be a family in Q(D^{n}) such that

f_{h} ≅ ∑_{σ∈Λ} F_{σ,h}(x + iΓ_{σ}0), F_{σ,h} ∈ Õ(D^{n} + iI), h ∈ Γ, σ ∈ Λ.  (5)

A necessary and sufficient condition that f_{h} converges in Q(D^{n}) when h ∈ Γ, ∥h∥ → ∞, is that f_{h} ∗ g converges in Q(D^{n}), h ∈ Γ, ∥h∥ → ∞, for every g = [G] ∈ Q^{−δ}(D^{n}) and for every δ > 0. More precisely:

a) If f_{h} → u = [U] in Q(D^{n}), h ∈ Γ, ∥h∥ → ∞, then f_{h} ∗ g → u ∗ g in Q(D^{n}), h ∈ Γ, ∥h∥ → ∞; g ∈ Q^{−δ}(D^{n}), δ > 0.

b) If f_{h} ∗ g → v_{g} in Q(D^{n}), h ∈ Γ, ∥h∥ → ∞, for every g ∈ Q^{−1}(D^{n}), then there exists an elliptic local operator J_{0}(D) such that f_{h} → J_{0}(D)v_{g} in Q(D^{n}), h ∈ Γ, ∥h∥ → ∞.

Proof. a) By Proposition 1 we can suppose that {F_{h}; h ∈ Γ} is such that F_{h} → F ∈ [U] in Õ((D^{n} + iI) ≠ D^{n}), h ∈ Γ, ∥h∥ → ∞. By Proposition 8.4.3 in [4] there exists f_{h} ∗ g, f_{h} ∗ g ∈ Q(D^{n}), h ∈ Γ, and

(f_{h} ∗ g)(x) = ∫_{R^{n}} f_{h}(x − ζ)g(ζ)dζ = ∑_{σ,σ̃∈Λ} F_{σ,h}(x + iΓ_{σ}0) ∗ Gσ̃(x + iΔσ̃0)

where

F_{σ,h}(x + iΓ_{σ}0) ∗ Gσ̃(x + Δσ̃0) = ∫_{R^{n}} F_{σ,h}(z − ξ − iησ̃)Gσ̃(ξ + iησ̃)dξ. (6)

ησ̃ ∈ Iσ̃, Δσ̃ is also an open σ̃ orthant in R^{n} and z be in R^{n} + i(I_{σ} + Iσ̃).

Let us consider only (6) for a fixed σ, σ̃ ∈ Λ. We will prove that

lim_{h∈Γ,|h|→∞} F_{σ,h}(x + iΓ_{σ}0) ∗ Gσ̃(x + iΔσ̃0) =

= F_{σ}(x + iΓ_{σ}0) ∗ Gσ̃(x + iΔσ̃0). in O(D^{n} + iI_{σ}).

Let ε < δ, K_{σ} be a compact set in I_{σ} and let ησ̃ be chosen such that y−ησ̃ ∈ I_{σ} when y ∈ K_{σ}. Suppose that for ω > 0 we have chosen h_{0} so that P_{K_{σ},ε}(F_{σ,h}−F_{σ}) < ω, |h| ≥ |h_{0}|, h ∈ Γ. Then

sup_{z∈R^{n}+iK_{σ}} e^{−ε|x|}| ∫_{R^{n}} F_{σ,h}(z−ξ−iησ̃)Gσ̃(ξ+iησ̃)dξ−∫_{R^{n}}F_{σ}(z−ξ−iησ̃)Gσ̃(ξ+iησ̃)dξ|

≤ ∫_{R^{n}} sup_{z∈D^{n}+iK_{σ}} e^{−ε|x|−ε|ξ|}|F_{σ,h}(z − ξ − iη) − F_{σ}(x − ξ − iησ̃)|e^{ε|ξ|}|G(ξ + iησ̃)|dξ

≤ ω ∫_{R^{n}} e^{−(δ−ε)|ξ|}dξ, |h| ≥ |h_{0}|, h ∈ Γ.

b) By Corollary 1) of Proposition 2 there exists a g_{0} ∈ Q^{−1}(D^{n}) such that

δ = J_{0}(D)g_{0}, g_{0} ∈ Q^{−1}(D^{n}).

Consequently

f_{h} = (J_{0}(D)g_{0}) ∗ f_{h} = J_{0}(D)(g_{0} ∗ f_{h}).

Since J_{0}(D) acts continuously on Q(D^{n}), f_{h} → J_{0}(D)v_{g} in Q(D^{n}), h ∈ Γ, |h| → ∞.

Applications. S-Asymptotics of Fourier hyperfunctions

We shall apply the convergence structures that we proved to the asymptotic behaviour of Fourier hyperfunctions. There are many definitions of asymptotic behaviour of generalized functions. We mention two of them, which are most used: the quasi-asymptotics and S-asymptotics.

S-asymptotics was defined for distributions [14] for ultradistributions [9, 10] and for some other generalized functions [17]. It was applied in the quantum field theory [1], for Abelian and Tauberian type theorems, for solutions of partial differential equations,... [11]–[14]. It is easy to extend it to Fourier hyperfunctions [18].

In fact, the S-asymptotics of Fourier hyperfunctions extends the S-asymptotics of tempered distributions, because we have the continuous inclusion S’ ↪ Q(D^{n}) (see [3]).

Definition 1. Suppose that c is a positive continuous function defined on R^{n} and f ∈ Q(D^{n}). f is said to have the S-asymptotics related to c in the cone Γ ∈ R^{n} if there exists

lim_{h∈Γ,|h|→∞} \frac{f(· + h)}{c(h)} = u in Q(D^{n}), u ≠ 0.

For short, f(x + h) ∼^{s} c(h) · u(x), h ∈ Γ, |h| → ∞, in Q(D^{n}). We shall prove two simple properties of the S-asymptotics.

Proposition 3. a) Let P (D) be a local operator and f ∈ Q(D^{n}). If f(x + h)

∼^{s} c(h) · u(x), h ∈ Γ, |h| → ∞ in Q(D^{n}), then P (D)f(x + h) ∼^{s} c(h) · P (D)u(x),

h ∈ Γ, |h| → ∞ in Q(D^{n}), as well.

b) Let supp f ⊂ K, where K is a compact set in R^{n}. Then f ∈ Q(D^{n}) and f(x + h) ∼^{s} c(h) · 0, h ∈ Γ, |h| → ∞ in Q(D^{n}) for every positive function c and every cone Γ.

Proof. a) This property follows from the property of a local operator; P (D) maps Q(D^{n}) into Q(D^{n}) and this mapping is continuous.

b) Let supp f ⊂ K, where K ⊂ R^{n} is a compact set. We take γ_{0} large enough and such that h + K ⊂ R^{n} \ K, |h| ≥ γ_{0}. By definition of the support of a hyperfunction, f(x + h) = 0, |h| ≥ γ_{0}. □

The next examples show that Definition 1 is not a trivial extension of the S-asymptotics of distributions. Let P (D) be a local operator

∑_{|α|≥0} b_{α}D^{α}, b_{α} ≠ 0.

The Fourier hyperfunction f = 1 + P (D)δ has the S-asymptotics related to c = 1 in any cone Γ and with the limit u = 1 but f is not a distribution. For the S-asymptotics of f it is enough to prove that

lim_{h∈Γ,|h|→∞} P (D)δ(x + h) = 0 in Q(D^{n}).

Since P (D) maps continuously Q(D^{n}) into Q(D^{n}), by Proposition 3. b) the above limit follows.

Since P (D)δ = ∑_{|α|≥0} b_{α}D^{α}δ is a distribution if and only if b_{α} ≠ 0 for a finite

number of α, the Fourier hyperfunction 1 + P (D)δ is not a distribution, but it has the S-asymptotics related to c = 1.

We can also find the coefficients b_{α} of the local operator P (D) such that f = 1+P (D)δ is not defined by an ultradistribution belonging to D^{(M_{p})′} or D^{{M_{p}}′} (Beurling or Roumieu type) when M_{p} = (p!)^{s}, s > 1 (see [18]). For ultradistributions see [6].

A direct consequence of Theorem 1 is

Theorem 2. A necessary and sufficient condition that f = [F] ∈ Q(D^{n}) has the S-asymptotics in Q(D^{n}) related to c in the cone Γ is that f ∗ g has the same property, for every g = [G] ∈ Q^{−δ}(D^{n}), δ > 0. More precisely:

a) If f(x + h) ∼^{s} c(h) · u(x), h ∈ Γ, |h| → ∞, in Q(D^{n}), then (f ∗ g)(x + h) ∼^{s} c(h) · (u ∗ g)(x), h ∈ Γ, |h| → ∞, in Q(D^{n}) for every g ∈ Q^{−δ}(D^{n}), δ > 0.

b) If (f ∗ g)(x + h) ∼^{s} c(h) · v_{g}(x), h ∈ Γ, |h| → ∞, in Q(D^{n}) for every g ∈ Q^{−1}(D^{n}), then there exists an elliptic local operator J_{0}(D) such that f(x+h) ∼^{s} c(h) · J_{0}(D)v_{g}, h ∈ Γ, |h| → ∞, in Q(D^{n}).

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Institut za matematiku (Received 23 03 1998) 21000 Novi Sad Yugoslavia