Bulletin T.CXXII de l’Académie Serbe des Sciences et des Arts - 2001 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 26

STRUCTURAL THEOREMS FOR FAMILIES OF FOURIER HYPERFUNCTIONS

B. STANKOVIĆ

(Presented at the 7th Meeting, held on October 27, 2000)

A b s t r a c t. A structural characterization of convergent and bounded families of Fourier hyperfunctions is given.

AMS Mathematics Subject Classification (2000): 46F15

Key Words: Fourier hyperfunctions, families of hyperfunctions

1. Introduction

Let {f_{h}; h ∈ H} be a family of Fourier hyperfunctions which is convergent or bounded. It is of interest for the theory or applications to know whether this family can be given by a unique differential operator J(D) and a family of continuous or smooth functions {p_{h}; h ∈ H} such that f_{h} = J(D)p_{h}, h ∈ H, where {p_{h}; h ∈ H} is convergent or bounded but in some space of functions.

This kind of results for distributions one can find already by Schwartz [9] and in [1], [4], [5], [8] for ultradistributions. In [2] some results have been proved which relate to convergent sequences of hyperfunctions with supports belonging to a compact set K. In [6], [7] convergent sequences of Fourier hyperfunctions have been treated and in [11], Fourier hyperfunctions having the S-asymptotics. In this paper we prove a theorem for any convergent or bounded net without new conditions, which generalizes the results in [6], [7] and [11].

2. Notation and definitions

Let O be the sheaf of analytic functions defined on C^{n}.

We denote by D^{n} the radial compactification of R^{n}, and supply it with the usual topology. The sheaf Õ^{−δ}, δ ≥ 0, on D^{n} +iR^{n} is defined as follows: For any open set U ⊂ D^{n} + iR^{n}, and δ ≥ 0, Õ^{−δ}(U) consists of those elements F of O(U ∩ C^{n}) which satisfy |F (z)| ≤ C_{V,ε} exp(−(δ − ε)|Rez|) uniformly for any open set V ⊂ C^{n}, V̄ ⊂ U, and for every ε > 0. By Õ we denote the sheaf on D^{n} + iR^{n}, Õ(U) = Õ^{0}(U). 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}.

Let I be a convex neighbourhood of 0 ∈ R^{n} and U_{j} = {(D^{n} + iI) ∩ {Imz_{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}),  (1)

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}. Similarly, Q^{−δ}, δ > 0 is defined using Õ^{−δ} instead of Õ (cf. Definition 8.2.5. in [3]).

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

Recall the topological structure of Q(D^{n}). Let f = [F], and K be a compact set in R^{n} then by P_{K,ε}(F) = sup_{z∈R^{n}+iK} |F (z) exp(−ε|Rez|)|, ε >

0, K ⊂⊂ I \{0}, is defined as the family of semi-norms in Õ((D^{n}+iI)#D^{n}); Õ((D^{n} + iI)#D^{n}) is a Fréchet and Montel space, as well as the quotient space Q(D^{n}) with the family of seminorms p_{K,ε}([F]) = inf_{G} P_{K,ε}(F + G), where G belongs to the denominator in (1). In Q(D^{n}) a weak bounded set is bounded. We associate to f = [F^{∗}]

f(x) ≅ ∑_{σ∈Λ} F_{σ}(x + iΓ_{σ}0), F_{σ} ∈ Õ(D^{n} + iI_{σ}), F_{σ} = sgnσF_{σ}^{∗}.  (2)

Let P_{∗} = ind lim_{I∋0} ind lim_{δ↓0} Õ^{−δ}(D^{n} + iI). P_{∗} and Q(D^{n}) are topologically dual to each other ([3, Theorem 8.6.2]).

The Fourier transform on 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 f be given by (2). The Fourier transform of f is defined by

F(f) ≅ ∑_{σ∈Λ} ∑_{σ̃∈Λ} F(χσ̃F_{σ})(ξ − iΓσ̃0),  (3)

where F(χσ̃F_{σ}) ∈ Õ(D^{n} − iIσ̃) and F(χσ̃F_{σ})(z) = O(e^{−w|x|}) for a suitable w > 0 along the real axis outside the closed σ−orthant (cf. Proposition 8.3.2 in [3]).

A function v defined on R^{n} (on C^{n}) is of infra-exponential type if for every ε > 0 there exists C_{ε} > 0 such that |v(z)| ≤ C_{ε}e^{ε|x|}, z ∈ R^{n} (z ∈ C^{n}). A local operator J(D) = ∑_{|α|≥0} b_{α}D^{α} with lim_{|α|→∞} ^{|α|} \sqrt{|b_{α}|α!} = 0 acts on Q(D^{n})

as a sheaf homomorphism and continuously on Q(D^{n}).

3. Main results

Theorem 1.Let f_{h} = [F_{h}^{∗}] ∈ Q(D^{n}), F_{h}^{∗} ∈ Õ((D^{n} + iI)#D^{n}), h ∈ H. If:

a) The net {f_{h}}_{h∈H} converges in Q(D^{n}) or

b) {f_{h}; h ∈ H} is a bounded set in Q(D^{n}).

Then there exist an elliptic local operator J(D) and nets of functions {q_{h,s}}_{h∈H}, s ∈ Λ, such that:

1. q_{h,s}(x), h ∈ H, s ∈ Λ, are smooth functions and of exponential type on R^{n}.

2. q_{h,s}(z) ∈ Õ(D^{n} + iI_{s}), s ∈ Λ, h ∈ H, where I_{s}, s ∈ Λ, does not depend on h ∈ H.

3. f_{h} = J(D) ∑_{s∈Λ} q_{h,s}, (x + iεs), h ∈ H, 0 < ε ≤ ε_{0}.

4. There exists ϵ_{0} > 0 such that for any compact sets K_{1} ⊂⊂ R^{n} and K_{2} ⊂⊂ (0, ϵ_{0}) :

In case a) nets {q_{h,s}(x + iϵs)}_{h∈H}, s ∈ Λ converge uniformly in x ∈ K_{1} and ϵ ∈ K_{2};

In case b) sets {q_{h,s}(x + iϵs)}_{h∈H}, s ∈ Λ, are uniformly bounded for x ∈ K_{1} and ϵ ∈ K_{2}.

Pr o o f. The idea of the proof is the same as in [11]. Let f_{h} = [F_{h}^{∗}] be given by (2) and their Fourier transform by (3). Let φ be a monotone increasing continuous, positive valued function φ(r), r ≥ 0, which satisfies φ(0) = 1, φ(r) → ∞, r → ∞. By Lemma 1.2 in [2] there exists an elliptic local operator J(D) whose Fourier transform J(ζ) satisfies the estimate:

|J(ζ)| ≥ C exp(|ζ|/φ(|ζ|), |Imζ| ≤ 1.  (4)

By (4), J^{−2}(ζ) ∈ Õ(D^{n} + i{|μ| < 1}). Denote by g = F^{−1}(1/J^{2}). By Theorem 8.2.6 in [3], g ∈ Q^{−1}(D^{n}). Consequently δ = J_{0}(D)g, J_{0} = J^{2}, and

f_{h} = J_{0}(D)(g ∗ f_{h}), h ∈ H.  (5)

By the properties of the Fourier transform, cited properties of χσ̃, σ̃ ∈ Λ, and supposition on F_{h}^{∗}, h ∈ H, we have for every h ∈ H :

(a) F(F_{h,σ}χσ̃)J^{−2}) ∈ Õ(x−iIσ̃) and decreases exponentially outside any cone containing \overline{Γ}_{σ} as a proper subcone.

(b) F(F_{h,σ}χσ̃)J^{−2}χ_{s} ∈ Õ(x − iIσ̃) and decreases exponentially outside any cone containing \overline{Γ}_{σ} and \overline{Γ}_{s} as proper subcones.

(c) F^{−1}(F(F_{h,σ}χσ̃)J^{−2}χ_{s}) ∈ Õ(x+i(I_{σ}∪I_{s})) and decreases exponentially outside any cone containing \overline{Γ}σ̃ as a proper subcone. We shall use these properties considering Fourier hyperfunctions f_{h} ∗ g, h ∈ H, given in (5). The analysis of f_{h} ∗ g is very similar to the analysis of f ∗ g in [11]. However we give it because of the integrity of the proof.

\begin{aligned}f_{h} ∗ g & = F^{−1}(F(f_{h})F(g)) \\ & ≅ \frac{1}{(2π)^{n}} ∑_{σ∈Λ} ∑_{σ̃∈Λ} ∫_{R^{n}} e^{iz_{σ}ζσ̃}F(χσ̃F_{h,σ})(ζσ̃)/J^{2}(ζσ̃)dξ, h ∈ H, \end{aligned}

where ζσ̃ = ξ + iησ̃, ησ̃ ∈ −Iσ̃ and z_{σ} ∈ R^{n} + iI_{σ}.

For fixed σ, for all σ̃ ∈ Λ and z_{σ} ∈ R^{n} + iI_{σ}

S_{h,σ,σ̃}(z_{σ}) = \frac{1}{(2π)^{n}} ∫_{R^{n}} e^{iz_{σ}ζσ̃}F(χσ̃F_{h,σ})(ζσ̃)/J^{2}(ζσ̃)dξ;

|S_{h,σ,σ̃}(z_{σ})| ≤ \frac{1}{(2π)^{n}} ∫_{R^{n}} e^{−xησ̃−y_{σ}ξ}|F(χσ̃F_{h,σ})(ζσ̃)/J^{2}(ζσ̃)|dξ, h ∈ H.

One can see that S_{h,σ,σ̃}(z_{σ}), h ∈ H, are continuable to the real axis. The obtained functions S_{h,σ,σ̃}(x) are continuous and of infra exponential type on R^{n}. By Lemma 8.4.7 in [3], S_{h,σ,σ̃}(x) ≅ S_{h,σ,σ̃}(x + iΓ_{σ}0), σ̃ ∈ Λ, h ∈ H and

(f_{h} ∗ g)(x) = ∑_{σ∈Λ} ∑_{σ̃∈Λ} S_{h,σ,σ̃}(x), h ∈ H.  (6)

The functions S_{h,σ,σ̃}(z_{σ}) can be written in the following form

S_{h,σ,σ̃}(z_{σ}) = \frac{1}{(2π)^{n}} ∑_{s∈Λ} ∫_{R^{n}} e^{iz_{σ}ζσ̃}F(χσ̃F_{h,σ})(ζσ̃)χ_{s}(ζσ̃)/J^{2}(ζσ̃)dξ, h ∈ H.

Denote by

S_{h,σ,σ̃,s}(z_{σ}) = \frac{1}{(2π)^{n}} ∫_{R^{n}} e^{iz_{σ}ζσ̃}F(χσ̃F_{h,σ})(ζσ̃)χ_{s}(ζσ̃)/J^{2}(ζσ̃)dξ, h ∈ H.

Functions S_{h,σ,σ̃,s}(z_{σ}), σ, σ̃, s ∈ Λ, h ∈ H, are also continuable to the real axis and the obtained functions S_{h,σ,σ̃,s}(x) are continuous and of infra exponential type on R^{n}. Moreover, for every h ∈ H

S_{h,σ,σ̃,s}(x) ≅ S_{h,σ,σ̃,s}(x + iΓ_{σ}0) and S_{h,σ,σ̃}(x) = ∑_{s∈Λ} S_{h,σ,σ̃,s}(x). (7)

Let us analyse the functions

I_{s,ϵ}(ζ) = J^{−2}(ζ)e^{−ϵsζ}χ_{s}(ζ), ζ ∈ R^{n} + i{|η| < 1},

where 0 < ϵ < 1. These functions are elements of P_{∗} because of

|I_{s,ϵ}(ζ)| = |J^{−2}(ζ)| exp(−ϵ ∑_{i=1}^{n} s_{i}ξ_{i}) ∏_{i=1}^{n} |χ_{s_{i}}(ζ_{i})|

≤ |J^{−2}(ζ)| ∏_{i=1}^{n} |χ_{s_{i}}(ζ_{i})| exp(−ϵs_{i}ξ_{i})

≤ C exp(−ϵ ∑_{i=1}^{n} |ξ_{i}|), |η| < 1, ζ = ξ + iη, s ∈ Λ.

Therefore, I_{s,ϵ} ∈ Õ^{−ϵ}(D^{n} + i{|η| < 1}), s ∈ Λ. Since the Fourier transform maps P_{∗} onto P_{∗}, there exists ψ_{s,ϵ} ∈ P_{∗} such that F(ψ_{s,ϵ}) = I_{s,ϵ}, s ∈ Λ. By Proposition 8.2.2 in [3],

ψ_{s,ϵ} ∈ Õ^{−1}(D^{n} + i{|y| < ϵ}), s ∈ Λ.  (8)

Denote by

\begin{aligned}q_{h,s}(x) & = ∑_{σ∈Λ} ∑_{σ̃∈Λ} S_{h,σ,σ̃,s}(x) \\ & ≅ ∑_{σ∈Λ} ∑_{σ̃∈Λ} F^{−1}(F(F_{h,σ}χσ̃)J^{−2}χ_{s})(x + i(Γ_{σ} ∪ Γ_{s})0), s ∈ Λ, h ∈ H. \end{aligned}

(9)

Let us prove that the functions q_{h,s}, s ∈ Λ, h ∈ H have properties 1. - 4. cited in Theorem.

Property 1 follows from (9) and (c). Property 2 is satisfied because of (6) and (7). Property 3 follows by (5), (6) and (9). It remains only the property 4. Let us prove it.

If f_{h} ∈ Q(D^{n}), h ∈ H, and φ ∈ P_{∗}, then, because of the supposition on F_{h}^{∗}, h ∈ H, f_{h} ∗ φ ∈ Õ(D^{n} + iI′) (cf. [10)], where I′ is an interval containing zero. We shall use this fact and the properties of the functions I_{s,ϵ}, we analysed.

For a fixed s ∈ Λ and h ∈ H there exists ϵ_{0} > 0, such that ϵs belongs to all infinitesimal wedges of the form R^{n} + i(Γ_{σ} ∪ Γ_{s})0 which apear in (9). For ϵ, 0 < ϵ ≤ ϵ_{0} we have

q_{h,s}(x + iϵs) =

= ∑_{σ∈Λ} ∑_{σ̃∈Λ} \frac{1}{(2π)^{n}} ∫_{R^{n}} e^{i(x+iϵs)ζσ̃}F(F_{h,σ}χσ̃)(ζσ̃)J^{−2}(ζσ̃)χ_{s}(ζσ̃)dξ

= ∑_{σ∈Λ} ∑_{σ̃∈Λ} \frac{1}{(2π)^{n}} ∫_{R^{n}} e^{ixζσ̃}F(F_{h,σ}χσ̃)(ζσ̃)F(ψ_{s,ϵ})(ζσ̃)dξ

= ∑_{σ∈Λ} ∑_{σ̃∈Λ}((F_{h,σ}χσ̃) ∗ ψ_{s,ϵ})(x) = (( ∑_{σ∈Λ} F_{h,σ}) ∗ ψ_{s,ϵ})(x)

= (f_{h} ∗ ψ_{s,ϵ})(x) = ⟨f_{h}(t), ψ_{s,ϵ}(x − t)⟩, s ∈ Λ, h ∈ H

(10)

Now, 4. a) and 4. b) follows from (10).

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