Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces

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Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through

sequence spaces

Antoine Delcroix, Maximilian F. Hasler, Stevan Pilipovic, Vincent Valmorin

To cite this version:

Antoine Delcroix, Maximilian F. Hasler, Stevan Pilipovic, Vincent Valmorin. Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces. Mathematical Proceedings, Cambridge University Press (CUP), 2004, 137 (3), pp.697-708.

�10.1017/S0305004104007923�. �hal-00758568�

Printed in the United Kingdom c 2004Cambridge Philosophical Society

Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces

By Antoine Delcroix Antoine.Delcroix@univ-ag.fr

andMaximilian F. Hasler†

MHasler@martinique.univ-ag.fr and Stevan Pilipovi´c pilipovic@im.ns.ac.yu andVincent Valmorin Vincent.Valmorin@univ-ag.fr

(5 May 2003)

Abstract

In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz spaces into the Colombeau algebraG are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable or quasiana- lytic), embedded into corresponding algebras, is the ordinary multiplication.

MSC: 46A45 (sequence spaces), 46F30 (generalized functions for nonlinear analysis);

secondary: 46E10, 46A13, 46A50, 46E35, 46F05.

1. Introduction

Differential algebras of generalized functions containing embedded distributions are a convenient framework for the analysis of problems with singular coefficients and/or singular data, especially for non linear problems, since multiplication and other non linear operations are in general not defined in classical generalized function spaces. Nowadays, there is a considerable literature concerning such algebras. (For example see [1, 2, 3, 7, 12, 15, 5, 6] and the references therein.)

We have proved in [4] that these algebras, here referred to as Colombeau type algebras, can be reconsidered as a class of sequence spaces algebras, and we gave a purely topological description of them.

In analogy to embeddings of Schwarz’ distributions, we show in this paper that

† corresponding author:

Univ. Antilles–Guyane, D.S.I., B.P. 7209, 97275 Schoelcher cedex (Martinique, F.W.I.)

A. Delcroix, M. Hasler, S. Pilipovi´ c, V. Valmorin

some classes of ultradistributions and periodic hyperfunctions can be embedded into well chosen sequence algebras. Moreover, we show that the product of enough regular elements, ultradifferentiable functions or quasianalytic periodic functions of appro- priate classes, is the ordinary multiplication.

The problem of embedding of classical spaces into corresponding sequence spaces algebra is closely related to the choice of sequences of mollifiers, sequences of appro- priately smooth functions converging to the delta distribution. While such a prob- lem is easy for embedding of Schwartz distributions defined on an open subset of Rⁿ, it is essential for ultradifferentiable functions and ultradistributions, considered in section 3. The same holds for periodic quasianalytic functions and corresponding periodic hyperfunctions of section 4.

Colombeau ultradistributions corresponding to a general non-quasianalytic se- quence were introduced and analyzed in [14]. Although we consider here the Gevrey sequence (p!^m)_p, m >1, we give sharper estimates and improve results of [14]: The construction of appropriate mollifiers enables us to give more precise results con- cerning embeddings. Colombeau periodic hyperfunctions introduced in this paper are more closely related to the global theory of generalized functions than those of [16]. In this sense, we improve results of [16].

The novelty of results related to both cited papers and the embedding of both classes of algebras into corresponding sequence space algebras, and so their topolog- ical description, are the main results of this paper.

2. General construction[4]

We use the standard notations N={0,1, ...},N^∗ ={1,2, ...}andR⁺ = [0,+∞).

We recall [4] that the algebra of Colombeau complex numbers is given by C=E0/N0≡ {x∈C^N

∗: lim sup|xn|^log1n <∞}/{x∈C^N

∗: lim sup|xn|^log1n = 0}. The passage from this description to Colombeau’s original construction is given by the following chain of equivalences, for a complex sequence (xn)_n∈C^N

∗: lim sup|xn|^1/logn<∞ ⇐⇒ ∃C ∈R+: lim sup|xn|^1/logn =C

⇐⇒ ∃B >0, ∃n0,∀n > n0:|xn| ≤B^logn=n^logB

⇐⇒ ∃γ ∈R:|xn|=o(n^γ) .

On the other hand, lim sup|xn|^1/logn = 0 (for the ideal) corresponds to takingC= 0 and thus∀B >0 resp. ∀γ∈R in the last lines.

Let us also recall our construction from [4] for the case ofE=C,rn =n^−1/m, n∈ N^∗, wherem >0 is fixed. Let

E₀^m=

c= (cn)_n ∈C^N

∗

|||c|||_|·|,n^−1/m = lim sup

n→∞

|cn|^n−1/m <∞

,

N₀^m=ⁿc= (cn)_n∈C^N

∗

|||c|||_|·|,n^−1/m = 0^o . The factor algebra C

m = E₀^m/N₀^m, m > 0, is called the ring of Colombeau ultra- complex numbers for m >1 and the ring of Colombeau hypercomplex numbers for m≤1.

Now we come to the general construction. Let (E_ν^µ, p^µ_ν)_µ,ν∈

N^∗ be a family of semi-

normed algebras overR orC such that

∀µ, ν ∈N^∗: E_ν^µ+1,→E_ν^µ and E_ν+1^µ ,→E_ν^µ (resp. E_ν^µ ,→E_ν+1^µ ),

where ,→ means continuously embedded (i.e., for the ν index we consider inclu- sions in the two directions). Then let←−

E= proj lim

µ→∞

proj lim

ν→∞

E^µ_ν = proj lim

ν→∞

E_ν^ν, (resp.

−

→E= proj lim

µ→∞

ind lim

ν→∞ E_ν^µ). Such projective and inductive limits are usually considered with norms instead of seminorms, and with the additional assumption that in the projective case sequences are reduced, while in the inductive case for every µ ∈ N^∗ the inductive limit is regular, i.e. a set A⊂ind lim

ν→∞ E_ν^µ is bounded iff it is contained in some E_ν^µ and bounded there.

Consider a positive sequence r = (rn)_n ∈ (R⁺)^N∗ decreasing to zero and define (withp≡(p^µ_ν)_ν,µ):

|||f|||_p^µ_ν,r= lim sup

n→∞

p^µ_ν(fn)^rn for f∈(E_ν^µ)^N∗ ,

←−

Fp,r =ⁿf ∈←−

E^N∗ ∀µ, ν∈N^∗:|||f|||_p^µ_{ν, r}<∞^o ,

←−

Kp,r =ⁿf ∈←−

E^N∗ ∀µ, ν∈N^∗:|||f|||_p^µ_{ν, r}= 0^o (resp. −→

Fp,r = ^\

µ∈N^∗

−

→F_p,r^µ , −→

F_p,r^µ = ^[

ν∈N^∗

n

f ∈(E_ν^µ)^N∗ |||f|||p^µ_ν,r <∞^o ,

−

→Kp,r = ^\

µ∈N^∗

−

→K_p,r^µ , −→

K^µ_p,r = ^[

ν∈N^∗

n

f ∈(E_ν^µ)^N

∗

|||f|||p^µ_ν,r= 0^o ). Recall [4]:

Writing←→· for both,←−· or−→· , we have that←→

Fp,r is an algebra and←→

Kp,r is an ideal of←→

Fp,r; thus, ←→

G p,r =←→ Fp,r/←→

Kp,r is an algebra.

For everyµ, ν∈N^∗, dp^µ_ν : (E_ν^µ)^N

∗

×(E_ν^µ)^N

∗

→R+ defined bydp^µ_ν(f, g) =|||f−g|||p^µ_ν,r

is an ultrapseudometric on (E_ν^µ)^N∗. Moreover, (d_p^µ_ν)µ,ν induces a topological algebra¹ structure on←−

Fp,r such that the intersection of the neighborhoods of zero equals←− Kp,r. From the properties above, the factor space←−

Gp,r =←− Fp,r/←−

Kp,r is a topological algebra over generalized numbers C^r = G|·|,r (constructed with the sequence r = (rn) as above for the Colombeau ultracomplex numbers). The topology of←−

Gp,r is defined by the family of ultrametrics ( ˜d_p^µ_ν)µ,ν where ˜d_p^µ_ν([f],[g]) =d_p^µ_ν(f, g), [f] standing for the class of f.

If τµ denotes the inductive limit topology on F_p,r^µ = ^S_ν∈N∗(( ˜Eν^µ)^N∗, dµ,ν), µ ∈ N^∗, then −→

Fp,r is a topological algebra¹ for the projective limit topology of the family (F_p,r^µ , τµ)_µ.

Remark 1. The two multiplicative sets H = [0,1] and I = [0,1) verify the rela- tionsH·H =H, I·H =I, I·I =I, just like the sets[0,∞)and{0}. Thus, similar constructions can also be made with||| · ||| ≤1and||| · |||<1instead of||| · |||<∞and

||| · |||= 0. This is used in the setting of infra-exponential algebras and also appears in the context of periodic hyperfunctions.

1 over (C^N

∗,||| · ||||·|,r), not overC: scalar multiplication is not continuous.

A. Delcroix, M. Hasler, S. Pilipovi´ c, V. Valmorin

3·1. Ultradistributions of Gevrey class

We refer to [9] for definitions of the spacesE^(m), D^(m), E^{m}, D^{m} (m >1), and their duals, Beurling and Roumieu type ultradistribution spaces. Here we construct Colombeau ultradistribution algebras corresponding toMp =p!^m,m >1. We apply the construction of section 2.

For the function space E = C^∞(R^s), we define for all µ, ν ∈ R+ and m >1 the seminorms

p^m,µ_ν (f) = sup

|x|≤µ,α∈N^s

ν^|α|

α!^m|f^(α)(x)| and q^m,µ_ν =p^m,µ_1/ν .

Then let, for µ, ν ∈ N^∗, E_ν^µ = E_p^m,µ_ν (resp. E_ν^µ = E_q^m,µ_ν ) be the subset of E on which the given seminorm is finite. For the first case, we clearly have E_ν^µ+1 ,→E^µ_ν, E^µ_ν+1 ,→ E_ν^µ and for the second case, we have E_ν^µ+1 ,→ E_ν^µ, E_ν^µ ,→ E^µ_ν+1 for any µ, ν ∈N^∗. Letm >1,m⁰ >0 and rn=n^−1/m0.

Definition 3·1. The sets of exponentially growth order ultradistribution nets and null nets of Beurling type are defined, respectively, by

E^(p!m,p!m

0)

exp =←−

Fp^m,r , N^(p!m,p!m

0)=←− Kp^m,r .

The sets of exponentially growth order ultradistribution nets and null nets of Roumieu type are defined, respectively, by

E^{p!m,p!m

0}

exp =−→

Fq^m,r , N^{p!m,p!m

0}

=−→ Kq^m,r .

Proposition 3·2.

(i) E_exp^(p!m,p!m0) (resp. E_exp^{p!m,p!m0}) are algebras under pointwise multiplication, and N^(p!m,p!m0) (resp. N^{p!m,p!m0}) are ideals of them.

(ii) The pseudodistances induced by ||| · |||p^m,µ_ν ,m⁰ (resp.||| · |||q_ν^m,µ,m⁰) are ultrapseu- dometrics on respective domains.

Proof. With Definition 3·1, this is just a particular case of of the general construc- tion recalled in section 2.

The Colombeau ultradistribution algebra G^(p!m,p!m0) (resp. G^{p!m,p!m0}) is defined by

G^(p!m,p!m

0)

=←−

Gp,r =E^(p!m,p!m

0)

exp /N^(p!m,p!m

0)

(resp. G^{p!m,p!m

0}

=−→

Gp,r =E^{p!m,p!m

0}

exp /N^{p!m,p!m

0}

).

These topological algebras are invariant under the actions of ultradifferential opera- tors of respective classes (m) and{m}, see e.g. [9].

3·2. Embeddings of ultradifferentiable functions and ultradistributions.

In what follows, mollifiers will be constructed by elements of spaces Σ^pow (resp.

Σder), which consist of smooth functions ϕ on R with the property that for some b >0,

σ^b(ϕ) = sup

β∈N,x∈R

|x^βϕ(x)|

b^ββ! <∞ (resp. σb(ϕ) = sup

α∈N,x∈R

|ϕ^(α)(x)|

b^αα! <∞) .

Both spaces are endowed with the respective inductive topologies.

Definition 3·3. Let (φⁿ)_n∈

N^∗ be a bounded net in Σ^pow (resp. Σder) such that

∀n∈N^∗:^R

Rt^jφⁿ(t) dt=δj,0 for j ∈0,1,2, . . . ,[n^1/m] + 1 , m >1. Then(φn)_n∈N∗

with φn =n φⁿ(n·) is called a net of{m,pow}(resp.{m,der})–mollifiers generated by(φⁿ)_n.

The following important lemma gives an explicit net of {m,pow}– and {m,der}–

mollifiers:

Lemma 3·4. For alln∈N^∗ and x∈R, let hn(x) = expn²− √ⁿ

n²ⁿ+x²ⁿ , kn(x) = exp −x²ⁿ . Then, for all n∈N^∗, hn(0) =kn(0) = 1 and

∀α∈ {1, ...,2n−1}: h^(α)_n (0) =k^(α)_n (0) = 0 , and there existr >0andC >0such that

sup

n∈N^∗

σr(hn)< C , sup

n∈N^∗

σ^r(kn)< C . (3·1) Moreover, for given m >1, the nets²

φⁿ= 1

2πFT(hg(n)) and φⁿ = 1

2πFT(kg(n)) ,

whereg(n) = ¹₂[n^1/(m−1)] + 1 for n∈N^∗, generate a net of {m,pow}–mollifiers and a net of {m,der}–mollifiers, respectively.

Proof. The first claims, h^(α)_n (0) = k^(α)_n (0) = δα,0 are easily verified, and imply obviously ^Rx^pFT(hn) = ^R x^pFT(kn) = 2π δp,0 ∀p ∈ {0, ...,2n−1}, which gives the second condition on{m,der}resp. {m,pow}–mollifiers for φⁿ.

So let us show (3·1), i.e. hn ∈ Σder, kn ∈ Σ^pow with constants independent of n.

Consider firsthn.

The function C 3 z 7→ √ⁿ

n²ⁿ+z²ⁿ has singularities at z = n e^{i π}(2k+1)/(2n). The nearest one to the real axis has the imaginary part nsin_2n^π ≥ 1 ∀n ∈ N^∗. So for every x ∈ R, the open disc { |z−x|<1} lies in the domain of analyticity of hn. Applying Cauchy’s integral formula, we have

∀x∈R,∀n∈N^∗: |h^(α)_n (x)|=

α!

2πi Z

|ζ−x|=¹₂

hn(ζ) dζ (ζ−x)^α+1

≤2^αα! max

θ∈[0,2π]

hn(x+¹₂e^iθ) .

Thus we have σ2(hn)≤C and therefore (3·1), if max|hn(x+ ¹₂e^iθ)|< C. So let us show that there existsC >0 such that

∀n∈N^∗, x∈R:<e

n²− ^qn n²ⁿ+ (x+ ¹₂e^iθ)²ⁿ

<lnC . (3·2) Letx+¹₂e^{i θ}=ρ e^{i φ} withρ∈R, |φ|< ^π₂. Consider first|ρ| ≥ ³₄n. Then, sinφ≤ _3n², thus 2n φ≤2narcsin_3n² < ^π₂ ∀n ≥1. Therefore<e1 + _n¹ρ e^iφ2n>1 and (3·2)

2 we denote the Fourier transform byFT(·) to avoid confusion with spacesFp,r etc.

A. Delcroix, M. Hasler, S. Pilipovi´ c, V. Valmorin

with lnC= 0. Next, if|ρ|< ³₄n, then

<e

n²− ⁿ q

n²ⁿ+ (ρ e^iφ)²ⁿ

< n²−n²ⁿ q

1− ³₄²ⁿ<1.

(The second function is decreasing forn≥2.) Again, this implies (3·2), with lnC = 1. So we have shown that∀n∈N^∗, σ₂(hn)<3, which proves (3·1) for hn. With all that precedes, it is easy to see that the givenφⁿgenerate a net of{m,pow}–mollifiers.

Now turn tokn ∈Σ^pow. Estimatingx^βkn(x) separately for|x| ≤2 and|x|>2 one can easily prove (3·1). Once again, this allows to conclude that the givenφⁿgenerate a net of{m,der}–mollifiers.

The embedding of ultradistributions into the corresponding weighted algebra of se- quences is realized through the first part of the next theorem. Its second part deals with the representatives of ultradifferentiable functions implying that the multipli- cation of regular enough elements within corresponding algebras is the ordinary multiplication.

Theorem 3·5. Assumem >1.

(i) Letψ∈ D^(m)(resp.ψ∈ D^{m−ρ})withρ >0such thatm−ρ >1) be compactly supported, and(φⁿ)_n generate a net of{m,pow}–mollifiers. Then

ψ∗φn−ψ∈ N^(p!m,p!m) , (φn=n φⁿ(n·)) ( resp. ψ∗φn−ψ∈ N^(p!m,p!m) ⊂ N^{p!m,p!m} ) .

(ii) Let f ∈ E^0(m) (resp. f ∈ E^{0 {m}}) with compact support; and (φⁿ)n gener- ate a net of {m,der}–mollifiers. Then f∗φn ∈ E_exp^{(p!m,p!m−1)}, (resp. f ∗φn ∈ E_exp^{(p!m,p!m−1)}⊂ E_exp^{{p!m,p!m−1}}).

If(φⁿ)n and(φ⁰ⁿ)n generate nets of {m,pow}–mollifiers, then

∀ψ∈ D^(m):hf∗φn−f∗φ⁰_n, ψi ∈ N₀^m , ( resp. ∀ψ∈ D^{m−ρ}:hf∗φn−f∗φ⁰_n, ψi ∈ N₀^m ). Remark 2. If ψ ∈ D^(m), m > 1, then (ψ)n ∈ E^(p!m,p!m

0)

for all m⁰ > 0. Fix a net of {m,pow}–mollifiers (φn)_n. The embedding D^(m) → E^(p!m,p!m) can be realized through ψ 7→ (ψ∗φn)_n as well as through ψ → (ψ)_n. This is a consequence of assertion (i). The similar conclusion follows forD^{m−ρ}. Thus, the product ofϕ, ψ∈ D^(m) (resp.ϕ, ψ ∈ D^{m−ρ}) is the usual one in E^(p!m,p!m) (resp. inE^{p!m,p!m}).

Assertion (ii) characterizes the embedding of elements inE^0(m)(resp.E^{0 {m}}) into the corresponding algebra by regularizations by {m,der}–mollifiers. Moreover, we have that the regularization of elements in E^0(m) (resp. E^{0 {m}}) with {m,pow}–mollifiers are weakly equal in the sense of ultracomplex numbers.

Note that D^(m1) ,→ D^{m1} ,→ D^(m2), m2 > m1 > 1, where the left space is dense in the right one. This implies D^0(m2) ,→ D^{0 {m1}} ,→ D^0(m1). With these relations theorem 3·5 implies various embedding results depending on the parameterm >1.

Proof. (i) Assume suppψ ⊂ [−µ, µ]. Sinceψ∗φn−ψ = 0 for |x| > µ, n > n0, we assume in this proof x∈ [−µ, µ], n > n0. First, we prove the assertion for the Beurling case; the Roumieu case is treated in a similar way.

Lets∈N. We have

(ψ∗φn−ψ)^(s)(x) = Z

R

ψ^(s)(x+t/n)−ψ^(s)(x)φⁿ(t) dt

= Z

R N−1

X

p=0

t^p

n^pp!ψ^(p+s)(x) + t^N

n^NN!ψ^(N+s)(ξ)−ψ^(s)(x)

!

φⁿ(t) dt ,

wherex≤ξ≤x+t/n. ForN = [n^1/m]+1 as in the definition of{m,pow}–mollifiers, (ψ∗φn−ψ)^(s)(x) =

Z

R

t^N

n^NN!ψ^(N+s)(ξ)φⁿ(t) dt . Letb >1 such thatσ^b(φⁿ)<∞. Then

ν^s

s!^m(ψ∗φn−ψ)^(s)(x)

≤ Z

R

1 (N+s)!^m

ψ^(N+s)(ξ)ν^s(N+s)!^m

n^Ns!^mN! t^N|φⁿ(t)|dt . We useN!^m≤(N^N)^m, (N+s)!≤e^N+sN!s! and _n¹N ≤ _N²N m^N , to get

ν^s

s!^m(ψ∗φn−ψ)^(s)(x)

≤ Z

R

(2e(ν+b))^N+s (N+s)!^m

ψ^(N+s)(ξ) N!^m N^mN

|t|^N

b^NN!|φⁿ(t)|dt . Let` >1. Insertinge<sup>−`N</sup>e<sup>`N</sup>,withν0= 2` e(ν+b), we have

r^s

s!^m(ψ∗φn−ψ)^(s)(x)

≤2^−`Np^m,µ_ν0 (ψ)σ^b(φⁿ).

Now we usee^{−` N</sup> ∼e<sup>−` n1/m} asn→ ∞. This implies for everyν >0 and` >0 there existC >0 so that

ν^s

s!^m(ψ∗φn−ψ)^(s)(x)

≤C e^{−` n1/m} . Taking the supremum over alls andx, we obtain that

|||ψ∗φn−ψ|||_p^m,µ_{ν ,n}−1/m= 0 .

Roumieu case:Letd >1 such thatσ^d(φⁿ)<∞and h >0 such thatp^m−ρ,µ_em−ρh(ψ)<

∞. With arbitraryν >0, as above, we have

ν^s

s!^m(ψ∗φn−ψ)^(s)(x)

≤ Z

R

|ψ^(N+s)(ξ)|

(N+s)!^m−ρ

ν^s(N+s)!^m−ρ

n^Ns!^mN! t^N|φⁿ(t)|dt .

≤ Z

R

(he^m−ρ)^N+s|ψ^(N+s)(ξ)|

(N+s)!^m−ρ

N!^m N^{N m}

(hν)^ss!^m−ρ(2dh)^N s!^mN!^ρ

|t|^N

d^NN!|φⁿ(t)|dt . Let` >1. Note

sup{(hν)^ss!^m−ρ

s!^m , s∈N}<∞, sup{(2dhe^`)^N

N!^ρ , N∈N}<∞.

As above we have, with suitable C >0, (insertinge^{−`N</sup>e<sup>`N}),

ν^s

s!^m(ψ∗φn−ψ)^(s)(x)

≤C e^−`Np^m−ρ,µ_em−ρh(ψ)σ^d(φⁿ) .

A. Delcroix, M. Hasler, S. Pilipovi´ c, V. Valmorin

(ii) We will give the proof in the Beurling case. The proof in the Roumieu case is similar. Recall [9], if f ∈ E^0(m), then there exists an ultradifferential operator of class (m), P(D) =^P_k∈NakD^k, µ0 >0 and continuous functions (Fk)_k∈N, with the property suppFk ⊂ [−µ0, µ0], sup

k∈N,x∈R

|Fk(x)| ≤ M, such that f = ^P_k∈NakD^kFk. This implies

∀x∈R:f∗φn(x) =

∞

X

k=0

(−1)^kakn^k Z

R

Fk(x+t/n)D^kφⁿ(t) dt ,

where (φn)_n is a net of {m,der}–mollifiers such that σb(φⁿ) < ∞ and ak, k ∈ N satisfy

∃h, B >0 :∀k∈N:|ak|< Bh^k/k!^m .

As in the part (i), we takex∈[−µ, µ], µ > µ0 andn > n0. Letν >1 be given and s∈N. We have

ν^s s!^m

f^(s)∗φn(x)=

∞

X

k=0

(−1)^kakn^k+s ν^s s!^m

Z

R

Fk(x+t/n)D^k+sφⁿ(t) dt

≤

∞

X

k=0

Bν^sh^kn^k+s k!^ms!^m

Z

R

|Fk(x+t/n)| |D^k+sφⁿ(t)|dt

≤

∞

X

k=0

B(νhe)^s+kn^k+s (k+s)!^m

Z

R

|Fk(x+t/n)| |D^k+sφⁿ(t)|dt

≤

∞

X

k=0

B 2^k

(2ebνh)^s+kn^k+s (k+s)!^m−1

Z

R

|Fk(x+_n^t)|

b^k+s(k+s)!

D^k+sφⁿ(t)dt

≤C e^{(2ebνhn)1/(m−1)}σb(φⁿ). This proves thatf∗φn∈ E_exp^{(p!m,p!m−1)}.

Let us prove (for the Beurling case) that D

f,( ˇφn−φˇ⁰_n)∗ψ^E∈ N₀^m .

By continuity, we know that there existµ∈N^∗, ν >0 and C >0 such that

|hf,( ˇφn−φˇ⁰_n)∗ψi| ≤C p^µ,m_ν (( ˇφn−φˇ⁰_n)∗ψ)

≤C ^hp^µ,m_ν ( ˇφn∗ψ−ψ) +p^µ,m_ν ( ˇφ⁰_n∗ψ−ψ)ⁱ . (3·3) By the first part of the theorem we have that

ψ∗φn−ψ, ψ∗φ⁰_n−ψ∈ N^(p!m,p!m) .

This implies that for everyk >0 there existsC >0 such that for everyn∈N^∗ both addends in (3·3) are≤C e^{−k n1/m}.

4. Generalized hyperfunctions on the circle

For λ >1, let Ωλ =z∈C| ¹_λ <|z|< λ and Oλ the Banach space of bounded holomorphic functions on Ωλ. We denote by E(T) (resp. A(T) := ind limλ→1Oλ) the space of smooth (resp. analytic) functions on the unit circle T = {z ∈ C |

|z| = 1} and by E⁰(T) (resp. B(T)) the corresponding space of distributions (resp.

hyperfunctions), cf. [11]. For f ∈ A(T), the coefficient Tb(k) of ek(z) = z^k in the Laurent expansion of f is its k-th Fourier coefficient. Complex numbers ck, k ∈ Z, are the Fourier coefficients of some analytic function (resp. some hyperfunction) if and only if|||(c_±k)_k∈

N^∗ |||_|·|,1/k<1 (resp.≤1).

Let m ∈ [0,1) and ν >0. We denote by Am,ν(T) the set of functions f ∈ A(T) such thatq_ν^m,∞(f) := sup_t∈R,α∈N^|f_ν^˜(α)αα!^(t)|m <∞where ˜f(t) =f(e^it), t∈R. We set

Am(T) = ind lim

ν→∞ Am,ν(T) and A1(T) = ind lim

m→1⁻ Am(T).

Clearly A1(T) a subalgebra ofA(T) whose elements are holomorphic inC^∗. To prove the following theorem, we establish

Lemma 4·1. Let m∈(0,1)and ρ > e/2. Then the function ϕ:t7→ρ^−tt^m(t+12)e^−mt , t∈¹₂,∞

reaches its minimum in a unique point tρ such that ¹₂ < tρ< ρ^m1 −¹₂. Moreover, ϕ is strictly increasing on [tρ,∞) and ϕ(ρ^1/m+¹₂)<√

e ρ e^−mρ1/m. Proof. The derivative ofψ = lnϕ, given byψ⁰(t) =−lnρ+m lnt+_2t¹,is strictly increasing fort∈(¹₂,∞) and verifies

ψ⁰(tρ) = 0 ⇐⇒ tρe^1/2tρ =ρ^1/m .

This yields ρ^1/m−tρ = tρ(e^1/2tρ −1), and, using x < e^x−1 < x e^x for x 6= 0, the claimed inequalities ontρ. Writing ln(ρ^1/m+ ¹₂) = _m¹ lnρ+ ln(1 + _2ρ1/m¹ ) gives ψ(ρ^1/m+¹₂) = ¹₂lnρ+m(ρ^1/m+ 1) ln(1 +_2ρ1/m¹ )−m(ρ^1/m+¹₂). Since ln(1 +_2ρ1/m¹ )≤

1

2ρ^1/m it follows that ψ(ρ^1/m+ ¹₂) ≤ ¹₂lnρ+m(_2ρ1/m¹ −ρ^1/m). Usingρ > e/2, and m∈(0,1), we find _2ρ^m1/m < ¹₂ and thusϕ(ρ^1/m+ ¹₂)≤√

e ρ e^−mρ1/m.

Previously, we defined the ||| · ||| norm only for sequences indexed byN. Here it is convenient to use it also for the nets ˆf(k) indexed byZ, for which we take it to be the maximum of the norms of ( ˆf(k))_k∈N and ( ˆf(−k))_k∈N, or equivalently

|||( ˆf(k))_k|||^±_(·)−1/m = lim sup

k→∞

max(|f(k)|,ˆ |f(−k)|)ˆ ^k

−1/m

.

Theorem 4·2. Let f∈ A(T) and m∈(0,1).

(i) If f∈ Am,ν(T) then:

|||( ˆf(k))_k|||^±_(·)−1/m≤e^−m/ν1/m.

Conversely if the above condition holds, thenf ∈ Am,ν⁰(T) for all ν⁰> ν. (ii) f ∈ Am(T) if and only if

|||( ˆf(k))_k|||^±_(·)−1/m<1.

(iii) f ∈ A0,ν(T) if and only if fˆ(k) = 0 for |k|> ν.

(iv)f ∈ A0(T) if and only if ( ˆf(k))k∈Z has finite support.

(v) For all f ∈ A1(T) there exists g∈ O(C^∗) such that g|_T=f.

Proof. Let f ∈ Am,ν(T) with 0 < m < 1. For all α ∈ N, ˜f^(α)(t) =

A. Delcroix, M. Hasler, S. Pilipovi´ c, V. Valmorin

P

p∈Z(ip)^αf(p)ˆ e^ipt. It follows that ^R_−π^π f˜^α(t)e^−iktdt = 2π(ik)^αf(k), and then con-ˆ sequently there is a positive constantC1 such that|k|^α|f(k)| ≤ˆ C1ν^αα!^m.

Using Stirling’s formula: α! =α^α+1/2e^−α√

2π(1 +εα), εα & 0, we find a positive constantC2such that:

∀α∈N^∗, ∀k∈Z, |k|^α|f(k)| ≤ˆ C2ν^αα^m(α+1/2)e^−mα. It follows that:

∀α∈N^∗,∀k∈Z^∗, |f(k)| ≤ˆ C2

ν

|k|

α

α^m(α+1/2)e^−mα.

Using the notations of Lemma 4·1 by taking ρ = ^|k|_ν with |k| > e ν, yields

|fˆ(k)| ≤ C2ϕ(t) for all t ∈ N^∗. Following the Lemma, we have ϕ(ρ^1/m+ ¹₂) ≤

√e ρ e^−mρ1/m. Since ϕ increases on [ρ^1/m − ¹₂, ρ^1/m + ¹₂] which contains a posi- tive integer αρ, then |f(k)| ≤ˆ C2ϕ(αρ) ≤ C2

√e ρ e^−mρ1/m for |k| > eν, that is

|fˆ(k)| ≤C2(^e|k|_ν )^1/2e^−m/ν1/m for|k|> eν from which inequality of (i)follows.

Conversely assume that f satisfies the condition of (i). Let ν⁰ > ν. Choose ν⁰⁰ such thatν⁰ > ν⁰⁰> ν and setβ⁰=m/(ν⁰)^1/m, β⁰⁰=m/(ν⁰⁰)^1/m. Sincee^−β00 > e^−m/ν1/m, there exists a positive constant C >|f(0)|ˆ such that|fˆ(k)| ≤Ce^{−β00|k|1/m} for every k∈ Z. For every α∈N, we have ˜f^(α)(t) =^P_k∈Z(ik)^αfˆ(k)e^ikt. Then, using the last inequality we find

kf˜^(α)k∞≤C ^X

k∈Z

e^{−(β00−β0)|k|1/m}

! sup

k∈Z

|k|^αe^−β0|k|1/m.

Letφ(t) =t^αe^−β0t1/m;t≥0. A simple study ofφshows that sup_t≥0φ(t) =φ(ν⁰α^m) = (ν⁰)^αα^mαe^−mα. Using Stirling’s formula, we get a positive constantC1such that for allα∈N, f˜^(α)

∞≤C1ν^0αα!^m, showing that f∈ Am,ν⁰(T) and proving(i).

Letf ∈ Am(T). Thenf ∈ Am,ν(T) for someν >0 and the inequality follows from (i) and e^−m/ν1/m < 1. Conversely if |||( ˆf(k))_k|||^±_(·)−1/m < 1, there exists ν > 0 such that|||( ˆf(k))_k|||^±_(·)−1/m≤e^−m/ν1/m. From(i), it follows thatf ∈ Am,ν⁰(T) forν⁰> ν.

Hence f∈ Am(T) proving(ii).

Let f ∈ A0,ν(T). The previous shows that there exists C1 > 0 such that

|k|^α|fˆ(k)| ≤C1ν^α. It follows that |f(k)| ≤ˆ C1

ν

|k|

^α

for allk ∈Z^∗ and all α∈N. If

|k|> ν, then _|k|^ν <1, and makingα→ ∞yields ˆf(k) = 0.

Conversely, assume that ˆf(k) = 0 for |k| > ν. Then we have ∀z ∈ C^∗, f(z) = P

|k|≤νf(k)ˆ z^k. It follows that for allα ∈N,

f˜^(α)

∞≤^P_|k|≤νfˆ(k) ν^α, that is f ∈ A0,ν(T) proving(iii).

Claim(iv)follows from (iii)straightforwardly.

Claims (ii) and (iv) show that for f ∈ A1(T), the series ^P_k∈Zfˆ(k)z^k converges absolutely for anyz ∈C, proving(v).

Now let r = (rn)_n with rn > 0 and rn & 0. For n ∈ N, we set ψn(z) = P

|k|≤1/rnz^k. We have ψn ∗ψn = ψn and limn→∞ψn = δ in E⁰(T). If H ∈ B(T), H ∗ψn = ^P_|k|≤1/rnH(k)zˆ ^k (where S∗T = z 7→ ^P_k∈ZS(k)b Tb(k)z^k) and conse- quently lim_n→∞H∗ψn =H inB(T).

For f ∈ Oλ we set q^λ(f) = kfk_L∞(Ω_λ) and ˆq^λ(f) = sup

k∈Z

λ^|k||f(k)|.ˆ Recall that, for