ON A CLASS OF TRANSLATION-INVARIANT SPACES OF QUASIANALYTIC

Vol. 45, No. 1, 2015, 143-175

ON A CLASS OF TRANSLATION-INVARIANT SPACES OF QUASIANALYTIC

ULTRADISTRIBUTIONS

Pavel Dimovski², Bojan Prangoski³, and Jasson Vindas⁴ Dedicated to Professor B. Stankovi´c on the occasion of his 90th birthday and

to Professor J. Vickers on the occasion of his 60th birthday Abstract. A class of translation-invariant Banach spaces of quasi- analytic ultradistributions is introduced and studied. They are Banach modules over a Beurling algebra. Based on this class of Banach spaces, we define corresponding test function spaces D^∗E and their strong du- alsDE^′∗∗′ of quasianalytic type, and study convolution and multiplicative products on DE^′∗_∗^′. These new spaces generalize previous works about translation-invariant spaces of tempered (non-quasianalytic ultra-) dis- tributions; in particular, our new considerations apply to the settings of Fourier hyperfunctions and ultrahyperfunctions. New weighted D_L^′∗p_η spaces of quasianalytic ultradistributions are analyzed.

AMS Mathematics Subject Classification(2010): 46F05, 46H25, 46F10, 46F15, 46E10

Key words and phrases: Quasianalytic ultradistributions; Convolution of ultradistributions; Translation-invariant Banach space of ultradistribu- tions; Tempered ultradistributions; Beurling algebra; Hyperfunctions

1. Introduction

Recently, the authors and Pilipovi´c have constructed and studied new classes of distribution and non-quasianalytic ultradistribution spaces in connection with translation-invariant Banach spaces [2, 4]. Those spaces generalize the concrete instances of weighted DL^′p and DL^′∗p spaces [1, 14] and have shown usefulness in the study of boundary values of holomorphic functions [3] and the convolution of generalized functions [4].

The aim of this article is to extend the theory of ultradistribution spaces associated to translation-invariant Banach spaces by considering mixed quasi- analytic cases. We have been able here to transfer all results from [4] to this new

1J. Vindas gratefully acknowledges support by Ghent University, through the BOF-grant 01N01014.

2Faculty of Technology and Metallurgy, University Ss. Cyril and Methodius, Ruger Boskovic 16, 1000 Skopje, Macedonia, e-mail: dimovski.pavel@gmail.com

3Faculty of Mechanical Engineering, University Ss. Cyril and Methodius, Karpos II bb, 1000 Skopje, Macedonia, e-mail: bprangoski@yahoo.com

4Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, 9000 Gent, Belgium, e-mail: jvindas@cage.UGent.be

setting with the aid of various new important results for quasianalytic ultradis- tribution spaces of typeS_†^′∗(R^d) (see Subsection 1.1 for the notation) from [10]

concerning the construction of parametrices and the structure of these spaces.

Such technical results will be stated in Section 2 without proofs, as details will be treated in [10]. Although our results in the present paper are analogous to those from [4], new arguments and ideas have had to be developed here in order to deal with the quasianalytic case and achieve their proofs.

In Section 3 we study the class of translation-invariant Banach spaces of ultradistributions of class∗ − †. These are translation-invariant Banach spaces satisfying S_†^∗(R^d) ,→ E ,→ S_†^′∗(R^d) and having ultrapolynomially bounded weight function of class†. Here∗ and †stand for the Beurling and Roumieu cases of sequences Mp and Ap, respectively. We would like to emphasize that our considerations apply to hyperfunctions and ultra-hyperfunctions, which correspond to the symmetric choices Mp = Ap = p!; but more generally, our weight sequence Mp, measuring the ultradifferentiability, is allowed to satisfy the mild conditionp!^λ⊂Mpwithλ >0. The growth assumption onApis just p! ⊂ Ap, which also allows us to deal with Banach spaces whose translation groups may have exponential growth.

Section 4 contains our main results. In analogy to [4], we introduce the test function spaces D^(ME ^p), DE^{Mp}, and ˜DE^{Mp}. We prove that the following continuous and dense embeddings hold S_†^∗(R^d) ,→ DE^∗ ,→ E ,→ S_†^′∗(R^d) and that D^∗E are topological modules over the Beurling algebra L¹_ω, where ω is the weight function of the translation group of E. We also prove the dense embeddingD^∗E,→ O^∗_†,C(R^d), where the spacesO_†^∗,C(R^d) are defined in a similar way as in [4]. The spaceD^′∗E_∗^′ is defined as the strong dual ofDE and various structural and topological properties ofD^′∗E^′_∗ are obtained via the parametrix method (Lemma 2.2). We also prove thatDE^{Mp}= ˜D^{E^Mp}, topologically.

As an application of our theory, we extend the theory of D_L^′∗p_η, Bη^′∗, and B˙^′∗η spaces not only by considering quasianalytic cases of∗ but also by allow- ing ultrapolynomially bounded weightsη which may growth exponentially. We establish relations among them and make a detailed investigation of their topo- logical properties. We would like to point out that applications of such results to the study of the general convolvability in the setting of quasianalytic ultra- distributions will appear elsewhere [10]. We conclude this section with some results about convolution and multiplicative products onD^′∗E′

∗. 1.1. Notation

Let (M_p)_p∈Nand (A_p)_p∈N be two sequences of positive numbers such that M₀ =M₁ =A₀ = A₁ = 1. Throughout the article, we impose the following assumptions over these weight sequences. The sequenceMpsatisfies the ensuing three conditions:

(M.1)M_p²≤Mp−1Mp+1, p∈Z+; (M.2)Mp ≤c0H^p min

0≤q≤p{Mp−qMq},p, q∈N, for some c0, H≥1;

(M.5) there existss > 0 such that M_p^s is strongly non-quasianalytic, i.e.,

there exists c0≥1 such that

∑∞ q=p+1

M_q^s₋₁

M_q^s ≤c0p M_p^s

M_p+1^s , ∀p∈Z+.

It is clear that if M_p^s is strongly non-quasianalytic than for any s^′ > s, M_p^s′ is also strongly non-quasianalytic. One easily verifies that when Mp satisfies (M.5) there existsκ >0 such thatp!^κ⊂M_p, i.e., there exist c₀, L₀>0 such that p!^κ ≤c₀L^p₀M_p, p∈ N (cf. [7, Lemma 4.1]). Following Komatsu [7], for p∈Z+, we denotem_p =M_p/M_p−1 and for ρ≥0 letm(ρ) be the number of m_p≤ρ. As a consequence of [7, Proposition 4.4], by a change of variables, one verifies that M_p satisfies (M.5) if and only if

∫ _∞

ρ

m(λ)

λ^s+1dλ≤cm(ρ)

ρ^s , ∀ρ≥m1.

A sufficient condition forM_p to satisfy (M.5) is if the sequencem_p/p^λ,p∈Z+

is monotonically increasing for someλ >0.

We assume that A_p satisfies (M.1) and (M.2). Of course, without loss of generality, we can assume that the constants c₀ and H from the condition (M.2) are the same forM_p andA_p. Moreover, we also assume thatA_psatisfies the following additional hypothesis:

(M.6)p!⊂Ap; i.e., there existc0, L0>0 such thatp!≤c0L^p₀Ap,p∈N. Of course, the constantsc₀andL₀in (M.6) can be chosen such thatc₀, L₀≥ 1. Although it is not a part of our assumptions, we will primary be interested in the quasianalytic case, i.e.,

∑∞ p=1

M_p−1 Mp

=∞.

We denote byM(·) andA(·) the associated functions ofM_pandA_p, that is, M(ρ) := sup

p∈Nln+

ρ^p

M_p andA(ρ) := sup

p∈Nln+

ρ^p

A_p forρ >0, respectively. They are non-negative continuous increasing functions (cf. [7]). We denote byRthe set of all positive monotonically increasing sequences which tend to infinity. For (lp) ∈R, denote byNl_p and Bl_p(·) the associated functions of the sequences Mp

∏p

j=1lj andAp

∏p

j=1lj, respectively.

Forh >0 we denote byS_A^M_p^p_,h^,hthe Banach spaces (in short (B)-space from now on) of all φ∈C^∞(R^d) for which the norm

σ_h(φ) = sup

α

h^|α|e^A(h|·|)D^αφ

L^∞(R^d)

M_α

is finite. One easily verifies that forh1< h2 the canonical inclusionS_A^M_p^p_,h^,h₂² → SA^M_p^p,h^,h₁¹is compact. As l.c.s., we defineS_(A^(M_p^p₎⁾(R^d) = lim

h←−→∞

SA^M_p^p,h^,handS_{^{_A^M_p^p_}^}(R^d) = lim−→

h→0

SA^M_p^p,h^,h. Since for h₁ < h₂ the inclusion SA^M_p^p,h^,h₂² → SA^M_p^p,h^,h₁¹ is compact,

S_(A^(M_p^p₎⁾(R^d) is an (F S)-space andS_{^{_A^M_p^p_}^}(R^d) is a (DF S)-space. In particular, they are both Montel spaces.

For each (rp)∈R, byS_A^M_p^p_,(r^,(r_p^p₎⁾we denote the space of allφ∈C^∞(R^d) such that

σ(r_p)(φ) = sup

α

e^Brp(|·|)D^αφ

L^∞(R^d)

M_α∏_|α|

j=1r_j <∞.

Provided with the normσ_(rp), the spaceS_A^M_p^p_,(r^,(r_p^p₎⁾becomes a (B)-space. Simi- larly as in [1, 9], one can prove thatS_{^{_A^M_p^p_}^}(R^d) is topologically isomorphic to

lim

(rp)∈R

←−

S_A^M_p^p_,(r^,(r_p^p₎⁾.

In the future we shall employS_†^∗(R^d) as a common notation forS_(A^(M_p^p₎⁾(R^d) (Beurling case) andS_{^{_A^M_p^p_}^}(R^d) (Roumieu case). It is clear that for eachh >0 and (r_p) ∈ R, the spaces S_A^M_p^p_,h^,h and S_A^M_p^p_,(r^,(r_p^p₎⁾ are continuously injected into S(R^d) (the Schwartz space).

We will often make use of the following technical result from [11].

Lemma 1.1 ([11]). Let (kp)∈ R. There exists (k^′_p)∈ R such that k^′_p ≤ kp

and

p+q∏

j=1

k^′_j≤2^p+q

∏p j=1

k^′_j·

∏q j=1

k^′_j, for allp, q∈Z+.

We adopt the following notations. The symbol “,→” stands for a contin- uous and dense inclusion between topological vector spaces. For h∈ R^d and f ∈ S_†^′∗(R^d) we denote asThf translation byh, i.e.,Thf =f(·+h). We write

⟨x⟩= (1 +|x|²)^1/2,x∈R^d.

2. Some important auxiliary results on the space S

( R

)

We collect in this section some important results on the nuclearity ofS_†^∗(R^d), the existence of parametrices as well as a characterisation of bounded sets in S_†^′∗(R^d). These are essential tools in the rest of the article. We refer to [10] for the proofs. Unless explicitly stated, we deal with the Beurling and Roumieu cases simultaneously. We follow the ensuing convention. We shall first state assertions for the (M_p)−(A_p) case followed in parenthesis by the corresponding statements for the{M_p} − {A_p} case.

Proposition 2.1. The space S_†^∗(R^d)is nuclear.

Proposition 2.2. For everyt >0 there existG∈ S_A^M_p^p_,t^,t and an ultradifferen- tial operator P(D) of class(Mp)(for every (tp)∈R there exist G∈ S_A^M_p^p_,(t^,(t_p^p₎⁾ and an ultradifferential operator P(D)of class {Mp}) such thatP(D)G=δ.

Lemma 2.3. Let r >0 ((rp)∈R).

i) For eachχ, φ∈ S_†^∗(R^d)andψ∈ S_A^M_p^p_,r^,r(ψ∈ S_A^M_p^p_,(r^,(r_p^p₎⁾), one hasχ∗(φψ)∈ S_†^∗(R^d).

ii) Let φ, χ∈ S_†^∗(R^d) with φ(0) = 1 and ∫

R^dχ(x)dx = 1. For each n∈Z+

define χn(x) =n^dχ(nx) andφn(x) =φ(x/n). Then there existsk≥2r ((kp) ∈ R with (kp) ≤(rp/2)) such that the operators Q˜n : ψ 7→ χn∗ (φnψ), are continuous as mappings fromSA^M_p^p,k^,kintoSA^M_p^p,r^,r(fromS_A^M_p^p_,(k^,(k_p^p₎⁾ intoS_A^M_p^p_,(r^,(r_p^p₎⁾), for all n∈Z+. Moreover Q˜_n→Idin Lb

(SA^M_p^p,k^,k,SA^M_p^p,r^,r

) (Lb

(S_A^M_p^p_,(k^,(k_p^p₎⁾,S_A^M_p^p_,(r^,(r_p^p₎⁾) ).

In the next proposition, given t > 0 ((t_p) ∈ R), we denote as S^MA_p^p,t^,t

(as S^MAp^p,(t^,(tp^p)⁾) the closure of S_(A^(M_p^p₎⁾(R^d) in SA^M_p^p,t^,t (the closure of S_{^{_A^M_p^p_}^}(R^d) in S_A^M_p^p_,(t^,(t_p^p₎⁾).

Proposition 2.4. Let B be a bounded subset ofS_†^′∗(R^d). There exists k >0 ((k_p)∈ R) such that each f ∈ B can be extended to a continuous functional f˜on S^MA_p^p,k^,k (on S^MA_p^p,(k^,(k_p^p)⁾). Moreover, there existsl ≥k ((l_p)∈R with (l_p)≤ (kp)) such that S_A^M_p^p_,l^,l ⊆ S^MAp^p,k^,k (S_A^M_p^p_,(l^,(l_p^p₎⁾⊆ S^MAp^p,(k^,(kp^p)⁾) and ∗:S_A^M_p^p_,l^,l× S_A^M_p^p_,l^,l → S^MAp^p,k^,k (∗ :S_A^M_p^p_,(l^,(l_p^p₎⁾× S_A^M_p^p_,(l^,(l_p^p₎⁾ → S^MAp^p,(k^,(kp^p)⁾) is a continuous bilinear mapping.

Furthermore, there exist an ultradifferential operator P(D) of class∗ and u∈ S^MA_p^p,l^,l (u∈ S^MA_p^p,(l^,(l_p^p)⁾) such thatP(D)u=δandf = (P(D)u)∗f =P(D)(u∗f˜) for each f ∈ B, where u∗f˜ is the image of f˜ under the transpose of the continuous mapping φ7→uˇ∗φ,S_(A^(M_p^p₎⁾(R^d)→ S^MAp^p,k^,k (S_{^{_A^M_p^p_}^}(R^d)→ S^MAp^p,(k^,(kp^p)⁾).

For f ∈ B, u∗f˜∈ L^∞_eA(l|·|)∩C(R^d) (u∗f˜∈L^∞

e^Blp(|·|) ∩C(R^d)) and in fact u∗f˜(x) = ⟨f , u(x˜ − ·)⟩. The set {u∗f˜|f ∈ B} is bounded in L^∞_eA(l|·|) (in L^∞

e^Blp(|·|)).

Lemma 2.5. Let B⊆ S_†^′∗(R^d). The following statements are equivalent:

i) B is bounded inS_†^′∗(R^d);

ii) for eachφ∈ S_†^∗(R^d),{f∗φ|f ∈B} is bounded inS_†^′∗(R^d);

iii) for eachφ∈ S_†^∗(R^d)there existt, C >0(there exist(tp)∈RandC >0) such that|(f∗φ)(x)| ≤Ce^A(t|x|)(|(f∗φ)(x)| ≤Ce^Btp(|x|)) for allx∈R^d, f ∈B;

iv) there existC, t >0(there exist (tp)∈RandC >0) such that

|(f∗φ)(x)| ≤Ce^A(t|x|)σ_t(φ) (

resp. |f∗φ(x)| ≤Ce^Btp(|x|)σ_(tp)(φ) ) for allφ∈ S_†^∗(R^d),x∈R^d,f ∈B.

Lemma 2.6. Let f ∈ S_(p!)^′(Mp)(R^d) (f ∈ S_{^′{_p!^M_}^p}(R^d)). Then f ∈ S_†^′∗(R^d) if and only if there exists t > 0 (there exists (tp)∈ R) such that for every φ∈ S_(p!)^(Mp)(R^d)(for everyφ∈ S_{^{_p!^M_}^p}(R^d))

sup

x∈R^d

e^−A(t|x|)|(f∗φ)(x)|<∞ (

sup

x∈R^d

e^−Btp(|x|)|(f∗φ)(x)|<∞ )

.

3. Translation-invariant Banach spaces of quasianalytic ultradistributions

We extend here the theory of translation-invariant Banach spaces of ultra- distributions to the quasianalytic case. We closely follow the approach from [2, 4], where the distribution and non-quasianalytic ultradistribution cases were treated. We mention that some of the arguments below are similar to those from [4], but for the reader’s convenience we include all details about the adap- tations in the corresponding proofs.

LetEbe a (B)-space. We callEatranslation-invariant(B)-space of ultra- distributions of class ∗ − †if it satisfies the following three axioms:

(I) S_†^∗(R^d),→E ,→ S_†^′∗(R^d).

(II) Th(E)⊆E for eachh∈R^d.

(III) There existτ, C >0 (for everyτ >0 there existsC >0), such that

∥T_hg∥E≤C∥g∥Ee^A(τ|h|), ∀h∈R^d, ∀g∈E.

Notice that the condition (III) implicitly makes use of the continuity of Th. The next lemma shows that such a continuity is always ensured by the conditions (I) and (II).

Lemma 3.1. Let E be a (B)-space satisfying (I) and (II). The translation operatorsTh:E→E are bounded for all h∈R^d.

Proof. Observe thatT_h is continuous as a mapping fromE toS_†^′∗(R^d) since it can be decomposed asE−→ S^Id _†^′∗(R^d)−−→ S^Th _†^′∗(R^d) andTh:S_†^′∗(R^d)→ S_†^′∗(R^d) is continuous. Thus the graph of Th is closed in E× S_†^′∗(R^d) and, since its image is inE, its graph is also closed inE×E(E×Eis continuously injected into E× S_†^′∗(R^d) via the mapping Id×Id). As E is a (B)-space, the closed graph theorem implies thatTh is continuous.

Lemma 3.2. Let E be a translation-invariant (B)-space of ultradistributions of class ∗ − †. For every g ∈ E, lim

h→0∥Thg−g∥E = 0. In particular for each g ∈E the mapping h7→Thg, R^d →E, is continuous at 0 (hence everywhere continuous).

Proof. The proof is straightforward and we omit it.

Summarizing, Lemma 3.1 and Lemma 3.2 prove that a translation-invariant (B)-space of ultradistributionsE of class∗ − †satisfies the following stronger condition than (II):

(fII) for eachh >0,Th:E→Eis continuous and for eachg∈Ethe mapping h7→Thg,R^d→E, is continuous.

Clearly T₀ = Id_E, T_h1+h2 = T_h1 ◦T_h2 = T_h2 ◦T_h1. Next, we define the weight functionω(h)ofE as

(3.1) ω(h) =∥T_−h∥_L(E).

Obviously the weight function is positive and ω(0) = 1. Furthermore, since S_†^∗(R^d) is separable (it is an (F S)-space or a (DF S)-space, respectively), so is E. Thus ω(h) =∥T₋h∥_L(E) is the supremum of∥T₋hg∥E where g belongs to a countable dense subset of the closed unit ball of E. Since h7→ ∥T₋hg∥E

is continuous, ω is measurable. Clearly, the logarithm of ω is subadditive and there exist C, τ > 0 (for every τ > 0 there exists C > 0) such that ω(h)≤Ce^A(τ|h|).

Remark 3.3. In the Beurling case whenA_p = p!, the assumption (III) is su- perfluous. In fact, assuming only (I) and (II), Lemma 3.1 implies that for each h ∈ R^d, T_h : E → E is continuous. Additionally, one easily verifies that for each fixedφ∈ S_†^∗(R^d), one hasT_hφ→φash→0 inS_†^∗(R^d) and consequently in E. Hence, employing the same reasoning as above, we obtain that ω is a measurable positive function with subadditive logarithm. Therefore, there ex- ist C, h >0 such that ω(h) ≤Ce^k|h|, ∀h∈R^d (cf. [5, Sect 7.4]), which is in fact condition (III) in this case.

We will also give an alternative version of (III) in the Roumieu case which is sometimes easier to work with than (III). For this purpose we need the following technical result from [11].

Lemma 3.4 ([11]). Let g : [0,∞) → [0,∞) be an increasing function that satisfies the following estimate:

For everyL >0 there existsC >0 such thatg(ρ)≤A(Lρ) + lnC.

Then there exists a subordinate functionϵ(ρ)such thatg(ρ)≤A(ϵ(ρ)) + lnC^′, for some constant C^′>1.

See [7] for the definition of subordinate function.

Lemma 3.5. In the Roumieu case condition (III) is equivalent to the following one:

(III)g there exist (lp)∈R and C >0 such that ∥Thg∥E ≤C∥g∥Ee^Blp(|h|), for allg∈E,h∈R^d.

Proof. The proof is analogous to that of (c)⇔(˜c) in [4, Theorem 4.2].

The next theorem gives a weak criterion to conclude that a (B)-spaceEis a translation-invariant space of ultradistributions of class∗ − †.

Theorem 3.6. Let E be a (B)-space satisfying:

(I)^′ S_(p!)^(Mp)(R^d),→E ,→ S_(p!)^′(Mp)(R^d) (S_{^{_p!^M_}^p}(R^d),→E ,→ S_{^′{_p!^M_}^p}(R^d));

(II) Th(E)⊆E, for allh∈Rⁿ;

(III)^′ for any g∈E there exist C =C_g >0 andτ =τ_g >0 (for everyτ >0 there exists C=C_g,τ >0) such that∥T_hg∥E≤Ce^A(τ|h|),∀h∈R^d. ThenEis a translation-invariant(B)-space of ultradistributions of class∗ − †. Proof. Employing the same technique as in the proof of Lemma 3.1, one easily verifies that conditions (I)^′ and (II) imply the continuity ofT_h:E→E. The proof of (III) can be obtained by adapting the proof of (c) in [4, Theorem 4.2].

We now address (I). To proveS_†^∗(R^d),→E, by (I)^′, it is enough to prove thatS_†^∗(R^d) is continuously injected intoE. Pickψ1∈ D(R^d) such that

∑

m∈Z^d

ψ1(x−m) = 1, ∀x∈R^d, suppψ1∈[−1,1]^d

andψ₁is non-negative and even. Next, pickψ₂∈ S_(p!)^(Mp)(R^d) (ψ₂∈ S_{^{_p!^M_}^p}(R^d)), such that∫

R^dψ2(x)dx= 1 andψ2is even. Setψ=ψ1∗ψ2. One readily verifies that ∑

m∈Z^dψ(x−m) = 1 for allx∈R^d andψ ∈ S_(p!)^(Mp)(R^d) in the Beurling case and ψ ∈ S_{^{_p!^M_}^p}(R^d) in the Roumieu case, respectively. By (III), there existC, τ >0 (for everyτ >0 there existsC >0) such that

∥φT₋mψ∥E ≤Ce^−A(τ|m|)∥e^2A(τ|m|)ψTmφ∥E, ∀φ∈ S_†^∗,∀m∈Z^d. (3.2)

Form∈Z^d, consider the linear mapping

ρm,τ(φ) =e^2A(τ|m|)ψTmφ,S_(A^(M_p^p₎⁾(R^d)→ S_(p!)^(Mp)(R^d)(S_{^{_A^M_p^p_}^}(R^d)→ S_{^{_p!^M_}^p}(R^d)).

Clearly, it is well defined. Let B be a bounded subset of S_†^∗(R^d). Then for everyh >0 (there existsh >0) such that

sup

φ∈B

sup

α∈N^d

h^|α|e^A(h|·|)D^αφ

L∞

M_α <∞ (3.3)

Now, [7, Lemma 3.6] implies

e^2A(τ|m|)≤2c0e^A(2Hτ|x+m|)e^A(2Hτ|x|). (3.4)

In the Beurling case, let h1 > 0 be arbitrary but fixed. Choose h > 0 such thath≥max{2Hτ,2h₁} ande^{A(2Hτ λ)}≤C^′e^hλ for allλ≥0 (such anhexists becausep!⊂A_p). By (3.3) and (3.4) we have

h^|₁^α||D^α(ψ(x)Tmφ(x))|e^h1|x|

Mα ≤C2e^−2A(τ|m|), (3.5)

for all x∈R^d, m∈ Z^d, φ∈B. Hence {ρm,τ|m ∈Z^d} is uniformly bounded onB. In the Roumieu case there exist ˜h,C >˜ 0 such that ˜h^|α||D^αψ(x)|e^˜h|x|≤ CM˜ α for all x ∈ R^d, α ∈ N^d. For the h > 0 for which (3.3) holds choose 0 < τ ≤h/(2H) such that e^{A(2Hτ λ)} ≤C^′e^˜hλ/2 for all λ ≥0 (such aτ exists because p! ⊂ Ap). Choose h1 ≤ min{h/2,˜h/2}. Then, by using (3.3) and (3.4), similarly as in the Beurling case, we obtain (3.5), i.e.,{ρm,τ|m∈Z^d}is uniformly bounded onB. Now, (I)^′implies that∥ρm,τ(φ)∥E≤C₂^′ for allφ∈B, m∈Z^d. By using (3.2), we obtain that the sequence {∑

|m|≤NφT₋mψ }_∞

N=0

is a Cauchy sequence in E for each φ∈B. Since its limit isφ in S_(p!)^′(Mp)(R^d) (in S_{^′{_p!^M_}^p}(R^d)) it converges to φ∈ E. Also ∥φ∥E ≤ C for all φ∈ B. This implies that S_†^∗(R^d)⊆E and the inclusion maps bounded sets into bounded sets. AsS_†^∗(R^d) is bornological, the inclusion is continuous. It remains to prove E ⊆ S_†^′∗(R^d). By (I)^′ for a bounded set B in S_(p!)^(Mp)(R^d) (inS_{^{_p!^M_}^p}(R^d)) there exists D >0 such that|⟨g,φˇ⟩| ≤D∥g∥E for all g∈E andφ∈B. Then (III) implies that there existC, τ >0 (for everyτ >0 there existsC >0) such that

|(g∗φ)(y)| ≤D∥Tyg∥E ≤CDe^A(τ|y|), for ally ∈R^d, φ∈B, g∈E.

In the Beurling case Lemma 2.6 implies E ⊆ S_(A^′(M_p^p₎⁾(R^d). In the Roumieu case Lemma 2.6 together with Lemma 3.5 implies E ⊆ S_{^′{_A^M_p}^p}(R^d). Since E→ S_(p!)^′(Mp)(R^d) is continuous (E→ S_{^′{_p!^M_}^p}(R^d) is continuous) it has a closed graph. Thus the inclusionE→ S_†^′∗(R^d) has a closed graph. AsS_(A^′(M_p)^p)(R^d) is a (DF S)-space (S_{^′{_A^M_p}^p}(R^d) is an (F S)-space), it is a Pt´ak space (cf. [12, Sect.

IV. 8, p. 162]). Thus the continuity of E → S_†^′∗(R^d) follows from the Pt´ak closed graph theorem (cf. [12, Thm. 8.5, p. 166]).

Throughout the rest of the article we shall always assume that E is a translation-invariant (B)-space of ultradistributions of class ∗ − †. Our next concern is the study of convolution structures on E. We need three technical lemmas.

Lemma 3.7. Let φ∈ S_†^∗(R^2d). Then for eachy ∈ R^d, φ(·, y)∈ S_†^∗(R^d) and the function ψ(x) =

∫

R^d

φ(x, y)dy is an element of S_†^∗(R^d). Moreover, the function f :R^d→E,y7→φ(·, y), is Bochner integrable and ψ=

∫

R^d

f(y)dy.

Proof. The fact thatφ(·, y)∈ S_†^∗(R^d) for eachy∈R^d and thatψ∈ S_†^∗(R^d) is trivial. Thus f is well defined onR^d with values in E (in fact its values are in S_†^∗(R^d)). One easily verifies that f is continuous, hence strongly measurable.

To prove that it is Bochner integrable it remains to prove thaty7→ ∥f(y)∥Eis