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

**ON A CLASS OF TRANSLATION-INVARIANT** **SPACES OF QUASIANALYTIC**

**ULTRADISTRIBUTIONS**

^{1}

**Pavel Dimovski**^{2}**, Bojan Prangoski**^{3}**, and Jasson Vindas**^{4}
*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-
als*D**E*^{′∗}*∗**′* of quasianalytic type, and study convolution and multiplicative
products on *D**E*^{′∗}_{∗}* ^{′}*. 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}*spaces of quasianalytic ultradistributions are analyzed.*

_{η}*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 *D**L*^{′}* ^{p}* and

*D*

*L*

^{′∗}*spaces [1, 14] and have shown usefulness in the study of boundary values of holomorphic functions [3] and the convolution of generalized functions [4].*

^{p}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 type*S*_{†}* ^{′∗}*(R

*) (see Subsection 1.1 for the notation) from [10]*

^{d}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*

^{′∗}*) and having ultrapolynomially bounded weight function of class*

^{d}*†*. Here

*∗*and

*†*stand for the Beurling and Roumieu cases of sequences

*M*

*p*and

*A*

*p*, respectively. We would like to emphasize that our considerations apply to hyperfunctions and ultra-hyperfunctions, which correspond to the symmetric choices

*M*

*p*=

*A*

*p*=

*p!; but more generally, our*weight sequence

*M*

*p*, measuring the ultradiﬀerentiability, is allowed to satisfy the mild condition

*p!*

^{λ}*⊂M*

*p*with

*λ >*0. The growth assumption on

*A*

*p*is just

*p!*

*⊂*

*A*

*p*, 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*^{(M}*E* ^{p}^{)}, *D**E*^{{}^{M}^{p}* ^{}}*, and ˜

*D*

*E*

^{{}

^{M}

^{p}*. We prove that the following continuous and dense embeddings hold*

^{}}*S*

_{†}*(R*

^{∗}*)*

^{d}*,→ D*

*E*

^{∗}*,→*

*E ,→ S*

_{†}*(R*

^{′∗}*) and that*

^{d}*D*

^{∗}*E*are topological modules over the Beurling algebra

*L*

^{1}

*, where*

_{ω}*ω*is the weight function of the translation group of

*E. We also prove the dense*embedding

*D*

^{∗}*E*

*,→ O*

^{∗}

_{†}*,C*(R

*), where the spaces*

^{d}*O*

_{†}

^{∗}*,C*(R

*) are defined in a similar way as in [4]. The space*

^{d}*D*

^{′∗}*E*

_{∗}*is defined as the strong dual of*

^{′}*D*

*E*and various structural and topological properties of

*D*

^{′∗}*E*

^{′}*are obtained via the parametrix method (Lemma 2.2). We also prove that*

_{∗}*D*

*E*

^{{}

^{M}

^{p}*= ˜*

^{}}*D*

^{{}*E*

^{M}

^{p}*, topologically.*

^{}}As an application of our theory, we extend the theory of *D*_{L}^{′∗}^{p}* _{η}*,

*B*

*η*

*, and*

^{′∗}*B*˙

^{′∗}*η*spaces not only by considering quasianalytic cases of

*∗*but also by allow- ing ultrapolynomially bounded weights

*η*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 on

*D*

^{′∗}*E*

*′*

*∗*.
**1.1.** **Notation**

Let (M* _{p}*)

_{p}*and (A*

_{∈N}*)*

_{p}

_{p}*be two sequences of positive numbers such that*

_{∈N}*M*

_{0}=

*M*

_{1}=

*A*

_{0}=

*A*

_{1}= 1. Throughout the article, we impose the following assumptions over these weight sequences. The sequence

*M*

*p*satisfies the ensuing three conditions:

(M.1)*M*_{p}^{2}*≤M**p**−*1*M**p+1**, p∈*Z+;
(M.2)*M**p* *≤c*0*H** ^{p}* min

0*≤**q**≤**p**{M**p**−**q**M**q**}*,*p, q∈*N, for some *c*0*, H≥*1;

(M.5) there exists*s >* 0 such that *M*_{p}* ^{s}* is strongly non-quasianalytic, i.e.,

there exists *c*0*≥*1 such that

∑*∞*
*q=p+1*

*M*_{q}^{s}_{−}_{1}

*M*_{q}^{s}*≤c*0*p* *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*

^{′}*M*

*p*satisfies (M.5) there exists

*κ >*0 such that

*p!*

^{κ}*⊂M*

*, i.e., there exist*

_{p}*c*

_{0}

*, L*

_{0}

*>*0 such that

*p!*

^{κ}*≤c*

_{0}

*L*

^{p}_{0}

*M*

*,*

_{p}*p∈*N (cf. [7, Lemma 4.1]). Following Komatsu [7], for

*p∈*Z+, we denote

*m*

*=*

_{p}*M*

_{p}*/M*

_{p}

_{−}_{1}and for

*ρ≥*0 let

*m(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*

*satisfies (M.5) if and only if*

_{p}∫ _{∞}

*ρ*

*m(λ)*

*λ*^{s+1}*dλ≤cm(ρ)*

*ρ*^{s}*,* *∀ρ≥m*1*.*

A suﬃcient condition for*M** _{p}* to satisfy (M.5) is if the sequence

*m*

_{p}*/p*

*,*

^{λ}*p∈*Z+

is monotonically increasing for some*λ >*0.

We assume that *A** _{p}* satisfies (M.1) and (M.2). Of course, without loss of
generality, we can assume that the constants

*c*

_{0}and

*H*from the condition (M.2) are the same for

*M*

*and*

_{p}*A*

*. Moreover, we also assume that*

_{p}*A*

*satisfies the following additional hypothesis:*

_{p}(M.6)*p!⊂A**p*; i.e., there exist*c*0*, L*0*>*0 such that*p!≤c*0*L*^{p}_{0}*A**p*,*p∈*N.
Of course, the constants*c*_{0}and*L*_{0}in (M.6) can be chosen such that*c*_{0}*, L*_{0}*≥*
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}
*M**p*

=*∞*.

We denote by*M*(*·*) and*A(·*) the associated functions of*M** _{p}*and

*A*

*, that is,*

_{p}*M*(ρ) := sup

*p**∈N*ln+

*ρ*^{p}

*M** _{p}* and

*A(ρ) := sup*

*p**∈N*ln+

*ρ*^{p}

*A** _{p}* for

*ρ >*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 (l

*p*)

*∈*R, denote by

*N*

*l*

*and*

_{p}*B*

*l*

*(*

_{p}*·*) the associated functions of the sequences

*M*

*p*

∏*p*

*j=1**l**j* and*A**p*

∏*p*

*j=1**l**j*, respectively.

For*h >*0 we denote by*S*_{A}^{M}_{p}^{p}_{,h}* ^{,h}*the Banach spaces (in short (B)-space from
now on) of all

*φ∈C*

*(R*

^{∞}*) for which the norm*

^{d}*σ** _{h}*(φ) = sup

*α*

*h*^{|}^{α}^{|}*e*^{A(h}^{|·|}^{)}*D*^{α}*φ*

*L** ^{∞}*(R

*)*

^{d}*M*_{α}

is finite. One easily verifies that for*h*1*< h*2 the canonical inclusion*S*_{A}^{M}_{p}^{p}_{,h}^{,h}_{2}^{2} *→*
*S**A*^{M}_{p}^{p}*,h*^{,h}_{1}^{1}is compact. As l.c.s., we define*S*_{(A}^{(M}_{p}^{p}_{)}^{)}(R* ^{d}*) = lim

*h**←−**→∞*

*S**A*^{M}_{p}^{p}*,h** ^{,h}*and

*S*

_{{}

^{{}

_{A}

^{M}

_{p}

^{p}

_{}}*(R*

^{}}*) = lim*

^{d}*−→*

*h**→*0

*S**A*^{M}_{p}^{p}*,h** ^{,h}*. Since for

*h*

_{1}

*< h*

_{2}the inclusion

*S*

*A*

^{M}

_{p}

^{p}*,h*

^{,h}_{2}

^{2}

*→ S*

*A*

^{M}

_{p}

^{p}*,h*

^{,h}_{1}

^{1}is compact,

*S*_{(A}^{(M}_{p}^{p}_{)}^{)}(R* ^{d}*) is an (F S)-space and

*S*

_{{}

^{{}

_{A}

^{M}

_{p}

^{p}

_{}}*(R*

^{}}*) is a (DF S)-space. In particular, they are both Montel spaces.*

^{d}For each (r*p*)*∈*R, by*S*_{A}^{M}_{p}^{p}_{,(r}^{,(r}_{p}^{p}_{)}^{)}we denote the space of all*φ∈C** ^{∞}*(R

*) such that*

^{d}*σ*(r* _{p}*)(φ) = sup

*α*

*e*^{B}^{rp}^{(}^{|·|}^{)}*D*^{α}*φ*

*L** ^{∞}*(R

*)*

^{d}*M** _{α}*∏

_{|}*α*

*|*

*j=1**r*_{j}*<∞.*

Provided with the norm*σ*_{(r}_{p}_{)}, the space*S*_{A}^{M}_{p}^{p}_{,(r}^{,(r}_{p}^{p}_{)}^{)}becomes a (B)-space. Simi-
larly as in [1, 9], one can prove that*S*_{{}^{{}_{A}^{M}_{p}^{p}_{}}* ^{}}*(R

*) is topologically isomorphic to*

^{d}lim

(r*p*)*∈*R

*←−*

*S*_{A}^{M}_{p}^{p}_{,(r}^{,(r}_{p}^{p}_{)}^{)}.

In the future we shall employ*S*_{†}* ^{∗}*(R

*) as a common notation for*

^{d}*S*

_{(A}

^{(M}

_{p}

^{p}_{)}

^{)}(R

*) (Beurling case) and*

^{d}*S*

_{{}

^{{}

_{A}

^{M}

_{p}

^{p}

_{}}*(R*

^{}}*) (Roumieu case). It is clear that for each*

^{d}*h >*0 and (r

*)*

_{p}*∈*R, the spaces

*S*

_{A}

^{M}

_{p}

^{p}

_{,h}*and*

^{,h}*S*

_{A}

^{M}

_{p}

^{p}

_{,(r}

^{,(r}

_{p}

^{p}_{)}

^{)}are continuously injected into

*S*(R

*) (the Schwartz space).*

^{d}We will often make use of the following technical result from [11].

**Lemma 1.1** ([11]). *Let* (k*p*)*∈* R. There exists (k^{′}* _{p}*)

*∈*R

*such that*

*k*

^{′}

_{p}*≤*

*k*

*p*

*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*

^{′∗}*) we denote as*

^{d}*T*

*h*

*f*translation by

*h, i.e.,T*

*h*

*f*=

*f*(

*·*+

*h). We write*

*⟨x⟩*= (1 +*|x|*^{2})^{1/2},*x∈*R* ^{d}*.

**2.** **Some important auxiliary results on the space** *S*

_{†}

^{∗}## ( R

^{d}## )

We collect in this section some important results on the nuclearity of*S*_{†}* ^{∗}*(R

*), the existence of parametrices as well as a characterisation of bounded sets in*

^{d}*S*

_{†}*(R*

^{′∗}*). 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*

^{d}*)*

_{p}*−*(A

*) case followed in parenthesis by the corresponding statements for the*

_{p}*{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*(M*p*)*(for every* (t*p*)*∈*R *there exist* *G∈ S*_{A}^{M}_{p}^{p}_{,(t}^{,(t}_{p}^{p}_{)}^{)}
*and an ultradiﬀerential operator* *P(D)of class* *{M**p**}) such thatP*(D)G=*δ.*

**Lemma 2.3.** *Let* *r >*0 *((r**p*)*∈*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^{d}*χ(x)dx* = 1. For each *n∈*Z+

*define* *χ**n*(x) =*n*^{d}*χ(nx)* *andφ**n*(x) =*φ(x/n). Then there existsk≥*2r
*((k**p*) *∈* R *with* (k*p*) *≤*(r*p**/2)) such that the operators* *Q*˜*n* : *ψ* *7→* *χ**n**∗*
(φ*n**ψ), are continuous as mappings fromS**A*^{M}_{p}^{p}*,k*^{,k}*intoS**A*^{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}*→*Id*in* *L**b*

(*S**A*^{M}_{p}^{p}*,k*^{,k}*,S**A*^{M}_{p}^{p}*,r*^{,r}

)
*(L**b*

(*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*

^{M}*A*

_{p}

^{p}*,t*

^{,t}(as *S*^{M}*A**p*^{p}*,(t*^{,(t}*p** ^{p}*)

^{)}) the closure of

*S*

_{(A}

^{(M}

_{p}

^{p}_{)}

^{)}(R

*) in*

^{d}*S*

*A*

^{M}

_{p}

^{p}*,t*

*(the closure of*

^{,t}*S*

_{{}

^{{}

_{A}

^{M}

_{p}

^{p}

_{}}*(R*

^{}}*) in*

^{d}*S*

_{A}

^{M}

_{p}

^{p}

_{,(t}

^{,(t}

_{p}

^{p}_{)}

^{)}).

**Proposition 2.4.** *Let* *B* *be a bounded subset ofS*_{†}* ^{′∗}*(R

*). There exists*

^{d}*k >*0

*((k*

*)*

_{p}*∈*R) such that each

*f*

*∈*

*B*

*can be extended to a continuous functional*

*f*˜

*on*

*S*

^{M}*A*

_{p}

^{p}*,k*

^{,k}*(on*

*S*

^{M}*A*

_{p}

^{p}*,(k*

^{,(k}

_{p}*)*

^{p}^{)}

*). Moreover, there existsl*

*≥k*

*((l*

*)*

_{p}*∈*R

*with*(l

*)*

_{p}*≤*(k

*p*)) such that

*S*

_{A}

^{M}

_{p}

^{p}

_{,l}

^{,l}*⊆ S*

^{M}*A*

*p*

^{p}*,k*

^{,k}*(S*

_{A}

^{M}

_{p}

^{p}

_{,(l}

^{,(l}

_{p}

^{p}_{)}

^{)}

*⊆ S*

^{M}*A*

*p*

^{p}*,(k*

^{,(k}*p*

*)*

^{p}^{)}

*) and*

*∗*:

*S*

_{A}

^{M}

_{p}

^{p}

_{,l}

^{,l}*× S*

_{A}

^{M}

_{p}

^{p}

_{,l}

^{,l}*→*

*S*

^{M}*A*

*p*

^{p}*,k*

^{,k}*(∗*:

*S*

_{A}

^{M}

_{p}

^{p}

_{,(l}

^{,(l}

_{p}

^{p}_{)}

^{)}

*× S*

_{A}

^{M}

_{p}

^{p}

_{,(l}

^{,(l}

_{p}

^{p}_{)}

^{)}

*→ S*

^{M}*A*

*p*

^{p}*,(k*

^{,(k}*p*

*)*

^{p}^{)}

*) is a continuous bilinear mapping.*

*Furthermore, there exist an ultradiﬀerential operator* *P*(D) *of class∗* *and* *u∈*
*S*^{M}*A*_{p}^{p}*,l*^{,l}*(u∈ S*^{M}*A*_{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→u*ˇ

*∗φ,S*

_{(A}

^{(M}

_{p}

^{p}_{)}

^{)}(R

*)*

^{d}*→ S*

^{M}*A*

*p*

^{p}*,k*

^{,k}*(S*

_{{}

^{{}

_{A}

^{M}

_{p}

^{p}

_{}}*(R*

^{}}*)*

^{d}*→ S*

^{M}*A*

*p*

^{p}*,(k*

^{,(k}*p*

*)*

^{p}^{)}

*).*

*For* *f* *∈* *B,* *u∗f*˜*∈* *L*^{∞}_{e}_{A(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*

^{∞}

_{e}

_{A(l|·|)}*(in*

*L*

^{∞}*e*^{Blp}^{(|·|)}*).*

**Lemma 2.5.** *Let* *B⊆ S*_{†}* ^{′∗}*(R

*). The following statements are equivalent:*

^{d}*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*(t

*p*)

*∈*R

*andC >*0)

*such that|*(f

*∗φ)(x)| ≤Ce*

^{A(t}

^{|}

^{x}

^{|}^{)}

*(|*(f

*∗φ)(x)| ≤Ce*

^{B}

^{tp}^{(}

^{|}

^{x}

^{|}^{)}

*) for allx∈*R

^{d}*,*

*f*

*∈B;*

*iv)* *there existC, t >*0*(there exist* (t*p*)*∈*R*andC >*0) such that

*|*(f*∗φ)(x)| ≤Ce*^{A(t}^{|}^{x}^{|}^{)}*σ** _{t}*(φ)
(

*resp.* *|f∗φ(x)| ≤Ce*^{B}^{tp}^{(}^{|}^{x}^{|}^{)}*σ*_{(t}_{p}_{)}(φ)
)
*for allφ∈ S*_{†}* ^{∗}*(R

*),*

^{d}*x∈*R

^{d}*,f*

*∈B.*

**Lemma 2.6.** *Let* *f* *∈ S*_{(p!)}^{′}^{(M}^{p}^{)}(R* ^{d}*)

*(f*

*∈ S*

_{{}

^{′{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*)). Then*

^{d}*f*

*∈ S*

_{†}*(R*

^{′∗}*)*

^{d}*if*

*and only if there exists*

*t >*0

*(there exists*(t

*p*)

*∈*R) such that for every

*φ∈*

*S*

_{(p!)}

^{(M}

^{p}^{)}(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*^{−}^{B}^{tp}^{(}^{|}^{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.

Let*E*be a (B)-space. We call*E*a*translation-invariant*(B)-space of ultra-
*distributions of class* *∗ − †*if it satisfies the following three axioms:

(I) *S*_{†}* ^{∗}*(R

*)*

^{d}*,→E ,→ S*

_{†}*(R*

^{′∗}*).*

^{d}(II) *T**h*(E)*⊆E* for each*h∈*R* ^{d}*.

(III) There exist*τ, C >*0 (for every*τ >*0 there exists*C >*0), such that

*∥T*_{h}*g∥**E**≤C∥g∥**E**e*^{A(τ}^{|}^{h}^{|}^{)}*,* *∀h∈*R^{d}*,* *∀g∈E.*

Notice that the condition (III) implicitly makes use of the continuity of
*T**h*. 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
*operatorsT**h*:*E→E* *are bounded for all* *h∈*R^{d}*.*

*Proof.* Observe that*T** _{h}* is continuous as a mapping from

*E*to

*S*

_{†}*(R*

^{′∗}*) since it can be decomposed as*

^{d}*E−→ S*

^{Id}

_{†}*(R*

^{′∗}*)*

^{d}*−−→ S*

^{T}

^{h}

_{†}*(R*

^{′∗}*) and*

^{d}*T*

*h*:

*S*

_{†}*(R*

^{′∗}*)*

^{d}*→ S*

_{†}*(R*

^{′∗}*) is continuous. Thus the graph of*

^{d}*T*

*h*is closed in

*E× S*

_{†}*(R*

^{′∗}*) and, since its image is in*

^{d}*E, its graph is also closed inE×E*(E

*×E*is continuously injected into

*E× S*

_{†}*(R*

^{′∗}*) via the mapping Id*

^{d}*×*Id). As

*E*is a (B)-space, the closed graph theorem implies that

*T*

*h*is continuous.

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

*h**→*0*∥T**h**g−g∥**E* = 0. In particular for each
*g* *∈E* *the mapping* *h7→T**h**g,* 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 ultradistributions*E* of class*∗ − †*satisfies the following stronger
condition than (II):

(f*II) for eachh >*0,*T**h*:*E→E*is continuous and for each*g∈E*the mapping
*h7→T**h**g,*R^{d}*→E, is continuous.*

Clearly *T*_{0} = Id* _{E}*,

*T*

_{h}_{1}

_{+h}

_{2}=

*T*

_{h}_{1}

*◦T*

_{h}_{2}=

*T*

_{h}_{2}

*◦T*

_{h}_{1}. 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

*) is separable (it is an (F S)-space or a (DF S)-space, respectively), so is*

^{d}*E. Thus*

*ω(h) =∥T*

_{−}*h*

*∥*

*(E) is the supremum of*

_{L}*∥T*

_{−}*h*

*g∥*

*E*where

*g*belongs to a countable dense subset of the closed unit ball of

*E. Since*

*h7→ ∥T*

_{−}*h*

*g∥*

*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 when*A** _{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 fixed

*φ∈ S*

_{†}*(R*

^{∗}*), one has*

^{d}*T*

_{h}*φ→φ*as

*h→*0 in

*S*

_{†}*(R*

^{∗}*) and consequently in*

^{d}*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

*(cf. [5, Sect 7.4]), which is in fact condition (III) in this case.*

^{d}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* (l*p*)*∈*R *and* *C >*0 *such that* *∥T**h**g∥**E* *≤C∥g∥**E**e*^{B}^{lp}^{(}^{|}^{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)-space*E*is
a translation-invariant space of ultradistributions of class*∗ − †*.

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

*(I)*^{′}*S*_{(p!)}^{(M}^{p}^{)}(R* ^{d}*)

*,→E ,→ S*

_{(p!)}

^{′}^{(M}

^{p}^{)}(R

*)*

^{d}*(S*

_{{}

^{{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*)*

^{d}*,→E ,→ S*

_{{}

^{′{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*));*

^{d}*(II)* *T**h*(E)*⊆E, for allh∈*R^{n}*;*

*(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*_{h}*g∥**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 of

*T*

*:*

_{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 prove*S*_{†}* ^{∗}*(R

*)*

^{d}*,→E, by (I)*

*, it is enough to prove that*

^{′}*S*

_{†}*(R*

^{∗}*) is continuously injected into*

^{d}*E. Pickψ*1

*∈ D*(R

*) such that*

^{d}∑

*m**∈Z*^{d}

*ψ*1(x*−m) = 1,* *∀x∈*R^{d}*,* supp*ψ*1*∈*[*−*1,1]^{d}

and*ψ*_{1}is non-negative and even. Next, pick*ψ*_{2}*∈ S*_{(p!)}^{(M}^{p}^{)}(R* ^{d}*) (ψ

_{2}

*∈ S*

_{{}

^{{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*)), such that∫*

^{d}R^{d}*ψ*2(x)dx= 1 and*ψ*2is even. Set*ψ*=*ψ*1*∗ψ*2. One readily verifies
that ∑

*m**∈Z*^{d}*ψ(x−m) = 1 for allx∈*R* ^{d}* and

*ψ*

*∈ S*

_{(p!)}

^{(M}

^{p}^{)}(R

*) in the Beurling case and*

^{d}*ψ*

*∈ S*

_{{}

^{{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*) in the Roumieu case, respectively. By (III), there exist*

^{d}*C, τ >*0 (for every

*τ >*0 there exists

*C >*0) such that

*∥φT*_{−}*m**ψ∥**E* *≤Ce*^{−}^{A(τ}^{|}^{m}^{|}^{)}*∥e*^{2A(τ}^{|}^{m}^{|}^{)}*ψT**m**φ∥**E**,* *∀φ∈ S*_{†}^{∗}*,∀m∈*Z^{d}*.*
(3.2)

For*m∈*Z* ^{d}*, consider the linear mapping

*ρ**m,τ*(φ) =*e*^{2A(τ}^{|}^{m}^{|}^{)}*ψT**m**φ,S*_{(A}^{(M}_{p}^{p}_{)}^{)}(R* ^{d}*)

*→ S*

_{(p!)}

^{(M}

^{p}^{)}(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

*). Then for every*

^{d}*h >*0 (there exists

*h >*0) such that

sup

*φ**∈**B*

sup

*α**∈N*^{d}

*h*^{|}^{α}* ^{|}*e

^{A(h}

^{|·|}^{)}

*D*

^{α}*φ*

*L**∞*

*M*_{α}*<∞*
(3.3)

Now, [7, Lemma 3.6] implies

*e*^{2A(τ}^{|}^{m}^{|}^{)}*≤*2c0*e*^{A(2Hτ}^{|}^{x+m}^{|}^{)}*e*^{A(2Hτ}^{|}^{x}^{|}^{)}*.*
(3.4)

In the Beurling case, let *h*1 *>* 0 be arbitrary but fixed. Choose *h >* 0 such
that*h≥*max*{*2Hτ,2h_{1}*}* and*e*^{A(2Hτ λ)}*≤C*^{′}*e** ^{hλ}* for all

*λ≥*0 (such an

*h*exists because

*p!⊂A*

*). By (3.3) and (3.4) we have*

_{p}*h*^{|}_{1}^{α}^{|}*|D** ^{α}*(ψ(x)T

*m*

*φ(x))|e*

^{h}^{1}

^{|}

^{x}

^{|}*M**α* *≤C*2*e*^{−}^{2A(τ}^{|}^{m}^{|}^{)}*,*
(3.5)

for all *x∈*R* ^{d}*,

*m∈*Z

*,*

^{d}*φ∈B. Hence*

*{ρ*

*m,τ*

*|m*

*∈*Z

^{d}*}*is uniformly bounded on

*B. In the Roumieu case there exist ˜h,C >*˜ 0 such that ˜

*h*

^{|}

^{α}

^{|}*|D*

^{α}*ψ(x)|e*

^{˜}

^{h}

^{|}

^{x}

^{|}*≤*

*CM*˜

*α*for all

*x*

*∈*R

*,*

^{d}*α*

*∈*N

*. For the*

^{d}*h >*0 for which (3.3) holds choose 0

*< τ*

*≤h/(2H*) such that

*e*

^{A(2Hτ λ)}*≤C*

^{′}*e*

^{˜}

*for all*

^{hλ/2}*λ*

*≥*0 (such a

*τ*exists because

*p!*

*⊂*

*A*

*p*). Choose

*h*1

*≤*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.,

*{ρ*

*m,τ*

*|m∈*Z

^{d}*}*is uniformly bounded on

*B*. Now, (I)

*implies that*

^{′}*∥ρ*

*m,τ*(φ)

*∥*

*E*

*≤C*

_{2}

*for all*

^{′}*φ∈B,*

*m∈*Z

*. By using (3.2), we obtain that the sequence {∑*

^{d}*|**m**|≤**N**φT*_{−}*m**ψ*
}_{∞}

*N*=0

is a Cauchy sequence in *E* for each *φ∈B. Since its limit isφ* in *S*_{(p!)}^{′}^{(M}^{p}^{)}(R* ^{d}*)
(in

*S*

_{{}

^{′{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*)) it converges to*

^{d}*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. As

*S*

_{†}*(R*

^{∗}*) is bornological, the inclusion is continuous. It remains to prove*

^{d}*E*

*⊆ S*

_{†}*(R*

^{′∗}*). By (I)*

^{d}*for a bounded set*

^{′}*B*in

*S*

_{(p!)}

^{(M}

^{p}^{)}(R

*) (in*

^{d}*S*

_{{}

^{{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*)) there exists*

^{d}*D >*0 such that

*|⟨g,φ*ˇ

*⟩| ≤D∥g∥*

*E*for all

*g∈E*and

*φ∈B. Then (III)*implies that there exist

*C, τ >*0 (for every

*τ >*0 there exists

*C >*0) such that

*|*(g*∗φ)(y)| ≤D∥T**y**g∥**E* *≤CDe*^{A(τ}^{|}^{y}^{|}^{)}*,* for all*y* *∈*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*

^{}}*). Since*

^{d}*E→ S*

_{(p!)}

^{′}^{(M}

^{p}^{)}(R

*) is continuous (E*

^{d}*→ S*

_{{}

^{′{}

_{p!}

^{M}

_{}}

^{p}*(R*

^{}}*) is continuous) it has a closed graph. Thus the inclusion*

^{d}*E→ S*

_{†}*(R*

^{′∗}*) has a closed graph. As*

^{d}*S*

_{(A}

^{′}^{(M}

_{p}_{)}

^{p}^{)}(R

*) is a (DF S)-space (*

^{d}*S*

_{{}

^{′{}

_{A}

^{M}

_{p}

_{}}

^{p}*(R*

^{}}*) is an (F S)-space), it is a Pt´ak space (cf. [12, Sect.*

^{d}IV. 8, p. 162]). Thus the continuity of *E* *→ S*_{†}* ^{′∗}*(R

*) follows from the Pt´ak closed graph theorem (cf. [12, Thm. 8.5, p. 166]).*

^{d}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 each

*y*

*∈*R

^{d}*,*

*φ(·, y)∈ S*

_{†}*(R*

^{∗}*)*

^{d}*and*

*the function*

*ψ(x) =*

∫

R^{d}

*φ(x, y)dy* *is an element of* *S*_{†}* ^{∗}*(R

*). Moreover, the*

^{d}*function*

**f**:R

^{d}*→E,y7→φ(·, y), is Bochner integrable and*

*ψ*=

∫

R^{d}

**f**(y)dy.

*Proof.* The fact that*φ(·, y)∈ S*_{†}* ^{∗}*(R

*) for each*

^{d}*y∈*R

*and that*

^{d}*ψ∈ S*

_{†}*(R*

^{∗}*) is trivial. Thus*

^{d}**f**is well defined onR

*with values in*

^{d}*E*(in fact its values are in

*S*

_{†}*(R*

^{∗}*)). One easily verifies that*

^{d}**f**is continuous, hence strongly measurable.

To prove that it is Bochner integrable it remains to prove that*y7→ ∥***f**(y)*∥**E*is