Boundary values in ultradistribution spaces related to extended Gevrey regularity

Following the well-known theory of Beurling and Roumieu ultra-distributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. We prove that they can be represented as boundary values of analytic functions in corresponding infinitesimal wedge domains. The essential condition for that purpose, condition $(M.2)$ in the classical ultradistribution theory, is replaced by a new one, $\widetilde{(M.2)}$. For that reason, new techniques were performed in the proofs. As an application, we characterize the corresponding wave front sets.


INTRODUCTION
In this paper we describe certain intermediate spaces between the space of Schwartz distributions and (any) space of Gevrey ultradistributions as boundary values of analytic functions. More precisely, we continue to investigate a new class of ultradifferentiable functions and their duals ( [14][15][16][17]) following Komatsu's approach [8,9]. We refer to [1] and the references therein for another, equally interesting approach, Distributions as boundary values of analytic functions are investigated in many papers, see [2] for the historical background and the relevant references therein. We point out a nice survey for distribution and ultradistribution boundary values given in the book [7]. The essence of the existence of a boundary value is the determination of the growth condition under which an analytic function F (x + iy), observed on a certain tube domain with respect to y, defines an (ultra)distribution as y tends to 0. The classical result can be roughly interpreted as follows: if F (x + iy) ≤ C|y| −M for some C, M > 0 then F (x + i0) is in the Schwartz space D ′ (U ) in a neighborhood U of x. (see Theorem 3.1.15 in [7]). For ultradistributions, a suitable sub-exponential growth rate implies the boundary value (for example e k|y| −1/(t−1) for some or for every k > 0) corresponds to the Gevrey sequence p! t , t > 1 (in the Roumieu or Beurling case). In general, such representations are provided if test functions admit almost analytic extensions in the non-quasianalytic case related to Komatsu's condition (M.2) (see [12]).
Different results concerning boundary values in the spaces of ultradistributions can be found in [2,8,9,12,13]. Even now this topic for ultradistribution spaces is interesting cf. [3][4][5]22]. Especially, we have to mention [6]. At the end of this introduction we will shortly comment the approach in this paper and our approach.
Extended Gevrey classes E τ,σ (U ) and D τ,σ (U ), τ > 0, σ > 1, are introduced and investigated in [14]- [17], [20,21]. The derivatives of functions in such classes are controlled by sequences of the form M τ,σ p = p τ p σ , p ∈ N. Although such sequences do not satisfy Komatsu's condition (M.2), the corresponding spaces consist of ultradifferentiable functions, that is, it is possible to construct differential operators of infinite order and prove their continuity properties on the test and dual spaces.
Our main intention in this paper is to prove that the elements of dual spaces can be represented as boundary values of analytic functions. We follow the classical approach to boundary values given in [2] and carry out necessary modifications in order to use it in the analysis of spaces developed in [14]- [17]. Here, for such spaces, a plenty of non-trivial constructions are performed. In particular, we analyze the corresponding associated functions as a main tool in our investigations.
Moreover, we apply these results in the description of related wave front sets. The wave front set WF τ,σ (u), τ > 0, σ > 1, of a Schwarz ditribution u is analyzed in [15][16][17]20,21]. In particular, it is proved that they are related to the classes E τ,σ (U ). We extend the definition of WF τ,σ (u) to a larger space of ultradistributions by using their boundary value representations. This allows us to describe intersections and unions of WF τ,σ (u) (with respect to τ ) by using specific functions with logarithmic type behaviour.
Let us comment another, very interesting, concept of construction of a "large" class of ultradistribution spaces. In [6,10,18] and several other papers the authors consider sequences of the form k!M k , where they presume a fair number of conditions on M k and discuss in details their relations. For example, consequences to the composition of ultradifferentiable functions determined by different classes of such sequences are discussed. Moreover, they consider weighted matrices, that is a family of sequences of the form k!M λ k , k ∈ N, λ ∈ Λ (partially ordered and directed set) and make the unions, again considering various properties such as compositions and boundary values. Their analysis follows the approach of [1,11]. In essence, an old question of ultradistribution theory was the analysis of unions and intersections of ultradifferentiable function spaces. This is very well elaborated in quoted papers. Definition of an ultradistribution space in [6,10,18] is related to the property of k!M λ k so that conditions (M.2) ′ and (M.2) hold for every fixed λ. This is not the case in our considerations. From this point of view, our space is not covered with the constructions of quoted papers (see (2.3) below). Actually, the precise estimates of our paper can be used for the further extensions in matrix approach since the original idea for our approach is quite different and based on the relation between [n s ]! and n! s in the estimate of derivatives ([n s ] means integer value not exceeding n s , s ∈ (0, 1), cf. [14,15]).
The paper is organized as follows: We end the introduction with some notation. In Section 2 we introduce the necessary background on the spaces of extended Gevrey functions and their duals, spaces of ultradistributions. Our main result Theorem 3.1 is given in Section 3. Wave front sets in the framework of our theory are discussed in Section 4. Finally, in Appendix we prove a technical result concerning the associated functions T τ,σ,h (k), and recall the basic continuity properties of ultradifferentiable operators on extended Gevrey classes, in a certain sense analogous to stability under the ultradifferentiable operators in the classical theory.
1.1. Notation. We denote by N, Z + , R, C the sets of nonnegative integers, positive integers, real numbers and complex numbers, respectively. For a multi-index The open ball is B(x 0 , r) has radius r > 0 and center Throughout the paper we always assume τ > 0 and σ > 1.

TEST SPACES AND DUALS
We are interested in M τ,σ p , p ∈ N, sequences of positive numbers such that conditions (M.1) and (M.3) of [8] hold, and instead of (M.2) ′ and (M.2) of [8], In the sequel we consider the sequence M τ,σ p = p τ p σ , p ∈ N, which fulfills the above mentioned conditions (see [14,Lemma 2.2.]). This particular choice slightly simplifies our exposition. Clearly, by choosing σ = 1 and τ > 1 we recover the well known Gevrey sequence p! τ .
Recall [17], the associated function related to the sequence p τ p σ , is defined by For h, σ = 1 and τ > 1, T τ,1,1 (k) is the associated function to the Gevrey sequence p! τ .
In the next lemma we derive the precise asymptotic behaviour properties of the associated function T τ,σ,h the sequence p τ p σ . This in turn highlights the essential difference between T τ,σ,h and the associated functions determined by Gevrey type sequences.
Lemma 2.1. Let h > 0, and let T τ,σ,h be given by (2.1). Then there exists constants More precisely, if then, there exist constants A 1 , A 2 > 0 such that Proof. Lemma 2.1 can be proved by following the arguments used in the proof of [17, Theorem 2.1]. Details are left for the reader.
We define (following the classical approach [8]): It turns out that T * τ,σ,h (k) enjoys the same asymptotic behaviour as T τ,σ,h , cf. Lemma 4.1 a) in the Appendix. This is another difference between our approach and the classical ultradistribution theory, where T * plays an important role.
Next we recall the definition of spaces E τ,σ (U ) and D τ, We have where ֒→ denotes the strict and dense inclusion. The set of functions from E τ,σ,h (K) supported by K is denoted by D K τ,σ,h . Next, Spaces in (2.4) and (2.6) are called Roumieu type spaces while (2.5) and (2.7) are Beurling type spaces. Note that all the spaces of ultradifferentiable functions defined by Gevrey type sequences are contained in the corresponding spaces defined above.
For the corresponding spaces of ultradistributions we have: Topological properties of all those spaces are the same as in the case of Beurling and Roumieu type spaces given in [8].

MAIN RESULT
The condition (M.2) (also knows as the stability under the ultradifferentiable operators), essential for the boundary value theorems in the framework of ultradistribution spaces [6,8,12], is in our approach replaced by the condition (M.2). In the case of the sequence M τ,σ p = p τ p σ , p ∈ N, the asymptotic behaviour given in Lemma 2.1 is essentially used to prove our main result as follows.
and such that for some A, H > 0 (resp. for every H > 0 there exists A > 0). Then More precisely, if
Fix h > 0, and let and for j = 1, . . . , d, In order to use Stoke's formula (see [12]) we need to estimate Φ and its derivatives on Z Y . To that end we had to adjust the standard technique in a nontrivial manner.

By (3.3) and (3.4) it is sufficient to prove (3.8) for
(3.9) Note that for z ∈ Z Y we have We will show that there exists constant B h > 0 such that The estimates for S 2 and S 3 can be obtained in a similar way.
It remains to show that sup β∈N d I 1,β and sup β∈N d I 2,β are finite.
Note that for H = h, (3.2) and (3.7) imply that there exists A h > 0 such that (3.2) and (3.8) imply that there exists B h > 0 such that (3.14) Now (3.12), (3.13) and (3.14) implies for suitable constant B ′ h > 0. This completes the proof of the second part of theorem, and the first part follows immediately.

WAVE FRONT SETS
In this section we analyze wave front sets WF τ,σ (u) related to the classes E τ,σ (U ) introduced in Section 2. We refer to [15-17, 20, 21] for properties of WF τ,σ (u) when u is a Schwartz distribution.
We begin with the definition.
Moreover (cf. [16]), for 0 < τ 1 < τ 2 and σ > 1 we have where WF A denotes analytic wave front set. Let and For such wave front sets we have the following corollary which is an immediate consequence of Lemma 2.1.
We write u(x) = F (x + iΓ 0) if u(x) is obtained as boundary value of an analytic function F (x + iy) as y → 0 in Γ. Recall (cf. [7]) denotes the dual cone of Γ.
To conclude the paper we prove the following theorem.
Finally we discuss certain stability and embedding properties of E τ,σ (U ) given by (2.4) and (2.5). Analogous considerations hold when the spaces D τ,σ (U ) from (2.6) and (2.7) are considered instead.