On generalized definitions of ultradifferentiable classes

We show that the ultradifferentiable-like classes of smooth functions introduced and studied by S. Pilipovi\'c, N. Teofanov and F. Tomi\'c are special cases of the general framework of spaces of ultradifferentiable functions defined in terms of weight matrices in the sense of A. Rainer and the third author. We study classes"beyond geometric growth factors"defined in terms of a weight sequence and an exponent sequence, prove that these new types admit a weight matrix representation and transfer known results from the matrix-type to such a non-standard ultradifferentiable setting.


Introduction
Spaces of ultradifferentiable functions are sub-classes of smooth functions with certain restrictions on the growth of their derivatives. Two classical approaches are commonly considered, either the restrictions are expressed by means of a weight sequence M = (M j ) j , also called Denjoy-Carleman classes (e.g. see [11]), or by means of a weight function ω also called Braun-Meise-Taylor classes; see [3]. More precisely (in the one-dimensional case) for each compact set K, the sets are required to be bounded, where ϕ * ω denotes the Young-conjugate of t → ω(e t ). We shall mention that in the second situation the classes can be defined directly by using ω and controlling the decay of the Fourier transform f with growth factors t → exp(hω(t)), h > 0. In fact, this is the original description; see [1] and also the discussion in [3] where the original approach is transferred to the boundedness condition expressed in (1.1). In the literature standard growth and regularity conditions are assumed for M and ω and in both settings we can consider two different types of spaces: For the Roumieu-type the boundedness of the sets in (1.1) is required for some h > 0, whereas for the Beurling-type it is required for all h > 0. The most well-known examples are the Gevrey sequences of type α > 0 with G α j := j αj for j ∈ N (or equivalently use M α j := j! α ). Alternatively, one can use the function t → t 1/α =: ω α (t). It is then a natural question how both classical settings are related. In [2] this problem is studied and it has been shown that in general both approaches are mutually distinct. However, based on this work, in [27] and [23] A. Rainer and the third author have introduced the notion of weight matrices M = {M (x) : x > 0} which allows to treat both classical methods in a unified way and to transfer proofs from one context to the other. This can be achieved when considering M = {M } for the weight sequence and the so-called associated weight matrix W := {W (ℓ) : ℓ > 0} with W (ℓ) j := exp( 1 ℓ ϕ * ω (ℓj)) in the weight function case. But one is also able to describe more classes, e.g. take the Gevrey matrix G := {G α : α > 1}; see [23,Thm. 5.22].
A second recent generalization was presented by S. Pilipović, N. Teofanov and F. Tomić; see [19]. For given parameters τ > 0 and σ > 1 they consider the sequence M τ,σ j := j τ j σ . However, in their definition the geometric growth factor h j appearing in (1.1) is replaced by h j σ . Observe that the growth of j → h j σ is closely connected with j → M τ,σ j . The authors called their framework "beyond Gevrey regularity" because M τ,1 j = j τ j for σ = 1, i.e., the Gevrey sequence of type τ > 0. Since all the classes considered in this work are, in some sense, generalizations of Gevrey classes, these spaces will be called Pilipović-Teofanov-Tomić classes, or PTT-classes for short.
The difference between the growth of j → h j and j → h j σ suggests that the PTT-classes can be viewed as "non-standard ultradifferentiable classes" and one can ask how both generalizations are related. In the introduction of [21] it was claimed that the PTT-classes are not covered by the weight matrix approach which is due to the different growth of the factors mentioned before. However, the aim of this paper is to show that also the PTT-classes are contained in the weight matrix approach.
In fact, we treat a more abstract setting by considering an exponent sequence Φ = (Φ j ) j∈N and by replacing in (1.1) the growth j → h j by j → h Φj . This notion yields "ultradifferentiable classes beyond geometric growth factors" and we show that under mild regularity and growth assumptions on Φ such spaces admit a representation as weight matrix classes (as locally convex vector spaces) by involving the canonical matrix Applying this main result to the PTT-classes, we are also able to see that when both σ > 1 and τ > 0 are fixed then the corresponding space cannot be represented by a single weight sequence M or by a weight function ω; i.e., one requires the general weight matrix setting to describe these classes. In other words PTT-classes constitute genuine examples of ultradifferentiable classes defined by weight matrices.
On the other hand, in the very recent paper [33] it is shown that when only σ > 1 is fixed and when one considers matrix-type classes with parameter τ > 0, i.e. PTT-limit classes, then these spaces can alternatively be defined in terms of a weight function (in particular of a so-called associated weight function). We give an independent proof of this result by applying purely weight matrix techniques; see Theorem 6.8.
The paper is structured as follows: In Section 2 all necessary and relevant conditions on weight sequences, weight functions and weight matrices are given and the corresponding classes are defined. In Section 3 we introduce ultradifferentiable spaces "beyond geometric growth factors" and prove in Section 4 the main characterization results, i.e., Theorems 4.4, 4.7 and 4.9, showing that, in particular, the PTT-classes can be represented as weight matrix spaces. In Section 5 we apply this fact for fixed parameters τ > 0, σ > 1, and study properties of the relevant matrix M τ,σ in order to transfer known results from the matrix setting to PTT-classes. In Section 6 this is done analogously for so-called limit classes when fixing σ but letting τ → 0 resp. τ → +∞. It is shown that such spaces can be represented as Braun-Meise-Taylor classes (see Theorem 6.8) and satisfy additional properties since in this weight structure both mixed moderate growth conditions of the particular type are valid.
Acknowledgements. We wish to thank J. Vindas for pointing out additional results available for matrix classes; more precisely for bringing, what is now property (f) in Sections 5.2 and 6.3, to our attention. In addition he suggested to consider [4], whose implications are the content of Section 6.4. And we thank N. Teofanov and F. Tomić for forwarding their preprint of [33] and the subsequent helpful discussions.
If M is log-convex and normalized, then both M and j → (M j ) 1/j are nondecreasing. In this case we get M j ≥ 1 for all j ≥ 0 and Moreover we get M j M k ≤ M j+k for all j, k ∈ N; e.g. see [26,Lemma 2.0.4, Lemma 2.0.6]. If m is log-convex, then M is also log-convex and in this case we call M strongly log-convex and write that M is (slc).
For convenience we introduce the following set of sequences: M has moderate growth, denoted by (mg), if A weaker condition is derivation closedness, denoted by (dc), if In the literature (mg) is also known under stability of ultradifferential operators or (M.2), (dc) under (M.2) ′ and (nq) under (M.3) ′ ; see [11]. It is also known that for log-convex (normalized) weight sequences (nq) is equivalent to which holds by the so-called Carleman-inequality; see [26,Prop. 4.1.7] and the references therein.
M has (β 1 ) (named after [18] In [18, Proposition 1.1] it has been shown that for M ∈ LC both conditions are equivalent and in the literature (γ 1 ) is also called "strong nonquasianalyticity condition". In [11] it is denoted by (M.3) (in fact, there µj j is replaced by µj j−1 for j ≥ 2 but which is equivalent to having (γ 1 )). A weaker condition on M is (β 3 ) (named after [27], see also [2]) which reads as follows: For any s ≥ 0 we set G s := (j! s ) j∈N , so for s > 0 this denotes the classical Gevrey sequence of index/order s.
For an abstract introduction of the associated function we refer to [13, Chapitre I]; see also [11,Definition 3.1].
Moreover, under this assumption t → ω M (t) is a continuous nondecreasing function, which is convex in the variable log(t) and tends faster to infinity than any log(t j ), j ≥ 1, as t → +∞. lim j→+∞ (M j ) 1/j = +∞ implies that ω M (t) < +∞ for each finite t which shall be considered as a basic assumption for defining ω M .
Summarizing, if M ∈ LC, then ω M is a normalized pre-weight function (e.g. see [9, , j ∈ N. If M and N are both R-and B-equivalent, then we say for simplicity that they are equivalent.
We recall several growth and regularity assumptions on a given weight matrix: R-equivalence between matrices preserves all Roumieu-type conditions listed above and B-equivalence all Beurling-type conditions. Finally, let us recall j . A matrix is called non-quasianalytic if any sequence M (α) is non-quasianalytic; see [29,Sect. 4]. When dealing with Roumieu type classes then it suffices to assume that there exists α 0 ∈ I such that M (α0) is non-quasianalytic since smaller indices can be skipped; see also the discussion in [25, Sect. 5.1].
2.6. Ultradifferentiable classes. Let U ⊆ R d be non-empty open and for K ⊆ R d compact we write K ⊂⊂ U if K ⊆ U , i.e., K is in U relatively compact. We introduce now the following spaces of ultradifferentiable function classes. First, for weight sequences we define the (local) classes of Roumieu-type by and the classes of Beurling-type by where we denote

Ultradifferentiable classes beyond geometric growth factors
The main objective is to prove that the PTT-classes can be represented as classes defined by (suitable) weight matrices. Indeed, we can obtain a more general result by letting the exponents of the defining estimates be Φ j instead of j or j σ where Φ = (Φ j ) j is arbitrary and only satisfying some mild regularity property.
In particular every exponent sequence tends to infinity.
We introduce now Φ-ultradifferentiable function classes (defined in terms of a single weight sequence M ): The (local) class of Roumieu-type is given by and the class of Beurling-type by For a sufficiently regular compact set K is a Banach space and so we have the following topological vector space representations Similarly, we get Note that condition (3.1) means that the factor h Φ |α| in the seminorms is at least geometric.
Remark 3.2. In the literature, global ultradifferentiable classes, test function spaces, ultraholomorphic classes, spaces of weighted sequences of complex numbers and, PTT-test function spaces (see [19]) have been studied. In a completely analogous way, global Φ-ultradifferentiable classes, Φ-test function spaces, Φ-ultraholomorphic classes and Φ-spaces of weighted sequences of complex numbers (weighted with seminorms of the type (3.2)) can be defined ; i.e., when the symbol/functor E is replaced by B, D, A or Λ.

Notation:
When the open set U is not relevant in certain statement, we will simply write E {M} ,

Example 3.3. We have two important examples in mind:
(a) If Φ j = j for all j ∈ N, then we recover the classes from Section 2.6; i.e., the usual definition of ultradifferentiable classes defined in terms of weight sequences; e.g. see [11]. For reasons of simplicity we will skip the letter Φ in the definition.
Given an exponent sequence Φ ∈ R N ≥0 and a pair of sequences M, N ∈ R N >0 , we want to compare the classes E {M,Φ} and E {N,Φ} , or resp. E (M,Φ) and which is obviously never reflexive and symmetric and stronger than Φ . If M ⊳ Φ N , then E {M,Φ} ⊆ E (N,Φ) with continuous inclusion (and similarly for the other classes mentioned in Remark 3.2).

Classes beyond geometric factors versus weight matrices
The aim of this section is to verify that, under mild growth and regularity assumptions on Φ, the classes E {M,Φ} resp. E (M,Φ) can be represented (as locally convex vector spaces) by the matrix classes E {M} resp. E (M) for a suitable choice of the matrix M.

Preparatory results.
We introduce an appropriate matrix of sequences. Let M ∈ R N >0 and Φ ∈ R N ≥0 , then consider the set For any sequence Φ ∈ R N ≥0 , we have that M M,Φ is a weight matrix in the notion of Section 2.5. Next we show that the mild growth restriction (3.1) on Φ is equivalent to the fact that the matrix M M,Φ allows to absorb exponential growth.
>0 and Φ ∈ R N ≥0 be given. Then the following are equivalent: Conversely, let h, c ≥ 1 be given. Then (3.1) yields the existence of some ǫ > 0 and j ǫ ∈ N >0 such that for all j ≥ j ǫ we get If we assume more growth requirements on the sequence Φ, then we can deduce further regularity conditions for M M,Φ .
i.e., the sequences are even ordered w.r.t. their corresponding quotient sequences.
Proof. (i) and (iv) are direct consequences. For the last statement in (ii) let d > c and write Since d c ≥ 1 and the convexity of Φ precisely means that j → Φ j − Φ j−1 is non-decreasing we get that j → d Φj −Φj−1 µ j is non-decreasing as well.
Let us give an argument for (iii): From convexity since Φ 0 = 0, one can deduce that j → Φj j is non-decreasing: Since Φ is increasing we get Φ j > 0 for all large j and hence Φj j is bounded away from 0. ) 1/j → ∞ as j → ∞ for each c > 0 are desirable (standard) properties in the theory of ultradifferentiable (and ultraholomorphic) classes, statements (ii) and (iv) in Proposition 4.2 suggest that for applications the choices for Φ and M should not considered to be independent; cf. (3.3).
In concrete applications the requirement (M (c,Φ) j ) 1/j → ∞ as j → ∞ for each c > 0 might be checked easily. However, even if M is log-convex, Φ is convex and M and Φ are well related as in (3.3), in general as c → 0 one can only expect that condition (4.2) will be satisfied from some j c ∈ N >0 on (and j c → +∞ as c → 0). Nevertheless, in this situation one can replace each M (c,Φ) (for c < 1 small) by some equivalent sequence when changing M (c,Φ) at the beginning, i.e., only for finitely many j. This technical modification leaves the classes E {MM,Φ} and E (MM,Φ) unchanged.

4.2.
Comparison results. This section is devoted to formulate and prove the main comparison theorems. Using the preparation from the previous section we are in position to prove the first statement.
>0 be given and let Φ be an exponent sequence. Let M M,Φ be the matrix defined in (4.1), then as locally convex vector spaces we get By the analogous definitions of the spaces we see that (4.3) also holds for the other classes mentioned in Remark 3.2.
Both cases from Example 3.3 satisfy (3.1); if Φ j = j and so we are treating the classical situation, then the above result becomes trivial in the sense that Proof. The Roumieu case. By definition, we have the following estimate The Beurling case. Follows analogously, but in this case we use Lemma 4.1 to prove the (continuous) On the other hand, let us show now that condition (3.1) is also necessary to obtain (4.3) (or even more), when assuming mild extra assumptions on M . A crucial part of the proof of the Roumieu case is based upon the existence of so-called optimal functions in Roumieu classes: For any given normalized log-convex sequence N , we consider the function with ν j := Nj Nj−1 for j ≥ 1 and ν 0 := 1. It is known that There it has been commented that θ N / ∈ E (N ) (R, C) and the proof shows that in (4.4) we can replace ν j by ν j+1 . The proof of the Beurling case makes use of the following functional analytic result.
Proposition 4.5. Let E, F be Fréchet spaces, such that E is a linear subspace of F (not assuming continuous inclusion). Assume that both are continuously included in C(U ) (or even in any Proof. We want to show that the inclusion ι : E → F is continuous. By the closed graph theorem, it suffices to show that if f n → 0 in E (and thus in C(U )), and f n → g in F (and thus in C(U )), we have g = 0. But this is now clear since C(U ) is Hausdorff.  We immediately get the following consequence:

Proof. (i):
Choose c such that M (c,Φ) is log-convex and normalized. By applying the optimal functions θ N from (4.4) to N ≡ M (c,Φ) we get by the equality of the classes that there exist h, x 0 (w.l.o.g. greater than 1) such that j , and therefore we get (4.6) ∀x ≥ x 0 ∀j ∈ N : In addition we infer, again by working with optimal functions, but now for the sequences j → n j L (n) j , that for all n ∈ N there exists c n (w.l.o.g. increasing in n) such that ∀ j ∈ N : n j L (n) j ≤ c Φj n M j , and by taking roots we end up with then we can find a sequence of integers j n such that c Φj n /jn n ≤ 2.
Combining (4.6) and (4.7) we infer (for all n ∈ N with n ≥ x 0 ) which yields a contradiction as n → ∞.
(ii): By Proposition 4.5, we infer that the spaces are isomorphic as Fréchet spaces. Thus we get that for any compact set K ⊂⊂ R there exist h, x 0 > 0 and a compact set J ⊂⊂ R such that for all Due to log-convexity we can apply (2.2) and get from this estimate (since the sequences of L are pointwise ordered), that for all x ≤ x 0 and j ∈ N h j+1 L (x) j ≤ M j and finally, since w.l.o.g. h ≤ 1, we get that for all x ≤ x 0 and j ∈ N >0 Analogously we argue to get that for all n ∈ N there exists c n > 0 such that and therefore Combining (4.8) and (4.9) we infer which again gives the desired contradiction and thus finishes the proof.
In particular, if we choose for the matrix L the concrete matrix M M,Φ from (4.1), then we can draw the same conclusion i.e., that Φ already has to be an exponent sequence. Under somewhat milder conditions, we can actually show even more in this case. This is due to the fact that we can prove the desired implication directly, however by using the same techniques ((4.4), Proposition 4.5) as in the proof of Theorem 4.7 before. If the symmetric restriction from above is imposed on the sequence Φ, i.e., the growth of h Φj is at most geometric, then we recover the classical ultradifferentiable classes defined by a single weight sequence.  Finally, we can treat the converse statement.
Theorem 4.11. Let M ∈ R N >0 and Φ ∈ R N ≥0 be given. Then we get: (i) The Roumieu case. Assume that M (c,Φ) is log-convex and normalized for some c > 1, and let M ∈ LC or even M be only normalized and log-convex. Assume that is valid (as sets). Then Φ has to satisfy both (3.1) and (4.10).
Proof. We follow the proof of Theorem 4.7 with M = L (x) for any x > 0. In the Roumieu case the second part in (4.5) implies (4.10) for Φ. Then we consider j → h j M j for some arbitrary but fixed h > 1 instead of j → n j L Similarly, we consider for the Beurling case Accordingly, we introduce (cf. In both cases we can replace the symbol (functor) E by B, D, A or by Λ.
These classes will be relevant for the study of PTT-limit classes in Section 6. Theorem 4.12 and the matrix introduced in (4.11) should be compared with the matrix M σ , see (6.1); in particular, this result becomes relevant for the equalities in Remark 6.3.
On the other hand, take M ∈ R N >0 and let F := {Φ a : a > 0} be a family of sequences Φ a ∈ R N ≥0 such that . We introduce the following locally convex vector spaces Similarly, we set and Finally, let us introduce the matrix (4.14) Fix now h > 1 and by (4.13) we can assume that d ≥ c. Hence and so (4.16) is verified with (e.g.) ǫ : Then let h > 1 be given (large) and iterate the previous estimate n-times, with n ∈ N >0 chosen minimal such that e nǫ/2 ≥ h. This then yields choices d = c n+1 > c n > · · · > c 1 = c (since by assumption the value of ǫ is not depending on the choice for c i ) such that  ( * ) Assume that there exists some c 0 > 0 such that Φ c0 satisfies (3.1) with value ǫ 0 > 0.
Then, arguing as in the constant case before, we get (4.16) with ǫ := log(2)ǫ 0 for all choices c ≥ c 0 . Note that in the Roumieu case we can omit all c < c 0 without changing the corresponding function class. ( * ) If for all c > 0 we have that Φ c satisfies (3.1) uniformly in c, i.e., then (4.17) holds true with ǫ := log(2)ǫ 1 .

PTT-classes as spaces defined by weight matrices
Let the parameters τ > 0 and σ > 1 be given but from now on fixed and consider (with the convention 0 0 := 1) For these particular choices of M and Φ we write M (c,τ,σ) for M (c,Φ) . Thus the sequences and the matrix introduced in (4.1) have the form

5.1.
Properties of the matrix M τ,σ . Note that, in particular, Theorem 4.4 applies to this special situation. We thus have as a corollary, in accordance with the notation in the works of S. Pilipović, N. Teofanov, and F. Tomić, the following statement.
Therefore, we may apply certain results available in the weight matrix setting to PTT-classes. First we need to study the properties of the defining weight matrix M τ,σ . for all 0 < c 1 < c 2 .
(iv) First, by Stirling's formula we get for all c > 0 and j ≥ 1: Thus (M {rai} ) follows because we have for all 1 ≤ c ≤ c 1 , A ≥ e and 1 ≤ j ≤ k: Concerning (M (rai) ), let 0 < c 1 ≤ c < 1, A ≥ 1 and 1 ≤ j ≤ k, then So the desired estimate follows by choosing c 1 = c and A large enough.
Recall that each strongly log-convex sequence satisfying m 0 = M 0 = 1 has the property that j → (m j ) 1/j is nondecreasing, compare this with (ii).
(vii) We test conditions (M {mg} ) and (M (mg) ) for j = k ≥ 1. So let A ≥ 1 and c, c 1 > 0 be given, then: (M (c,τ,σ) 2j As j → ∞ the first summand on the right-hand side tends to 0, whereas the second one tends to −∞ and so does the right-hand side. This leads to a contradiction for any choice c, c 1 > 0.
(viii) For every 0 < c 1 < c 2 , we have that (c) Image of the Borel map ( [18], [32], [31], [16], [17]): We have the following description of the image of the Borel map: By (vi) we have that each M (c,τ,σ) has (γ 1 ) and the rest follows from the results of aforementioned papers. ) which is valid by (i). Finally, also in [5] a matrix M is called non-quasianalytic if each M ∈ M is non-quasianalytic and this is valid, in particular, by (vi).

PTT-limit classes as spaces defined by weight matrices
In [19], for fixed σ ≥ 1, the authors also consider limits with respect to the parameter τ , i.e., the spaces E ∞,σ and E 0,σ , presented in Subsection 2.7, which are endowed with the natural inductive resp. projective limit topology. The main reason to consider these classes is represented by the fact, that they are stable with respect to so-called ultradifferential operators. Observe here that a function f lies in E ∞,σ (U ) if and only if there exists a uniform τ such that f ∈ E {τ,σ} (K) for all K ⊂⊂ U and thus the limit classes E ∞,σ (U ) do not quite fit in the realm of Roumieu-type classes defined via weight matrices since quantifiers are exchanged. In the latter case it is allowed that τ is also depending on K; see (2.3).
In order to comment on this subtle difference, first for σ ≥ 1 let us from now on consider the matrix (using the notation from (5.2)) (6.1) We get the following connection with the respective matrix class defined in terms of M σ .
Proof. The Roumieu case. The first inclusion is clear from the definition of the respective spaces. For the second one, observe that any f ∈ E {M σ } (U ) lies in E {M (τ,τ,σ) } (V ) for some τ , i.e., there exists h ≥ 1 and A > 0 such that for all x ∈ V and α ∈ N d we have and for any τ ′ > τ , we find B > 0 such that for all j we have which finishes the Roumieu case.
The Beurling case. Since there is a universal quantifier for the compact set, the index τ , and the geometric factor h, we do not have to worry about interchanging those. The rest follows from Proposition 5.1.
Proposition 6.2. Let σ ≥ 1 and N = {N (x) : x > 0} be a (M sc ) and non-quasianalytic weight matrix. Suppose that we have Then it follows that Proof. Similarly as in (viii) in Theorem 5.2 we get The inclusion E ∞,σ (U ) ⊆ E {N } (U ) and the optimal function θ M (τ 1 ,τ 1 ,σ) (see (4.4)) implies the following: Let K j be a sequence of mutually disjoint compact sets with non-empty interior contained in U such that they accumulate at the boundary of U , i.e., for any compact set K ⊂⊂ U there exists j such that K j ∩ K = ∅. Let S j be also a sequence of compact sets such that S j ⊆ K • j . Finally, let x j ∈ S j . Then by [22,Cor. 3.12]  Then clearly h j ∈ D {N (j) } (K • j ), and thus h ∈ E {N } (U ). But h / ∈ E {τ,σ} for any τ (and therefore not in E ∞,σ (U )). To see this, take for given M (τ,τ,σ) some j big enough to get (6.3). By taking K = K j , one immediately gets h / ∈ E {τ,σ} (U ). Remark 6.3. After a private communication, in the very recent paper [33] the authors already have taken into account this fact and included the definition of the limit classes see [33, (2.12)]. Note that this difference might be considered negligible in the light of Theorem 6.1 and for the Beurling-type both notions coincide; i.e.
When considering the notion of germs of E ∞,σ -functions then also no difference occurs. For these equalities recall Theorem 4.12.
So far we have verified the desired properties on the diagonal (i.e., j = k). However, by the equivalence stated in (ii) before also M σ has both (M {mg} ) and (M (mg) ) verified on the diagonal. Thus [27,Thm. 9.5.1,Thm. 9.5.3] applied to M σ yields the conclusion and by the equivalence we are done with M σ , too.
(iv) (M {rai} ) follows by repeating the estimates from (iv) in Theorem 5.2. Note that we have for all 1 ≤ c ≤ c 1 , A ≥ e and 1 ≤ j ≤ k Let 0 < c 1 ≤ c < 1, A ≥ 1 and 1 ≤ j ≤ k, then We take c 1 = c and repeat the computation from (iv) in Theorem 5.2 when τ is replaced by c. This should be compared with (ii) and recall that each strongly log-convex sequence satisfying m 0 = M 0 = 1 has the property that j → (m j ) 1/j is nondecreasing.
(vi) For all h ≥ 1 (large) and all 0 < c < c 1 we can find some constant A ≥ 1 (large) such that for all j ∈ N >0 : (vii) The same estimate as given in (vi) also yields the following property for M σ : hence both desired properties.
6.2. PTT-limit classes as Braun-Meise-Taylor classes. Let σ > 1 be given. Then, on the one hand E [M σ ] cannot be described by a single weight sequence which follows by (vii) in Theorem 6.4. On the other hand, the aim of this section is to show that it actually can be understood as a Braun-Meise-Taylor class. This question has very recently been studied and solved in [33] (for the modified defined limit classes mentioned in Remark 6.3). There the authors give precise asymptotics of ω in terms of the so-called Lambert function W ; cf. [33,Prop. 3.1]. However, we give an independent proof of their main result by involving only weight matrix techniques.
Let us emphasize that for σ = 1 this statement is not true: By (6.1) the matrix M 1 := {M (τ,τ,1) : τ > 0} consists of sequences M (τ,τ,1) j = τ j j τ j and hence M 1 is equivalent to the Gevrey matrix G 0 = {(j! τ ) j∈N : τ > 0}. From (the first paragraph in the proof of) [23,Thm. 5.22] it follows that the corresponding weight matrix class cannot be described by a space given by a log-convex M (in particular M ∈ LC) or by a weight function ω.
We prove now an abstract result on the connection between weight sequences and their associated weight functions which is important in the ultradifferentiable setting on its own. Lemma 6.5. Let M, N ∈ LC. Then the following are equivalent: (i) M and N are related by (ii) The associated weight functions are related by Moreover, the following are equivalent: (i) ′ M and N are related by (ii) ′ The associated weight functions are related by Proof. (ii) ⇒ (i) We have ω M (t) ≤ cω N (t) + c for some c ≥ 1 (large) and all t ≥ 0. W.l.o.g. take c ∈ N >0 and then (2.2) yields for all j ∈ N: Thus (6.4) is shown with A = e.
(ii) ′ ⇒ (i) ′ For all c ∈ N >0 we can find D ≥ 1 such that ω M (t) ≤ 1 c ω N (t) + D for all t ≥ 0 and so, analogously as before, we obtain (6.5) with A := e D . (i) ′ ⇒ (ii) ′ Using the notation from above, (6.6) transfers into Then we follow the arguments in (i) ⇒ (ii) and combine (6.8) with the first half from (6.7) applied to N in order to get Thus ω M (t) = o(ω N (t)) is verified.
The importance of Lemma 6.5 is that it enables the possibility to express all requirements in [29,Cor. 3.17 (ii)] purely in terms of the given matrix N directly: Corollary 6.6. Let M be (M sc ). Then as locally convex vector spaces with ω being a weight function in the sense of Braun-Meise-Taylor (see [3], [29]) if and only if there exists a (M sc ) matrix N = {N (α) : α > 0} which is R-resp. B-equivalent to M and such that Remark 6.7. Any ω N (α) (for N (α) ∈ N ) is a valid choice for ω in Corollary 6.6. Note that (6.9) is clearly preserved under R-and B-equivalence of weight matrices.
6.3. Results for PTT-limit classes. As seen in the previous section the PTT-limit classes can be represented by certain Braun-Meise-Taylor classes (defined by the weight function ω (σ) := ω M (1,1,σ) ).
Let us now give additional properties available for PTT-limit classes that follow from this representation (and the properties listed in Theorem 6.4). (c) The image of the Borel map ( [18], [32], [31], [16], [17]): We have the following description of the image of the Borel map: . By (viii) we have that each M (τ,τ,σ) has (γ 1 ) and the rest follows from the results of the aforementioned papers. 6.4. A further possible result. In [4], the authors consider ultradistributional boundary values of constant coefficient hypoelliptic partial differential operators. Ultradistributional is understood in the framework of Denjoy-Carleman classes, i.e., classes defined via weight sequences. They require, apart from the normalization condition 1 = M 0 = M 1 as basic assumptions for M log-convexity, = 1 is violated for τ = 1 but which can be achieved by switching to an equivalent weight matrix.) Then one can try to compensate the failure of (mg) by applying (M {mg} ) resp. (M (mg) ) instead and which is valid by (iii) in Theorem 6.4.
However, by (vii) in Theorem 5.2 both generalized moderate-growth-type conditions fail for M τ,σ and so a generalization of the proofs from [4]