Equality of Ultradifferentiable Classes by Means of Indices of Mixed O-regular Variation

We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.


Introduction
In the theory of ultradifferentiable function spaces there exist two classical approaches in order to control the growth of the derivatives of the functions belonging to such classes: Either one uses a weight sequence M = (M j ) j or a weight function ω : [0, +∞) → [0, +∞). In both settings one requires several basic growth and regularity assumptions on M and ω and one distinguishes between two types, the Roumieu type spaces E {M } and E {ω} , and the Beurling type spaces E (M ) and E (ω) . In the following we write E [ ] for all arising classes ( * ) Study the effects of growth properties assumed for M on this construction. ( * * ) Describe the class E [M] alternatively (as a locally convex vector space) by E [ωM] . This enables the possibility to apply techniques from the classical (single) weight function setting to the space E [M] .
Concerning ( * * ) we can see the following "dual problem": ( * * ) Starting with an abstractly given weight function matrix M W := {ω x : x ∈ I}, can we describe the class E [MW ] alternatively (as a locally convex vector space) by E [N ] for some weight matrix N = {N x : x ∈ I} ?
For treating ( * ) and ( * * ), in [14,Sect. 9.3] and [15] the following properties for M have become relevant, see also [11,Sect. 4.1] and [14,Sect. 7.2]): All these conditions are automatically valid for Ω, when ω is satisfying standard properties. (M {mg} ) and (M (mg) ) are the natural generalizations of condition moderate growth arising frequently in the weight sequence setting. In [14,Sect. 9.3] and [15] only the sufficiency of (M {L} ) and (M (L) ) has been applied for the study of question ( * * ) and these conditions seemed to be Vol. 77 (2022) Equality of Ultradifferentiable Classes by Means of Indices Page 3 of 32 28 too strong, see also the discussion in (ii) in Remark 2.2. Note that (M [L] ) is indispensable to have E [Ω] = E [ω] , see (2.4).
The main aim of this article is to give a complete solution to ( * * ) and to ( * * ) expressed in terms of growth properties for M. In order to do so, instead of (M {L} ) we consider the weaker requirement ∀ x ∈ I ∃ y ∈ I : lim sup and similarly ∀ x ∈ I ∃ y ∈ I : lim sup instead of (M (L) ). (1.1) and (1.2) are the mixed versions of the standard assumption ω(2t) = O(ω(t)) as t → +∞ on weight functions (denoted by (ω 1 ) in this work). These types of mixed conditions, for either weight sequences or weight functions, have frequently appeared in the literature in the study of mixed extension results in the ultradifferentiable or ultraholomorphic settings, see [3,6,7,10,16].
In [4] the main goal has been to give connections between standard/ frequently used assumptions for weight sequences and weight functions in the ultradifferentiable and ultraholomorphic framework and the concept of O-regular variation. More precisely, this has been done for moderate growth for weight sequences and (ω 1 ) for (associated) weight functions. Note that analogous growth properties are also showing up when dealing with different areas of weighted spaces in Functional Analysis, see the introduction in [4]. We have been able to characterize these conditions in terms of growth properties, positivity and finiteness of weight indices.
Inspired by these known results and the mixed conditions mentioned before, for weight sequences and (their associated) weight functions we introduce in this paper new mixed growth indices which are related to mixed O-regular variation. Moreover, we study and compare these indices under the construction M → ω M → Ω M .
Summarizing, it turns out that we can give an answer to problem ( * * ) by assuming (M {mg} ) and (1.1) resp. (M (mg) ) and (1.2), and problem ( * * ) can by reduced to ( * * ). Alternatively, we have the possibility to express problem ( * * ) (and hence ( * * ) ) purely in terms of these new concept of mixed indices. In particular, when considering M = {M } or M W = {ω}, then we get the results from [1] expressed in the language of O-regular variation. Finally, by using these indices one is able to understand the difference between conditions (M {L} ) and (1.1) resp. between (M (L) ) and (1.2) in a more quantitative way.
The paper is structured as follows: In Sect. 2 we gather all relevant information on weight sequences, functions and matrices and we recall the multiindex construction. In Sect. 3, first we give an exhaustive characterization of (M {L} ) and (M (L) ) resp. of (1.1) and (1.2) by means of growth conditions expressed in terms of the weight structures M, ω M and of the multi-index construction, see Lemma 3.1 and Theorem 3.2. It turns out that for the weight matrix Ω there is no difference between these requirements of the particular, Roumieu or Beurling, type, see Corollary 3.3. Based on these characterizations, in Sect. 3.1 we introduce mixed growth indices for weight sequences and (associated) weight functions, see Proposition 3.6 and Theorem 3.7.
In Sect. 4 the analogous procedure is done for the mixed moderate growth conditions (M {mg} ) and (M (mg) ), see Proposition 4.2 and Theorem 4.4.
In the final Sect. 5 we apply the developed theory and prove the main results, Theorem 5.4 which is solving ( * * ), and Theorem 5.6 answering problem ( * * ) . The special (and classical) cases M = {M } and M W = {ω} are discussed as well.

Weight Sequences
If M is log-convex and normalized, then both j → M j and j → (M j ) 1/j are nondecreasing and (M j ) 1/j ≤ μ j for all j ∈ N >0 . For our purposes it is convenient to consider the following set of sequences We see that M ∈ LC if and only if (μ j ) j is nondecreasing and lim j→+∞ μ j = +∞ (e.g. see [11, p. 104]). Moreover, there is a one-to-one correspondence between M and μ = (μ j ) j by taking M j := j i=0 μ i . M has the condition moderate growth, denoted by (mg), if In [8] both are normalized, then M N does precisely mean M j ≤ C j N j for some C ≥ 1 and all j ∈ N.

Associated Weight Function
For an abstract introduction of the associated function we refer to [9, Chapitre I], see also [8,Definition 3.1]. If lim inf j→+∞ (M j ) 1/j > 0, then ω M (t) = 0 for 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 .
(2.1) If ω satisfies in addition ω(t) = 0 for all t ∈ [0, 1], then we call ω a normalized weight function. For convenience we will write that ω has (ω 0 ) if it is a normalized weight. Moreover we consider the following conditions, this list of properties has already been used in [14].
We recall the following known result, e.g. see [15,Lemma 2.8] resp. [5, Lemma 2.4 (i)] and the references mentioned in the proofs there.

Weight Matrices
For the following definitions and conditions see also [11,Section 4].
Let I = R >0 denote the index set (equipped with the natural order), a weight matrix M associated with I is a (one parameter) family of weight We call a weight matrix M standard log-convex, denoted by (M sc ), if A matrix is called constant if M x ≈ M y for all x, y ∈ I. On the set of all weight matrices we consider the following relations, see [11, p. 111

Weight Matrices Obtained by Weight Functions and Multi-index Weight Matrices
We summarize some facts which are shown in [11,Section 5] and are needed in this work. All properties listed below are valid for ω ∈ W 0 , except (2.4) for which (ω 1 ) is necessary.
(i) The idea was that to each ω ∈ W 0 we can associate a standard log-convex weight matrix Ω : (iv) Equivalent weight functions yield equivalent associated weight matrices.
(v) (ω 6 ) holds if and only if some/each W l satisfies (mg) if and only if W l ≈ W n for each l, n > 0. Consequently (ω 6 ) is characterizing the situation when Ω is constant.
In particular, given M ∈ LC we denote by Ω M the weight matrix associated with ω M .
In [15, Section 2.7] the new weight function matrix ω M := {ω M x : x ∈ I} has been considered and ultradifferentiable classes E {ωM} and E (ωM) have been introduced.
Let us recall now this approach. First let M : Based on this information, in [15, Section 3.1] the following multi-index weight matrix construction has been introduced: where for x ∈ I, l j ∈ R >0 , j ∈ N >0 , and i ∈ N we put M x;l1,...,lj+1 i for t > 0, ω M x;l 1 ,...,l j (0) := 0.
In this notation, for any x ∈ I we get Note that by normalization of M we have ω M (t) = 0 for 0 ≤ t ≤ 1, e.g. see the known integral representation formula for ω M , [9, 1.8. III] and also [8, (3.11)]. The similar statement is valid for the higher multi-index sequences. For a given (M sc ) matrix M := {M x : x ∈ I} we put M (2) := {M x;l : x ∈ I, l > 0}. An abstractly given matrix of weight functions denotes a family of weight functions {ω x : x ∈ I}, such that ω y ≤ ω x for any x, y ∈ I with x ≤ y.

Ultradifferentiable Classes
Let U ⊆ R d be non-empty open. We write K ⊂⊂ U if K is compact and K ⊆ U . We introduce now the following spaces of ultradifferentiable functions classes. First, for weight sequences we define the (local) classes of Roumieu type by and the classes of Beurling type by where we denote For a compact set K with sufficiently regular (smooth) boundary is a Banach space and so we have the following topological vector spaces Similarly, we get For any ω ∈ W 0 let the Roumieu type class be defined by and the space of Beurling type by .
For compact sets K with sufficiently regular (smooth) boundary is a Banach space and we have the following topological vector spaces Similarly, we get Next, we consider classes defined by weight matrices of Roumieu type E {M} and of Beurling type E (M) as follows, see also [11, 4.2]. For all K ⊂⊂ U with sufficiently smooth boundary we put and so for U ⊆ R d non-empty open Similarly we get for the Beurling case Finally, for the matrix ω M we put Thus we obtain the topological vector spaces representations and Since all arising limits are equal to countable ones, in each setting the Beurling type class is a Fréchet space. Ultradifferentiable classes of multi-index matrices can be defined in a similar way; for our purposes we will only need M (2) , see Sect. 5.

The Mixed (ω 1 ) Conditions
Concerning (M {L} ) and (M (L) ) we start with the following characterization.
(II) The following conditions are equivalent: Vol. 77 (2022) Equality of Ultradifferentiable Classes by Means of Indices Page 11 of 32 28 (iv) The matrix ω M does satisfy Proof. We will only treat the Roumieu case, the Beurling case is analogous.
For all x ∈ I we can find D > 0 and y ∈ I such that for all j ∈ N and t ≥ 0. This yields the conclusion by definition of associated weight functions.
(iii) ⇒ (iv) This is clear for C ≤ 2 (since each associated weight function is nondecreasing). If C > 2, then we take n ∈ N chosen minimal such that C ≤ 2 n is valid and apply iteration: Given x ∈ I, there exists y 1 ∈ I and as desired.
(iv) ⇒ (i) We apply (2.1) and get for all j ∈ N: and so we are done.
For abstractly given weight matrices M, a connection to mixed "(ω 1 )conditions" has been established in [15,Prop. 3.12,Cor. 3.15]. The different equivalent conditions in the following result make intervene both the associated weight functions ω M x and the matrices associated with the latter. In particular, we generalize the main characterizing result [12, Thm. 3.1] from the weight sequence to the weight matrix setting. x ∈ I, l > 0}. Then we get: (I) The following conditions are equivalent: (II) The following conditions are equivalent: In Proof. Again, we treat the Roumieu case in detail, the Beurling setting is completely analogous.
(ii) ⇒ (iii) Let x ∈ I be given, so ω M y (2t) ≤ Lω M x (t) + L for some index y ∈ I, L ∈ N >0 and all t ≥ 0. Then, by applying (2.1), we get for all j ∈ N: which proves (iii).
(iii) ⇒ (iv) Let r 0 > 1 be the value given in (iii). If r ∈ (1, r 0 ], then nothing is to prove. If r > r 0 , then we choose n ∈ N minimal to have r ≤ r n 0 and apply n iterations. Namely, given x ∈ I, there exists y 1 ∈ I and L 1 ∈ N >0 such that Recursively, for i = 2, . . . , n there exist y i ∈ I and L i ∈ N >0 such that The choice y = y n and L = n i=1 L i clearly fulfills the requirements. (iv) ⇒ (v) Recall that the matrix {M x;a : a > 0} associated with the weight ω M x is given by 14) and M x ≡ M x;1 by (2.5). Moreover, if L ∈ N >0 we have for every j ∈ N, Let x ∈ I and C > 1 be given. By the assumption (reasoning with r = C) we have for some y ∈ I, L ∈ N >0 , A ≥ 1, and all j ∈ N. Hence, for all t > 0 and j ∈ N, and we obtain by definition ω M y;   In conclusion, we use (3.14) in order to obtain as desired.
(v) ⇒ (vi) It suffices to take, for any a > 0, b = Ba, where B is given in (v).
(vi) ⇒ (vii) If we choose a = 1 in (3.7) we get j , what leads to the conclusion in view of (3.14).
(vii) ⇒ (viii) Note that the mapping j → 1 jb ϕ * ω M x (jb) is nondecreasing for any fixed b > 0 and x ∈ I. Let j 1 ∈ R >1 be given and take j ∈ N ≥2 with j − 1 < j 1 ≤ j and then for all t > 0 and j ∈ N, and we obtain ω M y;b (ht) ≤ ω M x (t) + log(A) for all t ≥ 0. We conclude by using (3.15) again.
We gather now the information for weight matrices associated with given Braun-Meise-Taylor weight functions ω and get the following characterization. So, the mixed (ω 1 ) conditions of Roumieu and/or Beurling type are equivalent to (ω 1 ) for ω.

Remark 3.4.
We summarize now some facts concerning the arising lim infconditions in the previous results.
(a) (3.1) implies (3.5) (with L = 1), but the latter condition is weaker than the first one since j → (M x j ) 1/j is nondecreasing by log-convexity and normalization for each index x fixed.
From this, the desired statement follows for the Roumieu type immediately.
In the Beurling case, when 0 < y < x, then we choose L 1 ∈ N >0 such that L 1 ≥ x y and so (W x j ) 1/j ≤ (W y L1j ) 1/(L1j) holds true for any j ∈ N >0 . Thus we can estimate by

Mixed Growth Indices Based on Mixed (ω 1 ) Conditions
We start this section with some observations. In (3.5) and (3.11) in the arising lim inf condition in order to make sense we have to assume L ∈ N >0 , whereas only b > 0 is required in the conditions from (vii) and (viii) in Theorem 3.2. Moreover recall that, since M x;a ∈ LC, by (2.1) we get = exp(ϕ * ω M x;a (j)), and the last expression makes even sense for any real j ≥ 0. Based on the characterizations shown in Theorem 3.2 we introduce now two mixed growth indices. Let M, N ∈ LC with M ≤ N and let ω, σ be weight functions with σ ≥ ω.
Note that, if we write Ω N = {W q = (W q j ) j∈N : q > 0} for the weight matrix associated with ω N , so in particular W 1 = N (see Sect. 2.6), then the previous condition reads what explains the notation used. It is immediate to see that if the condition is satisfied for some given b > 0, then also for all 0 < b < b (with the same choice for q).
Similarly, given a > 0 we write (σ, ω) ω1,a if which shall be compared with [4,Thm. 2.11 (iv)]. Again it is clear that if the condition is satisfied for some given a > 0, then also for all a > a (with the same choice of K). According to these growth restrictions we put If there does not exist any b > 0, resp. a > 0, such that (M, Ω N ) L,b , resp. (σ, ω) ω1,a , holds true, then we put β(M, Ω N ) = 0, resp. α(σ, ω) = ∞.
A first immediate consequence is the following: Proof. Suppose β(M, Ω N ) > 0 and let 0 < b < β(M, Ω N ), so (3.16) is valid for b, and then there exists some q > 1 and C ≥ 1 such that for all t ≥ 0 we get ϕ * ωN (tq) ≥ qϕ * ωM (t) + tq log(q b ) − C. We apply the Young conjugate to this inequality. Hence for all s ≥ log(q b ) we have Consequently there does exist Hence we have shown So we have verified (M, Ω N ) L,1/a with the choice q : One may easily conclude that the stated equality holds in any case (with the conventions 1/0 = ∞, 1/∞ = 0), even if one of the indices is zero or infinity.
Thus, by involving the notation of mixed indices in this section and Proposition 3.6, we can now reformulate Theorem 3.2 as follows.
Analogously, the following are equivalent: (i) Any of the equivalent mixed (ω 1 )-conditions of Beurling type in (II) of Theorem 3.2 hold true, Proof. Again we limit ourselves to the Roumieu case.
(i) ⇒ (ii) It suffices to take into account (3.3), which easily shows that for every x there exists y such that (ω M x , ω M y ) ω1,a holds for a suitable value of a.
(ii) ⇒ (i) By hypothesis, for every x ∈ I there exists y ∈ I and K > 1 such that After a finite iteration (if necessary), we can guarantee that (3.4) holds, and we are done.
We close this section with the following observations for given M = {M x : x ∈ I} being (M sc ): (a) Lemmas 3.1 and 3.5 imply that condition (M {L} ), respectively (M (L) ), yields that for every x ∈ I there exists y ∈ I such that α( In particular, if Ω is the matrix associated with some ω ∈ W 0 , by Corollary 3.3 we know that, as soon as ω has in addition (ω 1 ), we are in the situation described in (a).

Mixed Moderate Growth Conditions
In this section we study the mixed moderate growth conditions for a given weight matrix. The first three equivalent conditions in the next result are stated in [15] and [14], new conditions (iv) and (v) deal with replacing the constant 2 in (iii) by some q > 0.
Moreover, in the Beurling setting, we have the following equivalences: ). Then (iii) implies the fact that for all x ∈ I we find y ∈ I such that the (4.1) is valid with the same indices x and y and the universal choice q = 2.
(iv) ⇒ (v) For any j ∈ R, j ≥ 1, we get , since q → 1 jq ϕ * ω M x (jq)) is nondecreasing for any x ∈ I and j > 0 (fixed) and since j ≤ j + 1 ≤ 2 j for all j ≥ 1. Thus (4.1) with the choice q > 0 does imply (4.2) with the same indices x and y and with the parameter q := q/2.
Hence, after applying sufficiently many iterations again depending only on given q (choose n ∈ N minimal to have 2 n q ≥ 2), we get (4.2) with q ≥ 2 and some index z ∈ I and are again able to conclude.
Assertion (ii) in the previous result is the mixed (ω 6 )-condition of the particular, Roumieu or Beurling, type. Using resp. applying iterations as in the proofs of the previous section it is straight-forward to extend the list(s) of equivalences in Proposition 4.1 by replacing in (ii) the value 2 by any C > 1 not depending on x and y.

Mixed Growth Indices Based on Mixed Moderate Growth Conditions
Let M, N ∈ LC with M ≤ N be given and a > 0. We write (N, Ω M ) mg,a , if which should be compared with [4,Thm. 3.16 (v)]. Note that, if we write Ω M = {W q = (W q j ) j∈N : q > 0} for the weight matrix associated with ω M , so in particular W 1 = M (see Sect. 2.6), then the previous condition reads Given weight two functions ω and σ with σ ≥ ω we write (ω,  Proof. Let a > α(N, Ω M ), so (4.3) is valid for a > 0. Then there exists some C ≥ 1 such that for all t ≥ 0 we get ϕ * ωM (tq) ≤ qϕ * ωN (t) + tq log(q 1/a ) + C. We apply the Young conjugate to this inequality, hence for all s ≥ log(q a ) we have Consequently, there does exist C 1 ≥ 1 such that for all t ≥ 0 we have ωN (t) ≥ K 1/a with K := q a > 1. So (ω N , ω M ) ω6,1/a is valid with this choice K for any a > a which proves β(ω N , ω M ) ≥ 1/a . Since a can be chosen arbitrarily close to α(N, Hence applying the Young-conjugate yields for all s ≥ 0: Thus we have shown that there exists C ≥ 1 such that ϕ * Note that this proof also shows that if one of the indices is zero the other is infinity, so also in this situation the equality holds with the usual convention.
Analogously, for the Beurling case, the following are equivalent:

Consequences for Ultradifferentiable Classes
The aim of this final section is to extend the characterizing results from [1] to the matrix setting, to refine the consequences shown in [15] and to give the new mixed growth indices introduced in the previous sections an interpretation in terms of the characterization for the equivalence of ultradifferentiable classes defined by (abstractly given) weight matrices. The aim of this section is to show that (5.1) is characterized in terms of the growth properties studied before. More precisely we will see that assumption (M [L] ) is too strong.

Classes Defined by Abstractly Given Weight Matrices
The proof of Theorem 5.1 has been split into two parts. The first one [15,Theorem 3.4] deals with the mixed moderate growth conditions and can be reformulated as follows. The second part has been treated in [15, Section 3.10] by studying consequences of the assumption (M [L] ) on M. However, in order to have equality between the classes E [M (2) ] and E [ωM] it is sufficient to have one of the equivalent but weaker conditions from Theorem 3.2 above. Proof. In the Roumieu case, (ii) and (iii) are equivalent, by Theorem 3.7, to the fact that one/each of the conditions from (I) in Theorem 3.2 holds true. These conditions are shown to imply (i) in [15,Section 3.10]. The same arguments apply for these implications in the Beurling case. So, it is only pending the proof that (i) in the particular Roumieu or Beurling case implies one of the equivalent conditions from (I), resp. (II), in Theorem 3.2.
Then recall that for any given N ∈ LC we can define (recall ν k := N k N k−1 , ν 0 := 1): We get that θ N ∈ E {N } (R, C) (in fact θ N does admit global estimates on whole R) and θ (j) We refer to [17,Theorem 1], for a detailed proof see also [13,Prop. 3.1.2] and [11,Lemma 2.9]. However, it is not difficult to see that θ N does not belong to the Beurling type class E (N ) (R, C).
Each M x;a ∈ M (2) does belong to the class LC. Let h > 1 be arbitrary, but from now on fixed. Let also x ∈ I be arbitrary but fixed and put M hence one may easily deduce that (3.8) is verified. The Beurling case. We follow the ideas from the characterization of the inclusion relations, see [11,Prop. 4.6] and also [15,Prop. 3.9]. By assumption the nontrivial inclusion E (ωM) (R, C) ⊆ E (M (2) ) (R, C) is valid and the inclusion mapping is continuous by the closed graph theorem. Note that both spaces are Fréchet spaces. This means that ∀ K ⊆ R compact ∀ x ∈ I ∀ a > 0 ∀ h > 0 ∃ K 1 ⊆ R compact ∃ y ∈ I ∃ b > 0 ∃ D > 0 ∀ f ∈ E (ωM) (R, C) : f M x;a ,K,h ≤ D f ω M y ,K1,b .