On the class of almost subadditive weight functions

We answer a question from A. V. Abanin and P. T. Tien about so-called almost subadditive weight functions in the sense of Braun-Meise-Taylor. Using recent knowledge of a growth index for functions, crucially appearing in the ultraholomorphic setting, we are able to show the existence of weights such that there does not exist an equivalent almost subadditive function and hence almost subadditive weights are a proper subclass.


Introduction
In [1] it has been shown that almost subadditive weight functions are sufficient to describe the theory of ultradistributions in the sense of Braun-Meise-Taylor; see [4].Thus the authors are dealing with non-quasianalytic weight functions ω; see Section 2.2 for precise definitions and conditions.All relevant information about growth and regularity properties for weight functions and weight sequences can be found in Section 2.
The notion of almost subadditivity is more recent and more general than the classical (standard) assumption subadditivity appearing in the Beurling-Björck framework; see e.g.[2].On the other hand we mention that in the literature different "subadditivity-like" conditions for (associated) weight functions have been studied and used; see e.g.[17] and the citations there.
The aim of this article is to answer this question and to prove that in general there does not exist an equivalent almost subadditive weight.This fact shows that the class of almost subadditive weight functions is a proper subclass and hence the main result [1,Thm. 1.4] means that this particular subclass of weight functions is sufficient to describe the theory of ultradistributions in the sense of Braun-Meise-Taylor.Our statements can be found in Section 3. In this work, since we are mainly focusing on functions and their growth properties, it is not required to introduce the associated ultradifferentiable function spaces explicitly; we refer to [4], [3], [9], [13].
In order to treat this problem, crucially we involve recent knowledge about the growth index γ(ω) from [6]; see Section 2.3 for precise definitions and explanations.This value becomes relevant when proving extension results in the ultraholomorphic setting; see [7], [8].We establish a connection between the notion of almost subadditivity and γ(ω) (see Lemma 3.1 and Remark 3.2).Involving this information, intrinsic properties of γ(ω), in particular using the fact that this growth index is preserved under equivalence of weights, we show the main result Theorem 3.3.Note that using this method we are able to give more information: On the one hand, we are working within a more general setting of weight functions; see Definition 2.1.Thus our basic assumptions on ω are weaker than being a weight in the sense of Braun-Meise-Taylor, but also weaker than the standard requirements assumed in [2] and even slightly weaker than in [7], [8].This is possible by the general setting studied in [6,Sect. 2.3 & 2.4].On the other hand, instead of almost subadditivity, we treat a more general condition (see (3.1)) and obtain that the above problem is not depending on the notion of (non-)quasianalyticity.So the main result is valid for quasianalytic weights as well; see Remark 3.5 for more detailed comments.
However, Theorem 3.3 provides abstract information and the aim in Theorem 3.6 and Theorem 3.7 is to construct explicitly weights with the desired property.There we are focusing on so-called associated weight functions ω M with M ∈ R N >0 being a weight sequence; see Section 2.4.The advantage in this case is that the desired growth properties for ω M can be expressed in terms of M and so one is able to proceed by constructing an appropriate weight sequence M .Such explicit constructions of (counter-)examples can be useful for further applications in different contexts.
Here, the growth index γ(M ) originally introduced by V. Thilliez in [19] is becoming relevant; see again [6] for more details and its relation to γ(ω M ).These are the basic assumptions resp.the notion for being a weight which is used in [6,Sect. 2.3 & 2.4].These requirements are sufficient to introduce and work with the crucial index γ(ω); see Section 2.3 for details.
According to [1,Def. 1.1] we say that a weight function ω This should be compared with the more original condition subadditivity in the Beurling-Björck framework (see [2]): (ω sub ) ω(s + t) ≤ ω(s) + ω(t) for all s, t ≥ 0. Obviously (ω sub ) implies (2.1) with the uniform choice C q = 0 for all q > 1. Note: If the inequality in (2.1) is valid for some q 0 > 1, then for all q ≥ q 0 as well when choosing C q = C q0 .So, in order to verify almost subadditivity, one has to study q → 1.Note that by definition neither (ω sub ) nor (2.1) is preserved under equivalence of weight functions in general.
Moreover, by [17,Prop. 3.4] it follows that if M has in addition (mg), then ω M has (ω 1 ) if and only if (2.4) appeared crucially in [3] and there it also has been shown that (2.4) implies (ω 1 ) for ω M even without having (mg); see [

Main results
We formulate and prove now the main results of this article.In order to proceed we crucially involve the information about the growth index γ(ω).Section 3.1 deals with an abstract statement and in Sections 3.2 and 3.3 we are investigating in more detail the (non-)quasianalytic weight sequence case.
3.1.General result for weight functions.The next technical and crucial result illustrates the intimate connection between the notion of almost subadditivity and the value of γ(ω).Indeed, in this context we study a more general growth relation.

Lemma 3.1.
Let ω be a weight function.
Using this information we are able to show the following main statement.

the following property:
There does not exist a weight function σ being equivalent to τ a and such that σ is almost subadditive (see (2.1)).In particular, each τ a itself is violating (2.1).( * ) If ω is a (normalized) BMT-weight function, then each τ a , too.
Roughly speaking this result means that "many" BMT-weight functions are not equivalent to an almost subadditive function in the sense of [1]; to each weight with finite γ(ω) we can assign an uncountable infinite family of pair-wise different equivalence classes of weights such that in each of these classes no almost subadditive weight exists.Proof.Take any a > 0 with γ(ω) < 1 a , then set τ a := ω 1/a and by (2.3) we get (3.4) γ(τ a ) = γ(ω 1/a ) = aγ(ω) < 1.
Finally, let a 1 > a 2 > 0 satisfy γ(ω) < 1 a1 < 1 a2 and let these choices be arbitrary but from now on fixed.We prove that ω 1/a1 and ω 1/a2 are not equivalent: On the one hand, since ω is assumed to be non-decreasing, we clearly have ω 1/a1 (t) ≤ ω 1/a2 (t) ⇔ ω(t 1/a1 ) ≤ ω(t 1/a2 ) for any 0 < a 2 < a 1 and all t ≥ 1.But, on the other hand, if the weights are equivalent, i.e. if ω 1/a1 ∼ω 1/a2 , then and so We show that this is equivalent to Finally, we verify that (3.6) implies γ(ω) = +∞ for any weight function: Fix some K > 1 arbitrary and take any γ > 0. (As we are going to see the choice for K can be taken uniformly for all γ > 0.) Then we can find t γ,K := K γ > 0 such that ω(K γ t) ≤ ω(t 2 ) for all t ≥ t γ,K since ω is non-decreasing and so ω( Each ω s is a BMT-weight function; if s > 1 then ω s is non-quasianalytic and for all 0 < s ≤ 1 the weight is quasianalytic.It is straight-forward to see that γ(ω s ) = s.Fix s > 0 and by Theorem 3.3 for each a ∈ (0, s −1 ), and so as < 1, the weight τ a = ω s (t 1/a ) = t 1/(sa) has the desired property.
Note that for each s ≥ 1 we have that (ω sub ) is valid and for each s > 1 even (ω snq ); see (iii) in Remark 2.3.Summarizing, for any s ≥ 1 the weight ω s is (almost) subadditive but, if 0 < s < 1, then there does not exist a weight function τ such that τ is almost subadditive and τ ∼ω s .Compare this also with (iii) in Lemma 3.1.However, since for each such small s the weight is quasianalytic, formally this example does not solve the question from [1, Rem.1.5].
Remark 3.5.The (standard) assumptions on the weight ω in Theorem 3.3 are more general than in [1] and, in particular, this statement applies to quasianalytic weights as well since by (i) in Remark 2.4 in this case γ(ω) ≤ 1 is valid.Moreover, if instead of almost subadditivity we consider the more general condition (3.1), then Theorem 3.3 transfers to this situation with the analogous proof; only the interval I ω changes.If q 0 > 1 2 , then we get I ω = (0, γ(ω) < 1 a and so by (2.3) .
The rest follows as before.If q 0 = 1 2 , then we have I ω = (0, +∞) and any choice for a > 0 is valid.3.2.The non-quasianalytic weight sequence case.From now on we focus on the case ω = ω M for a specific sequence M ∈ R N >0 .The proofs are involved and use technical preparations and therefore we split it into several steps.However, the ideas and constructions can be useful for further applications.

Theorem 3.6. There exist uncountable infinite many pair-wise different (i.e. non-equivalent) weight sequences M ∈ LC and normalized and non-quasianalytic BMT-weight functions ω = ω M such that each of these functions has the following property:
There does not exist a weight function τ such that τ is equivalent to ω M and τ is almost subadditive.Note that ω s in Example 3.4 corresponds, up to equivalence, to ω G s with G s := (j! s ) j∈N .But for 0 < s ≤ 1 the weights ω s , ω G s (and the sequences G s ) are quasianalytic and hence not an appropriate choice.

Proof.
Step I -Comments on sufficient growth and regularity properties.By Lemma 3.1 in order to conclude it suffices to construct a normalized BMT-weight function ω which is ( * ) non-quasianalytic and ( * ) such that 0 < γ(ω) < 1.
In order to achieve this goal the idea is now to consider ω ≡ ω M with M ∈ LC satisfying explicit growth and regularity requirements and to define M explicitly.
Step II -Sufficient growth and regularity properties for the weight sequence.
We want to find M ∈ LC such that (a) M satisfies (2.4),(b) M satisfies (nq), (c) M satisfies (mg), (d) M satisfies Because for such M we get the following properties; for the precise definition of the growth index γ(M ), its comparison with γ(ω M ) and more details we refer to [6, Sect.( * ) Note that in [6] a weight sequence denotes a sequence satisfying all requirements from the class LC except necessarily M 0 ≤ M 1 (see [6,Sect. 3.1]).Moreover, concerning the quotients an index-shift appears; i.e. m p in [6] is corresponding to µ p+1 .However, this does not effect the value of the index and hence the conclusion; see [6,Rem. 3.8] and the comments after [6, Cor.3.12].Altogether, ω M is a (normalized) non-quasianalytic BMT-weight function satisfying γ(ω M ) = γ(M ) < 1 and by Step I this function provides a convenient example.Thus it remains to construct explicitly M ∈ LC such that (a) − (d) holds.
Step III -Definition of the sequence M .
We introduce M in terms of its quotient sequence µ = (µ k ) k∈N (with µ 0 := 1) and so set M j := j k=0 µ k , j ∈ N.For this, first fix an arbitrary real constant A > 2. Second, we consider three auxiliary sequences (of positive integers) (a j ) j∈N>0 , (b j ) j∈N>0 and (d j ) j∈N>0 with a j < b j < a j+1 for all j ∈ N >0 and being defined as follows: For the sequence (d j ) j∈N>0 we assume that it is strictly increasing with d 1 ≥ 1 and satisfying the following growth restriction: .
This choice for d j is possible since A > 2.
For notational reasons we split each interval (a j , b j ], j ∈ N >0 , into d j -many subintervals To be formally precise, we only consider all integers belonging to these intervals.We define µ on a 1 and on the intervals (a j , b j ], j ∈ N >0 , as follows: (3.10) µ a1 := 2(= 2a 1 ), This means that iteratively on I aj ,i , i ≥ 1, the sequence of quotients is given by multiplying the already defined quotients on I aj ,i−1 with the constant A and µ is constant on each I aj ,i .And the values on I aj ,0 are obtained by multiplying µ aj with A. In particular, we get Next, on each (b j , a j+1 ], j ∈ N >0 , we introduce µ as follows: (3.12) Note: When considering in (3.12) the choices i = j − 1 and k = 2 i+1 b j = 2 j b j = a j+1 this gives the value of µ aj+1 and so the definition of M is complete. Since µ k for all k ∈ N >0 gives a product of k-many factors, (3.12) yields and which implies by iteration In particular, the choice i = j − 1 gives Let us now show by induction the following consequence of (3.9) which is needed in the estimates below: The case j = 1 follows by (3.10).Moreover, combining (3.8), (3.11) (3.15) with the induction hypothesis and finally with (3.9) gives for all j ∈ N >0 Step IV -Verification of the desired properties for M .We divide this step into several claims.

Claim b: M satisfies (nq).
On the one hand, by recalling definition (3.10) and by (3.16) we have for all j ∈ N >0 : On the other hand, by (3.8), (3.12) and by combining (3.9) and (3.16) we get for all j ∈ N >0 : .
Summarizing, we obtain and the claim is shown.
We verify (2.4) for the choice Q = 4 and use the equalities and explanations from (i) − (v) in Claim c above.(In fact even Q = 2 is sufficient to treat all cases except (iii).)For all k as considered in (ii) in Claim c we get with Q = 2 the same equality and for k = 1 in case (i), too.The equality in (iv) in Claim c yields Finally, let k be as in case (iii) in Claim c.Then 4k ∈ I bj ,1 and so . Summarizing, even inf j→+∞ µ4j µj ≥ 2 1/4 > 1 is verified and thus the claim is shown.Claim e: M satisfies (3.7).

.
Step V -There exist uncountable infinite many pair-wise different weight sequences/functions satisfying the required properties.
Given a parameter A > 2, then in view of (3.9) for all A ′ > A we can take the same auxiliary sequence (d j ) j∈N>0 and hence also the sequences (a j ) j∈N>0 and (b j ) j∈N>0 remain unchanged; see (3.8).So, when fixing A 0 > 2 and varying A in [A 0 , +∞), then in the construction all claims stay valid and it is easy to compare the quotient sequences since the positions where the growth behavior of this sequence is changing remain invariant.Consider now A ′ > A > 2 arbitrary but from now on fixed, then we write M A and M A ′ for the corresponding sequences and similarly µ A and µ A ′ for the corresponding sequences of quotients.By (3.11) and (3.15) one has Iterating this procedure we immediately get Next, we claim that this property implies that M A and M A ′ are not equivalent: Both sequences satisfy (mg) (see Claim c) and since Consequently, since M A has (mg), we see that the estimate has to be violated: Otherwise M A ′ j ≤ B(M A cj ) 1/c ≤ BC j M A j for some B, C ≥ 1 and all j ∈ N and this contradicts (3.17).(3.18) is precisely [5, (6.4)] for N ≡ M A ′ and M ≡ M A .Therefore, the first part in [5,Lemma 6.5] implies that ω M A (t) = O(ω M A ′ (t)) as t → +∞ fails and so ω M A and ω M A ′ are not equivalent.
3.3.The quasianalytic weight sequence case.The aim is to transfer Theorem 3.6 also to quasianalytic BMT-weight functions; so we illustrate independently that the question whether there does always exist an equivalent almost subadditive BMT-weight function is not depending on the notion of (non-)quasianalyticity.First we comment in more detail on the growth of j → µj j in the example of Theorem 3.6.Given M ∈ LC, then ω M satisfies (ω 5 ) if and only if lim j→+∞ (M j /j!) 1/j = +∞, see e.g.[16,Lemma 2.8] and [7,Lemma 2.4] and the references mentioned in the proofs there.Moreover, lim j→+∞ (M j /j!) 1/j = +∞ if and only if lim j→+∞ µj j = +∞; see e.g.[13, p. 104] and recall that by Stirling's formula j → j! 1/j and j → j grow comparable up to some positive constant.For M ∈ LC it is known that non-quasianalyticity implies lim j→+∞ µj j = +∞; see e.g.[16,Prop. 4.4].The growth of j → If 0 < s < 1, then for ω s in Example 3.4 corresponding to ω G s with G s := (j! s ) j∈N we get that lim j→+∞ (j! s /j!) 1/j = 0.If s = 1, then the limit is equal to 1.
Let now, more generally, the parameter A > 1 be arbitrary (but fixed) and set a j+1 := 2 cj b j for all j ∈ N >0 ; i.e. j in the exponent is replaced by a fourth auxiliary sequence of integers (c j ) ∈N>0 .Note that the conclusion of Claim e in Step IV in Theorem 3.6 stays valid when replacing j by any arbitrary non-decreasing (c j ) j∈N>0 satisfying c j ≥ j.
( * ) On each interval I aj ,i the mapping k → µ k k is clearly non-increasing.( * ) If k = 2 i a j for some j ∈ N >0 and 0 ≤ i ≤ d j − 1, i.e. positions where a jump occurs, then This estimate is valid if A ≥ 2 for all k ∈ N >0 (as in Theorem 3.6) and, if A > 1, for all k sufficiently large depending on chosen A.
Moreover,  Proof.For all cases we use the analogous constructions and definitions as in Theorem 3.6 but apply some technical modifications/generalizations and involve the comments on monotonicity before.
Case a: We take the sequences (a j ) j∈N>0 , (b j ) j∈N>0 and (d j ) j∈N>0 as considered in (3.8) but then let us fix some 1 < A 0 < 2 and write q := 2 A0 .For (d j ) j∈N>0 we assume that the sequence is (strictly) increasing with d 1 ≥ 1.We also set µ a1 = µ 1 := 1 as "initial condition" instead of µ a1 = 2 (which is required to prove (3.16) for j = 1 but in the recent setting this property fails in any case as we are going to see).Next let us show that is non-decreasing on (1, +∞) for any fixed k ∈ N and hence in (3.19) it suffices to consider q = 2 A0 .Then we use induction on j: The case j = 1 gives a1 µa 1 q d 1 −1 q−1 = q d 1 −1 q−1 ≥ 1 which is clear.For the induction step we have where the last desired estimate is equivalent to having 2 j 2 (q dj+1 − 1) ≥ 1 − 1 q d j and this clearly holds.(However, note that for parameters A ≤ 2 the choice (3.9) and hence the estimate (3.16) is impossible.)Then fix a real parameter A with 1 < A ≤ A 0 and follow the proof of Step IV.All parts except Claim b follow by a word-by-word repetition of the given arguments there.Concerning Claim a note that normalization is still valid by µ 1 = µ a1 = 1 and in Claim d we have lim inf j→+∞ µ4j µj ≥ min{2 1/4 , A} > 1; so the value 2 1/4 has to be replaced by A if A < 2 1/4 which is excluded by the choice A > 2 in the proof before.However, (nq) fails: By (3.19) we estimate as follows for all j ∈ N >0 : Since all appearing summands in the series under consideration are non-negative the previous estimate immediately verifies k≥1 1 µ k = +∞ as desired.
Step V follows again by varying the parameter A ∈ (1, A 0 ]; note that the choice of the numbers d j is not depending on given A and so a direct comparison of the quotients is again possible.Finally, we get and so by taking into account the comments just before the statement in this section it follows that lim k→+∞ µ k k = 0 (for any parameter A ∈ (1, A 0 ]).Case b: First let us fix a value A > 2. Second, we introduce four auxiliary sequences (of positive integers) (a j ) j∈N>0 , (b j ) j∈N>0 , (c j ) j∈N>0 and (d j ) j∈N>0 with a j < b j < a j+1 for all j ∈ N >0 and being defined as follows:

All properties from
Step IV except Claim b dealing with (nq) follow analogously as shown in the claims there; for Claim e note that c j ≥ j for all j.Concerning the failure of (nq) by definition, (3.24) (and recall (3.14)) we estimate as follows for all j ∈ N >0 : .
Since all appearing summands in the series under consideration are non-negative the previous estimate immediately verifies k≥1

Example 3 . 4 .
K .Hence it suffices to choose K := C − ǫ with C denoting the constant from (3.6) and ǫ > 0 small and fixed.But this is a contradiction to the basic assumption γ(ω) < +∞.(This step of the proof should be compared with [8, (A.1), p. 2171] and the proof of [8, Lemma A.1, p. 2173]; formally there we have dealt with more specific weight functions but the additional assumptions on the weights are superfluous.)Consider the Gevrey weights ω s (t) := t 1/s , s > 0.

µjj
is crucial to see if the real-analytic functions are (strictly) contained in the corresponding ultradifferentiable class; more precisely lim j→+∞ µj j = +∞ if and only if the real-analytic functions are contained in the function classes under consideration, see e.g.again [13, Prop.2.12, p. 104].
for all j ∈ N >0 and 0 ≤ i ≤ d j − 1. ( * ) By (3.8) and (3.11) we see if A ≥ 2. ( * ) In view of (3.12) we have for all 2 i b j

1 µ 1 j
k = +∞ as desired.Let us see that here we get lim k→+∞ µ k k = +∞ as well: By definition µ b j bj = µa j aj A 2 dj → +∞ as j → +∞, whereas (3.20), (3.21) and (3.23) yield + 1) → +∞ as j → +∞.By the comments made at the beginning of this section we are done.Case c: Here we take the same construction and definition as in Case b but for the sequences (d j ) j∈N>0 and (c j ) j∈N>0 , we assume the following: Again we require (3.22) but instead of (3.23) and (3.24) iteratively we choose (d j ) j∈N>0 and (c j ) j∈N>0 to grow fast enough to ensure (3.25) ∀ j ∈ N >0 : 25) implies lim sup k→+∞ µ k k = +∞ and (3.26) yields, by taking into account (3.20) and (3.Now let us see that (3.26) already implies (3.24) for all j large and this is sufficient to verify the quasianalyticity as in Case b: We have √ 2 cj −1 − 1 ≥ j √ 2 µ b j bj − 1 and clearly, in view of (3.25), log(j+1) ≥ 1 for all j large enough.
see e.g.again [14, Lemma 2.2].Moreover, by log-convexity and normalization we clearly have