On Orlicz classes defined in terms of associated weight functions

N-functions and their growth and regularity properties are crucial in order to introduce and study Orlicz classes and Orlicz spaces. We consider N-functions which are given in terms of so-called associated weight functions. These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence. We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter-) examples. Moreover, we study how for abstractly given N-functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.


Introduction
Let us start by recalling briefly the basic definitions of Orlicz classes and Orlicz spaces, we refer to [11], [17] and to the informative summary presented in [1].For this let F be a so-called N-function, see Definition 3.1 below and [11,  , by revisiting a result by de la Vallée Poussin, it has been shown how such a growth restriction expressed in terms of certain convex functions F is arising naturally.In the literature the aforementioned sets are occasionally also defined in an even more general measure theoretic setting and slightly different assumptions on F are used; e.g. using Young functions in [17], see Remark 3.15 for more details.
In order to study these classes several growth and regularity assumptions for F and F c are considered frequently in the literature.Most prominent are the so-called ∆ 2 , ∆ 3 , ∆ 2 and ∆ ′ condition for F , see e.g.[11, Chapter I, §4- §6], [17, Chapter II], [1, Sect.2.2] and Section 7 in this work.If F c satisfies a "∆-type" property then by convention usually one writes that F has the corresponding "∇-type" condition (and vice versa).
The aim of this paper is to introduce and study N-functions F M which are given in terms of a given sequence M ∈ R N >0 , see Definition 3.14, via the so-called associated weight function ω M (see Section 2.2).Here the sequence is expressing the growth of F M and M is assumed to satisfy mild standard and growth assumptions, see Section 2.1.Recall that ω M is appearing frequently in the theory of classes of ultradifferentiable (and ultraholomorphic) functions defined in terms of weight sequences and it serves also as an example for an abstractly given weight function ω in the sense of Braun-Meise-Taylor, see [2].Consequently, F M contains additional information expressed in the underlying sequence M and the idea is to exploit this fact, to "combine" the ultradifferentiable-type and the Orlicz-type setting and to treat the following questions/problems: ( * ) Study for F M the aforementioned known and important growth properties for abstractly given N-functions in terms of M .When given two N-functions F M , F L expressed in terms of sequences M and L, then study the crucial relation between N-functions (see (3.4)) in terms of a growth comparison between M and L. ( * ) Use this knowledge in order to construct (counter)-examples illustrating the relations and connections between the different growth conditions for N-functions.( * ) Compare the (partially) new growth properties for weight sequences with known conditions appearing in the ultradifferentiable setting.( * ) Check if these properties and conditions can be transferred from given M , L to related constructed sequences, e.g. the point-wise product M • L and the convolution product M ⋆ L (see (2.1)).( * ) Let G be an abstractly given N-function.Is it then possible to associate with G a weight sequence, say M G , and to apply the derived results in order to get information for G itself (via using F M G )? ( * ) When given M and F M study and establish the connection between the notions of the complementary N-function F c M and the dual sequence D (w.r.t.M ) which has been introduced in [5, Def. 2. 1.40, p. 81].This question has served as the main motivation for writing this article.The relevance of D is given by the fact that the so-called orders and Matuszewska indices for M and D are "reflected/inverted" as it has been shown in the main result [5,Thm. 2.1.43].Concerning these notions we refer to [5, Sect.2.1.2],[7] and the citations there for more details and precise definitions.However, it turns out that in the weight sequence setting we cannot expect that the relevant function t → ϕ ωM (t) := ω M (e t ) (see (3.14)) directly is an N-function, see Remark 3.7 for more explanations.One can overcome this technical problem by using the fact that ϕ ωM is the so-called principal part of an N-function F M , see Definition 3.8 and Corollary 3.12.On the other hand we mention that ϕ ωM also allows to compare different used notions for being a weight in the "Orlicz-setting", see Remark 3.15 for more details.
Note that the crucial conditions for M in order to ensure the desired growth properties for F M are partially (slightly) different compared with the known ones used in the ultradifferentiable setting.This is mainly due to the fact that the relevant function under consideration is given by ϕ ωM and not by ω M directly.For example, the prominent ∆ 2 -property for N-functions (see Section 7.1) is also appearing as a known growth condition in the ultradifferentiable weight function setting (abbreviated by (ω 1 ) in this work) but the crucial condition for M is different (see Theorem 7.2 and the comments there).
The paper is structured as follows: In Section 2 all relevant definitions concerning weight sequences and (associated) weight functions are given and in Section 3 we recall and introduce the notions of (associated) N-functions.In Section 4 we focus on the study of the comparison between associated N-functions, see Theorems 4.1, 4.2, 4.4 and Corollary 4.5, and give in Section 4.2 several sufficient conditions on the sequences to ensure equivalence between the associated N-functions.Section 5 is dedicated to the study of the meaning of the associated weight sequence M G when G is an abstractly given N-function, see Theorems 5.2 and 5.5.In Section 6 we introduce and study the complementary N-function F c M (see Theorem 6.4 and Corollary 6.5) and establish the connection between F c M and the dual sequence D, see the main statement Theorem 6.9.Finally, in Section 7 we provide a detailed study of growth and regularity conditions for N-functions in the weight sequence setting, see Theorems 7.2, 7.7, 7.12, 7.18 and Proposition 7.25.Some (counter-)examples and their consequences are mentioned as well, see (7.17) and Corollary 7.20.
M is called log-convex if ∀ j ∈ N >0 : M 2 j ≤ M j−1 M j+1 , equivalently if (µ j ) j is non-decreasing.If M is log-convex and normalized, then both j → M j and j → (M j ) 1/j are non-decreasing and (M j ) 1/j ≤ µ j for all j ∈ N >0 .
M (with M 0 = 1) has condition moderate growth, denoted by (mg), if In [10] this is denoted by (M.2) and also known under the name stability under ultradifferential operators.
For our purpose it is convenient to consider the following set of sequences We see that M ∈ LC if and only if 1 = µ 0 ≤ µ 1 ≤ . . ., lim j→+∞ µ j = +∞ (see e.g.[15, p. 104]) and there is a one-to-one correspondence between M and µ = (µ j ) j by taking (i) The Gevrey-sequences G s , s > 0, given by G s j := j! s .(ii) The sequences M q,n , q, n > 1, given by M q,n j := q j n .If n = 2, then M q,2 is the so-called q-Gevrey-sequence.

Associated weight function. Let
For an abstract introduction of the associated function we refer to [12, Chapitre I], see also [10,Definition 3.1].Note that ω M is here extended to whole R in a symmetric (even) way.
If lim inf j→+∞ (M j ) 1/j > 0, then ω M (t) = 0 for sufficiently small t, since log t j Mj < 0 ⇔ t < (M j ) 1/j holds for all j ∈ N >0 .(In particular, if M j ≥ 1 for all j ∈ N, then ω M is vanishing on [0, 1].)Moreover, under this assumption t → ω M (t) is a continuous non-decreasing 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 .
For given M ∈ LC we define the counting function i.e.Σ M (t) is the maximal positive integer j such that µ j ≤ t (and Σ M (t) = 0 for 0 ≤ t < µ 1 ).It is known that ω M and Σ M are related by the following integral representation formula, see e.g.[12, 1.8.III] and [10, (3.11)]: Consequently, ω M vanishes on [0, µ 1 ], in particular on the unit interval.By definition of ω M the following formula is immediate: In [10,Lemma 3.5] for given M, L ∈ LC it is shown that Finally, if M ∈ LC, then we can compute M by involving ω M as follows, see [12,Chapitre I,1.4,1.8] and also [10,Prop. 3.2]: (2.5) , j ∈ N.
Remark 2.2.Let M ∈ LC be given, we comment on the surjectivity of Σ M .( * ) Obviously Σ M (t) ∈ N for all t ≥ 0 and Σ M is surjective if and only if µ j < µ j+1 for all j ∈ N >0 , i.e. if j → µ j is strictly increasing: In this case we have Σ M (t) = j for all µ j ≤ t < µ j+1 , j ∈ N >0 , and Σ M (t) = 0 for t ∈ [0, µ 1 ).( * ) Note that µ j < µ j+1 for all j does not hold automatically for all sequences belonging to the set LC.However, when given M ∈ LC, then we can always find M ∈ LC such that M and M are equivalent and such that the corresponding sequence of quotients ( µ j ) j≥1 is strictly increasing, see [6,Lemma 3.18].This formal switch allows to avoid technical complications resp.to simplify arguments.
More precisely, in [6, Lemma 3.18] it has been shown that even which clearly implies M ≈ M .We write M ∼ = N if (2.6) holds for the corresponding sequences of quotients µ, ν.
( * ) The convexity and F (0) = 0 imply that see [11, (1.14)].This holds since by convexity we have for all 0 ≤ t ≤ 1 and x, y ≥ 0 and then set y = 0. ( * ) Finally, let us recall see [11, (1.15), (1.16)], and which follows from (II) resp.(III) for f .When given two N-functions (or even arbitrary functions) (ii) We have that (iii) We have that (iv) We have that In particular, the above characterization applies if both F 1 and F 2 are N-functions.
This motivates the following definition, see [11,Chapter I,§3].Definition 3.4.We call two functions F 1 and F 2 equivalent, written In particular, for any N-function F we have that all F k : t → F (kt), k > 0, are equivalent.In [11,Thm. 13.2] it has been shown that Remark 3.5.We comment on relation ∼ c for given N-functions F 1 , F 2 and their corresponding right-derivatives f 1 , f 2 appearing in (3.1): (i) On [11, p. 15] it is mentioned that if Indeed, this implication holds for any non-decreasing functions F 1 , F 2 : [0, +∞) → [0, +∞) such that either F 1 or F 2 is assumed to be convex and normalized: For any 0 < a ≤ 1 we clearly have aF 2 (u) ≤ F 2 (u) and if a > 1, then as in the proof of Lemma 3.3 the estimate (3.2) applied to t := a −1 gives aF 2 (u) ≤ F 2 (au) for all u ≥ 0. The proof for F 1 is analogous.
In particular, (3.8) holds (with b = 1) if F 1 (t) = F 2 (t) for all t large.(ii) Moreover, if lim t→+∞ F i (t) = +∞, i = 1, 2, then (3.8) holds with b = 1 if Note: f 1 c f 2 implies the above relation and since f 2 is non-decreasing, equivalently we can use in (3.11) On the one hand, for all x ≥ t 1 with t 1 denoting the value in (3.11) ) is verified with C := 2F 1 (t 1 ) and K := k.Conversely, let x > 0 with x ≥ t 0 /2, t 0 from (3.4), and estimate as follows: The sufficiency of (3.12) for having the inclusion Note that this relation ∼ is precisely [11, (8.6)] and it is also the crucial one for the characterization of inclusions (resp.equalities) of classes in the ultradifferentiable weight function setting, see [15,Sect. 5].
Moreover, let us write In view of (3.2), for given N-functions F 1 , F 2 we have that F 1 F 2 implies F 2 c F 1 .For this implication one only requires that either F 1 or F 2 is normalized and convex, see the proof of (ii) ⇒ (i) in Lemma 3.3.Consequently, if N-functions F 1 and F 2 are related by F 1 ∼ F 2 , then they are also equivalent.
For the sake of completeness let us summarize more relations between (N-)functions mentioned in [17, Sect.2.2, Def.1]: and write F 1 c F 2 for this relation which is, of course, stronger than F 1 c F 2 .When taking K := 1 it also implies F 2 F 1 .Lemma 3.3 transfers to this relation and using this analogously to (iv) in Remark 3.5 one can prove that The choice ǫ := 1 implies again F 2 F 1 .
( * ) F 2 is increasing more rapidly than F 1 , if If F 2 ⊳ F 1 is valid, then F 2 is increasing more rapidly than F 1 by the uniform choice δ := 1 for all ǫ > 0.
Remark 3.7.However, requirement (II) cannot be achieved for Σ M • exp for any M ∈ LC: If M 1 = M 0 (= 1), and so Finally remark that, if M is log-convex with lim j→+∞ (M j ) 1/j = +∞ but such that normalization fails, then 0 < µ 1 < 1 and so Σ M (e t ) ≥ 1 for any t ≥ 0. Thus also in this case the first requirement in (II) is violated.This failure is related to the fact that the first property in (3.3) and ϕω M (t) t > 0 for all t > 0 are not satisfied automatically for ϕ ωM , see the proofs and arguments in [11, Chapter I, §1, 5, p. [8][9].Thus ϕ ωM is formally not an N-function according to Definition 3.1.
In order to overcome this technical problem we recall the following notion, see [11, Chapter I, §3, 3, p. 16]: We have the following result, see [11,Thm. 3.3] and the proof there: = +∞.Then there exists an N-function F such that Q is the principal part of F .More precisely, we even get that with f denoting the function appearing in (3.1) of F and q denoting the non-decreasing and rightcontinuous function appearing in the representation see [11, (1.10)].Here a ≥ 0 is such that Q(a) = 0 and we have t 0 > a.
Proposition 3.10.Let Q be a convex function such that lim t→+∞ Q(t) t = +∞ and let F be the N-function according to Theorem 3.9.Then we get cf. (3.9).This relation implies lim t→+∞ F (t) Proof.More generally, when for given functions for all t large then we have In particular, when applying these results to Q = ϕ ωM we get the following consequence: Corollary 3.12.Let M ∈ LC be given.Then there exists an N-function F M such that ϕ ωM is the principal part of F M and so This implies ϕ ωM ∼ F M and hence also ϕ ωM ∼ c F M .Moreover, if f M denotes the function appearing in the representation (3.1) of F M , then we even get In view of this equality we call Σ M • exp the principal part of f M (see [11, p. 18]).
We close this section by commenting on the relation between ϕ ωM and other notions of defining functions in the Orlicz setting.
Remark 3.15.As seen above, for any given M ∈ LC we cannot expect that ϕ ωM is formally an N-function.On the other hand ϕ ωM can be used to illustrate the differences between appearing definitions for Orlicz classes in the literature.In [14] an exhaustive study is provided and the different notions and conditions for the defining functions are compared, see also the literature citations there. (

Comparison between associated N-functions
The goal of this Section is to give a connection resp.comparison between the growth relation for weight sequences, crucially appearing in the theory of ultradifferentiable and ultraholomorphic functions, and the previously defined relations c and for (associated) N-functions.
4.1.Main statements.The first main result establishes a characterization of relation F M c F L in terms of a growth comparison between M and L. However, the characterization is not given via but expressed in terms of the corresponding sequences of quotients µ = (µ j ) j∈N and λ = (λ j ) j∈N .In explicit applications and for constructing weight sequences M it is often convenient to start with µ.Note that when involving µ we get automatically growth conditions for the counting function Σ M as well.
Theorem 4.1.Let M, L ∈ LC be given.Then the following are equivalent: (a) We have that The proof shows that we can take A = B = k in (a) and (b) and the result becomes trivial for The analogous characterization for F M c F L is obtained when replacing (4.1) by (3.13) and so (4.2) by the condition be minimal to ensure µ j0 ≥ t 0 (note that lim j→+∞ µ j = +∞).For any t ≥ µ j0 we have µ j ≤ t < µ j+1 for some j ∈ N >0 , j ≥ j 0 .Then by assumption j A and so t k ≥ λ ⌈j/A⌉ (note that Σ L (t k ) ∈ N).When taking t := µ j we get (4.2) with B := A and the same k > 0 for all j ≥ j 0 with µ j < µ j+1 .If j ≥ j 0 with µ j = µ j+1 , then and so t k ≥ µ k j ≥ λ ⌈j/B⌉ which gives Σ M (t) = j and ⌈j/B⌉ ≤ Σ L (t k ).Thus (4.1) follows when j ≤ Aj/B ≤ A⌈j/B⌉ is ensured.So we can take A := B, the same k > 0 and t 0 := µ j0 .
(a) ⇔ (c) First, (4.1) precisely means that Σ M (e s ) ≤ AΣ L (e sk ) for some A ≥ 1, k > 0 and all sufficiently large s.Thus the desired equivalence holds by taking into account the representation (3.14), following the estimates in (iv) in Remark 3.5 with f 1 being replaced by Σ M • exp and f 2 by Σ L • exp ((4.1) is precisely (3.11) for these choices), and finally using Corollary 3.12.
(a) ⇔ (d) This holds by (3.19).We continue with the following result providing a complete characterization for the relation between associated N-functions.
(ii) ⇔ (iii) First note that ϕ ωM ϕ ωL precisely means for some h > 1 and all t ≥ 0. Hence since ϕ ωL is non-decreasing and s + log(h) ≤ sh ⇔ log(h) ≤ s(h − 1) for all s large enough.This verifies ϕ ωM c ϕ ωL and Corollary 3.12 implies again F M c F L .(iii) Consequently, equivalent weight sequences yield equivalent associated N-functions and, in particular, this applies to the situation described in Remark 2.2.(iv) However, the converse implication in (iii) is not true in general.For this recall that by (2.4) we get Then, by following the arguments in (i) in Remark 3.5, we see that hence by Corollary 3.12 also F M ∼ c F M ℓ for any ℓ > 0. However, for ℓ = 1 the sequences M and M ℓ are not equivalent since lim j→+∞ (M j ) 1/j = +∞.(v) In particular, (iv) applies to the Gevrey-sequence The next result provides a second characterization for F M c F L in terms of a growth relation between M and L directly.
Theorem 4.4.Let M, L ∈ LC be given.Consider the following assertions: The associated N-functions F M and F L (see Corollary 3.12) satisfy The sequences M and L satisfy Then (i) ⇒ (ii) ⇒ (iii) holds.If either M or L has in addition (mg), then also (iii) ⇒ (ii).
By (iv) in Remark 4.3 the implication (i) ⇒ (ii) cannot be reversed in general.
and so by Corollary 3.12 (take w.l.o.g.
We set s := e t and hence this is equivalent to Recall that M, L ∈ LC implies (by normalization) ω M (s) = ω L (s) = 0 for all s ∈ [0, 1] and so the previous estimate holds for any s ≥ 0 (with the same constants).Thus (2.5) yields for all j ∈ N: and we are done when taking c := K and A := e D .
(iii) ⇒ (ii) Assume that (mg) holds for M .By assumption (4.5) and an iterated application of (mg) we get thus by definition of associated weight functions ω M c (t) ≤ ω L (Bt) + log(A) for all t ≥ 0. Consequently, since ϕ ωL (t) → +∞ and by (3.2) we get for all t large enough hence ϕ ω M c c ϕ ωL .By (4.4) we have that ϕ ω M c ∼ c ϕ ωM and so ϕ ωM c ϕ ωL is verified.Corollary 3.12 yields the conclusion.If L has in addition (mg), then by (4.5) and iterating (mg) first we get Thus (4.6) is verified for all k ∈ N with k = cj, j ∈ N arbitrary.For the remaining cases let k with cj < k < cj + c for some j ∈ N and then, since both M and L are also non-decreasing, we get for some C ≥ 1 Summarizing, (4.6) is verified for all k ∈ N and the rest follows as above.
Thus we have the following characterization: The proof of (ii) ⇒ (iii) in Theorem 4.4 transfers to relation We finish by comparing the characterizing conditions for M and L in the previous results.
( * ) On the other hand, recall that by (3.2) we get that and the last implication can be reversed provided that either M or L has in addition (mg).
4.2.On sufficiency conditions.We want to find some sufficient conditions for given sequences M , L in order to ensure F M ∼ c F L .More precisely, the aim is to ensure an asymptotic behavior of the counting functions near infinity (see (4.8)) which might be important for applications in the ultradifferentiable setting as well.
Lemma 4.7.Let M, L ∈ LC be given.Assume that Then we get F M ∼ c F L (equivalently ϕ ωM ∼ c ϕ ωL ). Note: Proof.For all t ≥ µ j0 we find j ≥ j 0 such that µ j ≤ t < µ j+1 .Then Σ M (t) = j and by (4.9) we get cj ≤ Σ L (t) < cj + d j .Thus and since lim j→+∞ dj j = 0 we get (4.8) with b := 1 c .Note that it is enough to require the existence of a sequence d j for j such that µ j < µ j+1 : If j ≥ j 0 and µ j = µ j+1 = • • • = µ j+ℓ < µ j+ℓ+1 for some ℓ ∈ N >0 , then by (4.9) we get and so for t with µ j+ℓ ≤ t < µ j+ℓ+1 we have Σ M (t) = j + ℓ and cj + cℓ = c(j + ℓ) ≤ Σ L (t) < cj + cℓ + d j+ℓ = c(j + ℓ) + d j+ℓ yielding the same estimate as above.(On the other hand, by switching to an equivalent sequence, we can assume µ j < µ j+1 for all j ∈ N >0 , see Remark 2.2.) We prove now the following characterization for (4.8).Lemma 4.9.Let M, L ∈ LC be given such that µ j < µ j+1 for all j ∈ N >0 .Then the following are equivalent: with d j ∈ N >0 such that lim j→+∞ dj j = 0.The proof shows that in all assertions the value b is the same.
Proof.(i) ⇒ (ii) By assumption, thus we find j ǫ ∈ N >0 large such that µ jǫ ≥ t ǫ so that for all t with µ j ≤ t < µ j+1 for some j ≥ j ǫ we get The first estimate in (4.13) yields Let now c 1 < b be arbitrary but fixed and take ǫ 1 small enough to ensure Note that the choice of ǫ 1 is only depending on chosen c 1 but not on j.Thus Σ L (t) < ⌈ j c1 ⌉ and so t < λ ⌈ j c 1 ⌉ .We take t := µ j and get µ j < λ ⌈ j c 1 ⌉ , i.e. the second half of (4.11) for all j ≥ j ǫ1 (since µ j < µ j+1 for all j) with d := 0.
Similarly, the second estimate in (4.13) gives ⌊ j b+ǫ ⌋ ≤ j b+ǫ < Σ L (t) and so t ≥ λ ⌊ j b+ǫ ⌋ .We let c 2 > b be arbitrary but fixed and choose ǫ 2 > 0 small enough to ensure Take t := µ j to get the first half of (4.11) for all j ≥ j ǫ2 with d := 1. Summarizing, we are done when taking j 0 := max{j ǫ1 , j ǫ2 } and note that d can be taken uniformly and not depending on chosen c 1 , c 2 .
(iii) ⇒ (i) Let b ∈ (0, +∞) be given and take arbitrary (close) 0 < c 1 < b < c 2 but from now on fixed.Let t ≥ µ j0 and so µ j ≤ t < µ j+1 for some j ≥ j 0 .Then (4.12) gives Summarizing, for all such t we obtain Then note that

From N-functions to associated weight sequences
The aim is to reverse the construction from Section 3.2; i.e. we start with an abstractly given Nfunction G, associate to it a sequence M G and study then the relation between G and the associated N-function Let F be an N-function and first we introduce By the properties of F we have that ω F belongs to the class W 0 : Concerning (ω 4 ) note that ϕ ωF (t) = ω F (e t ) = F (t) for all t ≥ 0, concerning (ω 3 ) we remark that ωF (t) log(t) = ωF (e s ) s = F (s) s for all t > 1 ⇔ s > 0 and the last quotient tends to +∞ as s → +∞, see (3.3).Note: When given ω ∈ W 0 , then one can put (5.2) F ω satisfies all properties to be formally an N-function (see Definition 3.1) except necessarily the first part in (3.3) and also F ω (t) > 0 for all t = 0 is not clear.Thus the set of all N -functions does not coincide with the class W 0 .However, one can overcome this technical problem for F ω by passing to an equivalent (associated) N-function when taking into account Theorem 3.9 analogously as it has been done before with ϕ ωM .
We consider the so-called Legendre-Fenchel-Young-conjugate of ϕ ωF which is given by , j ∈ N.
The second equality holds by normalization and this formula should be compared with (2.5).Hence for any j ∈ N we get = exp sup hence by (5.1) In order to avoid confusion let from now in this context G be the given N-function, M G the sequence defined in (5.4) and F M G the associated N-function from Corollary 3.12 (applied to the sequence M G ).
Definition 5.1.Let G be an N-function.Then the sequence M G is called the associated weight sequence.
Theorem 5.2.Let G be an N-functions and let F M G be the N-function associated with the sequence M G .Then we get and this implies both Proof.Corollary 3.12 applied to M G and (5.6) yield and These estimates prove the first two parts in (5.7) and the last one there follows by (3.2), so by convexity and normalization of F M G , see also the estimate in (i) in Remark 3.5 applied to a := 2. The desired relations follow now by (5.7), Lemma 3.3 and Corollary 3.12.The next result gives the (expected) equivalence when starting with a weight sequence.Proposition 5.3.Let L ∈ LC be given and let F L be the associated N-function.Then we get This estimate implies M FL ≈L, Proof.We apply the previous constructions to the associated N-function F L .For all t ≥ 1 we have ω FL (t) = F L (log(t)) and so by (3.18) Because L ∈ LC we have ω L (t) = 0 for all t ∈ [0, 1] (normalization) and ω FL (t) = 0 holds for all t ∈ [0, 1] by (5.1).Thus (5.8) is valid for any t ≥ 0. By combining (5.8) with (5.4) and (2.5) we arrive at ∃ C, D ≥ 1 ∀ j ∈ N : e −D L j ≤ M FL j ≤ e C L j , hence the conclusion.This relation clearly implies M FL ≈L and so, by Theorem 4.4 (recall also Remark 4.3) the equivalence F L ∼ c F M F L .Moreover, (4.3) is verified with c = 1 (pair-wise) and hence Theorem 4.2 gives that F L ∼ F M F L as well.We continue with the following observations: ( * ) Theorem 5.2 suggests that for abstractly given N-functions G it is important to have information about F M G resp. ϕ ω M G and to study the associated weight sequence M G .In particular, in view of (3.14) and (3.19) the knowledge about the counting function Σ M G is useful and this amounts to study the sequence of quotients µ G . ( * ) On the other hand, when desired growth behaviours are expressed in terms of µ G , it is an advantage not to compute first M G via given G and then the corresponding quotient sequence µ G but to come up with a property for G directly.( * ) This can be achieved by relating µ G directly to G as follows: Put and so (5.4) takes alternatively the following form: This should be compared with [19,Sect. 6], [22, Sect.2.6, 2.7].Indeed, M G coincides with the crucial associated sequence M vG in [19], [22].By the assumptions on G we get that v G is a (normalized and convex) weight in the notion of [19], [22], i.e. in the setting of weighted spaces of entire functions, see also the literature citations in these papers.( * ) Let now (t j ) j∈N be the sequence such that t j ≥ 0 is denoting a/the global maximum point of the mapping t → t j v G (t). ( * ) The advantage when considering v G is that in [19, (6.5)] we have derived the following relation between the sequence of quotients µ G (set µ G 0 := 1) and (t j ) j∈N : (5.10) ∀ j ∈ N : t j ≤ µ G j+1 ≤ t j+1 .Note: For j = 0 we have put t 0 := 1 and so equality with µ G 0 .In fact for j = 0 we can choose any t ∈ [0, 1] as t 0 since v G is non-increasing and normalized, and by the convention 0 0 := 1.So t j ≥ 1 for any j ∈ N because j → t j is non-decreasing and since lim j→+∞ µ G j = +∞ we also get t j → +∞.For concrete given G (belonging to C 1 ) the concrete computation for the values of t j might be not too difficult.(5.12) Using Σ G we introduce Note that, analogously to the explanations for ϕ ωM given in Remark 3.7 in Section 3.2 we have that ϕ G is formally not an N-function since t j ≥ 1 for all j ≥ 1 and so by definition We summarize the whole information in the final statement of this section: Theorem 5.5.Let G be an N-function, M G the associated weight sequence, F M G the associated N-function and ϕ G given by (5.13).Then Proof.The first two parts are shown in Theorem 5.2.The last one follows by Proposition 5.4: We use (5.12) and the representations (5.13) and (3.14) in order to get: The second last estimate in (5.14) holds since lim t→+∞ ϕ G (t) t = +∞ and the last one by (3.2) applied to ϕ G .Both requirements on ϕ G are valid by the representation (5.13), see Remark 3.11.Then (5.14) implies both ϕ ω M G ∼ c ϕ G and ϕ ω M G ∼ ϕ G .Theorem 5.5 shows that, up to equivalence, the whole information concerning growth and regularity properties of G is already expressed by involving a certain associated weight sequence M G and its related/associated N-function F M G .

On complementary N-functions and dual sequences
We give a connection between the so-called complementary N-functions F c and the dual sequences D in the weight sequence setting.Note that in the theory of N-functions one naturally has the pair (F, F c ) and thus also (F M , F c M ).Similarly for each M ∈ LC one can naturally assign the dual sequence D ∈ LC and we show that F c M is closely related to D via an integral representation formula using the counting-function log •Σ D .
6.1.Complementary N-functions in the weight sequence setting.Let F be an N-function given by the representation (3.1) with right-derivative f .First, introduce (6.1) In [11, Chapter I, (2.9), p. 13] it is mentioned that and this formula can be considered as an equivalent definition for F c .Remark 6.2.We comment on the comparison of growth relations c and between (associated) N-functions F , G and their complementary N-functions F c , G c . ( see also the proof of [15,Lemma 5.16].This relation implies (3.7) and so Lemma 3.3 yields that F c c G c .When one can choose C = 1, then also G c F c follows but in general this implication is not clear; see also Lemma 7.1.
By combining (5.3) and (6.3) we get ϕ * ωF = F c , and since M F j = exp(ϕ * ωF (j)), for this see (5.4) and the computations below this equation, it follows that ∀ j ∈ N : M F j = exp(F c (j)).Let now M ∈ LC be given, then write F c M and f c M for the functions considered before w.r.t. to the associated N-function F M and the corresponding right-derivative f M .Moreover, in view of (6.1), let us introduce (6.4) Γ M (s) := sup{t ≥ 0 : Σ M (e t ) ≤ s}, s ≥ 0.
Finally we set (6.5) By (3.19) it follows that (6.6) , because both functions are non-decreasing and f M (t) = Σ M (e t ) for all t ≥ t 0 , with t 0 the value appearing in (3.19).So (6.6) is valid for all s ≥ s 0 := f M (t 0 ).Using this identity we can prove the analogous result of Corollary 3.12 for the functions F c M and ϕ c ωM .Proposition 6.3.Let M ∈ LC be given.Then we get Proof.We use (6.6) and the representations (6.2) and (6.5) (recall also the arguments in the proof of Proposition 3.10).For all s ≥ s 0 (with s 0 from (6.6)) we get hence ϕ c ωM (s) ≤ F c M (s) + C for all s ≥ 0 when choosing C := ϕ c ωM (s 0 ).Similarly, for all s ≥ s 0 hence F c M (s) ≤ ϕ c ωM (s) + D for all s ≥ 0 when choosing D := F c M (s 0 ).On the other hand, formula (6.3) applied to ϕ ωM yields (6.7) the Legendre-Fenchel-Young-conjugate of ϕ ωM .By combining (3.18), (6.3) applied to F M and (6.7) we get see the estimate in (ii) in Remark 6.2.Hence F c M ∼ c ϕ * ωM and F c M ∼ ϕ * ωM hold.When combining this with Proposition 6.3 we arrive at the following result: Theorem 6.4.Let M ∈ LC be given.Then Finally, we apply Theorem 6.4 to the associated weight sequence M G .Corollary 6.5.Let G be an N-function, let M G ∈ LC be the associated sequence defined via (5.4) and ϕ G be given by (5.13).Then Proof.Concerning ∼ c , the first and the second equivalence hold by Theorem 6.4, the third and fourth one by taking into account (5.7), Theorem 5.5 (see (5.14)) and (i) in Remark 6.2.Here (ϕ G ) c is given in terms of (6.3).For ∼ we use the same results, however are not clear in general: In order to conclude we want to apply (6.3) and so in the relations only an additive constant should appear, see (ii) in Remark 6.2.However, in both (5.7) and (5.14) we also have the multiplicative constant 2.

Complementary associated N-functions versus dual sequences.
In this section the aim is to see that Γ M in (6.4) and crucially appearing in the representation (6.5) is closely connected to the counting function Σ D , with D denoting the so-called dual sequence of M .Moreover, (6.4) should be compared with the formula for the so-called bidual sequence of M in [5, Def.2.1.41,Thm.
If µ j = µ j+1 , then we distinguish: Either there exist ℓ, c ) by the same reasons as in Case I before.
Finally, if this choice is not possible, then this precisely means Summarizing, we have shown the following statement: Proposition 6.6.Let M ∈ LC be given with quotient sequence (µ j ) j .
If not, then δ j = d ∈ N ≥2 for the minimal integer j with j ≥ µ 1 + 1 (case II).This occurs if for this minimal integer already j ≥ µ d + 1 is valid.Note that d is finite since lim j→+∞ µ j = +∞ and let d be such that µ d + 1 ≤ j < µ d+1 + 1 for j minimal such that j ≥ µ 1 + 1.
( * ) Consider 1 ≤ t < 2 and distinguish: In the first case Σ D (t) is the largest integer j with µ 1 + 1 ≤ j < µ 2 + 1 and in the second case Σ D (t) coincides with the largest integer j with j < µ 1 + 1.Since µ 1 + 1 ≥ 2 in both cases the existence of such an integer j is ensured.( * ) More generally, in case I if n ≤ t < n + 1 with n ∈ N >0 , then Σ D (t) is the largest integer j such that j < µ n+1 + 1. ( * ) In case II, if n ≤ t < n + 1 for some n ∈ N >0 with n + 1 ≤ d, then Σ D (t) is (still) the largest integer j such that j < µ 1 + 1 since for the minimal integer j ≥ µ 1 + 1 we have The last equality holds since µ d < µ d+1 by the choice of d.

and:
(a) If for the minimal integer j such that j ≥ µ 1 + 1 one has When combining Propositions 6.6 and 6.7 we are able to establish a connection between the counting functions Γ M and Σ D .
Proof.First, by Proposition 6.7 we get that Σ D (t) coincides with the maximal integer j < µ n+1 +1 for all t ≥ 0 having n ≤ t < n + 1 and all n ≥ d with d ∈ N >0 such that for the minimal integer j ≥ µ 1 + 1 we get µ d + 1 ≤ j < µ d+1 + 1.In particular, for all such n we get Then recall that by Proposition 6.6 we get Γ M (t) = log(µ n+1 ) for all t such that n ≤ t < n + 1 with n ∈ N such that µ n+1 > 1.In particular, as mentioned before, this holds for all n ≥ d.So let n ∈ N >0 be such that n ≥ d.Take t with n ≤ t < n + 1, then one has Σ D (t) < µ n+1 + 1 and showing the second part of (6.10).
On the other hand for all such t we get Σ D (t) ≥ µ n+1 = exp(Γ M (t)), in fact even Σ D (t) ≥ ⌈µ n+1 ⌉ holds, and which proves the first estimate in (6.10).
Using the counting function Σ D we set (6.11) Γ D is non-decreasing and right-continuous and tending to infinity.Recall that Σ D (s) ∈ N >0 for all s ≥ 1 and this technical modification is unavoidable: In (6.11) we cannot consider log(Σ D (s)) directly for all s ≥ 0 because Σ D (s) = 0 for 0 ≤ s < 1 by definition.Then in view of (6.10) we get (6.12) Note that F ΓD is formally not an N-function by analogous reasons as in Remark 3.7: We have Γ D (s) = 0 for (at least) all 0 ≤ s < 1, see the proof of Proposition 6.7.Now we are able to prove the main statement of this section.
Theorem 6.9.We have the following equivalences: (i) Let M ∈ LC be given and D its dual sequence, then (ii) Let G be an N-function, M G ∈ LC the associated sequence defined via (5.4) and D G the corresponding dual sequence.Finally, let ϕ G be given by (5.13).Then Proof.(i) In view of Theorem 6.4 only F ΓD ∼ c ϕ c ωM resp.F ΓD ∼ ϕ c ωM has to be verified.This follows by using (6.12) and the representations (6.11) and (6.5).Note that these representations imply lim t→+∞ (ii) By Corollary 6.5 it suffices to verify Both relations follow by applying the first part to M G and D G .Concerning ∼ note that here both functions are given directly by the representations (6.11) resp.(6.5), so we are not involving formula (6.3) and since in (6.12)only additive constants appear the problem described in the proof of Corollary 6.5 (see (ii) in Remark 6.2) does not occur.
We remark that F ΓD does not coincide with F D directly but nevertheless the dual sequence D can be used to get an alternative (equivalent) representation and description of the complementary N-function F c M .

Growth and regularity conditions for associated N-functions
In the theory of Orlicz classes and Orlicz spaces several conditions for (abstractly given) N-functions appear frequently.The aim of this section is to study these known growth and regularity assumptions in the weight sequence setting in terms of given M .
We remark that all appearing conditions are naturally preserved under ∼ c for (associated) Nfunctions, see the given citations in the forthcoming sections resp.Remark 7.6.However, by inspecting the proofs one can see that this fact also holds for a wider class of functions, e.g. when having normalization, convexity, being non-decreasing and tending to infinity (in particular for ϕ ωM ).Therefore recall that the technical failure of ϕ ωM to be formally an N-function occurs at the point 0 (see Remark 3.7) whereas for all crucial conditions under consideration large values t ≥ t 0 > 0 are relevant.
Of course, it makes also sense to consider the conditions in this section for arbitrary functions F : [0, +∞) → [0, +∞).
Proof.(i) ⇔ (ii) This is clear by Corollary 3.12 since F M ∼ c ϕ ωM .
(ii) ⇔ (iii) This is immediate.( * ) According to [3, Lemma 6.5] we know that any M ∈ LC cannot satisfy (mg) and (7.4) simultaneously.In particular, the Gevrey sequences G s := (j! s ) j∈N , s > 0, are violating (7.4).( * ) Consider the sequences M q,n := (q j n ) j∈N with q, n > 1.Then (7.Proof.By assumption M and L both have (7.4) and so it is immediate that M •L has this property as well.
Concerning the convolution note that we get ω M⋆L = ω M + ω L and so ω M⋆L has (7.3) because both ω M and ω L have this property.Finally we apply Theorem 7.2 to M = M G and get the following.(vi) ω G (equivalently ω M G ) satisfies (7.3).
Proof.Everything is immediate from Theorem 7.2 applied to M G : (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) holds by Theorem 5.5 and since ∆ 2 is preserved under equivalence.Concerning (v) and (vi) note that condition (7.2) resp.(7.3) holds equivalently for ω G and ω M G (by (5.5) and since these conditions are preserved under ∼).
When iterating ∇ 2 we find (since ℓ > 1) We choose d ∈ N >0 such that ℓ d ≥ k 2 and n ∈ N >0 such that 2 n−1 ≥ ℓ d .Then we estimate for all t ≥ max{t 0 , t 1 } as follows: i.e. ∇ 2 for G holds with ℓ n+d and for all t ≥ t 2 := max{t 0 , t 1 }.(ii) Similarly, we show that ∇ 2 is preserved under relation ∼: Assume that F has ∇ 2 and let G be another N-function such that F ∼ G.So and then we take n ∈ N >0 such that k 2 ≤ 2 n−1 .Iterating n-times property ∇ 2 gives for all t sufficiently large that Since G is an N-function we get k 2 G(ℓ n t) ≤ G(k 2 ℓ n t) (recall (3.2)) and 2k 2 ℓ n ≤ (2ℓ) n by the choice of n.Consequently, 2k 2 ℓ n G(t) ≤ G(k 2 ℓ n t) is verified for all sufficiently large t, i.e. ∇ 2 for G with k 2 ℓ n .
In the weight sequence setting we are interested in having (7.5) for F M resp.for ϕ ωM .Since ∇ 2 is preserved under equivalence, via Corollary 3.12 this condition transfers into The aim is to characterize now (7.7) in terms of M .
Theorem 7.7.Let M ∈ LC be given.Then the following are equivalent: (i) The associated N-function F M satisfies the ∇ 2 -condition.
(iv) ω M satisfies The sequence M satisfies (7.9) The proof shows that in (iv) and (v) we can take the same choice for ℓ and the correspondence between C and A is given by A = e 2ℓC .Consequently, if (7.8) holds for ℓ > 1 then also for all ℓ ′ ≥ ℓ (even with the same choice for C).
(iv) ⇒ (v) By using (2.5) we get for all j ∈ N: j , so (7.9) is verified with A := e 2ℓC and the same ℓ.
Note that (7.13) is not well-related to (2.5) and in general it seems to be difficult to obtain a characterization for ∆ 2 in terms of M by using this formula.However, we give the following characterization of ∆ 2 in the weight sequence setting.First we have to prove a technical result: Lemma 7.11.Let M ∈ LC be given.Then the following are equivalent: i.e. [11, (6.9)] for p = Σ M • exp (this is the ∆ 2 -condition for Σ M • exp).(ii) The sequence of quotients µ satisfies The proof shows the correspondence A = K.Proof.(i) ⇒ (ii) Write s := e t and so Σ M (s) 2 ≤ Σ M (s K ) is satisfied for all s ≥ s 0 := e t0 .Let j 0 ∈ N >0 be minimal such that µ j0 ≥ s 0 .Let j ≥ j 0 be such that µ j < µ j+1 and take s with µ j ≤ s < µ j+1 .Then j 2 = Σ M (s) 2 ≤ Σ M (s K ) follows which implies µ j 2 ≤ s K .In particular, when taking s := µ j we have shown (7.15) with A := K and all j ≥ j 0 such that µ j < µ j+1 .If j ≥ j 0 with µ j = • • • = µ j+ℓ < µ j+ℓ+1 for some ℓ ∈ N >0 , then following the previous step we get µ (j+ℓ) 2 ≤ µ K i for all j ≤ i ≤ j + ℓ.Since (j + ℓ) 2 ≥ i 2 for all such indices i and since by log-convexity j → µ j is non-decreasing we are done for all j ≥ j 0 .
(ii) ⇒ (i) Let s ≥ µ j0 and so µ j ≤ s < µ j+1 for some j ≥ j 0 .Then Σ M (s) = j and s A ≥ µ A j ≥ µ j 2 which implies Σ M (s A ) ≥ j 2 = Σ M (s) 2 .Thus (7.14) is shown with K := A and t 0 := log(µ j0 ).Using this we get the following characterization.Theorem 7.12.Let M ∈ LC be given.Then the following are equivalent: (i) Using the second part of this and since x for all x ≥ 0 we estimate as follows for all x > 0 with x ≥ t 0 : If x ≥ max{1, t 0 }, then this estimate and the first part in (7.16) applied to t := 2kx(> t 0 ) give i.e. the ∆ 2 -condition for f M is verified with t 1 := max{1, t 0 } and k 1 := 2k 2 .By (3.19) this is equivalent to the fact that Σ M • exp satisfies the ∆ 2 -condition; i.e.Recall that in order to verify (7.15) for µ G for abstractly given N-functions G the formula (5.10) can be used.Proof.First, Theorem 7.12 applied to M G yields (i) ⇔ (iii).By Theorem 5.5 we have that F M G , G and ϕ G are equivalent and since ∆ 2 is preserved under equivalence we are done.
Proof.(i) ⇒ (ii) This is contained in the proof of [11,Thm. 6.1].First, for any N-function F we get for all t ≥ 1 that In fact this holds for any non-decreasing and non-negative F .On the other hand, by using ∆ 3 for some k > 1 and all t sufficiently large one has s)ds ≤ F (t)t ≤ F (kt).
(ii) ⇒ (i) By the equivalence we get F (t) ≤ F (kt) for some k > 1 and all t sufficiently large.Thus for all such large t we estimate by thus xf (x) ≤ 2kf (2kx) for all x ≥ t 0 /2, with k > 1, t 0 > 0 from (7.19).We use this estimate and apply it also to y := 2kx(> t 0 ) in order to get for all x ≥ t 1 := max{t 0 , 2k}.Thus ∆ 3 for f is verified with t 1 and 1 := 4k 2 .
(iv) ⇒ (iii) This follows by replacing in (ii) ⇒ (i) the function F by f and F by F .We apply this characterization to the weight sequence setting.Proof.(i) ⇔ (ii) ⇔ (iii) follows by Lemma 7.17 applied to F M , by (3.19) and the fact that ∆ 3 is preserved under equivalence, see Corollary 3.12.
By [11,Lemma 5.1]  i.e. this corresponds to the weight ω s (t) := max{0, log(t) s } and hence to the sequences M q,n such that 1 s + 1 n = 1, see the proof of Corollary 7.20.A direct check of (7.24) seems to be quite technical resp.hardly possible since due to the multiplicative nature this estimate is not well-related w.r.t.formula (2.5).The same comment applies to ∇ ′ and to the counting function Σ M .In [11,Thm. 5.1] a sufficiency criterion for ∆ ′ is shown: Theorem 7.23.Let F (t) = |t| 0 f (s)ds be a given N-function (recall the representation (3.1)).Then F satisfies the ∆ ′ -condition provided that f has the following growth property which we abbreviate by (∆ ′ f ) from now on: There exists some t 0 > 1 such that for every fixed t ≥ t 0 the function h f given by h f (s) := f (st) f (s) is not increasing on [t 0 , +∞).
In the weight sequence setting this result takes the following form: Proof.In view of (3.19), we get (∆ ′ ΣM •exp ) if and only if (∆ ′ fM ) when enlarging t 0 sufficiently if necessary.Then Theorem 7.23 applied to f = f M and F = F M yields the conclusion.However, we show now that in general (∆ ′ ΣM •exp ) fails in the weight sequence setting.Proposition 7.25.Let M ∈ LC be given such that 1 ≤ µ 1 < µ 2 < . . ., i.e. the sequence of quotients is strictly increasing (see Remark 2.2).Then (∆ ′ ΣM •exp ) (resp.equivalently (∆ ′ fM )) is violated.

Theorem 4 . 2 .
Let M, L ∈ LC be given.Then the following are equivalent: (i) The associated N-functions satisfy F M F L .(ii) The functions ϕ ωM and ϕ ωL satisfy ϕ ωM ϕ ωL .(iii) The sequences M and L are related by

Corollary 4 . 5 .
Let M, L ∈ LC be given.Assume that either M or L has in addition (mg), then the following are equivalent: (i) The associated N-functions F M and F L are equivalent.(ii) The functions ϕ ωM and ϕ ωL are equivalent.(iii) The sequences M and L satisfy

Proposition 5 . 4 .
11)Σ G (t) := |{j ∈ N >0 : t j ≤ t}|, t ≥ 0, and Σ G : [0, +∞) → N is a right-continuous non-decreasing function with Σ G (t) = 0 for all 0 ≤ t < 1 and Σ G (t) → +∞ as t → +∞.By taking into account (5.10) and the definition of Σ M G and Σ G we get: Let G be an N-function and let M G be the associated weight sequence.Then the counting functions Σ G and Σ M G are related by

4 )
is valid (with A = B = 1 and k such that k ≥ 2 1/(n−1) ) as it is shown in [3, Example 6.6 (1)].( * ) Combining Lemma 7.1, Theorem 7.2 and Remark 4.6 yields the following: Let M, L ∈ LC be given and assume that either M or L has (mg) and that either M or L has (7.4).Then (4.3) ⇐⇒ F M F L ⇐⇒ F L c F M ⇐⇒ (4.5), i.e. in (4.7) the second implication can also be reversed.Corollary 7.4.Let M, L ∈ LC be given.Assume that both F M and F L satisfy the ∆ 2 -condition.Then F M•L and F M⋆L satisfy the ∆ 2 -condition, too.

Corollary 7 . 5 .
Let G be an N-function.Let ω G be the function from (5.1), M G ∈ LC the associated sequence defined via (5.4) and F M G the associated N-function.Then the following are equivalent: (i) G satisfies the ∆ 2 -condition.(ii) The associated N-function F M G satisfies the ∆ 2 -condition.(iii) ϕ ω M G satisfies the ∆ 2 -condition.(iv) The function ϕ G (see (5.13)) satisfies the ∆ 2 -condition.(v) ω G (equivalently ω M G ) satisfies (7.2).
(7.14)  for Σ M .Corollary 7.13.Let G be an N-function and let M G be the associated weight sequence (see (5.4)).Then the following are equivalent: (i) The sequence of quotients µ G satisfies (7.15).(ii) G satisfies the ∆ 2 -condition.(iii) The associated N-function F M G satisfies the ∆ 2 -condition.(iv) The function ϕ G (see (5.13)) satisfies the ∆ 2 -condition.

Lemma 7 . 17 . 0 F
Let F be an N-function.Then the following are equivalent: (i) F satisfies the ∆ 3 -condition.(ii) We have F ∼ c F with (7.20) F (t) := |t| (s)ds, t ∈ R. (iii) The function f from the representation (3.1) satisfies the ∆ 3 -condition.(iv)We have that F ∼ c f .
Example 2.1.Frequently we will consider the following important examples belonging to the class LC: < +∞.Sequences M and L are called equivalent, denoted by M ≈ L, if M L and L M .
Remark 3.2.In [17, Sect.1.3,Thm.1, Cor. 2; Sect.2.1] an analogous integral representation for N-resp.evenYoungfunctions has been obtained with the integrand f ("density") being nondecreasing and left-continuous.Thus the authors are working with the left-derivative of F .Since we are focusing on the weight sequence case, see(3.14)inSection 3.2, we have to involve the counting function Σ M from (2.2) which is right-continuous by the very definition and so we prefer to work within the above setting.
M ω L implies ϕ ωM ϕ ωL as well.Consequently, the desired equivalence (ii) ⇔ (iii) follows by the first part in [4, Lemma 6.5].Next let us gather some more immediate consequences concerning relation c .Remark 4.3.Let M, L ∈ LC be given.In view of Corollary 3.12 we get F M c F L .(ii)More generally, if L M , then by definition and since M 0 1 by normalization of M and L we get ω L (t) ≤ Kω M (t)+ D for all t ≥ 0, i.e. ω L (t) = O(ω M (t)) and so ω M ω L .Similarly, ω (i) If L ≤ M , then by definition ω M (t) ≤ ω L (t) for all t ≥ 0 and so ϕ ωM (t) ≤ ϕ ωL (t) for all t ≥ 0.
property (4.8) holds trivially with b = 1.Lemma 4.8.Let M, L ∈ LC be given.Assume that Sect. 2.2, Thm. 2 (a)] in fact this is an equivalence.By [17, Sect.2.2, Thm. 2 (b)] the corresponding statements holds w.r.t.relation c as well.In particular, two N-functions are equivalent if and only if their complementary N-functions so are.
Let M ∈ LC be given, we analyze now Γ M : 7.1.The ∆ 2 -condition.The most prominent property is the so-called ∆ 2 -condition, see e.g.[11, Chapter I, §4, p. 23] and [17, Sect.2.3, Def. 1, p. 22], which reads as follows: [20,a 7.1.Let F 1 and F 2 be two N-functions such that eitherF 1 or F 2 has ∆ 2 .Then F 1 c F 2 if and only if F 2 F 1 .In fact, in order to conclude, we only require that either F 1 or F 2 is normalized and convex and that either F 1 or F 2 has ∆ 2 .Proof.By (3.2) we have that F 1 F 2 implies F 2 c F 1 (see Remark 3.6).The converse implication holds by an iterated application of ∆ 2 for either F 1 or F 2 .In the weight sequence setting in view of Corollary 3.12 we require (7.1) (i.e.(ω 1 )) not for ω M directly but for ϕ ωM .Note that in[20, Thm.3.1]we have already given a characterization of (ω 1 ) for ω M in terms of M but ∆ 2 for ϕ ωM (resp.equivalentlyforF M ) does precisely mean The associated N-function F M (see Corollary 3.12) satisfies the ∆ 2 -condition.(ii)ϕωM satisfies the ∆ 2 -condition.(iii)ω M satisfies (7.2).