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+arXiv:2301.11594v1  [math.FA]  27 Jan 2023
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT
+FUNCTIONS
+GERHARD SCHINDL
+Abstract. N-functions and their growth and regularity properties are crucial in order to in-
+troduce 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.
+1. Introduction
+Let us start by recalling briefly the basic definitions of Orlicz classes and Orlicz spaces, we refer to
+[11] 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, Chapter 1, §1, 3], [1, Def. 2.1]. Moreover, let Ω be a bounded and
+closed set in Rd and consider on Ω the usual Lebesgue measure. Then the Orlicz class is given by
+LF (Ω) := {u : Ω → R, measurable :
+�
+Ω
+F(u(x))dx < +∞},
+and the Orlicz space by
+L∗
+F (Ω) := {u : Ω → R, measurable :
+�
+Ω
+u(x)v(x)dx < +∞,
+∀ v ∈ LF c(Ω)},
+with F c denoting the so-called complementary N-function which is again an N-function, see Section
+6.1 and [11, Chapter 1, §2], [1, Def. 2.2]. If we do not need to specify the set Ω we omit it and only
+write LF resp. L∗
+F . We mention that sometimes in the literature slightly different assumptions on
+F are used, see Remark 3.14.
+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], [1, Sect. 2.2] and Section 7. 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 FM which are given in terms of a
+given sequence M ∈ RN
+>0, see Definition 3.13, via the so-called associated weight function ωM (see
+Date: January 30, 2023.
+2020 Mathematics Subject Classification. 26A12, 26A48, 26A51, 46E30.
+Key words and phrases. Orlicz classes and Orlicz spaces, N-functions, associated weight functions, weight se-
+quences, dual sequence.
+G. Schindl is supported by FWF-Project P33417-N.
+1
+
+2
+G. SCHINDL
+Section 2.2). Here the sequence is expressing the growth of FM and M is assumed to satisfy mild
+standard and growth assumptions, see Section 2.1. Recall that ωM 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, FM 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 FM the aforementioned known and important growth properties for abstractly
+given N-functions in terms of M. When given two N-functions FM, FL 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 FMG)?
+(∗) When given M and FM study and establish the connection between the notions of the com-
+plementary 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 �→ ϕωM (t) := ωM(et) (see (3.12)) directly is an N-function, see Remark 3.6 for more explanations.
+One can overcome this technical problem by using the fact that ϕωM is the so-called principal part
+of an N-function FM, see Definition 3.7 and Corollary 3.11. On the other hand we mention that
+ϕωM also allows to compare different used notions for being a weight in the ”Orlicz-setting”, see
+Remark 3.14 for more details.
+Note that the crucial conditions for M in order to ensure the desired growth properties for FM 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 ϕωM and
+not by ωM 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 (ω1) 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
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+3
+N-functions, see Theorems 4.1, 4.3 and Corollary 4.4, 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 7.24. Some (counter)-examples
+and their consequences are mentioned as well, see (7.16) and Corollary 7.20.
+2. Weights and conditions
+2.1. Weight sequences. We write N = {0, 1, 2, . . .} and N>0 := {1, 2, . . .}.
+Given a sequence M = (Mj)j ∈ RN
+>0 we also use µ = (µj)j defined by µj :=
+Mj
+Mj−1 , µ0 := 1, and
+analogously for all other appearing sequences. M is called normalized if 1 = M0 ≤ M1 holds true.
+For any ℓ > 0 we put M ℓ := (M ℓ
+j )j∈N, i.e. the ℓ-th power, and write M · L = (MjLj)j∈N. Finally,
+let us introduce the convolved sequence M ⋆ L by
+(2.1)
+M ⋆ Lj := min
+0≤k≤j MkLj−k,
+j ∈ N,
+see [10, (3.15)].
+M is called log-convex if
+∀ j ∈ N>0 : M 2
+j ≤ Mj−1Mj+1,
+equivalently if (µj)j is non-decreasing. If M is log-convex and normalized, then both j �→ Mj and
+j �→ (Mj)1/j are non-decreasing and (Mj)1/j ≤ µj for all j ∈ N>0.
+M (with M0 = 1) has condition moderate growth, denoted by (mg), if
+∃ C ≥ 1 ∀ j, k ∈ N : Mj+k ≤ Cj+kMjMk.
+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
+LC := {M ∈ RN
+>0 : M is normalized, log-convex,
+lim
+j→+∞(Mj)1/j = +∞}.
+We see that M ∈ LC if and only if 1 = µ0 ≤ µ1 ≤ . . . , limj→+∞ µj = +∞ (see e.g. [15, p. 104])
+and there is a one-to-one correspondence between M and µ = (µj)j by taking Mj := �j
+k=0 µk. If
+M, L ∈ LC, then M · L, M ⋆ L ∈ LC (for the convolution see [10, Lemma 3.5]).
+Let M, L ∈ RN
+>0 be given, then write M ≤ L if Mj ≤ Lj for all j ∈ N and M ⪯ L if
+supj∈N>0
+�
+Mj
+Lj
+�1/j
+< +∞. Sequences M and L are called equivalent, denoted by M ≈ L, if M⪯L
+and L⪯M.
+Example 2.1. Frequently we will consider the following important examples belonging to the class
+LC:
+(i) The Gevrey-sequences Gs, s > 0, given by Gs
+j := j!s.
+(ii) The sequences M q,n, q, n > 1, given by M q,n
+j
+:= qjn. If n = 2, then M q,2 is the so-called
+q-Gevrey-sequence.
+
+4
+G. SCHINDL
+2.2. Associated weight function. Let M ∈ RN
+>0 (with M0 = 1), then the associated function
+ωM : R → R ∪ {+∞} is defined by
+ωM(t) := sup
+j∈N
+log
+�|t|j
+Mj
+�
+for t ∈ R, t ̸= 0,
+ωM(0) := 0.
+For an abstract introduction of the associated function we refer to [12, Chapitre I], see also [10,
+Definition 3.1]. Note that ωM is here extended to whole R in a symmetric (even) way.
+If lim infj→+∞(Mj)1/j > 0, then ωM(t) = 0 for sufficiently small t, since log
+�
+tj
+Mj
+�
+< 0 ⇔ t <
+(Mj)1/j holds for all j ∈ N>0. (In particular, if Mj ≥ 1 for all j ∈ N, then ωM is vanishing on
+[0, 1].) Moreover, under this assumption t �→ ωM(t) is a continuous non-decreasing function, which
+is convex in the variable log(t) and tends faster to infinity than any log(tj), j ≥ 1, as t → +∞.
+limj→+∞(Mj)1/j = +∞ implies that ωM(t) < +∞ for each finite t which shall be considered as a
+basic assumption for defining ωM.
+For given M ∈ LC we define the counting function ΣM : [0, +∞) → N by
+(2.2)
+ΣM(t) := |{j ∈ N>0 :
+µj ≤ t}|,
+i.e. ΣM(t) is the maximal positive integer j satisfying µj ≤ t (and ΣM(t) = 0 for 0 ≤ t < µ1). It is
+known that ωM and ΣM are related by the following integral representation formula, see e.g. [12,
+1.8. III] and [10, (3.11)]:
+(2.3)
+ωM(t) =
+� t
+0
+ΣM(u)
+u
+du =
+� t
+µ1
+ΣM(u)
+u
+du.
+Consequently, ωM vanishes on [0, µ1], in particular on the unit interval.
+By definition of ωM the following formula is immediate:
+(2.4)
+∀ ℓ > 0 ∀ t ≥ 0 :
+ℓωM(t1/ℓ) = ωMℓ(t).
+In [10, Lemma 3.5] for given M, L ∈ LC it is shown that
+∀ t ≥ 0 :
+ΣM⋆L(t) = ΣM(t) + ΣL(t),
+which implies by (2.3)
+∀ t ≥ 0 :
+ωM⋆L(t) = ωM(t) + ωL(t).
+Finally, if M ∈ LC, then we can compute M by involving ωM as follows, see [12, Chapitre I, 1.4,
+1.8] and also [10, Prop. 3.2]:
+(2.5)
+Mj = sup
+t≥0
+tj
+exp(ωM(t)),
+j ∈ N.
+Remark 2.2. Let M ∈ LC be given, we comment on the surjectivity of ΣM.
+(∗) Obviously ΣM(t) ∈ N for all t ≥ 0 and ΣM is surjective if and only if µj < µj+1 for all
+j ∈ N>0, i.e. if j �→ µj is strictly increasing: In this case we have ΣM(t) = j for all
+µj ≤ t < µj+1, j ∈ N>0, and ΣM(t) = 0 for t ∈ [0, µ1).
+(∗) Note that µj < µj+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 (�µj)j≥1 is strictly
+increasing, see [6, Lemma 3.18]. This formal switch allows to avoid technical complications
+resp. to simplify arguments.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+5
+More precisely, in [6, Lemma 3.18] it has been shown that even
+(2.6)
+0 < inf
+j∈N
+µj
+�µj
+≤ sup
+j∈N
+µj
+�µj
+< +∞,
+which clearly implies M≈�
+M. We write M ∼= N if (2.6) holds for the corresponding se-
+quences of quotients µ, ν.
+2.3. Growth properties for abstractly given weight functions. Let ω : [0, +∞) → [0, +∞),
+we introduce the following growth and regularity conditions
+(ω1) :
+ω(2t) = O(ω(t))
+t → +∞,
+(ω3) :
+log(t) = o(ω(t))
+t → +∞,
+(ω4) :
+ϕω : t �→ ω(et) is a convex function on R.
+These conditions are named after [18]. (ω1), (ω3) and (ω4) are standard assumptions in the theory
+of ultradifferentiable functions defined by so-called Braun-Meise-Taylor weight functions ω, see [2].
+We write that ω has (ω0) if ω is continuous, non-decreasing, ω(t) = 0 for all t ∈ [0, 1] (normalization)
+and limt→+∞ ω(t) = +∞. Finally let us put
+W0 := {ω : [0, ∞) → [0, ∞) : ω has (ω0), (ω3), (ω4)}.
+If M ∈ LC then ωM ∈ W0, see e.g. [9, Lemma 3.1] and the citations there.
+3. N-functions in the weight sequence setting
+3.1. Basic definitions and abstractly given N-functions. We revisit the basic definitions
+from [11, Chapter I, §1, 3] and [1, Def. 2.1]. Consider f : [0, +∞) → [0, +∞) with the following
+properties:
+(I) f is right-continuous and non-decreasing;
+(II) f(0) = 0 and f(t) > 0 for all t > 0;
+(III) limt→+∞ f(t) = +∞.
+Then we give the following definition, see e.g. [11, Chapter I, §1, 3, p. 6].
+Definition 3.1. Let f : [0, +∞) → [0, +∞) be satisfying (I), (II) and (III).
+The function
+F : R → [0, +∞) defined by
+(3.1)
+F(x) :=
+� |x|
+0
+f(t)dt,
+is called an N-function.
+Every N-function F satisfies the following properties, see [11, Chapter I, §1, 4, p. 7]:
+(∗) F(0) = 0 (normalization) and F(x) > 0 for all x ̸= 0,
+(∗) F is even, non-decreasing, continuous, and convex.
+(∗) The convexity and F(0) = 0 imply that
+(3.2)
+∀ 0 ≤ t ≤ 1 ∀ u ≥ 0 :
+F(tu) ≤ tF(u),
+see [11, (1.14)]. This holds since by convexity we have F(tx+(1−t)y) ≤ tF(x)+(1−t)F(y)
+for all 0 ≤ t ≤ 1 and x, y ≥ 0 and then set y = 0.
+
+6
+G. SCHINDL
+(∗) Finally, let us recall
+(3.3)
+lim
+t→0
+F(t)
+t
+= 0,
+lim
+t→+∞
+F(t)
+t
+= +∞,
+see [11, (1.15), (1.16)], and which follows from (II) resp. (III) for f.
+When given two N-functions (or even arbitrary functions) F1, F2 : [0, +∞) → [0, +∞), then write
+F1 ⪯c F2 if
+(3.4)
+∃ K > 0 ∃ t0 > 0 ∀ t ≥ t0 :
+F1(t) ≤ F2(Kt).
+If either F1 or F2 is non-decreasing then w.l.o.g. we can restrict to K ∈ N>0 and this relation
+is clearly reflexive and transitive. In [11], F1 and F2 are called comparable, if either F1⪯cF2 or
+F2⪯cF1.
+For this relation we are gathering several equivalent reformulations.
+Lemma 3.2. Let F1, F2 : [0, +∞) → [0, +∞) be non-decreasing. Assume that either F1 or F2 is
+normalized, convex and tending to infinity as t → +∞. Then the following are equivalent:
+(i) F1⪯cF2 holds true.
+(ii) We have that
+(3.5)
+∃ K, K1 ≥ 1 ∃ t0 > 0 ∀ t ≥ t0 :
+F1(t) ≤ K1F2(Kt).
+(iii) We have that
+(3.6)
+∃ C > 0 ∃ K ≥ 1 ∀ t ≥ 0 :
+F1(t) ≤ F2(Kt) + C.
+(iv) We have that
+(3.7)
+∃ K, K1 ≥ 1 ∃ C > 0 ∀ t ≥ 0 :
+F1(t) ≤ K1F2(Kt) + C.
+In particular, the above characterization applies if both F1 and F2 are N-functions.
+Proof. (i) ⇒ (ii) is trivial and (ii) ⇒ (i) follows by (3.2): If F2 is normalized and convex, then
+when given K1 > 1 we put t := K−1
+1
+in (3.2) and hence K1F2(u) ≤ F2(K1u) for all u ≥ 0.
+Similarly, if F1 is normalized and convex, then the assumption gives K−1
+1 F1(t) ≤ F2(Kt) and so
+F1(t) ≤ K−1
+1 F1(K1t) ≤ F2(KK1t) for all t ≥ t0 holds true which shows F1⪯cF2 when choosing
+K2 := KK1.
+(i) ⇒ (iii) is clear, since F2(t) ≥ 0 and F1 is non-decreasing take e.g. C := F1(t0).
+(iii) ⇒ (ii) When given C ≥ 1 then we have F2(Kt) + C ≤ 2F2(Kt) for all sufficiently large t if
+limt→+∞ F2(t) = +∞. Thus (3.5) is verified with K1 := 2 and the same K. If limt→+∞ F1(t) = +∞,
+then by (3.6) also limt→+∞ F2(t) = +∞ and the rest follows as before.
+(iii) ⇒ (iv) is trivial and (iv) ⇒ (iii) holds as (ii) ⇒ (i).
+□
+This motivates the following definition, see [11, Chapter I, §3].
+Definition 3.3. We call two functions F1 and F2 equivalent, written F1 ∼c F2, if F1⪯cF2 and
+F2⪯cF1.
+In particular, for any N-function F we have that all Fk : t �→ F(kt), k > 0, are equivalent.
+In [11, Thm. 13.2] it has been shown that F1∼cF2 if and only if L∗
+F1 = L∗
+F2.
+Remark 3.4. We comment on relation ∼c for given N-functions F1, F2 and their corresponding
+right-derivatives f1, f2 appearing in (3.1):
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+7
+(i) On [11, p. 15] it is mentioned that if
+(3.8)
+∃ b ∈ (0, +∞) :
+lim
+t→+∞
+F1(t)
+F2(t) = b,
+then F1∼cF2 holds true. Indeed, this implication holds for any non-decreasing functions
+F1, F2 : [0, +∞) → [0, +∞) such that either F1 or F2 is assumed to be convex and normal-
+ized:
+For any 0 < a ≤ 1 we clearly have aF2(u) ≤ F2(u) and if a > 1, then as in the proof of
+Lemma 3.2 the estimate (3.2) applied to t := a−1 gives aF2(u) ≤ F2(au) for all u ≥ 0. The
+proof for F1 is analogous.
+In particular, (3.8) holds true (with b = 1) if F1(t) = F2(t) for all t large.
+(ii) Moreover, if limt→+∞ Fi(t) = +∞, i = 1, 2, then (3.8) holds with b = 1 if
+(3.9)
+∃ C, D ≥ 1 ∀ t ≥ 0 :
+F1(t) − C ≤ F2(t) ≤ F1(t) + D.
+(iii) In [11, Lemma 3.1] it has been shown that f1⪯cf2 implies F1⪯cF2 and in [11, Lemma 3.2]
+that
+(3.10)
+∃ b ∈ (0, +∞) :
+lim
+t→+∞
+f1(t)
+f2(t) = b
+implies F1∼cF2.
+Remark 3.5. In the theory of Orlicz classes also the following slightly different relation between
+functions is considered:
+(∗) Let F1, F2 be N-functions. In [11, Thm. 8.1], see also [1, Thm. 3.3], it has been shown that
+LF1(Ω) ⊆ LF2(Ω) (as sets) if and only if
+(3.11)
+∃ t0 > 0 ∃ K ≥ 1 ∀ t ≥ t0 :
+F2(t) ≤ KF1(t),
+i.e.
+F2(t) = O(F1(t)) as t → +∞.
+The sufficiency of (3.11) for having the inclusion
+LF1(Ω) ⊆ LF2(Ω) is clear.
+(∗) Given F1, F2 : [0, +∞) → [0, +∞) we write F1 ⪯ F2 if (3.11) holds true and F1 ∼ F2 if
+F1 ⪯ F2 and F2 ⪯ F1.
+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].
+(∗) In view of (3.2), for given N-functions F1, F2 we have that F1 ⪯ F2 implies F2⪯cF1. For
+this implication one only requires that either F1 or F2 is normalized and convex, see the
+proof of (ii) ⇒ (i) in Lemma 3.2. Consequently, if N-functions F1 and F2 are related by
+F1 ∼ F2, then they are also equivalent.
+3.2. From weight sequences to associated N-functions. Let M ∈ LC be given. When rewrit-
+ing (2.3) we obtain for all t ≥ 0 (note that µ1 ≥ 1):
+(3.12)
+ϕωM (t) = ωM(et) =
+� et
+µ1
+ΣM(s)
+s
+ds =
+� t
+log(µ1)
+ΣM(eu)
+s
+sdu =
+� t
+log(µ1)
+ΣM(eu)du =
+� t
+0
+ΣM(eu)du,
+since ΣM(eu) = 0 for 0 ≤ u < log(µ1). This formula should be compared with [11, Thm. 1.1,
+(1.10)]. ΣM ◦ exp is right-continuous, non-decreasing and clearly limt→+∞ ΣM(et) = +∞.
+Recall that ωM ∈ W0 and so ϕωM is convex, t �→
+ϕωM (t)
+t
+is non-decreasing with limt→+∞
+ϕωM (t)
+t
+=
++∞ and finally ϕωM (0) = 0 is valid, see e.g. [2, Rem. 1.3, Lemma 1.5] and also [11, p. 7].
+
+8
+G. SCHINDL
+Remark 3.6. However, requirement (II) cannot be achieved for ΣM ◦ exp for any M ∈ LC: If
+M1 = M0(= 1), and so µ1 = 1, then (2.2) yields ΣM(e0) = ΣM(µ1) ≥ 1 ̸= 0. If M1 > M0 ⇔ µ1 >
+1, then ΣM(et) = 0 for all 0 ≤ t < log(µ1).
+Finally remark that, if M is log-convex with limj→+∞(Mj)1/j = +∞ but such that normalization
+fails, then 0 < µ1 < 1 and so ΣM(et) ≥ 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
+ϕωM (t)
+t
+> 0 for all t > 0 are
+not satisfied automatically for ϕωM , see the proofs and arguments in [11, Chapter I, §1, 5, p. 8-9].
+Thus ϕωM 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]:
+Definition 3.7. A convex function Q is called the principal part of an N-function F if
+∃ t0 > 0 ∀ t ≥ t0 :
+Q(t) = F(t).
+We have the following result, see [11, Thm. 3.3] and the proof there:
+Theorem 3.8. Let Q : [0, +∞) → [0, +∞) be a convex function such that limt→+∞
+Q(t)
+t
+= +∞.
+Then there exists an N-function F such that Q is the principal part of F.
+More precisely, we even get that
+(3.13)
+∃ t0 > 0 ∀ t ≥ t0 :
+f(t) = q(t),
+with f denoting the function appearing in (3.1) of F and q denoting the non-decreasing and right-
+continuous function appearing in the representation
+(3.14)
+Q(t) =
+� t
+a
+q(s)ds,
+see [11, (1.10)]. Here a ≥ 0 is such that Q(a) = 0 and we have t0 > a.
+Proposition 3.9. Let Q be a convex function such that limt→+∞
+Q(t)
+t
+= +∞ and let F be the
+N-function according to Theorem 3.8. Then we get
+(3.15)
+∃ C, D ≥ 1 ∀ t ≥ 0 :
+Q(t) − C ≤ F(t) ≤ Q(t) + D,
+cf. (3.9). This relation implies limt→+∞
+F (t)
+Q(t) = 1 and so both F∼cQ and F ∼ Q holds true.
+Proof. More generally, when given functions F1, F2 : [0, +∞) → [0, +∞) are satisfying F1(t) =
+F2(t) for all t large then we have F1(t) ≤ F2(t) + C, F2(t) ≤ F1(t) + D for all t ≥ 0 with
+C := max{F1(t) : 0 ≤ t ≤ t0} and D := max{F2(t) : 0 ≤ t ≤ t0}.
+Hence (3.15) follows by Theorem 3.8 (recall Definition 3.7).
+Note that Q is convex but normalization for Q (i.e. a = 0 in (3.14)) is not guaranteed in general.
+□
+Remark 3.10. Conversely, each convex function Q : [0, +∞) → [0, +∞) admitting the represen-
+tation (3.14) for a non-decreasing and right-continuous function q with q(t) → +∞ satisfies also
+limt→+∞
+Q(t)
+t
+= +∞. This holds since for all t ≥ a (see the proof of (1.16) in [11, Chapter I, §1,p.
+7]):
+Q(2t) =
+� 2t
+a
+q(s)ds ≥
+� 2t
+t
+q(s)ds ≥ q(t)t.
+In particular, when applying these results to Q = ϕωM we get the following consequence:
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+9
+Corollary 3.11. Let M ∈ LC be given. Then there exists an N-function FM such that ϕωM is the
+principal part of FM and so
+(3.16)
+∃ C, D ≥ 1 ∀ t ≥ 0 :
+ϕωM (t) − C ≤ FM(t) ≤ ϕωM (t) + D.
+This implies ϕωM ∼ FM and hence also ϕωM ∼cFM.
+Moreover, if fM denotes the function appearing in the representation (3.1) of FM, then we even get
+(3.17)
+∃ t0 > 0 ∀ t ≥ t0 :
+fM(t) = ΣM(et).
+In view of this equality we call ΣM ◦ exp the principal part of fM (see [11, p. 18]).
+Proof. We can apply Theorem 3.8 to Q ≡ ϕωM because limt→+∞
+ϕωM (t)
+t
+= lims→+∞
+ωM(s)
+log(s) = +∞
+by (ω3) (recall [9, Lemma 3.1] and the citations there). In fact (ω3) for ωM is precisely [11, (3.6)]
+for Q = ϕωM .
+(3.16) follows by Proposition 3.9, and (3.17) holds by taking into account the representation (3.12).
+Note that by normalization of M we get ϕωM (0) = ωM(1) = 0 and so in (3.14) we have a = 0 and
+q = ΣM ◦ exp.
+□
+The following is an immediate consequence of (3.16):
+Corollary 3.12. Let M, L ∈ LC be given. Then FM⪯cFL if and only if ϕωM ⪯cϕωL and FM ⪯ FL
+if and only if ϕωM ⪯ ϕωL.
+Definition 3.13. Let M ∈ LC be given. Then the N-function FM from Corollary 3.11 is called the
+associated N-function.
+We close this section by commenting on the relation between ϕωM and other notions of defining
+functions in the Orlicz setting.
+Remark 3.14. As seen above, for any given M ∈ LC we cannot expect that ϕωM is formally an
+N-function. On the other hand ϕωM 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.
+(∗) ϕωM coincides with an N-function (with FM) for sufficiently large values.
+(∗) For any M ∈ LC the function ϕωM is always a Young function, see [14, Def. 1.4]: ϕωM is
+convex, satisfies ϕωM (0) = ωM(1) = 0 by normalization and ϕωM (t) → +∞ as t → +∞.
+(∗) ϕωM is a strong Young function (see [14, Def. 1.7]) if and only if µ1 = 1: Continuity is
+clear and ϕωM (t) > 0 for all t > 0 follows if and only if ΣM(et) > 0 for all t ≥ 0, see
+(3.12). This is clearly equivalent to µ1 = 1.
+(∗) Finally, ϕωM is always an Orlicz function (see [14, Def. 1.9]), since ϕωM is never identically
+zero or infinity (follows again by (3.12)).
+(∗) Consequently, ϕωM provides (counter)-examples for the first two strict implications in [14,
+Cor. 2.7], see [14, Cor. 2.8]: Each N-function is a strong Young function and each strong
+Young function is an Orlicz function but each implication cannot be reversed in general;
+one can take M ∈ LC with µ1 = 1 for the first and M ∈ LC with µ1 > 1 for the second
+part.
+
+10
+G. SCHINDL
+4. 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. We start with some immediate (known) observations. First, we have the
+following result providing a complete characterization for the relation ⪯.
+Theorem 4.1. Let M, L ∈ LC be given. Then the following are equivalent:
+(i) The associated N-functions satisfy FM ⪯ FL.
+(ii) The functions ϕωM and ϕωL satisfy ϕωM ⪯ ϕωL.
+(iii) The sequences M and L are related by
+(4.1)
+∃ A ≥ 1 ∃ c ∈ N>0 ∀ j ∈ N :
+Mj ≤ A(Lcj)1/c.
+Proof. (i) ⇔ (ii) holds by Corollary 3.12.
+(ii) ⇔ (iii) First note that ϕωM ⪯ ϕωL precisely means
+∃ K, D ≥ 1 ∀ t ≥ 0 :
+ωL(et) = ϕωL(t) ≤ KϕωM(t) + D = KωM(et) + D.
+Since ωM(t) = ωL(t) = 0 for all 0 ≤ t ≤ 1 by normalization of M and L we get ωL(t) ≤ KωM(t)+D
+for all t ≥ 0, i.e. ωL(t) = O(ωM(t)) and so ωM ⪯ ωL. Similarly, ωM ⪯ ωL implies ϕωM ⪯ ϕωL as
+well.
+Consequently, the desired equivalence (ii) ⇔ (iii) follows by the first part in [4, Lemma 6.5].
+□
+Now let us proceed with relation ⪯c and gather some immediate consequences.
+Remark 4.2. Let M, L ∈ LC be given.
+(i) If L ≤ M, then by definition ωM(t) ≤ ωL(t) for all t ≥ 0 and so ϕωM (t) ≤ ϕωL(t) for all
+t ≥ 0. In view of Corollary 3.11 we get FM⪯cFL.
+(ii) More generally, if L⪯M, then by definition and since M0 = N0 = 1 we have ωM(t) ≤ ωL(ht)
+for some h > 1 and all t ≥ 0. Hence
+∃ sh > 0 ∀ s ≥ sh :
+ϕωM (s) = ωM(es) ≤ ωL(es+log(h)) = ϕωL(s + log(h)) ≤ ϕωL(sh),
+since ϕωL is non-decreasing and s + log(h) ≤ sh ⇔ log(h) ≤ s(h − 1) for all s large enough.
+This verifies ϕωM ⪯cϕωL and Corollary 3.11 implies again FM⪯cFL.
+(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
+∀ ℓ > 0 ∀ t ≥ 0 :
+ϕωMℓ(t) = ωMℓ(et) = ℓωM(et/ℓ) = ℓϕωM(t/ℓ).
+Then, by following the arguments in (i) in Remark 3.4, we see that
+(4.2)
+∀ ℓ > 0 :
+ϕωM ∼cϕωMℓ ,
+hence by Corollary 3.11 also FM∼cFMℓ for any ℓ > 0. However, for ℓ ̸= 1 the sequences
+M and M ℓ are not equivalent since limj→+∞(Mj)1/j = +∞.
+(v) In particular, (iv) applies to the Gevrey-sequence M ≡ Gs, s > 0. It is known that ωGs ∼
+t �→ t1/s, i.e. ϕωGs ∼ t �→ et/s (see the proof of Theorem 4.1). So (4.2) is verified but
+clearly Gs is not equivalent to Gs′ if s ̸= s′.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+11
+The aim of the next result is to characterize FM⪯cFL in terms of a growth relation between M and
+L.
+Theorem 4.3. Let M, L ∈ LC be given. Consider the following assertions:
+(i) L⪯M is valid.
+(ii) The associated N-functions FM and FL (see Corollary 3.11) satisfy
+FM⪯cFL,
+equivalently ϕωM ⪯cϕωL is valid.
+(iii) The sequences M and L satisfy
+(4.3)
+∃ c ∈ N>0 ∃ A ≥ 1 ∀ j ∈ N :
+Lj ≤ AMcj.
+Then (i) ⇒ (ii) ⇒ (iii) holds true. If either M or L has in addition (mg), then also (iii) ⇒ (ii).
+By (iv) in Remark 4.2 the implication (i) ⇒ (ii) cannot be reversed in general.
+Proof. (i) ⇒ (ii) This is shown in (i), (ii) in Remark 4.2.
+(ii) ⇒ (iii) By Lemma 3.2 relation FM⪯cFL is equivalent to
+∃ K ≥ 1 ∃ C ≥ 1 ∀ t ≥ 0 :
+FM(t) ≤ FL(Kt) + C,
+and so by Corollary 3.11 (take w.l.o.g. K ∈ N>0)
+∃ K ∈ N>0 ∃ D ≥ 1 ∀ t ≥ 0 :
+ωM(et) = ϕωM (t) ≤ ϕωL(Kt) + D = ωL(etK) + D.
+We set s := et and hence this is equivalent to
+∃ K ∈ N>0 ∃ D ≥ 1 ∀ s ≥ 1 :
+ωM(s) ≤ ωL(sK) + D.
+Recall that M, L ∈ LC implies (by normalization) ωM(s) = ωL(s) = 0 for all s ∈ [0, 1] and so the
+previous estimate holds true for any s ≥ 0 (with the same constants). Thus (2.5) yields for all
+j ∈ N:
+MKj = sup
+t≥0
+tKj
+exp(ωM(t)) ≥ 1
+eD sup
+t≥0
+tKj
+exp(ωL(tK)) = 1
+eD sup
+s≥0
+sj
+exp(ωL(s)) = 1
+eD Lj,
+and we are done when taking c := K and A := eD.
+(iii) ⇒ (ii) Assume that (mg) holds for M. By assumption (4.3) and an iterated application of
+(mg) we get
+(4.4)
+∃ c ∈ N>0 ∃ A, B ≥ 1 ∀ k ∈ N :
+Lk ≤ AMck ≤ ABkM c
+k,
+thus by definition of associated weight functions ωMc(t) ≤ ωL(Bt) + log(A) for all t ≥ 0. Conse-
+quently, since ϕωL(t) → +∞ and by (3.2) we get for all t large enough
+ϕωMc(t) ≤ ϕωL(t + log(B)) + log(A) ≤ ϕωL(2t) + log(A) ≤ 2ϕωL(2t) ≤ ϕωL(4t),
+hence ϕωMc⪯cϕωL. By (4.2) we have that ϕωMc∼cϕωM and so ϕωM ⪯cϕωL is verified. Corollary
+3.11 yields the conclusion.
+If L has in addition (mg), then by (4.3) and iterating (mg) first we get
+∃ c ∈ N>0 ∃ A, B1 ≥ 1 ∀ j ∈ N :
+Lcj ≤ Bcj
+1 Lc
+j ≤ AcBcj
+1 M c
+cj.
+
+12
+G. SCHINDL
+Thus (4.4) 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
+Lk ≤ Lcj+c ≤ Cc(j+1)LcLcj ≤ Cc(j+1)LcAcBcj
+1 M c
+cj ≤ (AC)cLc(B1C)kM c
+k.
+Summarizing, (4.4) is verified for all k ∈ N and the rest follows as above.
+□
+Thus we have the following characterization:
+Corollary 4.4. 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 FM and FL are equivalent.
+(ii) The functions ϕωM and ϕωL are equivalent.
+(iii) The sequences M and L satisfy
+∃ c, d ∈ N>0 ∃ A, B ≥ 1 ∀ j ∈ N :
+Mj ≤ ALcj,
+Lj ≤ BMdj.
+We close this section by comparing the characterizing conditions for M and L in the previous
+results.
+Remark 4.5. Let M, L ∈ LC be given.
+(∗) We have Lj ≥ 1 for all j ∈ N and so (4.1) implies (4.3) (with M, L interchanged).
+(∗) On the other hand, recall that by (3.2) we get that FM ⪯ FL implies FL⪯cFM. Summariz-
+ing,
+(4.5)
+(4.1) ⇐⇒ FM ⪯ FL =⇒ FL⪯cFM =⇒ (4.3),
+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 sufficient conditions for given sequences M, L
+in order to ensure FM⪯cFL resp. FM∼cFL.
+In explicit applications and for constructing weight sequences M it is often convenient to start
+with the corresponding sequence of quotients µ = (µj)j∈N. Note that when involving µ we get
+automatically growth conditions for the counting function ΣM as well.
+Proposition 4.6. Let M, L ∈ LC be given.
+(a) The following assertions are equivalent:
+(i) We have that
+(4.6)
+∃ A ≥ 1 ∃ k > 0 ∃ t0 > 0 ∀ t ≥ t0 :
+ΣM(t) ≤ AΣL(tk).
+(ii) We have that
+(4.7)
+∃ B ≥ 1 ∃ k > 0 ∃ j0 ∈ N>0 ∀ j ≥ j0 :
+λ⌈j/B⌉ ≤ µk
+j .
+(b) Any of the equivalent conditions listed in (a) implies that FM⪯cFL (equivalently ϕωM ⪯cϕωL)
+holds true.
+The proof shows that in (a) we can take A = B and the same k and note that the result becomes
+trivial for M = L (set A = B = k = 1).
+Proof. (a)(i) ⇒ (ii) Let j0 ∈ N>0 be minimal to ensure µj0 ≥ t0 (note that limj→+∞ µj = +∞).
+For any t ≥ µj0 we have µj ≤ t < µj+1 for some j ∈ N>0, j ≥ j0. Then by assumption j = ΣM(t) ≤
+AΣL(tk) ⇔ ΣL(tk) ≥
+j
+A and so tk ≥ λ⌈j/A⌉ (note that ΣL(tk) ∈ N). When taking t := µj we get
+(4.7) with B := A and the same k > 0 for all j ≥ j0 with µj < µj+1.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+13
+If j ≥ j0 with µj = µj+1, then µj = µj+1 = · · · = µj+d < µj+d+1 for some d ∈ N>0 and thus
+j + d = ΣM(t) for µj+d ≤ t < µj+d+1. This yields ΣL(tk) ≥ j+d
+A , i.e. tk ≥ λ⌈(j+d)/A⌉ ≥ λ⌈(j+i)/A⌉
+for all 0 ≤ i ≤ d. Put t := µj+d(= · · · = µj) in order to verify (4.7) with B := A and the same
+k > 0 for all j ≥ j0.
+(a)(ii) ⇒ (i) Let t ≥ 0 be such that µj ≤ t < µj+1 for some j ≥ j0. Then µk
+j ≤ tk < µk
+j+1
+and so tk ≥ µk
+j ≥ λ⌈j/B⌉ which gives ΣM(t) = j and ⌈j/B⌉ ≤ ΣL(tk). Thus (4.6) follows when
+j ≤ Aj/B ≤ A⌈j/B⌉ is ensured. So we can take A := B, the same k > 0 and t0 := µj0.
+(b) If (4.6) is valid, then ΣM(es) ≤ AΣL(esk) for some A ≥ 1 and all sufficiently large s. Thus by
+(3.17) we can find s0 > 0 such that fM(s) ≤ AfL(ks) for all s ≥ s0. Then FM⪯cFL holds similarly
+as shown in [11, Lemma 3.1] (with A = 1 there): For all s ≥ s0 we get
+FM(s) =
+� s
+0
+fM(u)du =
+� s0
+0
+fM(u)du +
+� s
+s0
+fM(u)du ≤
+� s0
+0
+fM(u)du + A
+� s
+s0
+fL(ku)du
+≤
+� s0
+0
+fM(u)du + A
+� s
+0
+fL(ku)du =
+� s0
+0
+fM(u)du + A
+k
+� ks
+0
+fL(v)dv = FM(s0) + A
+k FL(ks),
+hence (3.7) between FM and FL follows when choosing C := FM(s0). Then Lemma 3.2 yields the
+assertion.
+□
+Another sufficiency criterion expressed in terms of the counting functions ΣM, ΣL is the following
+result.
+Lemma 4.7. Let M, L ∈ LC be given. Assume that
+(4.8)
+∃ b ∈ (0, +∞) :
+lim
+t→+∞
+ΣM(t)
+ΣL(t) = b.
+Then we get FM∼cFL (equivalently ϕωM ∼cϕωL).
+Note: For M = L property (4.8) holds trivially with b = 1.
+Proof.
+By (3.17) and (4.8) we have limt→+∞
+fM (t)
+fL(t) = b > 0 and so [11, Lemma 3.2] yields
+FM∼cFL.
+□
+Let us now study condition (4.8) in more detail.
+Lemma 4.8. Let M, L ∈ LC be given. Assume that
+(4.9)
+∃ c, j0 �� N>0 ∀ j ≥ j0, s.th. µj < µj+1, ∃ dj ∈ N>0 :
+λcj ≤ µj < µj+1 ≤ λcj+dj,
+with dj ∈ N>0 satisfying limj→+∞
+dj
+j = 0. Then (4.8) holds true (with b = 1
+c).
+Proof. For all t ≥ µj0 we find j ≥ j0 such that µj ≤ t < µj+1. Then ΣM(t) = j and by (4.9) we
+get cj ≤ ΣL(t) < cj + dj. Thus
+j
+cj + dj
+< ΣM(t)
+ΣL(t) ≤ j
+cj ,
+∀ µj ≤ t < µj+1, j ≥ j0,
+and since limj→+∞
+dj
+j = 0 we get (4.8) with b := 1
+c.
+□
+Note that it is enough to require the existence of a sequence dj for j such that µj < µj+1: If j ≥ j0
+and µj = µj+1 = · · · = µj+ℓ < µj+ℓ+1 for some ℓ ∈ N>0, then by (4.9) we get
+λcj ≤ λcj+cℓ ≤ µj = µj+1 = · · · = µj+ℓ < µj+ℓ+1 ≤ λcj+cℓ+dj+ℓ,
+
+14
+G. SCHINDL
+and so for t with µj+ℓ ≤ t < µj+ℓ+1 we have ΣM(t) = j + ℓ and cj + cℓ = c(j + ℓ) ≤ ΣL(t) <
+cj + cℓ + dj+ℓ = c(j + ℓ) + dj+ℓ yielding the same estimate as above. (On the other hand, by
+switching to an equivalent sequence, we can assume µj < µj+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 µj < µj+1 for all j ∈ N>0. Then the following are
+equivalent:
+(i) We have that
+(4.10)
+∃ b ∈ (0, +∞) :
+lim
+t→+∞
+ΣM(t)
+ΣL(t) = b,
+i.e. (4.8).
+(ii) We have that
+(4.11)
+∃ b ∈ (0, +∞) ∃ d ∈ N ∀ 0 < c1 < b < c2 ∃ j0 ∈ N>0 ∀ j ≥ j0 :
+λ⌈j/c2⌉−d ≤ µj < λ⌈j/c1⌉+d.
+(iii) We have that
+(4.12)
+∃ b ∈ (0, +∞) ∀ 0 < c1 < b < c2 ∃ j0 ∈ N>0 ∀ j ≥ j0 ∃ dj ∈ N>0 :
+λ⌈j/c2⌉−dj ≤ µj < λ⌈j/c1⌉+dj,
+with dj ∈ N>0 satisfying limj→+∞
+dj
+j = 0.
+The proof shows that in all assertions the value b is the same.
+Proof. (i) ⇒ (ii) By assumption,
+∀ ǫ > 0 ∃ tǫ > 0 ∀ t ≥ tǫ :
+ΣL(t)(b − ǫ) < ΣM(t) < (b + ǫ)ΣL(t),
+thus we find jǫ ∈ N>0 large satisfying µjǫ ≥ tǫ so that for all t with µj ≤ t < µj+1 for some j ≥ jǫ
+we get
+(4.13)
+ΣL(t)(b − ǫ) < j = ΣM(t) < (b + ǫ)ΣL(t).
+The first estimate in (4.13) yields ΣL(t) <
+j
+b−ǫ ≤ ⌈
+j
+b−ǫ⌉ and since ΣL(t) ∈ N we have ΣL(t) ≤
+⌈
+j
+b−ǫ⌉ − 1. Let now c1 < b be arbitrary but fixed and take ǫ1 small enough to ensure
+1
+b−ǫ1 ≤
+1
+c1 ⇔
+c1 ≤ b − ǫ1. Note that the choice of ǫ1 is only depending on chosen c1 but not on j.
+Thus ΣL(t) < ⌈ j
+c1 ⌉ and so t < λ⌈ j
+c1 ⌉. We take t := µj and get µj < λ⌈ j
+c1 ⌉, i.e. the second half of
+(4.11) for all j ≥ jǫ1 (since µj < µj+1 for all j) with d := 0.
+Similarly, the second estimate in (4.13) gives ⌊
+j
+b+ǫ⌋ ≤
+j
+b+ǫ < ΣL(t) and so t ≥ λ⌊
+j
+b+ǫ ⌋. We let c2 > b
+be arbitrary but fixed and choose ǫ2 > 0 small enough to ensure
+1
+b+ǫ2 ≥
+1
+c2 ⇔ c2 ≥ b + ǫ2 and so
+⌊
+j
+b+ǫ2 ⌋ ≥ ⌈
+j
+b+ǫ2 ⌉ − 1 ≥ ⌈ j
+c2 ⌉ − 1. Take t := µj to get the first half of (4.11) for all j ≥ jǫ2 with
+d := 1. Summarizing, we are done when taking j0 := max{jǫ1, jǫ2} and note that d can be taken
+uniformly and not depending on chosen c1, c2.
+(ii) ⇒ (iii) This is clear.
+(iii) ⇒ (i) Let b ∈ (0, +∞) be given and take arbitrary (close) 0 < c1 < b < c2 but from now on
+fixed. Let t ≥ µj0 and so µj ≤ t < µj+1 for some j ≥ j0. Then (4.12) gives
+λ⌈j/c2⌉−dj ≤ µj ≤ t < µj+1 < λ⌈(j+1)/c1⌉+dj+1.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+15
+So ΣM(t) = j and the first estimate implies ΣL(t) ≥ ⌈ j
+c2 ⌉ − dj ≥
+j
+c2 − dj, whereas the last estimate
+yields ΣL(t) ≤ ⌈ j+1
+c2 ⌉ + dj+1 − 1 ≤ j+1
+c1 + dj+1. Summarizing, for all such t we obtain
+j
+j/c1 + 1/c1 + dj+1
+≤ ΣM(t)
+ΣL(t) ≤
+j
+j/c2 − dj
+.
+Then note that dj+1
+j
+= dj+1
+j+1
+j+1
+j
+→ 0 as j → +∞ ⇔ t → +∞. Hence, as c1, c2 → b we get that
+ΣM(t)
+ΣL(t) → b as t → +∞.
+□
+5. 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 N-
+function G, associate to it a sequence M G and study then the relation between G and the associated
+N-function FMG.
+Let F be an N-function and first we introduce
+(5.1)
+ωF (t) := F(log(t)),
+t ≥ 1,
+ωF (t) := 0,
+0 ≤ t < 1
+(normalization).
+By the properties of F we have that ωF belongs to the class W0: Concerning (ω4) note that
+ϕωF (t) = ωF (et) = F(t) for all t ≥ 0, concerning (ω3) we remark that ωF (t)
+log(t) = ωF (es)
+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 ∈ W0, then one can put
+(5.2)
+Fω(t) := ϕω(|t|) = ω(e|t|),
+t ∈ R.
+Fω satisfies all properties to be formally an N-function (see Definition 3.1) except necessarily the
+first part in (3.3) and also Fω(t) > 0 for all t ̸= 0 is not clear. Thus the set of all N-functions
+does not coincide with the class W0. However, one can overcome this technical problem for Fω by
+passing to an equivalent (associated) N-function when taking into account Theorem 3.8 analogously
+as it has been done before with ϕωM .
+We consider the so-called Legendre-Fenchel-Young-conjugate of ϕωF which is given by
+(5.3)
+ϕ∗
+ωF (s) := sup{|s|t − ϕωF (t) : t ≥ 0} = sup{|s|t − F(t) : t ≥ 0},
+s ∈ R.
+ϕωF is non-decreasing, convex by assumption, ϕωF (0) = ωF (e0) = F(0) = 0 and limt→+∞
+ϕωF (t)
+t
+=
+lims→+∞
+ωF (s)
+log(s) = +∞ as seen before. Thus we get, see e.g. [2, Rem. 1.3]:
+ϕ∗
+ωF is convex, ϕ∗
+ωF (0) = 0, s �→
+ϕ∗
+ωF (s)
+s
+is non-decreasing and tending to +∞ as s → +∞. Finally
+ϕ∗∗
+ωF (s) = ϕωF (s) holds true for all s ≥ 0.
+Next let us introduce the associated sequence M F = (M F
+j )j by
+(5.4)
+M F
+j := sup
+t>0
+tj
+exp(ωF (t)) = sup
+t≥1
+tj
+exp(ωF (t)) = sup
+t≥1
+tj
+exp(F(log(t))),
+j ∈ N.
+The second equality holds by normalization and this formula should be compared with (2.5). Hence
+for any j ∈ N we get
+M F
+j = sup
+t>0
+tj
+exp(ωF (t)) = exp sup
+t>0
+(j log(t) − ωF(t)) = exp sup
+s∈R
+(js − ωF (es)) = exp sup
+s≥0
+(js − ωF (es))
+= exp sup
+s≥0
+(js − ϕωF (s)) = exp(ϕ∗
+ωF (j)),
+
+16
+G. SCHINDL
+which implies M F ∈ LC by the properties of the conjugate. Note that by definition (normalization)
+one has ωF (es) = 0 for all −∞ < s ≤ 0.
+Finally, by [18, Thm. 4.0.3] applied to ω = ωF , see also [15, Lemma 5.7] and [8, Lemma 2.5] with
+weight matrix parameter x = 1 there, we obtain
+(5.5)
+∃ C ≥ 1 ∀ t ≥ 0 :
+ωMF (t) ≤ ωF (t) ≤ 2ωMF (t) + C,
+hence by (5.1)
+(5.6)
+∃ C ≥ 1 ∀ t ≥ 0 :
+ϕωMF (t) ≤ ωF (et) = F(t) ≤ 2ϕωMF (t) + C.
+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 FMG the associated N-function from Corollary 3.11 (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 FMG be the N-function associated with the sequence
+M G. Then we get
+(5.7)
+∃ A, B ≥ 1 ∀ t ≥ 0 :
+FMG(t) ≤ G(t) + A ≤ 2FMG(t) + B ≤ FMG(2t) + B,
+and this implies both G ∼ FMG ∼ ϕωMG and G∼cFMG∼cϕωMG .
+Proof. Corollary 3.11 applied to M G and (5.6) yield
+∃ C, C1 ≥ 1 ∀ t ≥ 0 :
+G(t) ≤ 2ϕωMG(t) + C ≤ 2FMG(t) + C + C1,
+and
+∃ C2 ≥ 1 ∀ t ≥ 0 :
+FMG(t) ≤ ϕωMG (t) + C2 ≤ G(t) + C2.
+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 FMG, see also the estimate in (i) in Remark 3.4 applied to a := 2.
+The desired relations follow now by (5.7), Lemma 3.2 and Corollary 3.11.
+□
+The next result gives the (expected) equivalence when starting with a weight sequence.
+Proposition 5.3. Let L ∈ LC be given and FL the associated N-function. Then we get
+∃ A, B ≥ 1 ∀ j ∈ N :
+1
+ALj ≤ M FL
+j
+≤ BLj.
+This estimate implies M FL≈L, FL∼cFMFL and FL ∼ FMFL .
+Proof. We apply the previous constructions to the associated N-function FL. For all t ≥ 1 we have
+ωFL(t) = FL(log(t)) and so by (3.16)
+(5.8) ∃ C, D ≥ 1 ∀ t ≥ 1 :
+ωL(t)−C = ϕωL(log(t))−C ≤ ωFL(t) ≤ ϕωL(log(t))+D = ωL(t)+D.
+Because L ∈ LC we have ωL(t) = 0 for all t ∈ [0, 1] (normalization) and ωFL(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−DLj ≤ M FL
+j
+≤ eCLj,
+hence the conclusion.
+This relation clearly implies M FL≈L and so, by Theorem 4.3 (recall also Remark 4.2) the equiva-
+lence FL∼cFMFL . Moreover, (4.1) is verified with c = 1 (pair-wise) and hence Theorem 4.1 gives
+that FL ∼ FMFL as well.
+□
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+17
+We continue with the following observations:
+(∗) Theorem 5.2 suggests that for abstractly given N-functions G it is important to have in-
+formation about FMG resp. ϕωMG and to study the associated weight sequence M G. In
+particular, in view of (3.12) and (3.17) the knowledge about the counting function ΣMG is
+useful and this amounts to study the sequence of quotients µG.
+(∗) On the other hand, when desired growth behaviours are expressed in terms of µG, it is
+an advantage not to compute first M G via given G and then the corresponding quotient
+sequence µG but to come up with a property for G directly.
+(∗) This can be achieved by relating µG directly to G as follows: Put
+(5.9)
+vG(t) := exp(−G(log(t))),
+t ≥ 1,
+vG(t) := 1,
+0 ≤ t < 1,
+and so (5.4) takes alternatively the following form:
+M G
+j = sup
+t>0
+tjvG(t),
+j ∈ N.
+This should be compared with [19, Sect. 6], [17, Sect. 2.6, 2.7]. Indeed, M G coincides with
+the crucial associated sequence M vG in [19], [17]. By the assumptions on G we get that vG
+is a (normalized and convex) weight in the notion of [19], [17], i.e. in the setting of weighted
+spaces of entire functions, see also the literature citations in these papers.
+(∗) Let now (tj)j∈N be the sequence such that tj ≥ 0 is denoting a/the global maximum point
+of the mapping t �→ tjvG(t).
+(∗) The advantage when considering vG is that in [19, (6.5)] we have derived the following
+relation between the sequence of quotients µG (set µG
+0 := 1) and (tj)j∈N:
+(5.10)
+∀ j ∈ N :
+tj ≤ µG
+j+1 ≤ tj+1.
+Note: For j = 0 we have put t0 := 1 and so equality with µG
+0 . In fact for j = 0 we can choose
+any t ∈ [0, 1] as t0 since vG is non-increasing and normalized, and by the convention 00 := 1.
+So tj ≥ 1 for any j ∈ N because j �→ tj is non-decreasing and since limj→+∞ µG
+j = +∞ we
+also get tj → +∞. For concrete given G (belonging to C1) the concrete computation for
+the values of tj might be not too difficult.
+Then put
+(5.11)
+ΣG(t) := |{j ∈ N>0 :
+tj ≤ t}|,
+t ≥ 0,
+and ΣG : [0, +∞) → N is a right-continuous non-decreasing function with ΣG(t) = 0 for all 0 ≤ t < 1
+and ΣG(t) → +∞ as t → +∞. By taking into account (5.10) and the definition of ΣMG and ΣG
+we get:
+Proposition 5.4. Let G be an N-function and M G the associated weight sequence.
+Then the
+counting functions ΣG and ΣMG are related by
+(5.12)
+∀ t ≥ 0 :
+ΣMG(t) ≤ ΣG(t) + 1 ≤ ΣMG(t) + 1,
+which yields
+lim
+t→+∞
+ΣMG(t)
+ΣG(t)
+= 1.
+Using ΣG we introduce
+(5.13)
+ϕG(x) :=
+� |x|
+0
+ΣG(et)dt,
+x ∈ R.
+
+18
+G. SCHINDL
+Note that, analogously to the explanations for ϕωM given in Remark 3.6 in Section 3.2 we have that
+ϕG is formally not an N-function since tj ≥ 1 for all j ≥ 1 and so by definition ΣG(e0) ≥ 1 ̸= 0 if
+t1 = 1 or ΣG(et) = 0 for all 0 ≤ t < log(t1) if t1 > 1. 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, FMG the associated
+N-function and ϕG given by (5.13). Then
+G∼cFMG∼cϕωMG ∼cϕG,
+G ∼ FMG ∼ ϕωMG ∼ ϕG.
+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 (3.12) in order to get analogously as in the proof of
+(b) in Proposition 4.6:
+(5.14)
+∃ C ≥ 1 ∀ t ≥ 0 :
+ϕG(t) ≤ ϕωMG(t) ≤ ϕG(t) + t ≤ 2ϕG(t) + C ≤ ϕG(2t) + C.
+The second last estimate in (5.14) holds since limt→+∞
+ϕG(t)
+t
+= +∞ and the last one by (3.2)
+applied to ϕG. Both requirements on ϕG are valid by the representation (5.13), see Remark 3.10.
+Then (5.14) implies both ϕωMG ∼cϕG and ϕωMG ∼ ϕG.
+□
+Theorem 5.5 shows that 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 FMG.
+6. 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 (FM, 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 ◦ΣD.
+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)
+f c(s) := sup{t ≥ 0 : f(t) ≤ s},
+s ≥ 0,
+thus f c is the right-inverse of f, i.e. it is the right-inverse of the right-derivative of F, see [11, (2.1)].
+It is known that f c satisfies (I) − (III) in Section 3.1.
+Definition 6.1. The complementary N-function F c, see [11, Chapter I, §2, p. 11], is defined by
+(6.2)
+F c(x) :=
+� |x|
+0
+f c(t)dt,
+x ∈ R.
+In [11, Chapter I, (2.9), p. 13] it is mentioned that
+(6.3)
+F c(s) = max
+t≥0 {|s|t − F(t)},
+s ∈ R,
+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, Gc.
+(i) In [11, Thm. 3.1, Thm. 3.2] it has been shown that F⪯cG yields Gc⪯cF c. In particular, if
+two N-functions are equivalent then their complementary N-functions as well.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+19
+(ii) If F ⪯ G, then G(t) ≤ CF(t) + D for some C, D ≥ 1 and all t ≥ 0 and so (6.3) gives for
+any s ∈ R
+Gc(s) = max
+t≥0 {|s|t − G(t)} ≥ max
+t≥0 {|s|t − CF(t)} − D = C max
+t≥0 {C−1|s|t − F(t)} − D
+= CF c(s/C) − D,
+see also the proof of [15, Lemma 5.16]. This relation implies (3.7) and so Lemma 3.2 yields
+that F c⪯cGc. When one can choose C = 1, then also Gc ⪯ F c follows but in general this
+implication is not clear.
+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 FM and the corresponding right-derivative fM. Moreover, in view of (6.1),
+let us introduce
+(6.4)
+ΓM(s) := sup{t ≥ 0 : ΣM(et) ≤ s},
+s ≥ 0.
+Finally we set
+(6.5)
+ϕc
+ωM (x) :=
+� |x|
+0
+ΓM(s)ds,
+x ∈ R.
+By (3.17) it follows that
+(6.6)
+∃ s0 > 0 ∀ s ≥ s0 :
+f c
+M(s) = ΓM(s),
+because both functions are non-decreasing and fM(t) = ΣM(et) for all t ≥ t0, with t0 the value
+appearing in (3.17). So (6.6) is valid for all s ≥ s0 := fM(t0). Using this identity we can prove the
+analogous result of Corollary 3.11 for the functions F c
+M and ϕc
+ωM .
+Proposition 6.3. Let M ∈ LC be given. Then we get
+∃ C, D ≥ 1 ∀ t ≥ 0 :
+ϕc
+ωM (t) − C ≤ F c
+M(t) ≤ ϕc
+ωM (t) + D.
+This implies that both ϕc
+ωM ∼cF c
+M and ϕc
+ωM ∼ F c
+M hold true.
+Proof. We use (6.6) and the representations (6.2) and (6.5) (recall also the arguments in the proof
+of Proposition 3.9). For all s ≥ s0 (with s0 from (6.6)) we get
+ϕc
+ωM (s) =
+� s
+0
+ΓM(t)dt =
+� s0
+0
+ΓM(t)dt +
+� s
+s0
+ΓM(t)dt =
+� s0
+0
+ΓM(t)dt +
+� s
+s0
+f c
+M(t)dt
+≤
+� s0
+0
+ΓM(t)dt +
+� s
+0
+fM(t)dt = ϕc
+ωM (s0) + F c
+M(s),
+hence ϕc
+ωM (s) ≤ F c
+M(s) + C for all s ≥ 0 when choosing C := ϕc
+ωM (s0). Similarly, for all s ≥ s0
+F c
+M(s) =
+� s
+0
+f c
+M(t)dt =
+� s0
+0
+f c
+M(t)dt +
+� s
+s0
+f c
+M(t)dt =
+� s0
+0
+f c
+M(t)dt +
+� s
+s0
+ΓM(t)dt
+≤
+� s0
+0
+f c
+M(t)dt +
+� s
+0
+ΓM(t)dt = F c
+M(s0) + ϕc
+ωM (s),
+
+20
+G. SCHINDL
+hence F c
+M(s) ≤ ϕc
+ωM (s) + D for all s ≥ 0 when choosing D := F c
+M(s0).
+□
+On the other hand, formula (6.3) applied to ϕωM yields
+(6.7)
+sup
+t≥0
+{|s|t − ϕωM (t)} = ϕ∗
+ωM (s),
+s ∈ R,
+the Legendre-Fenchel-Young-conjugate of ϕωM . By combining (3.16), (6.3) applied to FM and (6.7)
+we get
+(6.8)
+∃ C, D ≥ 1 ∀ s ∈ R :
+−C + F c
+M(s) ≤ ϕ∗
+ωM (s) ≤ F c
+M(s) + D,
+see the estimate in (ii) in Remark 6.2. Hence F c
+M∼cϕ∗
+ωM and F c
+M ∼ ϕ∗
+ωM hold true. When combining
+this with Proposition 6.3 we arrive at the following result:
+Theorem 6.4. Let M ∈ LC be given. Then
+ϕc
+ωM ∼cF c
+M∼cϕ∗
+ωM ,
+ϕc
+ωM ∼ F c
+M ∼ ϕ∗
+ωM .
+Finally, we apply Theorem 6.4 to the associated weight sequence M G.
+Corollary 6.5. Let G be an N-function, M G ∈ LC the associated sequence defined via (5.4) and
+ϕG given by (5.13). Then
+ϕ∗
+ωMG ∼cϕc
+ωMG∼cF c
+MG∼cGc∼c(ϕG)c,
+ϕ∗
+ωMG ∼ ϕc
+ωMG ∼ F c
+MG.
+Proof. Concerning ∼c, the first and the second equivalence hold by Theorem 6.4, the third and
+the fourth one by taking into account (5.7), Theorem 5.5 (see (5.14)) and (i) in Remark 6.2. Here
+(ϕG)c is given in terms of (6.3).
+For ∼ we use the same results, however F c
+MG ∼ Gc and Gc ∼ (ϕG)c 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.
+□
+6.2. Complementary associated N-functions versus dual sequences. In this section the aim
+is to see that ΓM in (6.4) and crucially appearing in the representation (6.5) is closely connected
+to the counting function ΣD, 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.
+2.1.42].
+For given M ∈ LC we introduce the dual sequence D = (Dj)j defined in terms of the corresponding
+quotient sequence δ as follows, see [5, Def. 2.1.40, p. 81]:
+(6.9)
+∀ j ≥ µ1(≥ 1) :
+δj+1 := ΣM(j),
+δj+1 := 1
+∀ j ∈ Z, −1 ≤ j < µ1.
+Then we set
+Dj :=
+j�
+k=0
+δk,
+j ∈ N,
+hence D ∈ LC with 1 = D0 = D1 follows. Recall that by [5, Def. 2.1.27] the function νm in
+[5] precisely denotes the counting function ΣM (see (2.2)) and note that concerning the sequence
+of quotients there exists an index-shift: more precisely we have mj ≡ µj+1 for all j ∈ N with
+m = (mj)j used in [5] and [7]. Moreover, a weight sequence in [5] means a sequence satisfying all
+requirements from class LC except M0 ≤ M1; see [5, Sect. 1.1.1, Def. 1.1.8].
+Let M ∈ LC be given, we analyze now ΓM:
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+21
+(∗) Obviously, lims→+∞ ΓM(s) = +∞ and since limj→+∞ µj = +∞ we have µj < µj+1 for
+infinitely many indices j.
+(∗) Then recall that ΣM(et) ∈ N and that ΣM(et) = 0 for all 0 ≤ t < log(µ1). By normalization
+we get log(µj) ≥ log(µ1) ≥ log(1) = 0 for all j ∈ N>0.
+(∗) Case I: Assume that 1 = µ1 = · · · = µd < µd+1 for some d ∈ N>0. Then et ≥ µd = 1 and
+so ΣM(et) ≥ d for all t ≥ 0. Hence the set of values t ≥ 0 satisfying ΣM(et) ≤ s in the
+definition of ΓM(s) is empty for all values s with 0 ≤ s < d. In this case we put ΓM(s) := 0.
+Note that d is finite because limj→+∞ µj = +∞.
+Let now s be such that j ≤ s < j + 1 for some j ∈ N>0 with j ≥ d. In view of Remark
+2.2 in general it is not clear that we can find t ≥ 0 such that ΣM(et) = j. In fact this holds
+true if and only if µj < µj+1 because for all t ≥ 0 with log(µj) ≤ t < log(µj+1) we get
+ΣM(et) = j. We have Σ(elog(µj+1)) ≥ j + 1 > s and consequently for all such indices j, in
+particular for j = d, we obtain ΓM(s) = log(µj+1).
+If j ≥ d is such that µj = µj+1, then j ≥ d + 1. Thus there exist ℓ, c ∈ N>0 with
+d ≤ ℓ ≤ j − 1 and such that µℓ < µℓ+1 = µℓ+2 = · · · = µℓ+c = µj = µj+1. For all t with
+log(µℓ) ≤ t < log(µℓ+1) we get ΣM(et) = ℓ < j and since ΣM(elog(µℓ+1)) = ΣM(elog(µj+1)) ≥
+j + 1 > s we have again ΓM(s) = log(µj+1).
+(∗) Case II: Assume that µ1 > 1. First we have ΓM(s) = log(µ1)(> 0) for all 0 ≤ s < 1 since
+ΣM(et) = 0 for all 0 ≤ t < log(µ1) and ΣM(elog(µ1)) ≥ 1.
+Let now j ≤ s < j + 1 for some j ∈ N>0. Similarly as above, if µj < µj+1 then we get
+ΓM(s) = log(µj+1).
+If µj = µj+1, then we distinguish: Either there exist ℓ, c ∈ N>0 with 1 ≤ ℓ ≤ j − 1 such
+that µℓ < µℓ+1 = µℓ+2 = · · · = µℓ+c = µj = µj+1. Then again ΓM(s) = log(µj+1) by the
+same reasons as in Case I before.
+Finally, if this choice is not possible, then this precisely means µ1 = · · · = µj = µj+1.
+Since µ1 > 1 we have ΣM(et) = 0 for all 0 ≤ t < log(µ1) and ΣM(et) ≥ j + 1 > s for all
+t ≥ log(µj+1) = log(µ1). Thus ΓM(s) = log(µj+1) = log(µ1) holds true.
+Summarizing, we have shown the following statement:
+Proposition 6.6. Let M ∈ LC be given with quotient sequence (µj)j.
+(a) If 1 = µ1 = · · · = µd < µd+1 for some d ∈ N>0, then
+ΓM(s) = 0,
+∀ 0 ≤ s < d,
+ΓM(s) = log(µj+1),
+∀ j ≤ s < j + 1, ∀ j ≥ d.
+(b) If µ1 > 1, then
+ΓM(s) = log(µj+1),
+∀ j ≤ s < j + 1, ∀ j ∈ N.
+Next let us study ΣD in detail, see also the proof of [5, Thm. 2.1.42]:
+(∗) Let t ≥ 0, then the value ΣD(t) is equal to the maximal integer j ∈ N>0 satisfying δj ≤ t
+if it exists and otherwise equal to 0. By definition in (6.9) we have δj ∈ N>0 for all j ∈ N
+and so ΣD(t) = 0 for 0 ≤ t < 1.
+(∗) In fact δj = 1 for all j ∈ N>0 satisfying j < µ1 + 1. The largest of those integers j coincides
+with ⌊µ1 + 1⌋ ≥ 2 if µ1 /∈ N>0 (in this case µ1 > 1) and with µ1 if µ1 ∈ N>0. Moreover,
+δj = 1 for all j satisfying µ1 + 1 ≤ j < µ2 + 1 if such an integer j exists (case I). In
+particular, this holds if µ2 ≥ µ1 + 1.
+If not, then δj = d ∈ N≥2 for the minimal integer j satisfying j ≥ µ1 + 1 (case II). This
+occurs if for this minimal integer already j ≥ µd + 1 is valid. Note that d is finite since
+
+22
+G. SCHINDL
+limj→+∞ µj = +∞ and let d be such that µd + 1 ≤ j < µd+1 + 1 for j minimal satisfying
+j ≥ µ1 + 1.
+(∗) Consider 1 ≤ t < 2 and distinguish: In the first case ΣD(t) is the largest integer j satisfying
+µ1 + 1 ≤ j < µ2 + 1 and in the second case ΣD(t) coincides with the largest integer j
+satisfying j < µ1 + 1. Since µ1 + 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 ΣD(t) is the largest integer j
+satisfying j < µn+1 + 1.
+(∗) In case II, if n ≤ t < n + 1 for some n ∈ N>0 with n + 1 ≤ d, then ΣD(t) is (still) the
+largest integer j such that j < µ1 + 1 since for the minimal integer j ≥ µ1 + 1 we have
+δj = ΣM(j − 1) ≥ ΣM(µd) = d > t. The last equality holds since µd < µd+1 by the choice
+of d.
+If n ≤ t < n + 1 for some n ∈ N>0 with n ≥ d, then δj coincides with largest integer j
+satisfying j < µn+1 + 1. For n = d recall that at least one integer j exists with µd + 1 ≤
+j < µd+1 + 1 and so δj = ΣM(j − 1) = d.
+Summarizing, we have shown the following statement (see also [5, Thm. 2.1.42]):
+Proposition 6.7. Let M ∈ LC be given and D its dual sequence. Then
+ΣD(t) = 0,
+∀ 0 ≤ t < 1,
+and:
+(a) If for the minimal integer j satisfying j ≥ µ1 + 1 one has µd + 1 ≤ j < µd+1 + 1 for some
+d ∈ N>0, d ≥ 2, then
+ΣD(t) = max{j ∈ N>0 : j < µ1 + 1},
+∀ n ∈ N>0, n < d,
+∀ n ≤ t < n + 1,
+ΣD(t) = max{j ∈ N>0 : j < µn+1 + 1},
+∀ n ∈ N>0, n ≥ d,
+∀ n ≤ t < n + 1.
+(b) If for the minimal integer j satisfying j ≥ µ1 + 1 one has µ1 + 1 ≤ j < µ2 + 1, then
+ΣD(t) = max{j ∈ N>0 : j < µn+1 + 1},
+∀ n ∈ N>0,
+∀ n ≤ t < n + 1.
+When combining Propositions 6.6 and 6.7 we are able to establish a connection between the counting
+functions ΓM and ΣD.
+Theorem 6.8. Let M ∈ LC be given and D its dual sequence. Then
+(6.10)
+∃ t0 > 0 ∀ t ≥ t0 :
+ΓM(t) ≤ log(ΣD(t)) ≤ ΓM(t) + 1,
+and this estimate implies
+lim
+t→+∞
+ΓM(t)
+log(ΣD(t)) = 1.
+In (6.10) we can take t0 := d ∈ N>0 such that for the minimal integer j ≥ µ1 + 1 we get µd + 1 ≤
+j < µd+1 + 1.
+Proof. First, by Proposition 6.7 we get that ΣD(t) coincides with the maximal integer j < µn+1+1
+for all t ≥ 0 satisfying n ≤ t < n + 1 and all n ≥ d with d ∈ N>0 such that for the minimal integer
+j ≥ µ1 + 1 we get µd + 1 ≤ j < µd+1 + 1. In particular, for all such n we get µn+1 ≥ µd+1 > µ1 ≥ 1.
+Then recall that by Proposition 6.6 we get ΓM(t) = log(µn+1) for all t satisfying n ≤ t < n + 1
+with n ∈ N such that µn+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 ΣD(t) < µn+1 + 1 and
+log(ΣD(t)) < log(µn+1 + 1) ≤ log(eµn+1) = log(µn+1) + 1 = ΓM(t) + 1,
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+23
+showing the second part of (6.10).
+On the other hand for all such t we get ΣD(t) ≥ µn+1 = exp(ΓM(t)), in fact even ΣD(t) ≥ ⌈µn+1⌉
+holds, and which proves the first estimate in (6.10).
+□
+Using the counting function ΣD we set
+(6.11)
+F�ΓD(x) :=
+� |x|
+0
+�ΓD(s)ds,
+x ∈ R,
+with
+�ΓD(s) := log(ΣD(s)),
+s ≥ 1,
+�ΓD(s) := 0,
+0 ≤ s < 1.
+�ΓD is non-decreasing and right-continuous and tending to infinity. Recall that ΣD(s) ∈ N>0 for
+all s ≥ 1 and this technical modification is unavoidable: In (6.11) we cannot consider log(ΣD(s))
+directly for all s ≥ 0 because ΣD(s) = 0 for 0 ≤ s < 1 by definition. Then in view of (6.10) we get
+(6.12)
+∃ C, D ≥ 1 ∀ t ≥ 0 :
+−C + ΓM(t) ≤ �ΓD(t) ≤ ΓM(t) + D,
+which implies
+lim
+t→+∞
+ΓM(t)
+�ΓD(t)
+= 1.
+Note that F�ΓD is formally not an N-function by analogous reasons as in Remark 3.6: We have
+�ΓD(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
+F�ΓD∼cϕc
+ωM ∼cF c
+M∼cϕ∗
+ωM ,
+F�ΓD ∼ ϕc
+ωM ∼ F c
+M ∼ ϕ∗
+ωM .
+(ii) Let G be an N-function, M G ∈ LC the associated sequence defined via (5.4) and DG the
+corresponding dual sequence. Finally, let ϕG be given by (5.13). Then
+F�ΓDG ∼cϕ∗
+ωMG ∼cϕc
+ωMG∼cF c
+MG∼cGc∼c(ϕG)c,
+F�ΓDG ∼ ϕ∗
+ωMG ∼ ϕc
+ωMG ∼ F c
+MG.
+Proof. (i) In view of Theorem 6.4 only F�ΓD∼cϕc
+ωM resp. F�ΓD ∼ ϕc
+ωM has to be verified. This
+follows analogously as in the proof of (b) in Proposition 4.6 (see also (5.14)) by using (6.12) and
+the representations (6.11) and (6.5).
+Note that these representations imply limt→+∞
+F�ΓD (t)
+t
+=
+limt→+∞
+ϕc
+ωM (t)
+t
+= +∞, see Remark 3.10.
+(ii) By Corollary 6.5 it suffices to verify F�ΓDG ∼cϕc
+ωMG and F�ΓDG ∼ ϕc
+ωMG. Both relations follow
+by applying the first part to M G and DG. 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 FD 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.
+
+24
+G. SCHINDL
+7. 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 assump-
+tions in the weight sequence setting in terms of given M.
+We remark that all appearing conditions are naturally preserved under ∼c for (associated) N-
+functions, 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
+satisfying normalization, convexity, being non-decreasing and tending to infinity (in particular for
+ϕωM ).
+Therefore recall that the technical failure of ϕωM to be formally an N-function occurs at the point
+0 (see Remark 3.6) whereas for all crucial conditions under consideration large values t ≥ t0 > 0
+are relevant.
+Of course, it makes also sense to consider the conditions in this section for arbitrary functions
+F : [0, +∞) → [0, +∞).
+7.1. The ∆2-condition. The most prominent property is the so-called ∆2-condition, see e.g. [11,
+Chapter I, §4, p. 23], which reads as follows:
+(7.1)
+lim sup
+t→+∞
+F(2t)
+F(t) < +∞.
+This growth-condition is precisely (ω1) for F and thus also frequently appears for so-called weight
+functions ω in the sense of Braun-Meise-Taylor, see [2]. It is straight-forward that ∆2 is preserved
+under relation ∼ and also under ∼c, see [11, p. 23].
+It is known, see e.g. [11, Thm. 8.2] and [1, Thm. 3.4], that LF is a linear space if and only if F
+satisfies the ∆2-condition. Moreover, we also get:
+Lemma 7.1. Let F1 and F2 be two N-functions such that either F1 or F2 has ∆2. Then F1⪯cF2
+if and only if F2 ⪯ F1.
+In fact, in order to conclude, we only require that either F1 or F2 is normalized and convex and
+that either F1 or F2 has ∆2.
+Proof. By (3.2) we have that F1 ⪯ F2 implies F2⪯cF1 (see Remark 3.5). The converse implication
+holds by an iterated application of ∆2 for either F1 or F2.
+□
+In the weight sequence setting in view of Corollary 3.11 we require (7.1) (i.e. (ω1)) not for ωM
+directly but for ϕωM . Note that in [20, Thm. 3.1] we have already given a characterization of (ω1)
+for ωM in terms of M but ∆2 for ϕωM (resp. equivalently for FM) does precisely mean
+(7.2)
+lim sup
+t→+∞
+ωM(t2)
+ωM(t) < +∞,
+which is obviously stronger than (ω1) for ωM.
+Concerning this requirement we formulate the
+following result.
+Theorem 7.2. Let M ∈ LC be given. Then the following are equivalent:
+(i) The associated N-function FM (see Corollary 3.11) satisfies the ∆2-condition.
+(ii) ϕωM satisfies the ∆2-condition.
+(iii) ωM satisfies (7.2).
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+25
+(iv) ωM satisfies
+(7.3)
+∃ C > 0 ∃ H > 0 ∀ t ≥ 0 :
+ωM(t2) ≤ CωM(Ht) + C.
+(v) M satisfies
+(7.4)
+∃ k ∈ N>0 ∃ A, B ≥ 1 ∀ j ∈ N :
+(Mj)2k ≤ ABjMkj.
+(7.3) has already appeared (crucially) in different contexts; it is denoted by (ω7) in [9] and in [18];
+by (ω8) in [15] and by Ξ in [3].
+Proof. (i) ⇔ (ii) This is clear by Corollary 3.11 since FM∼cϕωM .
+(ii) ⇔ (iii) This is immediate.
+(iii) ⇒ (iv) This is clear since ωM is non-decreasing, limt→+∞ ωM(t) = +∞ and w.l.o.g. H ≥ 1.
+(iv) ⇒ (iii) As mentioned at the beginning of [9, Appendix A] each non-decreasing function satis-
+fying (7.3) has already (ω1). By iterating this property we get ωM(t2) ≤ C1ωM(t) + C1 for some
+C1 ≥ 1 and all t ≥ 0, i.e. (7.2).
+(iv) ⇔ (v) This has been shown in [3, Lemma 6.3, Cor. 6.4].
+□
+Example 7.3. We list some examples and their consequences:
+(∗) According to [3, Lemma 6.5] we know that any M ∈ LC cannot satisfy (mg) and (7.4)
+simultaneously. In particular, the Gevrey sequences Gs := (j!s)j∈N, s > 0, are violating
+(7.4).
+(∗) Consider the sequences M q,n := (qjn)j∈N with q, n > 1. Then (7.4) is valid (with A = B = 1
+and k satisfying k ≥ 21/(n−1)) as it is shown in [3, Example 6.6 (1)].
+(∗) Combining Lemma 7.1, Theorem 7.2 and Remark 4.5 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
+(4.1) ⇐⇒ FM ⪯ FL ⇐⇒ FL⪯cFM ⇐⇒ (4.3),
+i.e. in (4.5) the second implication can also be reversed.
+Corollary 7.4. Let M, L ∈ LC be given. Assume that both FM and FL satisfy the ∆2-condition.
+Then FM·L and FM⋆L satisfy the ∆2-condition, too.
+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 ωM⋆L = ωM +ωL and so ωM⋆L has (7.3) because both
+ωM and ωL have this property.
+□
+Finally we apply Theorem 7.2 to M = M G and get the following.
+Corollary 7.5. Let G be an N-function. Let ωG be the function from (5.1), M G ∈ LC the associated
+sequence defined via (5.4) and FMG the associated N-function. Then the following are equivalent:
+(i) G satisfies the ∆2-condition.
+(ii) The associated N-function FMG satisfies the ∆2-condition.
+(iii) ϕωMG satisfies the ∆2-condition.
+(iv) The function ϕG (see (5.13)) satisfies the ∆2-condition.
+(v) ωG (equivalently ωMG) satisfies (7.2).
+(vi) ωG (equivalently ωMG) satisfies (7.3).
+(vii) M G satisfies (7.4).
+
+26
+G. SCHINDL
+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 true equivalently for ωG and ωMG (by (5.5) and since these
+conditions are preserved under ∼).
+□
+7.2. The ∇2-condition. Let F be an N-function and F c its complementary function. In [11, Thm.
+4.2] it has been shown that F c satisfies the ∆2-condition if and only if F has
+(7.5)
+∃ ℓ > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+2ℓF(t) ≤ F(ℓt),
+also known under the name ∇2 for F.
+Remark 7.6. We summarize some properties for ∇2. For (i) and (ii) it is sufficient to require
+that F, G : [0, +∞) → [0, +∞) are non-decreasing, normalized and convex.
+(i) ∇2 is preserved under relation ∼c; this follows either from (i) in Remark 6.2 and since ∆2
+is preserved under ∼c, or it can be seen directly as follows:
+Assume that F has ∇2 and let G be another N-function such that F∼cG. So
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+G(tk−1) ≤ F(t) ≤ G(tk).
+When iterating ∇2 we find (since ℓ > 1)
+∃ t1 > 0 ∀ n ∈ N>0 ∀ t ≥ t1 :
+F(t) ≤
+1
+(2ℓ)n F(ℓnt).
+We choose d ∈ N>0 such that ℓd ≥ k2 and n ∈ N>0 such that 2n−1 ≥ ℓd. Then we estimate
+for all t ≥ max{t0, t1} as follows:
+G(t) ≤ F(kt) ≤
+1
+(2ℓ)n F(ℓnkt) ≤
+1
+(2ℓ)n F(ℓn+dk−1t) ≤
+1
+(2ℓ)n G(ℓn+dt) ≤
+1
+2ℓn+d G(ℓn+dt),
+i.e. ∇2 for G holds with ℓn+d and for all t ≥ t2 := max{t0, t1}.
+(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
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+k−1G(t) ≤ F(t) ≤ kG(t),
+and then we take n ∈ N>0 such that k2 ≤ 2n−1. Iterating n-times property ∇2 gives for all
+t sufficiently large that
+1
+k(2ℓ)nG(t) ≤ (2ℓ)nF(t) ≤ F(ℓnt) ≤ kG(ℓnt).
+Since G is an N-function we get k2G(ℓnt) ≤ G(k2ℓnt) (recall (3.2)) and 2k2ℓn ≤ (2ℓ)n by
+the choice of n. Consequently, 2k2ℓnG(t) ≤ G(k2ℓnt) is verified for all sufficiently large t,
+i.e. ∇2 for G with k2ℓn.
+(iii) Let us prove that for any non-decreasing F : [0, +∞) → [0, +∞) with limt→+∞ F(t) = +∞
+we have that (7.5) is equivalent to
+(7.6)
+∃ C ≥ 1 ∃ ℓ > 1 ∀ t ≥ 0 :
+2ℓF(t) ≤ F(ℓt) + 2ℓC.
+(7.5) implies (7.6) with the same ℓ, e.g. take C := F(t0), because F is non-decreasing. For
+the converse first we iterate (7.6) and get 4ℓ2F(t) ≤ 2ℓF(ℓt)+4ℓ2C ≤ F(ℓ2t)+2ℓC +4ℓ2C.
+Then, since F(t) → +∞ as t → +∞ we can find t0 > 0 such that F(ℓ2t) ≥ 2ℓC + 4ℓ2C for
+all t ≥ t0. Thus 4ℓ2F(t) ≤ 2F(ℓ2t) for all t ≥ t0 is verified, i.e. (7.5) with the choice ℓ2.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+27
+In the weight sequence setting we are interested in having (7.5) for FM resp. for ϕωM . Since ∇2 is
+preserved under equivalence, via Corollary 3.11 this condition transfers into
+(7.7)
+∃ ℓ > 1 ∃ s0 > 1 ∀ s ≥ s0 :
+2ℓωM(s) ≤ ωM(sℓ).
+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 FM satisfies the ∇2-condition.
+(ii) ϕωM satisfies the ∇2-condition.
+(iii) ωM satisfies (7.7).
+(iv) ωM satisfies
+(7.8)
+∃ C ≥ 1 ∃ ℓ > 1 ∀ s ≥ 0 :
+2ℓωM(s) ≤ ωM(sℓ) + 2ℓC.
+(v) The sequence M satisfies
+(7.9)
+∃ A ≥ 1 ∃ ℓ > 1 ∀ j ∈ N :
+M2j ≤ AM 2ℓ
+j .
+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 = e2ℓC. Consequently, if (7.8) holds for ℓ > 1 then also for all ℓ′ ≥ ℓ
+(even with the same choice for C).
+Proof. (i) ⇔ (ii) follows from Corollary 3.11 and (i) in Remark 7.6. (ii) ⇔ (iii) is clear and
+(iii) ⇔ (iv) follows as in resp. by (iii) in Remark 7.6.
+(iv) ⇒ (v) By using (2.5) we get for all j ∈ N:
+M2j = sup
+t≥0
+t2j
+exp(ωM(t)) = sup
+t≥0
+t2jℓ
+exp(ωM(tℓ)) ≤ e2ℓC sup
+t≥0
+t2jℓ
+exp(2ℓωM(t))
+= e2ℓC
+�
+sup
+t≥0
+tj
+exp(ωM(t))
+�2ℓ
+= e2ℓCM 2ℓ
+j ,
+so (7.9) is verified with A := e2ℓC and the same ℓ.
+(v) ⇒ (iv) We have
+tj
+Mℓ
+j ≤
+√
+A
+tj
+(M2j)1/2 for all j ∈ N and t ≥ 0. This yields by definition of associated
+weight functions
+(7.10)
+∀ t ≥ 0 :
+ωMℓ(t) ≤ ω�
+M2(t) + A1,
+with A1 := log(A)
+2
+, M ℓ := (M ℓ
+j )j∈N and the auxiliary sequence �
+M 2 := (M 1/2
+2j )j∈N, see [21, Sect.3],
+[20, (2.5), (2.6)] and also [4, Lemma 6.5]. When taking c = 2 in [4, Lemma 6.5 (6.7)], then
+(7.11)
+∃ D ≥ 1 ∀ t ≥ 0 :
+ω�
+M2(t) ≤ 2−1ωM(t) ≤ 2ω�
+M2(t) + D.
+Using (2.4) and the first half of (7.11) we continue (7.10) and get
+∀ t ≥ 0 :
+ℓωM(t1/ℓ) = ωMℓ(t) ≤ ω�
+M2(t) + A1 ≤ 2−1ωM(t) + A1,
+hence (7.8) is verified with the same ℓ and C := A1
+ℓ = log(A)
+2ℓ
+.
+□
+Example 7.8. We provide some examples of sequences such that (7.9) holds true.
+
+28
+G. SCHINDL
+(i) Any M ∈ LC satisfying (mg) does also have (7.9): By [18, Thm. 9.5.1] or [18, Thm. 9.5.3]
+applied to the matrix M = {M} (see also [13, Thm. 1]) we know that (mg) is equivalent to
+(7.12)
+∃ B ≥ 1 ∀ j ∈ N :
+M2j ≤ BjM 2
+j .
+Then note that BjM 2
+j ≤ AM 2ℓ
+j
+⇔ B ≤ A1/jM 2(ℓ−1)/j
+j
+holds for all j ∈ N>0 if A ≥ 1 is
+chosen sufficiently large because limj→+∞(Mj)1/j = +∞ and ℓ > 1.
+Thus, in particular, the Gevrey sequences Gs, s > 0, have (7.9).
+(ii) However, the converse implication is not valid in general: Consider again the sequences
+M q,n := (qjn)j∈N with q, n > 1. Condition (7.9) is valid because for given n > 1 we choose
+ℓ ≥ 2n−1 and get for all j ∈ N (and q > 1) that
+q(2j)n = M q,n
+2j
+≤ (M q,n
+j
+)2ℓ = (qjn)2ℓ ⇔ 1 ≤ q2jn(ℓ−2n−1).
+But (7.12) is violated: This requirement means q(2j)n ≤ Bjq2jn and since n > 1 this is
+impossible for any choice of B as j → +∞.
+Corollary 7.9. Let M, L ∈ LC be given and assume that both FM and FL satisfy the ∇2-condition.
+Then FM·L and FM⋆L have the ∇2-condition as well.
+Proof. By assumption both sequences satisfy (7.9) and so it is immediate that M · L has this
+property, too.
+Concerning the convolution we have ωM⋆L = ωM + ωL and so ωM⋆L has (7.8) because both ωM
+and ωL have this property. (For this recall that if (7.8) holds for some ℓ > 1 then for all ℓ′ ≥ ℓ as
+well.)
+□
+Finally, let us apply Theorem 7.7 to M = M G from Section 5.
+Corollary 7.10. Let G be an given N-function. Let ωG be the associated weight function from
+(5.1), M G the associated weight sequence (see (5.4)) and finally FMG the N-function associated
+with M G. Then the following are equivalent:
+(i) G satisfies the ∇2-condition (equivalently the complementary N-function Gc satisfies the
+∆2-condition).
+(ii) FMG satisfies the ∇2-condition (which is equivalent to the fact that the complementary
+N-function F c
+MG satisfies the ∆2-condition).
+(iii) ϕωMG satisfies the ∇2-condition (equivalently the complementary function ϕc
+ωMG satisfies
+the ∆2-condition).
+(iv) The function ϕG (see (5.13)) satisfies the ∇2-condition.
+(v) ωG (equivalently ωMG) satisfies (7.7).
+(vi) ωG (equivalently ωMG) satisfies (7.8).
+(vii) M G satisfies (7.9).
+Proof. We apply Theorem 7.7 to M G. Again (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) follows by Theorem 5.5
+and the fact that both the ∆2 and the ∇2-condition are preserved under equivalence, recall (i) in
+Remark 7.6 for the latter one.
+Concerning (v) and (vi) note that by (ii) in Remark 7.6 both (7.7) and (7.8) are preserved under
+relation ∼ which holds between ωG and ωMG (recall (5.5)).
+□
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+29
+7.3. The ∆2-condition. According to [11, Chapter I, §6, 5] we say that an N-function F satisfies
+the ∆2-condition if
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+F(t)2 ≤ F(kt).
+As stated on [11, p. 41], the ∆2-condition is preserved under relation ∼c and ∆2 holds if and only if
+F α∼cF for some/any α > 1. In fact, this equivalence holds for any non-decreasing F : [0, +∞) →
+[0, +∞) with limt→+∞ F(t) = +∞. By [11, Thm. 6.8] we know that F has ∆2 if and only if the
+complementary function F c has
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+F c(t2) ≤ ktF c(t).
+In view of Corollary 3.11 the associated N-function FM satisfies the ∆2-condition if and only if
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+(ωM(et))2 = ϕωM (t)2 ≤ ϕωM (kt) = ωM(ekt),
+i.e.
+(7.13)
+∃ k > 1 ∃ s0 > 1 ∀ s ≥ s0 :
+ωM(s)2 ≤ ωM(sk).
+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 a sufficient condition to ensure ∆2 in the weight sequence setting. First we prove
+the following technical result:
+Lemma 7.11. Let M ∈ LC be given. Then the following are equivalent:
+(i) The counting function ΣM satisfies
+(7.14)
+∃ K > 0 ∃ t0 > 0 ∀ t ≥ t0 :
+ΣM(et)2 ≤ ΣM(etK),
+i.e. [11, (6.9)] for p = ΣM ◦ exp.
+(ii) The sequence of quotients µ satisfies
+(7.15)
+∃ A > 0 ∃ j0 ∈ N ∀ j ≥ j0 :
+µj2 ≤ µA
+j .
+The proof shows the correspondence A = K.
+Proof. (i) ⇒ (ii) Write s := et and so ΣM(s)2 ≤ ΣM(sK) is satisfied for all s ≥ s0 := et0. Let
+j0 ∈ N>0 be minimal satisfying µj0 ≥ s0. Let j ≥ j0 be such that µj < µj+1 and take s with
+µj ≤ s < µj+1. Then j2 = ΣM(s)2 ≤ ΣM(sK) follows which implies µj2 ≤ sK. In particular, when
+taking s := µj we have shown (7.15) with A := K and all j ≥ j0 such that µj < µj+1. If j ≥ j0 with
+µj = · · · = µj+ℓ < µj+ℓ+1 for some ℓ ∈ N>0, then following the previous step we get µ(j+ℓ)2 ≤ µK
+i
+for all j ≤ i ≤ j + ℓ. Since (j + ℓ)2 ≥ i2 for all such indices i and since by log-convexity j �→ µj is
+non-decreasing we are done for all j ≥ j0.
+(ii) ⇒ (i) Let s ≥ µj0 and so µj ≤ s < µj+1 for some j ≥ j0. Then ΣM(s) = j and sA ≥ µA
+j ≥ µj2
+which implies ΣM(sA) ≥ j2 = ΣM(s)2. Thus (7.14) is shown with K := A and t0 = log(µj0).
+□
+Using this result we get the following:
+Theorem 7.12. Let M ∈ LC be given. Assume that (7.15) holds true. Then the associated N-
+function FM (or equivalently ϕωM ) satisfies the ∆2-condition.
+Proof. By Lemma 7.11 we have that (7.14) is valid. Then in view of (3.17) the function fM
+appearing in the representation (3.1) of FM enjoys (7.14) (with the same K and for all t ≥ t1, t1
+possibly strictly larger than t0 appearing in (7.14)). Thus we can apply [11, Lemma 6.1, Thm. 6.4]
+
+30
+G. SCHINDL
+in order to conclude. (There the estimate in (7.14) is assumed to be strict; however the proof only
+requires ≤.)
+□
+Corollary 7.13. Let G be an N-function. If the associated weight sequence M G (see (5.4)) satisfies
+(7.15), then G and the associated N-function FMG and ϕG (see (5.13)) enjoy the ∆2-condition.
+Recall that in order to verify (7.15) for M G for abstractly given N-functions G the formula 5.10
+can be used.
+Proof. First, Theorem 7.12 applied to M G yields ∆2 for FMG. By Theorem 5.5 we have that
+FMG, G and ϕG are equivalent and since ∆2 is preserved under equivalence we are done.
+□
+Remark 7.14. We gather some observations:
+(i) (7.15) means that the sequence of quotients has to increase ”relatively slowly” and w.l.o.g.
+we can assume A ∈ N≥2 in this condition (since µj ≥ 1 for all j).
+(ii) (7.15) is preserved under relation ∼=: Let M, L ∈ LC such that M ∼= L and assume that M
+has (7.15). Then
+∃ A > 0 ∃ B ≥ 1 ∃ j0 ∈ N ∀ j ≥ j0 :
+1
+B λj2 ≤ µj2 ≤ µA
+j ≤ BλA
+j
+follows and since limj→+∞ λj = +∞ we get B2λA
+j
+≤ λA′
+j
+for any A′ > A and for all
+j ≥ jA′,B sufficiently large. Thus L satisfies (7.15) when choosing A′ > A and restricting
+to j ≥ max{j0, jA′,B}.
+(iii) In [11, (6.10), p. 43] another sufficiency criterion is given. Suppose that an N-function F
+has
+∃ α > 0 ∃ t0 > 0 :
+t �→ log(F(t))
+tα
+is not decreasing on [t0, +∞),
+then F satisfies the ∆2-condition. One verifies that this condition yields F 2∼cF and hence
+this implication holds for any non-decreasing F : [0, +∞) → [0, +∞) with limt→+∞ F(t) =
++∞.
+In the weight sequence setting this expression amounts to the study of
+log(ϕωM (t))
+tα
+=
+log(ωM(et))
+tα
+= log(ωM(s))
+log(s)α
+for all s ≥ s0 = et0. If there exists α > 0 such that s �→ log(ωM(s))
+log(s)α
+is non-decreasing for all large s, then ϕωM satisfies the ∆2-condition and so FM as well.
+Example 7.15. We comment on some examples.
+(∗) All Gevrey-sequences Gs, s > 0, satisfy (7.15): This condition amounts to (j2)s ≤ (js)A
+and so the choices A := 2 and j0 := 0 are sufficient (for any s > 0).
+(∗) If M, L ∈ LC both have (7.15), then also the product sequence M · L: The corresponding
+sequence of quotients is given by the product µ · λ and so (7.15) follows immediately. The
+same implication is not clear for the convolution product M ⋆ L.
+(∗) The sequence M q,2 does not satisfy this requirement because in this case the corresponding
+sequence of quotients is given by (q2j−1)j. Then (7.15) transfers into q2j2−1 ≤ q(2j−1)A but
+which is impossible for any choice A ≥ 1 if j → +∞.
+And in this case also (7.13) is violated: We have, see the proof of Corollary 7.20 for
+more details and citations, that ωMq,2 ∼ ω2 for all q > 1, i.e. ωMq,2(t) = O(ω2(t)) and
+ω2(t) = O(ωMq,2(t)) as t → +∞ for each q > 1 with ω2(t) := max{0, log(t)2}. Fix q > 1
+and so (7.13) gives for some C ≥ 1 and all s ≥ 1
+−1 + C−1 log(s)4 ≤ ωMq,2(s)2 ≤ ωMq,2(sk) ≤ C log(sk)2 + C = Ck2 log(s)2 + C,
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+31
+yielding a contradiction as s → +∞.
+Summarizing, by Examples 7.3, 7.8 and 7.15 we get the following consequences:
+(7.16)
+∆2 ∧ ∇2, ∆2, ∇2 ⇏ ∆2,
+∆2 ⇏ ∆2,
+∇2 ⇏ ∆2.
+Let us study how condition (7.15) is related to moderate growth.
+Lemma 7.16. Let M ∈ LC be given such that (mg) holds and
+(7.17)
+lim inf
+n→+∞(µ2n)1/(n+2) > 1.
+Then M satisfies (7.15) and hence the associated N-function FM (or equivalently ϕωM ) satisfies the
+∆2-condition.
+Proof. First, see e.g. [16, Lemma 2.2] and the citations there, (mg) is equivalent to supj∈N
+µ2j
+µj <
++∞. Consequently, we find some A > 1 such that µ2kj ≤ Akµj for each k, j ∈ N.
+Take now j ∈ N>0 (the case j = 0 is trivial), then 2n ≤ j < 2n+1 for some n ∈ N and 22n ≤ j2 <
+22n+2, so
+µj2 ≤ µ22n+2 = µ2n+22n ≤ An+2µ2n.
+By (7.17) we find n0 ∈ N and δ > 1 such that (µ2n)1/(n+2) ≥ δ for all n ≥ n0. Set B := log(A)
+log(δ) +1 > 1,
+so δ = A1/(B−1) and hence An+2 ≤ µB−1
+2n
+for all n ≥ n0. This implies µj2 ≤ An+2µ2n ≤ µB
+2n ≤ µB
+j
+and so (7.15) is verified (for j0 := 2n0 and choosing B as before).
+□
+We finish by commenting on requirement (7.17):
+(∗) (7.17) is a mild extra growth assumption: It follows when lim infj→∞
+µj
+j > 0 because then
+µj ≥ jǫ for some ǫ > 0 and all j ∈ N. Thus (µ2n)1/(n+2) ≥ 2n/(n+2)ǫ1/(n+2) for all n ∈ N
+and so (µ2n)1/(n+2) > 1 for all n sufficiently large (depending on ǫ).
+(∗) On the other hand note that limn→+∞(2ns)1/(n+2) = 2s > 1 for any s > 0 and hence each
+Gs satisfies (7.17). However, limj→+∞
+js
+j = 0 holds for all 0 < s < 1.
+7.4. The ∆3-condition. An N-function F satisfies the ∆3-condition, see [11, Chapter I, §6, (6.1)],
+if
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+tF(t) ≤ F(kt).
+Since F(t) ≥ t for all large t (recall the second part in (3.3)) we immediately have that ∆2 implies
+∆3; however the converse is not true in general, see [11, p. 41]. It is also known that ∆3 for F
+implies ∆2 for F c, i.e. F has ∇2, see [11, Thm. 6.5].
+∆3 is preserved under equivalence and since tF(t) ≥ F(t) for all t ≥ 1 we get that
+(∗) F has ∆3 if and only if
+(∗) F and t �→ tF(t) are equivalent,
+see [11, Chapter I, §6, 1, p. 35]. Moreover, we have the following reformulation for ∆3:
+Lemma 7.17. Let F be an N-function. Then the following are equivalent:
+(i) F satisfies the ∆3-condition.
+(ii) We have �F∼cF with
+(7.18)
+�F(t) :=
+� |t|
+0
+F(s)ds,
+t ∈ R.
+Consequently, if any of these equivalent conditions holds true, then �F has ∆3, too.
+
+32
+G. SCHINDL
+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
+�F(2t) =
+� 2t
+0
+F(s)ds ≥
+� 2t
+t
+F(s)ds ≥ F(t)t ≥ F(t).
+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
+�F(t) =
+� t
+0
+F(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
+F(k2t) ≥ �F(2t) =
+� 2t
+0
+F(s)ds =
+� t
+0
+F(s)ds +
+� 2t
+t
+F(s)ds ≥ �F(t) + F(t)t ≥ F(t)t,
+i.e. ∆3 with k′ := 2k.
+□
+We apply this characterization to the weight sequence setting.
+Theorem 7.18. Let M ∈ LC be given. Then the following are equivalent:
+(i) The associated N-function FM (equivalently ϕωM ) satisfies the ∆3-condition.
+(ii) �FM∼cFM holds true (with �FM given by (7.18)).
+(iii) �ϕωM ∼cϕωM holds true with
+�ϕωM (t) :=
+� |t|
+0
+ϕωM (s)ds,
+t ∈ R.
+(iv) We have that
+(7.19)
+∃ k > 1 ∃ s0 > 1 ∀ s ≥ s0 :
+ωM(s) ≤ �ωM(s2) ≤ ωM(sk),
+with
+�ωM(t) := �ϕωM (log(t)),
+t ≥ 1,
+�ωM(t) := 0,
+0 ≤ t < 1.
+The function �ωM admits the representation
+(7.20)
+�ωM(s) =
+� |s|
+0
+ωM(u)
+u
+du =
+� |s|
+1
+ωM(u)
+u
+du,
+s ∈ R.
+Proof. (i) ⇔ (ii) follows by Lemma 7.17 applied to FM and the fact that ∆3 is preserved under
+equivalence, see Corollary 3.11.
+(ii) ⇒ (iii) The estimate �ϕωM (2t) ≥ ϕωM (t) for t ≥ 1 holds as in (i) ⇒ (ii) in Lemma 7.17 and for
+the converse we estimate as follows for t sufficiently large:
+�ϕωM (t) =
+� t
+0
+ϕωM (s)ds ≤
+� t
+0
+(FM(s) + C)ds = �FM(t) + Ct ≤ FM(kt) + Ct
+≤ ϕωM (kt) + Ct + D ≤ 2ϕωM (kt) ≤ ϕωM (2kt).
+The first estimate follows for some C ≥ 1 (and all s ≥ 0) by (3.16), the second one since �FM∼cFM
+by assumption, the third one again by (3.16), the fourth since limt→+∞
+ϕωM (t)
+t
+= +∞, and finally
+the last one by convexity and normalization for ϕωM (see (3.2)).
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+33
+(iii) ⇒ (ii) �FM(2t) ≥ FM(t) for all t ≥ 1 is shown in (i) ⇒ (ii) in Lemma 7.17. Conversely, by
+assumption �ϕωM (t) ≤ ϕωM (kt) for some k > 1 and all t(≥ 1) large. Thus for all t sufficiently large:
+�FM(t) =
+� t
+0
+FM(s)ds ≤
+� t
+0
+(ϕωM (s) + D)ds = �ϕωM (t) + Dt ≤ ϕωM (kt) + Dt
+≤ FM(kt) + Dt + C ≤ 2FM(kt) ≤ FM(2kt).
+The first estimate follows by (3.16) (for all s ≥ 0), the second one by assumption, the third one
+again by (3.16), for the fourth estimate we have used the second part in (3.3) and finally the last
+one holds by (3.2).
+(iii) ⇔ (iv) First, for all t ≥ 0 we have
+�ωM(et) = �ϕωM (t) =
+� t
+0
+ωM(es)ds =
+� et
+1
+ωM(u)
+u
+du =
+� et
+0
+ωM(u)
+u
+du,
+because ωM(t) = 0 for 0 ≤ t ≤ 1. Thus �ωM(t) =
+� t
+0
+ωM(u)
+u
+du for all t ≥ 1 and in fact even for all
+t ≥ 0 (since �ωM(t) := 0 for 0 ≤ t ≤ 1). Thus (7.20) is verified.
+Moreover, �ϕωM ∼cϕωM holds if and only if
+(7.21)
+∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 :
+ϕωM (t) ≤ �ϕωM (2t) ≤ ϕωM (kt),
+since the first estimate holds for all t ≥ 1 as mentioned in (ii) ⇒ (iii). (7.21) is obviously equivalent
+to (7.19).
+□
+Using this characterization we give two applications.
+Corollary 7.19. Let M, L ∈ LC be given. If both FM and FL have the ∆3-condition, then FM⋆L,
+too.
+Proof. Recall that M ⋆ L ∈ LC and ωM⋆L = ωM + ωL. Then (7.20) implies �ωM⋆L = �ωM + �ωL and
+by assumption we have (7.19) for both ωM and ωL and so for ωM⋆L as well. Theorem 7.18 yields
+that FM⋆L satisfies the ∆3-condition, too.
+□
+Corollary 7.20. There exist N-functions satisfying ∆2 and ∇2 but not ∆3.
+Proof. Let us consider the sequence(s) M q,n with q, n > 1. As seen in Examples 7.3, 7.8 each
+sequence yields an associated N-function FMq,n satisfying both ∆2 and ∇2.
+We prove now that (7.19) is violated. For this first recall results from [16, Sect. 5.5] and [18,
+Sect.
+3.10]: Let n > 1 be arbitrary but from now on fixed, then each M q,n is an element of
+the weight matrix (i.e. the one-parameter family of weight sequences) associated with the weight
+ωs(t) := max{0, log(t)s}, s > 1, such that 1
+s + 1
+n = 1. Therefore, ωMq,n ∼ ωs for all q > 1, i.e.
+ωMq,n(t) = O(ωs(t)) and ωs(t) = O(ωMq,n(t)) as t → +∞ for each q > 1, see [18, Thm. 4.0.3] and
+[15, Lemma 5.7].
+For all x ≥ 1 we have
+� x
+0
+ωs(t)
+t
+dt =
+� x
+1
+log(t)s
+t
+dt =
+�log(t)s+1
+s + 1
+�t=x
+t=1
+= log(x)s+1
+s + 1
+,
+and so by the above (recall also (7.20))
+∀ q > 1 ∃ D ≥ 1 ∀ x ≥ 1 :
+D−1 log(x)s+1
+s + 1
+− log(x) ≤ �ωMq,n(x) ≤ Dlog(x)s+1
+s + 1
++ D log(x).
+
+34
+G. SCHINDL
+Fix now q > 1 and then, when (7.19) is valid, we obtain
+D−1 log(x2)s+1
+s + 1
+− log(x2) ≤ �ωMq,n(x2) ≤ ωMq,n(xk) ≤ D log(xk)s + D
+=⇒ 2s+1 log(x)s+1 ≤ D2(s + 1)ks log(x)s + D2(s + 1) + 2(s + 1)D log(x).
+But this is impossible as x → +∞ for any choices of D and k.
+□
+By applying Theorem 7.18 to the associated sequence M G and recalling that ∆3 is preserved under
+equivalence we formulate the last statement in this section:
+Corollary 7.21. Let G be an N-function. Then the following are equivalent:
+(i) G satisfies the ∆3-condition.
+(ii) The associated N-function FMG satisfies the ∆3-condition.
+(iii) ϕG (see (5.13)) satisfies the ∆3-condition.
+(iv) The other assertions listed in Theorem 7.18 are valid for the associated sequence M G.
+7.5. The ∆′-condition. According to [11, Chapter I, §5] we say that an N-function F satisfies the
+∆′-condition if
+∃ k > 0 ∃ u0 > 0 ∀ t, s ≥ u0 :
+F(ts) ≤ kF(t)F(s).
+In the weight sequence setting in view of Corollary 3.11 and since ∆′ is preserved under equivalence,
+see [11, Chapter I, §5, p. 30], this condition means that
+(7.22)
+∃ k > 0 ∃ u0 > 0 ∀ t, s ≥ u0 :
+ωM(tlog(s)) ≤ kωM(s)ωM(t).
+By [11, Lemma 5.1] we know that ∆′ implies ∆2 and on [11, p. 30-31] it is shown that in general
+this implication is strict. Moreover, by [11, Thm. 6.6] it follows that if F satisfies ∆2, then F c has
+∆′.
+A direct check of (7.22) seems to be quite technical resp. hardly possible since this estimate is not
+well-related w.r.t. formula (2.5).
+In [11, Thm. 5.1] a sufficiency criterion for ∆′ is shown:
+Theorem 7.22. 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 t0 > 1 such that for every fixed t ≥ t0 the function hf given by hf(s) := f(st)
+f(s) is
+not increasing on [t0, +∞).
+In the weight sequence setting this result takes the following form:
+Corollary 7.23. Let M ∈ LC be given. If ΣM ◦ exp satisfies (∆′
+ΣM ◦exp) then the associated N-
+function FM (resp. equivalently ϕωM ) satisfies the ∆′-condition.
+Proof. In view of (3.17), we get (∆′
+ΣM ◦exp) if and only if (∆′
+fM ) when enlarging t0 sufficiently if
+necessary. Then Theorem 7.22 applied to f = fM and F = FM yields the conclusion.
+□
+However, we show now that in general (∆′
+ΣM ◦exp) fails in the weight sequence setting.
+Proposition 7.24. Let M ∈ LC be given such that 1 ≤ µ1 < µ2 < . . . , i.e.
+the sequence of
+quotients is strictly increasing (see Remark 2.2). Then (∆′
+ΣM ◦exp) (resp. equivalently (∆′
+fM )) is
+violated.
+
+ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS
+35
+Proof. First, with u := es we get ΣM (ets)
+ΣM(es) = ΣM(ut)
+ΣM(u) and hence (∆′
+ΣM ◦exp) precisely means:
+∃ t0 > 1 ∀ t ≥ t0 :
+u �→ ΣM(ut)
+ΣM(u) is not increasing on
+[et0, +∞) =: [u0, +∞).
+Let now t ≥ t0 > 1 be arbitrary but fixed. Then µj0 ≤ u0 < µj0+1 and µk ≤ ut
+0 < µk+1 for some
+(large) j0, k ∈ N>0. Note that k is depending on t and ΣM(ut
+0)
+ΣM(u0) = k
+j0 . Since ut
+0 ≥ u0 we clearly have
+k ≥ j0.
+We show that assuming (∆′
+ΣM ◦exp) yields a contradiction. Let u ∈ [u0, +∞) increase and we split
+the argument in several steps:
+(∗) If u ≥ u0 is given with µj0 < u < µj0+1 and µk < ut < µk+1, then for all u′ ∈ [ǫ − u, u + ǫ]
+with ǫ > 0 sufficiently small the quotient appearing in (∆′
+ΣM ◦exp) remains constant, i.e. in
+this case the crucial expression is locally constant.
+(∗) Even k > j0 is valid: If k = j0, then in order to have (∆′
+ΣM◦exp) we need µj0 ≤ u < ut <
+µj0+1 for all u ≥ u0 with µj0 < u < µj0+1, a contradiction as u → µj0+1.
+(∗) If (∆′
+ΣM ◦exp) holds true, then for all u with µj0 ≤ u0 ≤ u < µj0+1 it follows that ut < µk+1:
+Otherwise, if ut ≥ µk+1, then ΣM(ut)
+ΣM (u) ≥ k+1
+j0
+> k
+j0 = ΣM (ut
+0)
+ΣM (u0), a contradiction.
+(∗) On the other hand, for all u ≥ u0 satisfying µk ≤ ut < µk+1 it is allowed that u ≥ µj0+d
+for some d ∈ N>0: In this case, if µj0+d ≤ u < µj0+d+1 and still µk ≤ ut < µk+1, then
+ΣM(ut)
+ΣM (u) =
+k
+j0+d < k
+j0 .
+(∗) Take u = (µk+1)1/t and so u > u0. We get that µj0+d ≤ u < µj0+d+1 for some d ∈ N. Thus
+ΣM(ut) = k + 1 holds (since µ is strictly increasing!) and ΣM(u) = j0 + d.
+We distinguish: If µj0+d < u < µj0+d+1, then we can find ǫ > 0 sufficiently small to
+ensure u − ǫ > µj0+d and µk ≤ (u − ǫ)t < µk+1. In this case ΣM (ut)
+ΣM(u) =
+k+1
+j0+d >
+k
+j0+d =
+ΣM((u−ǫ)t)
+ΣM (u−ǫ) , a contradiction to (∆′
+ΣM◦exp).
+In particular this case happens if d = 0 because then µj0 ≤ u0 < u by assumption.
+(∗) If now µj0+d = u < µj0+d+1 for some d ≥ 1 and ut = µk+1, then take ǫ > 0 sufficiently
+small to ensure u − ǫ > µj0+d−1 and µk ≤ (u − ǫ)t < µk+1. Both estimates are possible
+since the sequence µ is assumed to be strictly increasing.
+Thus ΣM(ut)
+ΣM(u) =
+k+1
+j0+d ≤
+k
+j0+d−1 = ΣM ((u−ǫ)t)
+ΣM (u−ǫ)
+and which verifies (∆′
+ΣM◦exp) in this case
+because this estimate is equivalent to j0 + d − 1 ≤ k and this is clear since µj0+d = u <
+ut = µk+1.
+(∗) Summarizing all the information, a necessary condition to ensure (∆′
+ΣM ◦exp) is that
+(7.23)
+∃ j0 ∈ N>0 ∃ t0 > 1 ∀ t ≥ t0 ∃ k ∈ N>0, k > j0, ∃ d ∈ N>0 :
+µk+1 = (µj0+d)t.
+(∗) The equality precisely means t =
+log(µk+1)
+log(µj0+d). However, the expression on the right-hand side
+can only take countable many values whereas t is required to belong to an uncountable set
+and therefore (7.23) is impossible.
+□
+We finish with the following consequence showing that in general [11, Thm. 5.1] does not provide
+a characterization.
+Corollary 7.25. There exist N-functions F such that F satisfies the ∆′-condition but (∆′
+f) fails.
+
+36
+G. SCHINDL
+Proof. Consider the function(s) ωs, s > 1. Then ϕωs(t) = ts for all t ≥ 0 and so the ∆′-condition
+is satisfied. Since ωMq,n ∼ ωs for all q > 1 and n > 1 such that 1
+s + 1
+n = 1 we get that ϕωMq,n ∼ ϕωs
+as well (see the proof of (ii) ⇔ (iii) of Theorem 4.1). It is immediate that the ∆′-condition is also
+preserved under ∼. Hence ϕωMq,n and finally FMq,n satisfy the ∆′-condition. However, for any q > 1
+the corresponding sequence of quotients is clearly strictly increasing (recall that µq,n
+j
+= qjn−(j−1)n,
+j ≥ 1) and so by Proposition 7.24 property (∆′
+fMq,n ) fails.
+□
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+G. Schindl: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien,
+Austria.
+Email address: gerhard.schindl@univie.ac.at
+