Solid hulls and cores of classes of weighted entire functions defined in terms of associated weight functions

In the spirit of very recent articles by J. Bonet, W. Lusky and J. Taskinen we are studying the so-called solid hulls and cores of spaces of weighted entire functions when the weights are given in terms of associated weight functions coming from weight sequences. These sequences are required to satisfy certain (standard) growth and regularity properties which are frequently arising and used in the theory of ultradifferentiable and ultraholomorphic function classes (where also the associated weight function plays a prominent role). Thanks to this additional information we are able to see which growth behavior the so-called"Lusky-numbers", arising in the representations of the solid hulls and cores, have to satisfy resp. if such numbers can exist.


Introduction
Spaces of weighted entire functions are defined as follows is usually assumed to be continuous, non-increasing and rapidly decreasing, i.e. lim r→+∞ r k v(r) = 0 for all k ≥ 0. In the recent publications [5] and [2] the authors have studied the so-called solid hulls and solid cores of such spaces, using the identification of f (z) = +∞ j=0 a j z j with its sequence of Taylor-coefficients (a j ) j∈N . For more references and historical background we refer to the introductions of these papers.
First, this question has been motivated by recognizing that the family of weights studied in [5,Sect. 3] corresponds (up to an equivalence of weight functions) to the associated weight functions coming from the Gevrey sequences (p! s ) p∈N , s > 0, arising frequently in the theory of ultradifferentiable and ultraholomorphic functions. Second, by a known characterizing result concerning so-called ultradifferentiable operators from [13] (see Proposition 3.1), classes of weighted entire functions with the weight v(r) = exp(−ω M (r)) are naturally arising also in the ultradifferentiable setting.
Based on these observations the main idea has then been to connect two areas of research and exploit the additional information on the (standard) growth and regularity properties of the underlying weight sequence in order to verify condition (b), more precisely: Apply this approach to compute, via alternative techniques, explicitly the Lusky numbers, get knowledge about their possible growth resp. see and decide that such numbers cannot exist. This has also led us to the following questions: How are the (standard) properties on a weight sequence, which are arising frequently in the ultradifferentiable and ultraholomorphic framework, related to the regularity condition (b)? Do "nice and very regular" sequences, e.g. strongly regular sequences (see [23, 1.1]), always admit the existence of Lusky numbers?
Summarizing, on the one hand we have been able to see several connections between the required growth properties and so the weight sequence setting helps to compute the Lusky numbers resp. to see that such numbers cannot exist. E.g. we have been able to see that too fast increasing weight sequences do not admit the existence of Lusky numbers, see Lemma 5.4. But, on the other hand, we have been able to construct (counter-)examples showing that, roughly speaking, the required resp. desired notions of regularity in the ultradifferentiable and ultraholomorphic world and in the weighted entire world fall apart.
We summarize now the content of this article.
After recalling and collecting all necessary basic definitions of weight sequences and their growth properties in Section 2, we are rewriting a known characterizing result of ultradifferentiable operators in terms of solid hulls and cores, see Section 3. In Section 4 we provide a deep study of the regularity condition (b) in the weight sequence setting, see Lemma 4.1, which enables us to reformulate the main results of the solid hull from [5] and the solid core from [2] involving the "Lusky numbers", see Theorems 4.3 and 4.4.
Then, in Section 4.2 we study the behavior of the Lusky numbers moving from M to its so-called rinterpolating sequence P M,r , which has been used to prove extension results in the ultraholomorphic setting, see [22], [11]. It turns out that this natural construction can be used to determine in Theorem 4.10 the solid hulls and cores of spaces defined in terms of ramified weights, see (4.14). Using the derived formulas for the condition (b), in Section 4.3 we are able to give examples of weight sequences M which have many good growth and regularity properties but do not admit the existence of Lusky numbers. A first new example, related to the weight v(r) = exp(−(log(1+r)) 2 ) which is given by the so-called q-Gevrey sequences (q p 2 ) p∈N , q > 1, is studied in detail in Section 4.4. We are able to compute the Lusky numbers and obtain closed explicit expressions for the solid hull and core, see Corollary 4.14.
In Section 5, by using an auxiliary sequence (δ p ) p , which is measuring the growth of the quotients µ p := M p /M p−1 , we are able to provide a more detailed and precise study of the connection between the growth of M and the (possibly) existing Lusky numbers.
On the one hand, in Corollary 5.7 we derive necessary growth restrictions for sequences of integers which can serve as Lusky numbers in the weight sequence setting, see (5.6). Conversely, in Proposition 5.3 we show that for each sequence of integers (a j ) j satisfying these growth restrictions, we can associate a weight sequence M such that (a j ) j can be used as Lusky numbers for this particular constructed sequence. So, roughly speaking, the mapping M → (a j ) j is surjective and the weight sequence setting is in this sense sufficiently large enough.
In Lemma 5.8 we construct a (counter-)example of a sequence being equivalent to a strongly regular sequence but not admitting Lusky numbers which underlines the different behavior of condition (b) compared with the notion of regularity in the ultradifferentiable and ultraholomorphic setting.
In Section 5.1, as a second concrete example, we compute the Lusky numbers for the Gevrey sequences and are proving the characterization given in [5,Theorem 3.1] by completely different methods.
Section 6 provides some information about the idea when starting with a given abstract weight function v, satisfying the technical condition (6.2), and then considering the associated weight sequence M v analogously as it has been done in the ultradifferentiable setting. The motivation behind this approach is that it shows how the results obtained in this article can help to get information for abstractly given weight functions v as well.
The solid hulls and solid cores of weighted holomorphic functions on the unit disk have been studied in [4] and [2] and it has turned out that also in this situation the regularity condition (b) becomes crucial for a precise description, again in terms of the Lusky numbers. For the sake of completeness, in Section 7 we investigate this notion also in the weight sequence setting, see Theorem 7.4. Unfortunately, here the arising expressions are becoming much more involved (see Lemma 7.3) and concrete computations for the Lusky numbers seem to be much more complicated and involved.
Acknowledgements. The author of this article wishes to thank Prof. José Bonet Solves from the Universitat Politècnica de València and Prof. Wolfgang Lusky from the University of Paderborn for fruitful and interesting discussions and for giving helping and clarifying explanations during the preparation of this work. Moreover, he wants to thank Armin Rainer from the University of Vienna for giving some suggestions during reading a preliminary version. Finally, the author expresses his deepest thank to his friend David N. Nenning, also from the University of Vienna, for carefully reading a preliminary version of this article and for giving several important hints and ideas.

Weight sequences. Given a sequence
holds true which can always be assumed without loss of generality. For any s > 0 we write M s = (M s j ) j∈N for the s-th power of M . In the following we collect several growth and regularity properties for M which will be used later on. These conditions arise frequently and are standard in the ultradifferentiable and ultraholomorphic weight sequence setting (e.g. see [13]).
If M is log-convex and normalized, then both j → M j and j → (M j ) 1/j are non-decreasing and (M j ) 1/j ≤ µ j for all j ∈ N >0 (e.g. see [19,Lemma 2.0.4]).
If the sequence m := Mj j! j is log-convex, then M is called strongly log-convex, denoted by (slc).
In the ultradifferentiable and ultraholomorphic setting the following conditions on M arise frequently in the literature. We say that M has derivation closedness, denoted by (dc), if and M has the stronger condition moderate growth, denoted by (mg), if It is known (e.g. see [18,Lemma 2.2]) that for any given M ∈ LC condition (mg) is equivalent to having sup j∈N µ2j µj < +∞.
A prominent example is G s := (j! s ) j∈N , the so-called Gevrey-sequence of index s > 0. If s > 1, then it is straightforward to check that G s ∈ SR (e.g. see again [23, 1.1]).
Let M ∈ R N >0 (with M 0 = 1) be given. Then the associated function ω M : R ≥0 → R ∪ {+∞} is defined by Moreover under this assumption t → ω M (t) is a continuous nondecreasing function, which is convex in the variable log(t) and tends faster to infinity than any log(t j ), j ≥ 1, as t → +∞. lim j→+∞ (M j ) 1/j = +∞ implies that ω M (t) < +∞ for any finite t which shall be considered as a basic assumption for defining ω M . We refer to [15, Chapitre I] and [13,Def. 3.1].
It is known that the equivalence of ultradifferentiable function classes defined by (Braun-Meise-Taylor) weight functions is characterized by this relation (see [17,Cor. 5.17]) and for any s > 0 the mapping t → t s is equivalent to ω G 1/s .

Solid hulls, cores and ultradifferentiable operators
We recall now briefly the notion of solid sub-and superspaces for spaces of (complex) sequences, e.g. see [1]. Let A be a vector space of sequences, then A is said to be solid if (a j ) j ∈ A does imply (b j ) j ∈ A for all sequences satisfying |b j | ≤ |a j |, ∀j ∈ N. In [1, Lemma 2] it has been shown that for any given sequence space A there does exist s(A), the largest solid subspace (or solid core) of A, and there does exist S(A), the smallest solid superspace (or solid hull), of A. We have We start by recalling [13,Prop. 4.5] where the following characterization has been shown and this has been a main motivation for writing this article: Proposition 3.1. Let M ∈ LC and an entire function P (ξ) = +∞ j=0 a j ξ j be given. (i) The following are equivalent ("Roumieu-type variant"): ). (ii) The following are equivalent ("Beurling-type variant"): In both cases ω M (Lξ) means ω M (L|ξ|) (radial extension to C). The proof shows that in (a) ⇒ (b) we have to replace L by 2L, in (b) ⇒ (a) we can take the same constant L (since we are considering the case dimension d = 1).
An operator of the form P (∂) := +∞ j=0 a j ∂ j with (a j ) j satisfying (i)(a) resp. (ii)(a) above is called an ultradifferential operator of Roumieu-resp. Beurling-type. Hence the previous result motivates also from the point of view of studying problems in the ultradifferentiable setting to consider spaces of weighted entire functions defined as follows: For any M ∈ LC and c > 0 we set For c = 1 we will write v M instead of v M,1 . We call v : [0, +∞) → (0, +∞) a weight function, if v is continuous, non-increasing and rapidly decreasing, i.e. lim r→+∞ r k v(r) = 0 for all k ≥ 0. This is the same notion to be a weight function as it has been considered in [5] and [2].
If we set for c > 0 then Proposition 3.1 tells us that the sequence of Taylor coefficients (a j ) j∈N ∈ C N of any f (z) = +∞ j=0 a j z j ∈ H ∞ vM,c (C) has to belong to U M,c and any sequence (a j ) j∈N ∈ U M,c yields an entire In [5, Proposition 1.1] a first characterization of the solid core s(H ∞ v (C)) has been obtained and this results takes in our setting the following form: Proposition 3.2. Let M ∈ LC be given and c > 0. Then the solid core of H ∞ vM,c is given by |b j |r j < +∞}.

Solid hulls and cores of weighted entire functions
In order to get more precise information on the solid core s(H ∞ vM,c (C)) and the solid hull S(H ∞ vM,c (C)) we have to study the regularity condition (b), see [5, (2.2)] and [2, Definition 2.1]. Recall that, as commented above, the weight z → exp(−|z| s ), s > 0, is corresponding (up to an equivalence) to v G 1/s ,1 , with G 1/s denoting the Gevrey sequence with index 1/s.
. h M is denoting another auxiliary function arising in extension theorems in the ultradifferentiable setting, e.g. see [8] or [7]. The connection to ω M is given by h M (t) = exp(−ω M (1/t)), hence h M is continuous and non-decreasing and h M (t) = 1 for all t > 0 sufficiently large.
c and for all p ≥ 1: Since h M is continuous, also G k M,c is so. With r k,c we are denoting the global maximum point of the function r → G k M,c (r) on [0, +∞). For all values r with µ p < cr ≤ µ p+1 ⇔ µp c < r ≤ µp+1 c , p ∈ N ≥1 , we have r k h M ( 1 cr ) = r k−p Mp c p and so on such an interval we get Next we introduce the following expressions for arbitrary 0 ≤ k < l, analogously to [5, (2.1)]: Lemma 4.1. Let M ∈ LC be given and 0 ≤ k < l (real numbers). Then for any c > 0 we have that Proof. Let 0 ≤ k < l and c > 0 be arbitrary, but from now on fixed. First, if k = 0, then G 0 M,c (r) = h M ( 1 cr ) and so the global maximum of G 0 M,c is attained at any r > 0 with 0 < r ≤ µ1 c . In this case the maximum value of G 0 M,c is equal to 1: Because ω M is vanishing , on such an interval the map r → r k−p Mp c p is strictly increasing for all p ∈ N with p < k and strictly decreasing for all p > k (and also strictly increasing on [0, µ1 c ] as explained before). In . If for given k ∈ N ≥1 we have µ k = µ k+1 , then the maximum value point coincides with µ k c . (However, one can prove that w.l.o.g. we can always assume that p → µ p is strictly increasing by changing to an equivalent sequence.) Thus, in the notation of [5], we can write which should be compared for µ k = k s , s > 1, with [5, p. 596] (the quotient k s corresponds to the Gevrey sequence G s ). Since and similarly In the weight sequence setting it will be convenient to assume that the numbers k and l are integers, say k = a j < a j+1 = l, and then we can simplify the arising expressions to One shall note that both expressions are not depending on the given parameter c > 0 anymore, which justifies the notation A M (a j , a j+1 ) and B M (a j , a j+1 ). For this recall that for any given Hence N ≈M follows and more precisely ν k = µ k c which immediately implies that the parameter c > 0 is cancelling out. Remark 4.2. Due to the discrete behavior of the weight sequence setting, we have some freedom when studying the expressions A M (k, l) and B M (k, l). As commented in the proof of Lemma 4.1, for integers k ∈ N >0 with r k,c := r k c any choice r k ∈ [µ k , µ k+1 ] can be used as a maximal point. Then, again by [15, 1.8. III], we get for any integers 1 ≤ k < l: and again there is no dependence on c > 0 anymore.
With this preparation, in our setting the main result [5, Theorem 2.5] takes the following form: Let M ∈ LC be given. Assume that there exists a strictly increasing sequence (of integers) (a j ) j∈N ≥1 , also called the "Lusky numbers", and constants b and K with K ≥ b > 2 such that i.e. the regularity condition (b) holds true. Then the solid hull of H ∞ vM,c (C) is given by Concerning the solid core we get, by applying [2, Theorem 2.4], the following characterization: Theorem 4.4. Let M ∈ LC be given. Assume that there exists a strictly increasing sequence (of integers) (a j ) j∈N ≥1 and constants b and K with K ≥ b > 2 such that (4.4) (the regularity condition (b) holds true). Then the solid core of H ∞ vM,c (C) is given by or equivalently by Recall that, by the above computations, the (non)-existence of the Lusky numbers is not depending on the parameter c > 0. Moreover, it is straightforward to see the following consequences: i.e. all solid hulls and cores are isomorphic as sets.
Second, let M, N ∈ LC be given with M ≈N and such that M or N satisfy (4.4) for some sequence (a j ) j . Then we get the following isomorphisms as sets: This together with (3.2) imply the second part.
If ω M ∼ω N , then we cannot see directly such an identity. However, when M, N ∈ LC with ω M ∼ω N are given and such that M or N have (mg), then by [13,Proposition 3.6] this property is equivalent for ω M or ω N to have Hence by equivalence ∼ both ω M and ω N have (4.8). Then the proof of [21, Lemma 3.18 (2)] implies M ≈N and Corollary 4.5 can be applied.
To get more precise information on S(H ∞ vM,c (C)) and s(H ∞ vM,c (C)) via Theorems 4.3 and 4.4, more precisely via (4.6) and (4.7), we have to study the arising condition (4.4), i.e. the regularity condition (b). Hence in our setting we are interested in asking: When M ∈ LC is given, can we (always) find a strictly increasing sequence (of integers) (a j ) j∈N ≥1 , denoted by "Lusky numbers", such that there exist constants Given such a sequence (a j ) j , then it is clear that also each forward-shifted sequence a s := (a j+s ) j≥1 , Related to this comment, on [5, p. 598] the following remark (translated into our weight sequence setting) has already been mentioned: Remark 4.6. Let M ∈ LC be given such that there exists a strictly increasing sequence (of integers) (a j ) j∈N ≥1 such that (4.4) holds true. Then we can replace in (4.5) v M,c (r aj ) by v M,c (r aj+1 ) and r aj by r aj+1 resp. in (4.6) a j+1 +1 and µ aj +1 by µ aj+1+1 (and similarly in Theorem 4.4). Note that in the representations of S(H ∞ vM,c (C)) and s(H ∞ vM,c (C)) equivalently we can start the summation at any a j0 , j 0 > 1.
Moreover we see the following observations: (i) It is immediate that we never can choose (a j ) j to be constant and it is also impossible to have a j+1 = a j + 1, since in this case B M (a j , a j+1 ) = 1 follows. Thus a necessary growth assumption is in general this will be not valid anymore since for small s > 0 the first estimate can fail). More generally, if M, N ∈ LC both are satisfying (4.9) with the same sequence (a j ) j , then (a j ) j can be used for M · N := (M p N p ) p as well: The quotients of this sequence are given by µ j ν j and so (iii) For the sequence Q := M/N := (M p /N p ) p we can only show the upper bound: One has ρ j = µ j /ν j and so A Q (a j , a j+1 ) = A M (a j , a j+1 )

4.2.
On the r-interpolating sequence and ramified weights. Given M ∈ LC satisfying (4.9) with (a j ) j , then (ii) in Remark 4.7 provides a first method to construct new sequences still satisfying (4.9) (with the same Lusky numbers (a j ) j ). Another possibility is the following approach: In [22, Lemma 2.3] for given M ∈ LC and r ∈ N ≥1 the so-called r-interpolating sequence P M,r has been introduced as follows, see also [12,Sect. 2.5]: We have P M,r rj = M j for all j ∈ N (i.e. we get P M,1 ≡ M ) and by denoting π M,r k := Hence M ∈ LC if and only if P M,r ∈ LC and by using (4.11) we prove the following observation.
Lemma 4.8. Let M ∈ LC be given and r ∈ N ≥1 . Assume that there exists a sequence of integers (a j ) j satisfying (4.9) (for M ). Then the r-interpolating sequence P M,r does satisfy (4.9) for the choice (ra j ) j , i.e. stretching the Lusky numbers by the factor r.
Similarly we have (µ aj +1 ) aj+1−aj = (π raj +1 ) r(aj+1−aj ) and since the product arising in the numerator of B M (a j , a j+1 ) is precisely the same as in the denominator of A M (a j , a j+1 ) we have B M (a j , a j+1 ) = B P M,r (ra j , ra j+1 ) as well.
In addition, the r-interpolating sequence can be used to get some information on the inclusion of spaces of weighted entire functions w.r.t. relation ω N ω M : Let M, N ∈ LC such that ω N ω M holds true, then ω M (t) ≤ rω N (t) + r for all t ≥ 0 and some r ≥ 1 and w.l.o.g. we can take r ∈ N ≥1 . One has for all t ≥ 0 that which implies Consequently, ω M ∼ω N yields Moreover, the sequence P M,r can be used to see how ramification of the complex variable is translated into the weight sequence setting.
Lemma 4.9. Let M ∈ LC be given, r ∈ N ≥1 and P M,r the according r-interpolating sequence. Then for all t ≥ 0 we get Proof. We use the following integral representation formula for ω M , see [15, 1.8.III]: with Σ M (t) := |{p ∈ N >0 : µ p ≤ t}| = max{p ∈ N >0 : µ p ≤ t}. Then by (4.11) we get for all t ≥ 0: If µ k+1 ≤ t < µ k+2 for some k ∈ N, then Σ M (t) = k + 1 and π M,r rk+j = (µ k+1 ) 1/r ≤ t 1/r < (µ k+2 ) 1/r = π M,r r(k+1)+j for all 1 ≤ j ≤ r does precisely give Σ P M,r (t 1/r ) = r(k + 1). Note that (µ 1 ) 1/r = π M,r j for all 1 ≤ j ≤ r, thus Σ P M,r (t) = 0 for 0 ≤ t < (µ 1 ) 1/r and so precisely for all t satisfying 0 ≤ t r < µ 1 , i.e. for all t satisfying Σ M (t r ) = 0. Using this we can calculate as follows: For any given weight function v and r > 0 we set It is immediate that each v r , r v is again a weight function because lim t→+∞ t k v r (t) = lim t→+∞ t k v(t r ) = lim s→+∞ s k/r v(s) = 0 for all k ≥ 0 and similarly for r v. We set and similarly for r v. When r ∈ N >0 and v ≡ v M,c , then the r-interpolating sequence P M,r can be used to determine the solid hull and solid core of H ∞ w r M,c (C), where we set w r M,c : t → exp(−r −2 ω M ((ct) r )).
Theorem 4.10. Let M ∈ LC be given and r ∈ N ≥1 . Assume that there exists a sequence of integers (a j ) j satisfying (4.9) (the regularity condition (b)), then Proof. First, Lemma 4.9 does imply H ∞ w r M,c (C) = H ∞ v P M,r ,c (C). Lemma 4.8 yields that (ra j ) j satisfies (4.9) for P M,r . Now write P instead of P M,r , then (4.6) yields and (4.7) yields Finally, (4.10) and (4.11) imply that and π raj +1 = (µ aj +1 ) 1/r . For the sake of completeness, we finish this section with the following comments on arbitrary weight functions v and ramification parameters r > 1: (i) For any p > 0, when t p is denoting the global maximum point of t → t p v(t), i.e. t p p v(t p ) ≥ t p v(t) for all t ≥ 0, we get t rp p (v(t p )) r ≥ t rp (v(t)) r for all t ≥ 0. So with s := t r and s p := t r p one has s p p · r v((s p ) 1/r ) ≥ s p · r v(s 1/r ) ⇔ s p p · r v 1/r (s p ) ≥ s p · r v 1/r (s) for all s ≥ 0, i.e. s p is the maximum point of s → s p · r v 1/r (s). (ii) Thus, for the expressions under consideration in the regularity condition (b) for v, see [5, (2.1)], we get for any 0 < m < n: Thus, if r > 1 and v does satisfy the regularity condition (b), then each weight r v 1/r as well with the same sequence of Lusky numbers as for v. (If 0 < r < 1, then in general the estimate from above will fail.) 4.3. First (counter)-example. Next let us see that not each sequence M ∈ LC does automatically have (4.9), i.e. not for each weight sequence there do exist the "Lusky numbers".
Lemma 4.11. There does exist M ∈ LC such that there does not exist a sequence of integers (a j ) j satisfying (4.9).
Proof. We define M in terms of the sequence of quotients (µ p ) p∈N , i.e. M p = p i=1 µ i , p ∈ N, with µ 0 := 1 and µ p → +∞ as p → +∞. We set µ 0 := 1 and let (b j ) j∈N>0 an arbitrary strictly increasing sequence in N such that b 1 = 1. Then put µ p := c j , b j ≤ p < b j+1 , with (c j ) j∈N>0 an arbitrary sequence of positive real numbers satisfying c j < c j+1 and lim j→+∞ cj+1 cj = +∞.
Now take a given strictly increasing sequence (a j ) j (of integers) with a 1 ≥ 1, and we analyze A M (a j , a j+1 ). Given a j , j ≥ 1, we have b lj ≤ a j + 1 < b lj+1 for some l j ∈ N. We distinguish now between two cases: If also b lj ≤ a j+1 + 1 < b lj +1 , then A M (a j , a j+1 ) = 1 since in the numerator and in the denominator we have the same product (c lj ) aj+1−aj . If a j+1 + 1 ≥ b lj +1 , then b lj +kj ≤ a j+1 + 1 < b lj +kj +1 for some k j ∈ N ≥1 . We distinguish now between two cases: Either b lj +kj ≤ a j+1 , then Since d j → +∞ as j → +∞ we see that for any choice (a j ) j at least one estimate in (4.9) has to fail. Choosing the sequences (b j ) j and (c j ) j in a more precise (convenient) way we see that such constructed sequences can satisfy several known growth and regularity properties used in the theory of ultradifferentiable (and ultraholomorphic) functions.
Corollary 4.12. There does exist M ∈ LC satisfying (β 1 ) and (dc), i.e. M is strongly nonquasianalytic and has derivation closedness, but such that there does not exist a sequence of integers (a j ) j satisfying (4.9).
Proof. Let Q ∈ N ≥2 and D > 1 and then take Hence b j+1 = Q j for j ∈ N ≥1 and c 1 = D, c j = D Q j−1 , j ∈ N ≥2 , and we see that cj for all p ∈ N with b j ≤ p < b j+1 and (dc) follows because µ p ≤ D p for all p ∈ N by definition.
However, M does not have (mg) since for this property it is required to have sup p∈N µ2p µp < ∞ (e.g. see [18,Lemma 2.2]) and this property is obviously violated. But in Lemma 5.8 below, by using the techniques developed in Section 5, we provide an analogous example even satisfying (mg).

4.4.
First example -q-Gevrey sequences. The aim is now to give a more concrete example for the representations obtained in Theorems 4.3 and 4.4. We study the (family of) sequences Recall that by [17,Lemma 5.7] and [18, Sect. 5.5] we get that each ω M q is equivalent (w.r.t. ∼) to the weight t → max{0, (log(t)) 2 }, alternatively also to t → (log(1 + t)) 2 . For these weights (which are violating (4.8)) the solid hull and solid core has not been computed before in terms of the Lusky numbers.
The goal is to prove the following result: Proposition 4.13. For M q property (4.9) is satisfied for any sequence (of integers) (a j ) j∈N ≥1 satisfying In particular, when given q > 1, then with c chosen large enough to guarantee q > 2 1/(c(c−1)) (and c ∈ N ≥2 ) we can take a j+1 = a j + c, a 1 := 1, i.e.

Thus Theorems 4.3 and 4.4 give the following characterizations:
Corollary 4.14. Let M q be given with q ≥ 2. Then the solid hull of H ∞ v M q ,c is given by the set or equivalently in a more compact form: Moreover we get Proof. Concerning the solid hull, the first identity is immediate by the possible choice of (a j ) j . For each sequence (b j ) j contained in the first set it is equivalent that there exists some D ≥ 1 such that for all j ≥ 1 we have The solid core follows analogously.

Alternative representations for the regularity condition (b)
In this section we derive alternative useful representations for the expressions A M (a j , a j+1 ) and B M (a j , a j+1 ). As we see this method is convenient to get more information on the existence and (possible) growth behavior of the Lusky numbers a j in the weight sequence setting resp. how the growth of the Lusky numbers is related to the growth of the sequence M . This approach has been inspired by the construction of (counter)-examples (see [ For any given M ∈ LC (recall µ 0 := 1) we put Such a choice of numbers δ p ≥ 0 is always possible since by log-convexity p → µ p is non-decreasing and (M p ) 1/p → +∞ is equivalent to having lim p→+∞ µ p = +∞ ⇔ +∞ j=1 δ j = +∞ (e.g. see [17, p.104]). Conversely, given an arbitrary sequence (δ p ) p≥1 with δ p ≥ 0 for all p ≥ 1 and +∞ j=1 δ j = +∞, then we can introduce a sequence M ∈ LC via (5.1): We have µ 0 = 1 (empty sum) and normalization follows by µ 1 = exp(δ 1 ) ≥ exp(0) = 1. The obtained sequence M ∈ LC is unique since δ 1 = log(µ 1 ) = log(M 1 /M 0 ) = log(M 1 ) determines the first quotient. However, note that (C p M p ) p∈N , C > 0 arbitrary, gives the quotients (Cµ p ) p , hence yields the same sequence (δ p ) p≥2 .
In this notation property (mg) holds if and only if Moreover we summarize: Remark 5.1.
Lemma 5.2. Let (a j ) j∈N>0 be an arbitrary sequence of positive integers with a j+1 ≥ a j + 2 for all j ≥ 1. Then we get Proof. We have Similarly we have By using (5.4) and (5.5) we see in a better and more precise way that and how the size resp. growth of the sequences (a j ) j , the "Lusky numbers", and (δ j ) j is connected. Recall that by Lemma 4.11 and Corollary 4.12 not each M ∈ LC admits a sequence (a j ) j satisfying (4.9), see also (i) in Lemma 5.4 below. But, on the other hand when starting with a sequence (a j ) j satisfying some necessary growth restrictions, then we can show the following.
Proposition 5.3. Let (a j ) j≥1 be a sequence of positive integers such that Then there does exist a sequence M ∈ LC such that the regularity condition (b), i.e. (4.9), holds true for this particular given (a j ) j .
Recall that a j+1 − a j ≥ 2 is a necessary condition to have (4.9) for (a j ) j (see (i) in Remark 4.7). In Lemma 5.6 and Corollary 5.7 we show that also +∞ j=1 1 aj+1−aj = +∞ is necessary for any (a j ) j to be considered in (4.9). Consequently, this result tells us that for each sequence of integers (a j ) j which could be possibly used for Lusky numbers in the weight sequence setting there does exist a sequence M ∈ LC such that (a j ) j are Lusky numbers for any weight v M,c . Proof. We introduce M by the sequence (δ j ) j as follows: This growth shall be compared with (ii) in Lemma 5.4 below and note that 0 which shows lim p→+∞ µ p = +∞, so M ∈ LC is verified. By (5.4) and (5.5) we get for all j ≥ 1 Thus B M (a j , a j+1 ) ≥ exp(1) > 2 holds true and so (4.9) is valid for (a j ) j . Without any further information on the growth of (a j ) j it seems to be not possible to obtain further information on M ; e.g. for having (mg) we have to assume that sup j≥1 a j+1 − a j < +∞ ⇔ inf j≥1 1 aj+1−aj > 0 (see (i) in 5.1 and also (ii) in Lemma 5.4).
(i) If sup p≥1 δ p = +∞, which contradicts (mg) for M , then there does not exist a strictly increasing sequence (of positive integers) (a j ) j satisfying (4.9) for M , see also the example in Lemma 4.11. (ii) If lim sup p→+∞ δ p < +∞, then each sequence (a j ) j enjoying (4.9) (for M ) has to satisfy sup j≥1 δ aj+1+1 (a j+1 − a j ) < +∞, thus the growth rate of (δ p ) p is limiting the maximal admissible growth of (a j ) j .
(iii) If there exist numbers 0 < d 1 ≤ d 2 < +∞ such that d 1 ≤ δ p ≤ d 2 for all p ∈ N >0 , then (4.9) is satisfied for any sequence of integers (a j ) j satisfying C 1 ≤ a j+1 − a j ≤ C 2 for some 2 ≤ C 1 ≤ C 2 and all j ≥ 1 provided that .
Roughly speaking this result shows that too fast increasing weight sequences do not admit the existence of Lusky numbers.
Proof. (i) By assumption we can find for any k ∈ N some p k ∈ N such that δ p k ≥ k. Take now (a j ) j≥1 , then by a j → +∞ we find j k ∈ N such that a j k ≤ p k ≤ a j k +1 −1. Because a j+1 −a j −1 ≥ 1 has to be satisfied (see (i) in Remark 4.7) and since δ p ≥ 0, by using (5.4) we can estimate for A M (a j , a j+1 ) as follows: Finally, if p k = a j k + 1, then In any case, as k → +∞ we see that the upper bound in (4.9) becomes impossible. Note: As seen in (i) in Remark 5.1, the assumption sup p≥1 δ p = +∞ violates (mg).
Finally, we are going to prove an upper growth restriction for the sequence (a j ) j showing that also the second assumption in (5.6) above is really necessary.
The reason for writing "restricting the growth of (a j ) j from above" is that the divergence of the series is not excluding the situation having a subsequence (j k ) k≥1 such that a j k +1 − a j k ≥ c k for all k ≥ 1, with values c k ≥ 1 as large as desired.
Using this new information we can also prove now the following result which shows that even "very regular and nice" sequences considered in the ultradifferentiable and ultraholomorphic setting do not admit automatically a sequence (a j ) j satisfying (4.9). (By "nice" we mean that the ultradifferentiable resp. ultraholomorphic function classes do satisfy several good stability properties.) Lemma 5.8. There does exist M ∈ LC having (mg) and (β 1 ) (i.e. M is equivalent to N ∈ SR) but such that there does not exist a sequence (of positive integers) (a j ) j≥1 satisfying (4.9).
Proof. Let M be defined by its sequence of quotients (µ p ) p≥1 as follows: We put µ 0 := 1 and with (c i ) i a sequence of strictly increasing positive real numbers such that ci > 2 and sup i∈N ci+1 ci < +∞. A straight-forward choice would be c i := Q i , with 2 < Q < +∞.
Case II: Second, if a j + 1 < 2 lj +1 , then a j+1 ∈ I lj +1 has to be valid for all j large: If a j+1 ∈ I lj +i with i ≥ 2, then we would have by (4.3) that which tends to infinity as j → +∞. Thus the upper estimate in (4.9) fails for large j. Similarly we see that in Case I above we have a j+1 ∈ I lj +2 because i ≥ 3 would imply B M (a j , a j+1 ) ≥ cj+2 cj+1 2 l j +3 −2 l j +2 > 2 2 l j +2 , again contradicting the upper estimate in (4.9) as j → +∞.
In the first case, when a j + 1 = 2 lj +1 and a j+1 ∈ I lj +2 , then again by (4.3) we have and in the second one, when a j + 1 < 2 lj+1 and a j+1 ∈ I lj +1 , then we get Thus, in order to guarantee the upper estimate in (4.9), we have to require that a j+1 − 2 lj+2 ≤ d and a j+1 − 2 lj+1 ≤ d for some d ∈ N >0 not depending on j ≥ 1. But then, since a j+1 , a j+2 ∈ I lj +2 in the first, and a j+1 , a j+2 ∈ I lj +1 in the second case, is not possible we have a j+2 − a j+1 ≥ Hence, when starting with a given a j ∈ I lj with some l j ∈ N >0 , then by repeating the previous arguments we have shown that, as the number d must not depend on j ≥ 1 and 2 l+1 −2 l = 2 l → +∞ as l → +∞, for all i ∈ N >0 large enough only the second case occurs. More precisely, in order not to violate (4.3), for all i ≥ i 0 , i 0 large enough, we have a j+i ∈ I lj +i , a j+i+1 ∈ I lj +i+1 and a j+i ≤ 2 j+i + d, hence a j+i+1 − a j+i ≥ 2 lj+i − d > 0. Finally we can estimate as follows: for some 1 < α < 2 and all i ≥ i 0 by increasing i 0 if necessary. But this estimate contradicts Corollary 5.7.
We close this section with the following observations for M ∈ LC.
(i) If there does exist a sequence (of integers) (a j ) j≥1 such that a j+1 ), B M (a j , a j+1 )}, then by iteration we can get the first estimate in (4.9): One has and similarly Iterating these estimates n times, n ∈ N >0 chosen minimal such that b n > 2, we are done. Consequently, the choice a ′ j := a nj yields a sequence satisfying the first estimate in (4.9). However, if a j+1 ), B M (a j , a j+1 )} ≤ K, then in general it is not clear that the upper estimate in (4.9) holds true for the sequence a ′ j := a nj .
(ii) Such an iteration, yielding a stretching of the Lusky numbers, is always possible if for (4.9) any sequence (a j ) j satisfying C 1 ≤ a j+1 − a j ≤ C 2 for some 2 ≤ C 1 ≤ C 2 can be used (e.g. for the q-Gevrey sequences, see Proposition 4.13): In this case a ′ j := a nj satisfies nC 1 ≤ a ′ j+1 − a ′ j ≤ nC 2 for all j ≥ 1 and is still "admissible". (iii) But in general, if (a j ) j≥1 is a strictly increasing sequence of positive integers satisfying (4.9), then an arbitrary subsequence a ′ k := (a j k ) k≥1 cannot be considered for satisfying (4.9). More precisely, given such (a j ) j≥1 , then a ′ j := a 2 j can never be considered: By using the necessary assumption 2 ≤ a j+1 − a j we obtain l it follows that C −1 p s ≤ µ s p ≤ Cp s for some C ≥ 1 and all p ∈ N, hence M s is equivalent to the Gevrey sequence G s = (p! s ) p∈N .
The equivalence between M s and G s does imply (see Corollary 4.5) , and ω M s (t) = O(ω G s (Ct)) and ω G s (t) = O(ω M s (Ct)) for some C ≥ 1 as t → +∞. Since each power weight t → at 1/s , s, a > 0 arbitrary, does have ω(2t) = O(ω(t)) (which is a standard assumption in the theory of ultradifferentialbe functions, see [6]) we get that ω M s ∼ω G s ∼t → at 1/s for each a > 0. Moreover, each arising weight does have (4.8) as well. Altogether we have obtained: ∀ a > 0 ∀ b, c, d > 0: with w 1/s d,a denoting the weight t → exp(−a(dt) 1/s ). Thus Proposition 5.9 gives an alternative proof for the representation obtained in [5, Theorem 3.1] (see also Remark 4.6). Proof. We see that c ≥ 1 and cs = s if s ≥ 1 and cs ≥ 1 ⇔ c ≥ 1 s if 0 < s < 1, hence in any case cs ≥ 1 holds true. Moreover we have cs ≤ ( 1 s + 1)s = 1 + s.

Then (5.4) turns into
In order to simplify the computations and since the first three claims below hold true for a j := c(j + 2) 2 we use now this sequence in the following computations. First, we have a j+1 − a j = c(j + 3) 2 − c(j + 2) 2 = c(2j + 5), j ≥ 1, and start with A M (a j , a j+1 ).
Claim III: B M (a j , a j+1 ) ≤ exp(8cs) ≤ exp(8(1 + s)) for all j ≥ 1. Since the second summand is always ≤ 0 we have to study the first one. We estimate as follows: for all j ≥ 1 with the choice A 1 := 8sc = 2A by Claim II above.

From weight functions to weight sequences
The aim of this section is to see how the weight sequence setting is becoming meaningful when starting with an abstractly given weight function v : [0, +∞) → (0, +∞) in the weighted holomorphic setting, i.e. v is continuous, non-increasing and rapidly decreasing. We call v normalized, when v(t) = 1 for all t ∈ [0, 1] which can be assumed w.l.o.g.: Otherwise replace v by w such that w is normalized and v(t) = w(t) for all t ≥ t 0 > 1, which yields H ∞ v (C) = H ∞ w (C). Lemma 6.1. Let v be a normalized weight function. Then  Proof. (i) follows immediately by definition and the properties for v.
(ii) We have to show that for each ε > 0 there does exist D ε ≥ 1 such that log(t) ≤ εω v (t) + D ε for all t > 0, which is equivalent to t ≤ (v(t)) −ε exp(D ε ) and so to v(t)t 1/ε ≤ exp(D ε /ε). This holds true because v is rapidly decreasing.
The next result shows that for given weight functions v satisfying all requirements from before we can associate a weight sequence such that the corresponding weight functions v and v M can be compared. Apart from the power-weights corresponding to the Gevrey-sequences we are pointing out the following two examples: First recall that, as commented in Sect. 4.4 above for each q-Gevrey sequence M q = (q p 2 ) p∈N , q > 1, we have that ω M q is equivalent to the weight max{0, log(t) 2 }. Hence this case corresponds on the unit disk (up to equivalence of weight functions) to v(r) := exp(−(log( 1 1 − r )) 2 ) = exp(−(log(s)) 2 ) = 1 s log(s) = (1 − r) − log(1−r) , 0 ≤ r < 1, by taking s := 1 1−r (and so s ∈ [1, +∞)). Second, for given α > 0 on D we can consider the "standard weight" v α (z) := (1 − |z|) α , see [2,Remark 2.2]. In this case we get hence w(s) = α log(s) for some weight w (taking again s := 1 1−r ). Thus, in order to apply the weight sequence approach, by (7.1) we are searching for some M ∈ LC such that ω M ∼t → α log(1 + t), equivalently that ω M ∼t → log(1 + t), t ≥ 0. However, we show now that we cannot find such a sequence M , hence these "standard weights" on the unit disk cannot be considered within the set LC.
(ii) Let M = (M p ) p be a sequence with 1 = M 0 satisfying ∃ q 0 ∈ N >0 ∀ p > q 0 : M p = +∞, and such that 1 ≤ µ p ≤ µ p+1 for 1 ≤ p ≤ q 0 with µ q0+1 = Mq 0 +1 Mq 0 = +∞, i.e. M p ∈ R >0 and M is log-convex for only finitely many indices p. Then ω M does have (7.2), when ω M is defined via (2.1) by using the conventions 1 +∞ = 0 and log(0) = −∞. The first part means that any weight a log(1 + t), a > 0, cannot occur in the equivalence class of associated weight functions coming from (standard) weight sequences. However, the second part yields that for "exotic sequences", i.e. M p = +∞ for all but finitely many p ∈ N, each weight a log(1 + t), a > 0, is equivalent to ω M .
(i) For all small k > 0 satisfying s k,1,c = k c(k+1) ≤ 1 c − 1 cµ1 ⇔ k ≤ µ 1 − 1, hence s k,p,c ∈ I 0,c for all p ≥ 1 as well holds true, each function f k,p,c is (strictly) decreasing on I p,c , p ≥ 1. Thus r k,D,c = 1 c − 1 cµ1 for all such small k > 0. (Recall that by normalization we have µ 1 ≥ 1.) (ii) For all larger values k > 0 we have that s k,q,c ∈ I p,c for at least one pair p, q ≥ 1.
If s k,q,c ∈ I p,c with q > p ≥ 1, then s k,q,c < s k,p,c and f k,p,c is (strictly) increasing at s k,q,c . Since f k,p,c ≡ G k M,D,c | Ip,c we get that also G k M,D,c is strictly increasing at s k,q,c . Similarly, if 1 ≤ q < p, then f k,p,c is (strictly) decreasing at s k,q,c and so G k M,D,c as well. Finally, if s k,p,c ∈ I p,c for some p ≥ 1, then s k,p,c is the maximum point for f k,p,c on [0, 1 c ) resp. for G k M,D,c on I p,c . (iii) Note that we always find at least one such value p ≥ 1 because lim p→+∞ s k,p,c = 0 and lim p→+∞