1 Introduction

Spaces of ultradifferentiable functions are sub-classes of smooth functions with certain restrictions on the growth of their derivatives. Two classical approaches are commonly considered; either the restrictions are expressed by means of a weight sequence \(M=(M_p)_{p\in {\mathbb {N}}}\), also called Denjoy–Carleman classes (e.g., see [10]), or by means of a weight function \(\omega \) also called Braun–Meise–Taylor classes; see [3]. In this work, we are exclusively dealing with the weight sequence approach.

More precisely (in the one-dimensional case) for each compact set K, the set

$$\begin{aligned} \left\{ \frac{f^{(p)}(x)}{h^pM_p}\,\,:\,\, p\in {\mathbb {N}},\,\, x\in K \right\} \end{aligned}$$
(1.1)

is required to be bounded. Naturally, one can consider two different types of spaces: For the Roumieu type, the boundedness of the set in (1.1) is required for some \(h>0\), whereas for the Beurling type, it is required for all \(h>0\).

In the literature, standard growth and regularity conditions are assumed for M; roughly speaking, one is interested in sufficiently fast growing sequences M to ensure that \(M_p\) is (much) larger than p! for all \(p\in {\mathbb {N}}\). This is related to the fact that for such sequences, the corresponding function spaces are lying between the real-analytic functions and the class of smooth functions. Therefore, classes being (strictly) smaller than the spaces corresponding to the sequence \((p!)_{p\in {\mathbb {N}}}\) are excluded due to these basic requirements. Moreover, the regularity condition log-convexity, i.e., (M.1) in [10], is more or less standard and even \(M\in {\mathcal{L}\mathcal{C}}\) is basic; see Sect. 2.2 for the definition of this set. (Formally, if log-convexity for M fails, then one might avoid technical complications by passing to its so-called log-convex minorant.) The analogous notion of log-concavity has not been used in the ultradifferentiable setting.

The (most) well-known examples are the so-called Gevrey sequences of type \(\alpha >0\) with \(G^{\alpha }_p:= p!^{\alpha }\) (or equivalently use \(M^{\alpha }_p:=p^{p\alpha }\)) and this one-parameter family illustrates this behavior when considering different values of the crucial parameter \(\alpha \): usually, in the literature, one is interested in \(\alpha >1\) and the limiting case \(\alpha =1\) for the Roumieu type precisely yields the real-analytic functions. Indices \(0<\alpha <1\) give a non-standard setting and the corresponding function classes are tiny (”small Gevrey setting”). At this point, let us make aware that we are using for the sequence M the notation “including the factorial term” in (1.1), since, in the literature, occasionally authors also deal with \(\frac{f^{(p)}(x)}{h^pp!M_p}\), e.g., in [24], and so M in these works corresponds to the sequence m in the notation used in this paper (see Example 2.5). On the other hand, the crucial conditions on the sequences appearing in this work illustrate the relevance of the difference between m and M; see the assumptions in Sect. 4.4.

However, from an abstract mathematical point of view, it is interesting and makes sense to study also ultradifferentiable classes defined by non-standard/small sequences and to ask the following questions:

  1. (i)

    What are the differences between such small classes and spaces defined in terms of “standard sequences”?

  2. (ii)

    What is the importance of such small spaces and for which applications can they be useful?

  3. (iii)

    Can we transfer known results from the standard setting, e.g., the characterization of inclusion relations for function spaces in terms of the corresponding weight sequences, to small spaces?

  4. (iv)

    Does there exist a close resp. canonical relation between standard and non-standard sequences, or more precisely: Can one construct from a given standard sequence a small one (and vice versa)?

The aim of this article is to focus on these problems. Indeed, question (iv) has served as the main motivation and the starting point for writing this work. Very recently, in [5], we have introduced the notion of the dual sequence. For each given standard M, e.g., if \(M\in {\mathcal{L}\mathcal{C}}\), it is possible to introduce the dual sequence D; see Appendix A for precise definitions and citations. In [5], this notion and the relation between M and D have been exclusively studied by considering growth and regularity indices (which are becoming relevant in the so-called ultraholomorphic setting). The aim is now to study further applications of this new notion and the conjecture is that for “nice large standard sequences” M, the corresponding dual sequence D is a “convenient small one” which allows to study a non-standard setting.

The literature concerning small ultradifferentiable function classes is non-exhaustive, and to the best of our knowledge, we have only found works by M. Markin treating the small Gevrey setting; see [12, 13], and [14]. More precisely, the goal there has been: given a Hilbert space H and a normal (unbounded) operator A on H, then consider the associated evolution equation

$$\begin{aligned} y'(t) = A y(t), \end{aligned}$$

and one asks the following question: Is a priori known smoothness of all (weak) solutions of this equation sufficient to get that the operator A is bounded? Markin has studied this problem within the small Gevrey setting, i.e., it has been shown that if each weak solution of this evolution equation belongs to some small Gevrey class, then the operator A is bounded. To proceed, Markin considers (small) Gevrey classes with values in a Hilbert space. Based on this knowledge, one can then study if, for different small classes, Markin’s results also apply and if one can generalize resp. strengthen his approach.

The paper is structured as follows: In Sect. 2, we introduce the notion of the so-called conjugate sequence \(M^{*}\) (see (2.3)), we collect and compare all relevant (non-)standard growth and regularity assumptions on M and \(M^{*}\), and we define the corresponding function classes.

In Sect. 3, we treat question (i) and show that classes defined by small sequences M are isomorphic (as locally convex vector spaces) to weighted spaces of entire functions; see the main result Theorem 3.4. Thus, we are generalizing the auxiliary result [14, Lemma 3.1] from the small Gevrey setting; see Sect. 3.2 for the comparison. The crucial weight in the weighted entire setting is given in terms of the so-called associated weight \(\omega _{M^{*}}\) (see Sect. 2.7) and so expressed in terms of the conjugate sequence \(M^{*}\).

As an application of this statement, concerning problem (iii) above, we characterize for such small classes the inclusion relations in terms of the defining (small) sequences; see Theorem 3.9. This is possible by combining Theorem 3.4 with the recent results for the weighted entire setting obtained by the second author in [20].

Section 4 is dedicated to problem (ii) and the study resp. the generalization of Markin’s results. We introduce more general families of appropriate small sequences and extend the sufficiency testing criterion for the boundedness of the operator A to these sets.

Finally, in Appendix A, we focus on (iv) and show that dual sequences are serving as examples for non-standard sequences, and hence, this framework is establishing a close relation between known examples for weight sequences in the literature and small sequences for which the main results in this work can be applied (see Theorem A.7 and Corollary A.8).

2 Definitions and notations

2.1 Basic notation

We write \({\mathbb {N}}:=\{0,1,2,\dots \}\) and \({\mathbb {N}}_{>0}:=\{1,2,\dots \}\). Given a multi-index \(\alpha =(\alpha _1,\dots ,\alpha _d)\in {\mathbb {N}}^d\), we set \(|\alpha |:=\alpha _1+\dots +\alpha _d\). With \(\mathcal {E}\), we denote the class of all smooth functions and with \(\mathcal {H}(\mathbb {C})\) the class of entire functions.

2.2 Weight sequences

Let \(M=(M_p)_p\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), and we introduce also \(m=(m_p)_p\) defined by \(m_p:=\frac{M_p}{p!}\) and \(\mu =(\mu _p)_p\) by \(\mu _p:=\frac{M_p}{M_{p-1}}\), \(p\ge 1\), \(\mu _0:=1\). M is called normalized if \(1=M_0\le M_1\) holds true. If \(M_0=1\), then \(M_p=\prod _{i=1}^p\mu _i\) for all \(p\in {\mathbb {N}}\).

M is called log-convex, denoted by \((\text {lc})\) and abbreviated by (M.1) in [10], if

$$\begin{aligned} \forall \;p\in {\mathbb {N}}_{>0}:\;M_p^2\le M_{p-1} M_{p+1}. \end{aligned}$$

This is equivalent to the fact that \(\mu \) is non-decreasing. If M is log-convex and normalized, then both M and \(p\mapsto (M_p)^{1/p}\) are non-decreasing. In this case, we get \(M_p\ge 1\) for all \(p\ge 0\) and

$$\begin{aligned} \forall \;p\in {\mathbb {N}}_{>0}:\;\;\;(M_p)^{1/p}\le \mu _p. \end{aligned}$$
(2.1)

Moreover, \(M_pM_q\le M_{p+q}\) for all \(p,q\in {\mathbb {N}}\).

In addition, for \(M=(M_p)_p\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), it is known that

$$\begin{aligned} \liminf _{p\rightarrow +\infty }\mu _p\le \liminf _{p\rightarrow +\infty }(M_p)^{1/p}\le \limsup _{p\rightarrow +\infty }(M_p)^{1/p}\le \limsup _{p\rightarrow +\infty }\mu _p. \end{aligned}$$
(2.2)

For convenience, we introduce the following set of sequences:

$$\begin{aligned} {\mathcal{L}\mathcal{C}}:=\left\{ M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}:\;M\;\text {is normalized, log-convex},\;\lim _{p\rightarrow +\infty }(M_p)^{1/p}=+\infty \right\} . \end{aligned}$$

We see that \(M\in {\mathcal{L}\mathcal{C}}\) if and only if \(1=\mu _0\le \mu _1\le \dots \) with \(\lim _{p\rightarrow +\infty }\mu _p=+\infty \) (see, e.g., [17, p. 104]) and there is a one-to-one correspondence between M and \(\mu =(\mu _p)_p\) by taking \(M_p:=\prod _{i=0}^p\mu _i\).

M has moderate growth, denoted by \((\text {mg})\), if

$$\begin{aligned} \exists \;C\ge 1\;\forall \;p,q\in {\mathbb {N}}:\;M_{p+q}\le C^{p+q+1} M_p M_q. \end{aligned}$$

A weaker condition is derivation closedness, denoted by \((\text {dc})\), if

$$\begin{aligned} \exists \;A\ge 1\;\forall \;p\in {\mathbb {N}}:\;M_{p+1}\le A^{p+1} M_p\Leftrightarrow \mu _{p+1}\le A^{p+1}. \end{aligned}$$

It is immediate that both conditions are preserved under the transformation \((M_p)_p\mapsto (M_pp!^s)_p\), \(s\in {\mathbb {R}}\) arbitrary. In the literature (mg) is also known under stability of ultradifferential operators or (M.2) and (dc) under \((M.2)'\); see [10].

M has \((\beta _1)\) (named after [16]) if

$$\begin{aligned} \exists \;Q\in {\mathbb {N}}_{>0}:\;\liminf _{p\rightarrow +\infty }\frac{\mu _{Qp}}{\mu _p}>Q, \end{aligned}$$

and \((\gamma _1)\) if

$$\begin{aligned} \sup _{p\in {\mathbb {N}}_{>0}}\frac{\mu _p}{p}\sum _{k\ge p}\frac{1}{\mu _k}<+\infty . \end{aligned}$$

In [16, Proposition 1.1], it has been shown that for \(M\in {\mathcal{L}\mathcal{C}}\), both conditions are equivalent, and in the literature, \((\gamma _1)\) is also called ”strong non-quasianalyticity condition”. In [10], this is denoted by (M.3). (In fact, there \(\frac{\mu _p}{p}\) is replaced by \(\frac{\mu _p}{p-1}\) for \(p\ge 2\) but which is equivalent to having \((\gamma _1)\).)

A weaker condition on M is \((\beta _3)\) (named after [22], see also [2]) which reads as follows:

$$\begin{aligned} \exists \;Q\in {\mathbb {N}}_{>0}:\;\liminf _{p\rightarrow +\infty }\frac{\mu _{Qp}}{\mu _p}>1. \end{aligned}$$

For two weight sequences \(M=(M_p)_{p\in {\mathbb {N}}}\) and \(N=(N_p)_{p\in {\mathbb {N}}}\), we write \(M\le N\) if \(M_p\le N_p\) for all \(p\in {\mathbb {N}}\) and \(M{\preccurlyeq }N\) if

$$\begin{aligned} \sup _{p\in {\mathbb {N}}_{>0}}\left( \frac{M_p}{N_p}\right) ^{1/p}<+\infty . \end{aligned}$$

M and N are called equivalent, denoted by \(M{\approx }N\), if

$$\begin{aligned} M{\preccurlyeq }N\quad \text {and}\quad N{\preccurlyeq }M. \end{aligned}$$

Finally, we write \(M{\vartriangleleft }N\), if

$$\begin{aligned} \lim _{p\rightarrow +\infty }\left( \frac{M_p}{N_p}\right) ^{1/p}=0. \end{aligned}$$

In the relations above, one can replace M and N simultaneously by m and n, because \(M{\preccurlyeq }N\Leftrightarrow m{\preccurlyeq }n\) and \(M{\vartriangleleft }N\Leftrightarrow m{\vartriangleleft }n\).

For any \(\alpha \ge 0\), we set

$$\begin{aligned} G^{\alpha }:=(p!^{\alpha })_{p\in {\mathbb {N}}}. \end{aligned}$$

Therefore, for \(\alpha >0\), this denotes the classical Gevrey sequence of index/order \(\alpha \).

2.3 Classes of ultradifferentiable functions

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), \(U\subseteq {\mathbb {R}}^d\) be non-empty open, and for \(K\subseteq {\mathbb {R}}^d\) compact, we write \(K\subset \subset U\) if \(\overline{K}\subseteq U\), i.e., K is in U relatively compact. We introduce now the following spaces of ultradifferentiable function classes. First, we define the (local) classes of Roumieu type by

$$\begin{aligned} \mathcal {E}_{\{M\}}(U):=\{f\in \mathcal {E}(U):\;\forall \;K\subset \subset U\;\exists \;h>0:\;\Vert f\Vert _{M,K,h}<+\infty \}, \end{aligned}$$

and the classes of Beurling type by

$$\begin{aligned} \mathcal {E}_{(M)}(U):=\{f\in \mathcal {E}(U):\;\forall \;K\subset \subset U\;\forall \;h>0:\;\Vert f\Vert _{M,K,h}<+\infty \}, \end{aligned}$$

where we denote

$$\begin{aligned} \Vert f\Vert _{M,K,h}:=\sup _{\alpha \in {\mathbb {N}}^d,x\in K}\frac{|f^{(\alpha )}(x)|}{h^{|\alpha |} M_{|\alpha |}}. \end{aligned}$$

For a sufficiently regular compact set K (e.g., with smooth boundary and such that \(\overline{K^\circ }=K\))

$$\begin{aligned} \mathcal {E}_{M,h}(K):=\{f\in \mathcal {E}(K): \Vert f\Vert _{M,K,h}<+\infty \} \end{aligned}$$

is a Banach space, and so, we have the following topological vector spaces:

$$\begin{aligned} \mathcal {E}_{\{M\}}(K):=\underset{h>0}{\varinjlim }\;\mathcal {E}_{M,h}(K), \end{aligned}$$

and

$$\begin{aligned} \mathcal {E}_{\{M\}}(U)=\underset{K\subset \subset U}{\varprojlim }\;\underset{h>0}{\varinjlim }\;\mathcal {E}_{M,h}(K)=\underset{K\subset \subset U}{\varprojlim }\;\mathcal {E}_{\{M\}}(K). \end{aligned}$$

Similarly, we get

$$\begin{aligned} \mathcal {E}_{(M)}(K):=\underset{h>0}{\varprojlim }\;\mathcal {E}_{M,h}(K), \end{aligned}$$

and

$$\begin{aligned} \mathcal {E}_{(M)}(U)=\underset{K\subset \subset U}{\varprojlim }\;\underset{h>0}{\varprojlim }\;\mathcal {E}_{M,h}(K)=\underset{K\subset \subset U}{\varprojlim }\;\mathcal {E}_{(M)}(K). \end{aligned}$$

The spaces \(\mathcal {E}_{\{M\}}(U)\) and \(\mathcal {E}_{(M)}(U)\) are endowed with their natural topologies w.r.t. the above representations. We write \(\mathcal {E}_{[M]}\) if we mean either \(\mathcal {E}_{\{M\}}\) or \(\mathcal {E}_{(M)}\) but not mixing the cases. We omit writing the open set U if we do not want to specify the set where the functions are defined and formulate statements on the level of classes.

Usually, one only considers real or complex-valued functions, but we can analogously also define classes with values in Hilbert or even Banach spaces (for simplicity, we assume in this case that the domain U is contained in \(\mathbb {R}\)) by simply using

$$\begin{aligned} \Vert f\Vert _{M,K,h}:= \sup _{p \in \mathbb {N}, x\in K}\frac{\Vert f^{(p)}(x)\Vert }{h^p M_p}, \end{aligned}$$

in the respective definition, i.e., only the absolute value of \(f^{(p)}(x)\) is replaced by the norm in the Banach space. Observe that the (complex) derivative of a function with values in a Banach space is defined in complete analogy to the complex-valued case. If we want to emphasize that the codomain is a Hilbert (or Banach) space H, we write \(\mathcal {E}_{[M]}(U,H)\). In analogy to that also \(\mathcal {E}(U,H)\) shall denote the H-valued smooth functions on U.

Remark 2.1

Let \(M,N\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), the following is well known, see, e.g., [17, Prop. 2.12]:

\((*)\):

The relation \(M{\vartriangleleft }N\) implies \(\mathcal {E}_{\{M\}}\subseteq \mathcal {E}_{(N)}\) with continuous inclusion. Similarly, \(M{\preccurlyeq }N\) implies \(\mathcal {E}_{[M]}\subseteq \mathcal {E}_{[N]}\) with continuous inclusion.

\((*)\):

If \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) is log-convex (and normalized) and \(\mathcal {E}_{\{M\}}({\mathbb {R}})\subseteq \mathcal {E}_{(N)}({\mathbb {R}})\) (as sets), then by the existence of so-called M-characteristic functions, see [17, Lemma 2.9], [25, Thm. 1] and the proof in [21, Prop. 3.1.2], we get \(M{\vartriangleleft }N\) as well.

2.4 Ultradifferentiable classes of entire functions

We shall tacitly assume that a holomorphic function on (an open subset of) \(\mathbb {C}\) may have values in a Hilbert or even Banach space. The main theorems of one variable complex analysis (Cauchy integral formula, power series representation of holomorphic functions, etc.) hold mutatis mutandis, by virtue of the Hahn–Banach theorem, just as in the complex-valued case.

First, let us recall that for any open (and connected) set \(U \subseteq \mathbb {R}\) the space \(\mathcal {E}_{(G^1)}(U, H)\) can be identified with \(\mathcal {H}(\mathbb {C}, H)\), the class of entire functions, and both spaces are isomorphic as Fréchet spaces. The isomorphism \(\cong \) is given by

$$\begin{aligned} E:\mathcal {E}_{(G^1)}(U,H) \rightarrow \mathcal {H}(\mathbb {C},H),\quad f \mapsto E(f):=\sum _{k=0}^{+\infty }\frac{f^{(k)}(x_0)}{k!}z^k, \end{aligned}$$

where \(x_0\) is any fixed point in U. The inverse is given by restriction to U, and its continuity follows easily from the Cauchy inequalities.

We apply the observation from Remark 2.1 to \(N\equiv G^1\).

Lemma 1.2

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given.

  1. (i)

    If \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\), then \(\mathcal {E}_{\{M\}}\subseteq \mathcal {E}_{(G^1)} (\cong \mathcal {H}(\mathbb {C}))\) with continuous inclusion.

  2. (ii)

    Let M be log-convex and normalized. Assume that

    $$\begin{aligned} \mathcal {E}_{\{M\}}({\mathbb {R}})\subseteq \mathcal {E}_{(G^1)}({\mathbb {R}}) (\cong \mathcal {H}(\mathbb {C})) \end{aligned}$$

    holds (as sets), then \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\) follows. In particular, this implication holds for any \(M\in {\mathcal{L}\mathcal{C}}\).

Moreover, in the situation of Lemma 2.2, the inclusion always has to be strict. Thus, spaces \(\mathcal {E}_{[M]}\) for sequences with \(\lim _{p\rightarrow +\infty }m_p^{1/p}=0\) form classes of entire functions. Subsequently, we show that those spaces are weighted classes of entire functions and the weight is given by the associated weight function of the conjugate weight sequence. We thoroughly define and investigate those terms in the following sections. We remark that the definition of the conjugate sequence has been inspired by the Gevrey case treated by M. Markin; see Example 2.5 and Sect. 3.2.

2.5 Conjugate weight sequence

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), then we define the conjugate sequence \(M^{*}=(M^{*}_p)_{p\in {\mathbb {N}}}\) by

$$\begin{aligned} M^{*}_p:=\frac{p!}{M_p}=\frac{1}{m_p},\;\;\;p\in {\mathbb {N}}, \end{aligned}$$
(2.3)

i.e., \(M^{*}:=m^{-1}\) for short. Hence, for \(p\ge 1\), the quotients \(\mu ^{*}=(\mu ^{*}_p)_p\) are given by

$$\begin{aligned} \mu ^{*}_p:=\frac{M^{*}_p}{M^{*}_{p-1}}=\frac{m_{p-1}}{m_p}=\frac{p!M_{p-1}}{(p-1)!M_p}=\frac{p}{\mu _p}, \end{aligned}$$
(2.4)

and we set \(\mu ^{*}_0:=1\). By these formulas, it is immediate that there is a one-to-one correspondence between M and \(M^{*}\).

2.6 Properties of conjugate weight sequences

We summarize some immediate consequences for \(M^{*}\). Let \(M,N\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given.

  1. (i)

    First, we immediately have

    $$\begin{aligned} \forall \;p\in {\mathbb {N}}:\;\;\;M^{**}_p=M_p,\qquad M^{*}_p\cdot M_p=p!, \end{aligned}$$

    that is

    $$\begin{aligned} M^{**}\equiv M,\qquad M^{*}\cdot M\equiv G^1. \end{aligned}$$

    Moreover (see also the subsequent Lemma 2.6),

    $$\begin{aligned} M^{*}{\preccurlyeq }M\Longleftrightarrow G^{1/2}{\preccurlyeq } M,\qquad M{\preccurlyeq }M^{*}\Longleftrightarrow M{\preccurlyeq }G^{1/2}, \end{aligned}$$

    and alternatively, the relation \(\preccurlyeq \) can be replaced by \(\le \). We also get \(M^{*}_0=M_0^{-1}\), i.e., \(M^{*}\) is normalized if and only if \(1=M_0\ge M_1\).

  2. (ii)

    \(M{\preccurlyeq }N\) holds if and only if \(N^{*}{\preccurlyeq }M^{*}\), and so, \(M{\approx }N\) if and only if \(M^{*}{\approx }N^{*}\).

  3. (iii)

    We get the following:

\((*)\):

\(\lim _{p\rightarrow +\infty }(M^{*}_p)^{1/p}=+\infty \) holds if and only if \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\) and this implies \(\mathcal {E}_{\{M\}}\subseteq \mathcal {E}_{(G^1)}\) (with strict inclusion). If, in addition, M is log-convex (and normalized), then all three assertions are equivalent; see Lemma 2.2.

\((*)\):

If \(\lim _{p\rightarrow +\infty }(M_p)^{1/p}=+\infty \), then by \(\mu ^{*}_p/p=\frac{1}{\mu _p}\), (2.2) and Stirling’s formula, we get both \(\lim _{p\rightarrow +\infty }\mu ^{*}_p/p=0\) and \(\lim _{p\rightarrow +\infty }(m^{*}_p)^{1/p}=0\).

\((*)\):

\(\lim _{p\rightarrow +\infty }(m^{*}_p)^{1/p}=+\infty \) holds if and only if \(\lim _{p\rightarrow +\infty }(M_p)^{1/p}=0\).

  1. (iv)

    \(M^{*}\) is log-convex, i.e., \(\mu ^{*}_{p+1}\ge \mu ^{*}_p\) for all \(p\in {\mathbb {N}}_{>0}\), if and only if m is log-concave, that is

    $$\begin{aligned} \forall \;p\in {\mathbb {N}}_{>0}:\;\;\;m_p^2\ge m_{p-1}m_{p+1}\Longleftrightarrow \mu ^{*}_{p+1}\ge \mu ^{*}_p, \end{aligned}$$
    (2.5)

    which in turn is equivalent to the map \(p\mapsto \frac{\mu _p}{p}\) being non-increasing.

    Analogously as in [21, Lemma 2.0.4], we get: If a sequence \(S\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) is log-concave and satisfies \(S_0=1\), then the mapping \(p\mapsto (S_p)^{1/p}\) is non-increasing.

    Consequently, if \(M^{*}\) is log-convex and if \(1=M^{*}_0=m_0=M_0\), then \(p\mapsto (m_p)^{1/p}\) is non-increasing.

  2. (v)

    If M is log-convex (and having \(M_0=1\)), then \(M^{*}\) has \(({\text {mg}})\): In this case by [21, Lemma 2.0.6] for all \(p,q\in {\mathbb {N}}\), we get \(M_pM_q\le M_{p+q}\Leftrightarrow m_pm_q\le \frac{(p+q)!}{p!q!}m_{p+q}\), and so, \(m_pm_q\le 2^{p+q}m_{p+q}\). Hence, \(M^{*}_{p+q}\le 2^{p+q}M^{*}_pM^{*}_q\) holds true.

  3. (vi)

    \(M^{*}\) has \(({\text {dc}})\) if and only if \(\mu ^{*}_p\le A^p\Leftrightarrow \frac{p}{\mu _p}\le A^p\), so if and only if

    $$\begin{aligned} \exists \;A\ge 1\;\forall \;p\in {\mathbb {N}}:\;\;\;\mu _p\ge \frac{p}{A^p}, \end{aligned}$$

    which can be considered as “dual derivation closedness”. Note that this property is preserved under the mapping \((M_p)_p\mapsto (M_pp!^s)_p\), \(s\in {\mathbb {R}}\) arbitrary, and it is mild: \(\liminf _{p\rightarrow +\infty }\mu _p/p>0\) is sufficient to conclude.

  4. (vii)

    \(M^{*}\) has \((\beta _1)\), i.e., \(\liminf _{p\rightarrow +\infty }\frac{\mu ^{*}_{Qp}}{\mu ^{*}_p}>Q\) for some \(Q\in {\mathbb {N}}_{\ge 2}\), if and only if \(\liminf _{p\rightarrow +\infty }\frac{\mu _{p}}{\mu _{Qp}}>1\); similarly \(M^{*}\) has \((\beta _3)\) if and only if \(\liminf _{p\rightarrow +\infty }\frac{\mu _p}{\mu _{Qp}}>\frac{1}{Q}\).

Using those insights, we may conclude the following.

Lemma 1.3

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given with \(1=M_0\ge M_1\) and let \(M^{*}\) be the conjugate sequence defined via (2.3). Then:

  1. (a)

    \(M^{*}\in {\mathcal{L}\mathcal{C}}\) if and only if m is log-concave and \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\).

  2. (b)

    \(M^{*}\in {\mathcal{L}\mathcal{C}}\) implies \(\mathcal {E}_{\{M\}}\subseteq \mathcal {E}_{(G^1)}\) with strict inclusion.

  3. (c)

    If, in addition, M is log-convex with \(1=M_0=M_1\), then the inclusion \(\mathcal {E}_{\{M\}}(\mathbb {R})\subseteq \mathcal {E}_{(G^1)}(\mathbb {R})\) gives \(\lim _{p\rightarrow +\infty }(M^{*}_p)^{1/p}=+\infty \). Moreover, \(M^{*}\) has moderate growth.

Remark 2.4

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given and we comment on the log-concavity and related conditions (for the sequence m):

  1. (a)

    If m is not log-concave but satisfies

    $$\begin{aligned} \exists \;H\ge 1\;\forall \;1\le p\le q:\;\;\;\frac{\mu _q}{q}\le H\frac{\mu _p}{p}, \end{aligned}$$

    i.e., the sequence \((\mu _p/p)_{p\in {\mathbb {N}}_{>0}}\) is almost decreasing, then the sequence L defined in terms of the corresponding quotient sequence \(\lambda =(\lambda _p)_{p\in {\mathbb {N}}}\) given by

    $$\begin{aligned} \lambda _p:=H^{-1}p\sup _{q\ge p}\frac{\mu _q}{q},\;\;\;p\ge 1,\qquad \lambda _0:=1, \end{aligned}$$
    (2.6)

    satisfies

    $$\begin{aligned} \forall \;p\ge 1:\;\;\;H^{-1}\frac{\mu _p}{p}\le \frac{\lambda _p}{p}\le \frac{\mu _p}{p}. \end{aligned}$$
    (2.7)

    Then, we get

    1. (i)

      L and M are equivalent, and so, \(L^{*}\) is equivalent to \(M^{*}\), too.

    2. (ii)

      \(p\mapsto \frac{\lambda _p}{p}\) is non-increasing, i.e., l is log-concave, and so \(L^{*}\) is log-convex.

    3. (iii)

      If \(1=M_0\ge M_1\), i.e., if \(\mu _1\le 1\), then \(1=L_0\ge L_1\) is valid, since \(L_1=\lambda _1\le \mu _1\le 1\) holds true. Thus, \(L^{*}\) is normalized.

    4. (iv)

      \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\) if and only if \(\lim _{p\rightarrow +\infty }(l_p)^{1/p}=0\) (with \(l_p:=L_p/p!\)).

    5. (v)

      Finally, if M is log-convex, then L shares this property: We have \(\lambda _p\le \lambda _{p+1}\) if and only if \(p\sup _{q\ge p}\frac{\mu _q}{q}\le (p+1)\sup _{q\ge p+1}\frac{\mu _q}{q}\) for all \(p\ge 1\). When \(p\ge 1\) is fixed, then clearly \(p\frac{\mu _q}{q}\le (p+1)\frac{\mu _q}{q}\) for all \(q\ge p+1\). If \(q=p\), then

      $$\begin{aligned} p\frac{\mu _q}{q}=\mu _p\le \mu _{p+1}=(p+1)\frac{\mu _{p+1}}{p+1}\le (p+1)\sup _{q\ge p+1}\frac{\mu _q}{q}, \end{aligned}$$

      and so, the desired inequality is verified.

    Summarizing, if \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) satisfies \(1=M_0\ge M_1\) and \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\), then \(L^{*}\in {\mathcal{L}\mathcal{C}}\); see (a) in Lemma 2.3. If M is in addition log-convex, then L has this property too.

    The definition (2.6) is motivated by [19, Lemma 8] and [9, Prop. 4.15].

  2. (b)

    If m is log-concave, then for any \(s\ge 0\), also the sequence \((m_p/p!^s)_{p\in {\mathbb {N}}}\) is log-concave, because the mapping \(p\mapsto \frac{\mu _p}{p^s}\) is still non-increasing (see (2.5)). However, for the sequence (\(p!^sm_p)_{p\in {\mathbb {N}}}\), this is not clear in general.

Example 2.5

Let \(M\equiv G^s\) for some \(0\le s<1\); see [14]. (In fact, in [14] instead of \(G^s\), the sequence \((p^{ps})_{p\in {\mathbb {N}}}\) is treated but which is equivalent to \(G^s\) by Stirling’s formula.) Then, \(m\equiv G^{s-1}\) with \(-1\le s-1<0\), and so, m corresponds to a Gevrey sequence with negative index. We get \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\) and m is log-concave. Moreover, \(M^{*}\equiv G^{1-s}\) and so clearly \(M^{*}\in {\mathcal{L}\mathcal{C}}\).

In particular, if \(s=\frac{1}{2}\), then \((G^{\frac{1}{2}})^{*}=G^{\frac{1}{2}}\) and we prove the following statement which underlines the importance of \(G^{\frac{1}{2}}\) (up to equivalence of sequences) w.r.t. the action \(M\mapsto M^{*}\).

Lemma 1.6

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given. Then, the following are equivalent:

  1. (i)

    We have \(M{\preccurlyeq }M^{*}\).

  2. (ii)

    We have

    $$\begin{aligned} \exists \;C,h\ge 1\;\forall \;p\in {\mathbb {N}}:\;\;\;M_p^2\le Ch^pp!, \end{aligned}$$

    i.e., \(M{\preccurlyeq }G^{1/2}\).

  3. (iii)

    We have \(G^{1/2}{\preccurlyeq }M^{*}\).

The analogous equivalences are valid if \(M^{*}{\preccurlyeq }M\) resp. if relation \(\preccurlyeq \) is replaced by \(\le \). Thus, \(M{\approx }M^{*}\) if and only if \(M{\approx }G^{1/2}\) and \(M=M^{*}\) if and only if \(M=G^{1/2}=M^{*}\).

In particular, \(G^{1/2}=(G^{1/2})^{*}\) holds true.

Proof

The equivalences follow immediately from the definition of \(M^{*}\) in (2.3). \(\square \)

2.7 Associated weight function

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) (with \(M_0=1\)), then the associated function \(\omega _M: {\mathbb {R}}_{\ge 0}\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is defined by

$$\begin{aligned} \omega _M(t):=\sup _{p\in {\mathbb {N}}}\log \left( \frac{t^p}{M_p}\right) \;\;\;\text {for}\;t>0, \qquad \omega _M(0):=0. \end{aligned}$$

For an abstract introduction of the associated function, we refer to [11, Chapitre I]; see also [10, Definition 3.1]. If \(\liminf _{p\rightarrow +\infty }(M_p)^{1/p}>0\), then \(\omega _M(t)=0\) for sufficiently small t, since \(\log \left( \frac{t^p}{M_p}\right)<0\Leftrightarrow t<(M_p)^{1/p}\) holds for all \(p\in {\mathbb {N}}_{>0}\). Moreover, under this assumption \(t\mapsto \omega _M(t)\) is a continuous non-decreasing function, which is convex in the variable \(\log (t)\) and tends faster to infinity than any \(\log (t^p)\), \(p\ge 1\), as \(t\rightarrow +\infty \). \(\lim _{p\rightarrow +\infty }(M_p)^{1/p}=+\infty \) implies that \(\omega _M(t)<+\infty \) for each \(t>0\) and which shall be considered as a basic assumption for defining \(\omega _M\).

Given \(M\in {\mathcal{L}\mathcal{C}}\), then by [11, 1.8 III], we get that \(\omega _M(t)=0\) on \([0,\mu _1]\).

Finally note that for \(M\in {\mathcal{L}\mathcal{C}}\), we have \(\lim _{p\rightarrow +\infty }\mu _p=+\infty \); see, e.g., [17, p. 104].

3 Ultradifferentiable classes as weighted spaces of entire functions

In Sect. 2.4, we saw that ultradifferentiable classes \(\mathcal {E}_{[M]}\) with \(\lim _{p\rightarrow +\infty }m_p^{1/p}=0\) are classes of entire functions. Now, we go further and identify those classes with weighted spaces of entire functions, where the weight is given by the associated weight function of the conjugate weight sequence \(M^{*}\). To this end, let us first recall some notation already introduced in [20] (to be precise, in [20], the weighted spaces of entire functions have only been defined for the codomain \(\mathbb {C}\), but everything can be done completely analogously for H instead of \(\mathbb {C}\)): Let H be a Hilbert space and let \(v:[0,+\infty ) \rightarrow (0,+\infty )\) be a weight function, i.e., v is

\((*)\):

continuous,

\((*)\):

non-increasing, and

\((*)\):

rapidly decreasing, i.e., \(\lim _{t\rightarrow +\infty }t^kv(t)=0\) for all \(k\ge 0\).

Then, introduce the space

$$\begin{aligned} \mathcal {H}^\infty _v(\mathbb {C},H):=\left\{ f \in \mathcal {H}(\mathbb {C},H): \Vert f\Vert _v:= \sup _{z \in \mathbb {C}} \Vert f(z)\Vert v(|z|) < +\infty \right\} . \end{aligned}$$

We shall assume w.l.o.g. that v is normalized, i.e., \(v(t)=1\) for \(t \in [0,1]\) (if this is not the case, one can always switch to another normalized weight w with \(\mathcal {H}^\infty _v(\mathbb {C},H) = \mathcal {H}^\infty _w(\mathbb {C},H)\)).

In the next step, we consider weight systems; see [20, Sect. 2.2] for more details. For a non-increasing sequence of weights \(\underline{\mathcal {V}}=(v_n)_{n\in {\mathbb {N}}_{>0}}\), i.e., \(v_n\ge v_{n+1}\) for all n, we define the (LB)-space

$$\begin{aligned} \mathcal {H}_{\underline{\mathcal {V}}}^\infty (\mathbb {C},H):=\varinjlim _{n\in \mathbb {N}_{>0}} \mathcal {H}^\infty _{v_n}(\mathbb {C},H), \end{aligned}$$

and for a non-decreasing sequence of weights \(\overline{\mathcal {V}}=(v_n)_{n\in {\mathbb {N}}_{>0}}\), i.e., \(v_n\le v_{n+1}\) for all n, we define the Fréchet space

$$\begin{aligned} \mathcal {H}_{\overline{\mathcal {V}}}^\infty (\mathbb {C},H):=\varprojlim _{n\in \mathbb {N}_{>0}} \mathcal {H}^\infty _{v_n}(\mathbb {C},H). \end{aligned}$$

Remark 3.1

In [20], the spaces are denoted by \(H^\infty _v(\mathbb {C})\) instead of \(\mathcal {H}^\infty _v(\mathbb {C},\mathbb {C})\). We use \(\mathcal {H}\) to avoid any confusion with the Hilbert space H. In addition, \(\mathcal {H}^\infty _v(\mathbb {C})\) shall denote \(\mathcal {H}^\infty _v(\mathbb {C},\mathbb {C})\).

The following Lemma can be used to infer statements for \(\mathcal {H}^\infty _v(\mathbb {C},H)\) from the respective statements for \(\mathcal {H}^\infty _v(\mathbb {C})\).

Lemma 1.8

Let H be a (complex) Hilbert space and v be a weight. Then

$$\begin{aligned} f \in \mathcal {H}^\infty _v(\mathbb {C},H) ~\Leftrightarrow z \mapsto \langle f(z), y \rangle \in \mathcal {H}^\infty _v(\mathbb {C}) \text { for all } y \in H. \end{aligned}$$

Proof

For the non-trivial part, take some \(f \in \mathcal {H}(\mathbb {C},H)\), such that \(|\langle f(z), y \rangle | \le C_y v(|z|)\) for every \(y \in H\). Then, this just means that \(\{\frac{f(z)}{v(|z|)}:~z \in \mathbb {C}\}\) is weakly bounded (in H) which implies boundedness and this just means that \(f \in \mathcal {H}_v(\mathbb {C},H)\). \(\square \)

Remark 3.3

Of course, the same argument holds for a family of weights \(\overline{\mathcal {V}}\) or \(\underline{\mathcal {V}}\).

For a given weight v and \(c>0\), we shall write \(v_c(t):= v(ct)\) and \(v^c(t):=v(t)^c\), and set

$$\begin{aligned} \underline{\mathcal {V}}_\mathfrak {c}= (v_c)_{c \in \mathbb {N}_{>0}}, \text { and } \overline{\mathcal {V}}_\mathfrak {c}= (v_{1/c})_{c \in \mathbb {N}_{>0}}, \end{aligned}$$

and

$$\begin{aligned} \underline{\mathcal {V}}^\mathfrak {c}= (v^c)_{c \in \mathbb {N}_{>0}}, \text { and } \overline{\mathcal {V}}^\mathfrak {c}= (v^{1/c})_{c \in \mathbb {N}_{>0}}, \end{aligned}$$

in particular \(\underline{\mathcal {V}}_\mathfrak {c}\) and \(\underline{\mathcal {V}}^\mathfrak {c}\) are non-increasing, and \(\overline{\mathcal {V}}_\mathfrak {c}\) and \(\overline{\mathcal {V}}^\mathfrak {c}\) are non-decreasing sequences of weights, see again [20, Sect. 2.2].

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given with \(M_0=1\), such that M is \(({\text {lc}})\) and satisfies \(\lim _{p\rightarrow +\infty }(M_p)^{1/p}=+\infty \) (see [20, Def. 2.4, Rem. 2.6]). Then, we denote by \(\underline{\mathcal {M}}_\mathfrak {c}, \underline{\mathcal {M}}^\mathfrak {c}, \overline{\mathcal {M}}_\mathfrak {c},\) and \(\overline{\mathcal {M}}^\mathfrak {c}\) the respective sequences of weights defined by choosing \(v(t):= v_M(t):= e^{-\omega _M(t)}\) (see [20, Rem. 2.7]). If we write \(\underline{\mathcal {N}}_\mathfrak {c}, \underline{\mathcal {N}}^\mathfrak {c}, \overline{\mathcal {N}}_\mathfrak {c},\) and \(\overline{\mathcal {N}}^\mathfrak {c}\), we mean the respective definition for another weight sequence N. Finally, we write (of course) \(\underline{\mathcal {M}^*}_\mathfrak {c},\dots \) for the systems corresponding to the conjugate sequence \(M^*\).

Theorem 1.10

Let \(M \in \mathbb {R}^\mathbb {N}_{>0}\) with \(M_0 = 1 \ge M_1\) be given, such that \(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=0\) and m is log-concave. Let \(I\subseteq {\mathbb {R}}\) be an interval, then

$$\begin{aligned} E:\mathcal {E}_{\{M\}}(I,H) \rightarrow \mathcal {H}^\infty _{\underline{\mathcal {M}^*}_\mathfrak {c}}(\mathbb {C},H), \quad f \mapsto E(f):=\sum _{k = 0}^{+\infty }\frac{f^{(k)}(x_0)}{k!}(z-x_0)^k \end{aligned}$$

is an isomorphism (of locally convex spaces) for any fixed \(x_0\in I\). Moreover, with the same definition for E, also

$$\begin{aligned} E:\mathcal {E}_{(M)}(I,H) \rightarrow \mathcal {H}^\infty _{\overline{\mathcal {M}^*}_\mathfrak {c}}(\mathbb {C},H) \end{aligned}$$

is an isomorphism.

Remark 3.5

Before proving this main statement, we give the following observations:

  1. (i)

    By Lemma 2.3, the assumptions on M imply \(M^{*}\in {\mathcal{L}\mathcal{C}}\). It is easy to check that any small Gevrey class, i.e., choosing \(M_j= j!^\alpha \) for some \(\alpha \in [0,1)\), satisfies the assumptions of Theorem 3.4.

  2. (ii)

    Moreover, we comment in detail on the basic requirements for the sequence M in Theorem 3.4:

\((*)\):

Note that both assumptions \(M_0=1\ge M_1\) and log-concavity of m are not preserved under equivalence of weight sequences.

On the other hand, both isomorphisms in Theorem 3.4 are clearly preserved under equivalence: Equivalent sequences yield the same ultradifferentiable function classes, equivalent conjugate sequences [recall (ii) in Sect. 2.6], and finally (by definition) also the same weighted entire function classes; see [20, Prop. 3.8].

\((*)\):

Thus, we can assume more generally that M is equivalent to \(L\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), such that \(L_0 = 1 \ge L_1\), \(\lim _{p\rightarrow +\infty }(l_p)^{1/p}=0\), and l is log-concave. In this situation, we replace in the proof below M by L, m by l and \(M^{*}\) by \(L^{*}\). Recall that the log-concavity for l can be ensured, e.g., if \((\mu _p/p)_{p\in {\mathbb {N}}_{>0}}\) is almost decreasing; see Remark 2.4.

\((*)\):

Finally, note the following: Assume that M is equivalent to \(L\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), such that \(L_0=1\ge L_1\) and \(\lim _{p\rightarrow +\infty }(l_p)^{1/p}=0\), but none of the sequences L being equivalent to M has the property that l is log-concave. Thus log-convexity for \(L^{*}\) fails for any L being equivalent to M. Then, both \(\mathcal {H}^\infty _{\underline{\mathcal {L}^*}_\mathfrak {c}}(\mathbb {C},H)\) and \(\mathcal {H}^\infty _{\overline{\mathcal {L}^*}_\mathfrak {c}}(\mathbb {C},H)\) coincide with the respective classes when \(L^{*}\) is replaced by its log-convex minorant \((L^{*})^{{\text {lc}}}\); see [20, Rem. 2.6]. In this situation, the first part of the proof stays valid; i.e., the operator E is still continuous. However, the second part fails in general; more precisely for the equality just below (3.1) in the subsequent proof, the log-convexity of the appearing conjugate sequence is indispensable, and without this property, we can only bound \(F^{(n)}\) in terms of \(\frac{n!}{(L_n^{*})^{{\text {lc}}}}=:\overline{L}_n\ge L_n\).

Proof of Theorem 3.4

We start with the Roumieu case and assume w.l.o.g. that \(x_0=0\). Let us take \(f \in \mathcal {E}_{M,h}(K,H)\) for some compact set \(K \subset \subset I\) with \(0\in K\) and some \(h>0\), i.e., there is \(A(=\Vert f\Vert _{M,K,h})\), such that for all \(x \in K\) and all \(k \in \mathbb {N}\), we have

$$\begin{aligned} \Vert f^{(k)}(x)\Vert \le A h^k M_k. \end{aligned}$$

Then, we infer immediately that

$$\begin{aligned} \Vert E(f)(z)\Vert \le A \sum _{k = 0}^{+\infty }\frac{h^kM_k}{k!}|z|^k=A\sum _{k = 0}^{+\infty }\frac{h^k}{M^{*}_k}|z|^k \le 2A \exp (\omega _{M^*}(2h|z|)). \end{aligned}$$

Therefore, E maps \(\mathcal {E}_{M,h}(K,H)\) continuously into \(\mathcal {H}^\infty _{v_{M^*, 2h}}(\mathbb {C},H)\) and this immediately implies continuity of E as a mapping defined on the inductive limit with respect to h.

In the Beurling case, a function \(f \in \mathcal {E}_{(M)}(I, H)\) lies in \(\mathcal {E}_{M,h}(K,H)\) for any \(h>0\), and thus, the above reasoning immediately gives that E is continuous as a mapping into \(\mathcal {H}^\infty _{\overline{\mathcal {M}^*}_\mathfrak {c}}(\mathbb {C},H)\).

Let us now show continuity of the inverse mapping, which is clearly given by restricting an entire function to the interval I. Take some \(F \in \mathcal {H}^\infty _{v_{M^*, k}}(\mathbb {C},H)\), then

$$\begin{aligned} \Vert F(z)\Vert \le A e^{\omega _{M^*}(k|z|)} \end{aligned}$$

for \(A=\Vert F\Vert _{v_{M^*,k}}>0\). Consider an arbitrary \(K \subset \subset I\) and let \(R\ge 1\) be such that \(K \subset [-R,R]\). Then, take \(r\ge 2R\), which ensures that \(K + B(0,r) \subset B(0,2r)\) and where B(0, r) denotes the ball around 0 of radius r. Then, by the Cauchy estimates, we infer for such r and all \(x \in K\) and \(n\in {\mathbb {N}}\)

$$\begin{aligned} \Vert F^{(n)}(x)\Vert \le An!\frac{e^{\omega _{M^*}(2kr)}}{r^n}. \end{aligned}$$
(3.1)

Since \(e^{\omega _{M^*}(r)}=\frac{r^n}{M^*_n}\) for \(r \in [\mu ^*_n, \mu ^*_{n+1})\) (see, e.g., [11, 1.8 III]), we may plug in some \(r \in [\mu ^*_n/(2k), \mu ^*_{n+1}/(2k))\) in (3.1); for all n large enough, such that \(\mu ^*_n/(2k) \ge 2R\) (thus depending on chosen compact K) and which is possible since \(M^* \in \mathcal {L}\mathcal {C}\) and so \(\lim _{n\rightarrow +\infty }\mu ^*_n=+\infty \). Hence, we get

$$\begin{aligned} \Vert F^{(n)}(x)\Vert \le An!\frac{(2kr)^n}{r^n M_n^*}=A (2k)^nM_n. \end{aligned}$$

For the remaining (finitely, say \(n_0\)) many integers n with \(\mu _n^*/(2k) < 2R\), we can estimate

$$\begin{aligned} \Vert F^{(n)}(x)\Vert \le CA (2k)^n M_n, \end{aligned}$$

where, e.g., \(C=n_0!e^{\omega _{M^*}(2kR)}\). Altogether, we have shown

$$\begin{aligned} \Vert F|_{I}\Vert _{M,K,2k} \le C \Vert F\Vert _{v_{M^*,k}}, \end{aligned}$$

which proves continuity of the inverse mapping in both the Roumieu and the Beurling case. \(\square \)

3.1 Comparison of \(\mathcal {H}^\infty _{\underline{\mathcal {M}^*}_\mathfrak {c}}\) and \(\mathcal {H}^\infty _{\underline{\mathcal {M}^*}^\mathfrak {c}}\) (resp. \(\mathcal {H}^\infty _{\overline{\mathcal {M}^*}_\mathfrak {c}}\) and \(\mathcal {H}^\infty _{\overline{\mathcal {M}^*}^\mathfrak {c}}\))

Let us quickly recall a recent result characterizing the equality of the two different types of weighted spaces of entire functions; see [20, Thm. 5.4]. To this end, we need one more condition for M

$$\begin{aligned} \;\exists L\in \mathbb {N}_{>0}:\quad \liminf _{j \rightarrow +\infty } \frac{(M_{Lj})^{1/(Lj)}}{(M_j)^{1/j}}>1. \end{aligned}$$
(3.2)

In [23, Thm. 3.1], it has been shown that \(M\in {\mathcal{L}\mathcal{C}}\) has (3.2) if and only if

$$\begin{aligned} \omega _M(2t)=O(\omega _M(t)) \text { as } t \rightarrow +\infty . \end{aligned}$$
(3.3)

Lemma 1.12

Let \(M \in {\mathcal{L}\mathcal{C}}\). Then, the following statements are equivalent:

  1. (i)

    M has \(({\text {mg}})\) and satisfies (3.2),

  2. (ii)

    \(\mathcal {H}^\infty _{\underline{\mathcal {M}}_\mathfrak {c}}(\mathbb {C},H) \cong \mathcal {H}^\infty _{\underline{\mathcal {M}}^\mathfrak {c}}(\mathbb {C},H)\),

  3. (iii)

    \(\mathcal {H}^\infty _{\overline{\mathcal {M}}_\mathfrak {c}}(\mathbb {C},H) \cong \mathcal {H}^\infty _{\overline{\mathcal {M}}^\mathfrak {c}}(\mathbb {C},H)\).

Proof

In [20, Thm. 5.4], the result is shown for \(H = \mathbb {C}\). To get that (i) implies (ii) and (iii) the proof of [20, Thm. 5.4] can be repeated and only the appearances of \(|\cdot |\) (the absolute value in \(\mathbb {C}\)) have to be substituted by \(\Vert \cdot \Vert \) (the norm in the Hilbert space H).

To get the other implications, i.e., that (ii) resp. (iii) implies (i), note that the respective equality in the Hilbert space-valued case implies the equality for the \(\mathbb {C}\)-valued case by observing that \(f \in \mathcal {H}^\infty _v(\mathbb {C}) (=\mathcal {H}^\infty _v(\mathbb {C},\mathbb {C}))\) if and only if, for any \(0 \ne x \in H\), we have \(z \mapsto f(z)x \in \mathcal {H}^\infty _v(\mathbb {C},H)\). Therefore, we may apply [20, Thm. 5.4] and infer (i). \(\square \)

Together with results from Sect. 2.6, we derive the following.

Corollary 1.13

Let \(M \in \mathbb {R}^\mathbb {N}_{>0}\) be given and assume the following:

\((*)\):

M is log-convex with \(1=M_0=M_1\) (i.e., both normalization and \(1 = M_0 \ge M_1\)),

\((*)\):

\(\lim _{p\rightarrow +\infty }m_p^{1/p}=0\),

\((*)\):

m is log-concave, and finally,

\((*)\):

for some \(Q\in {\mathbb {N}}_{\ge 2}\), we have \(\liminf _{p \rightarrow +\infty } \frac{\mu _p}{\mu _{Qp}} > \frac{1}{Q}\).

Then

$$\begin{aligned} \mathcal {H}^\infty _{\underline{\mathcal {M}^*}_\mathfrak {c}}(\mathbb {C},H) \cong \mathcal {H}^\infty _{\underline{\mathcal {M}^*}^\mathfrak {c}}(\mathbb {C},H), \quad \mathcal {H}^\infty _{\overline{\mathcal {M}^*}_\mathfrak {c}}(\mathbb {C},H) \cong \mathcal {H}^\infty _{\overline{\mathcal {M}^*}^\mathfrak {c}}(\mathbb {C},H), \end{aligned}$$

and E is an isomorphism between \(\mathcal {E}_{\{M\}}(I,H)\) and \(\mathcal {H}^\infty _{\underline{\mathcal {M}^*}^\mathfrak {c}}(\mathbb {C},H)\) resp. between \(\mathcal {E}_{(M)}(I,H)\) and \(\mathcal {H}^\infty _{\overline{\mathcal {M}^*}^\mathfrak {c}}(\mathbb {C},H)\).

Proof

By (v) in Sect. 2.6, it follows that \(M^*\) has \(({\text {mg}})\). By (vii) from Sect. 2.6, we infer that \(M^*\) has \((\beta _3)\), and thus, [23, Prop. 3.4] gives that \(M^*\) has (3.2). Finally, observe that \(M^*\in {\mathcal{L}\mathcal{C}}\) holds true: \(\lim _{p\rightarrow +\infty }m_p^{1/p}=0\) implies \(\lim _{p\rightarrow +\infty }(M^*_p)^{1/p}=+\infty \) (see (iii) in Sect. 2.6), log-convexity of \(M^*\) follows from log-concavity of m (see (iv) in Sect. 2.6) and normalization of \(M^*\) is immediate. Thus, we may apply Lemma 3.6 to \(M^*\). The rest follows from Theorem 3.4. \(\square \)

Remark 3.8

Observe that the conditions of Lemma 3.6 hold if and only if \(\mathcal {E}_{[M^*]} \cong \mathcal {E}_{[\omega _{M^*}]}\), cf. [2, Thm. 14], [17, Sect. 5] and [23, Prop. 3.4].

Note also that Corollary 3.7 applies, in particular, to all small Gevrey sequences \(G^{\alpha }\), \(0\le \alpha <1\); see the next section for its importance.

3.2 A result by Markin as a Corollary of Theorem 3.4

One of Markin’s core results in [14], Lemma 3.1, shows, in our setting the following: For any \(\alpha \in [0,1)\) and \(M^\alpha _j:=j^{j\alpha }\), which is equivalent to \(G^\alpha _j= j!^\alpha \) (i.e., the small Gevrey sequence of order \(\alpha \)) and with \(v(t):= e^{-t^{1/(1-\alpha )}}\), we obtain that

$$\begin{aligned} E: \mathcal {E}_{\{G^\alpha \}}(I,H)\rightarrow \mathcal {H}^\infty _{\underline{\mathcal {V}}^\mathfrak {c}}(\mathbb {C},H) \end{aligned}$$

is an isomorphism of locally convex vector spaces; and mutatis mutandis the same holds in the respective Beurling case. With our preparation, this now is a corollary of Theorem 3.4 together with the following observations:

  • Corollary 3.7 applies to \(M=G^\alpha \),

  • \((G^\alpha )^* = G^{1-\alpha }\),

  • \(\omega _{G^{1-\alpha }} \cong t^{\frac{1}{1-\alpha }}\), i.e., \(\omega _{G^{1-\alpha }}(t) = O( t^{\frac{1}{1-\alpha }}), ~ t^{\frac{1}{1-\alpha }} = O(\omega _{G^{1-\alpha }}(t))\) as \(t \rightarrow +\infty \).

3.3 Characterization of inclusion relations for small weight sequences

In the theory of ultradifferentiable functions, the characterization of the inclusion \(\mathcal {E}_{[M]}\subseteq \mathcal {E}_{[N]}\) in terms of a growth property expressed in terms of M and N is studied. Summarizing, we get the following, e.g., see [17, Prop. 2.12] and the literature citations there; similar/analogous techniques have also been applied to the more general and recent approaches in [17, Prop. 4.6] and [6, Sect. 4]:

\((*)\):

If \(M,N\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) with \(M{\preccurlyeq }N\), then \(\mathcal {E}_{\{M\}}\subseteq \mathcal {E}_{\{N\}}\) and \(\mathcal {E}_{(M)}\subseteq \mathcal {E}_{(N)}\) with continuous inclusion.

\((*)\):

If, in addition, M is normalized and log-convex, then \(\mathcal {E}_{\{M\}}({\mathbb {R}})\subseteq \mathcal {E}_{\{N\}}({\mathbb {R}})\) (as sets) yields \(M{\preccurlyeq }N\).

If \(M,N\in {\mathcal{L}\mathcal{C}}\), then \(\mathcal {E}_{(M)}({\mathbb {R}})\subseteq \mathcal {E}_{(N)}({\mathbb {R}})\) (as sets and/or with continuous inclusion; see the proof of [17, Prop. 4.6] and [6, Prop. 4.5, Rem. 4.6]) yields \(M{\preccurlyeq }N\).

Thus, for the necessity of \(M{\preccurlyeq }N\), standard regularity and growth assumptions for M are required, and so far, it is not known what can be said for (small) sequences M “beyond” this setting. Via an application of Theorem 3.4 and main results from [20], we now may actually prove as a corollary an analogous statement.

First, let us recall [20, Thm. 3.14], where the following characterization is shown (even under formally slightly more general assumptions on the weight N; see also [20, Rem. 2.6]).

Theorem 1.15

Let \(N\in {\mathcal{L}\mathcal{C}}\) and \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), such that M satisfies \(M_0=1\) and \(\lim _{p\rightarrow +\infty }(M_p)^{1/p}=+\infty \). Then, the following are equivalent:

  1. (a)

    We have \(N{\preccurlyeq }M\).

  2. (b)

    We have

    $$\begin{aligned} \mathcal {H}^{\infty }_{\underline{\mathcal {M}}_{\mathfrak {c}}}(\mathbb {C})\subseteq \mathcal {H}^{\infty }_{\underline{\mathcal {N}}_{\mathfrak {c}}}(\mathbb {C}). \end{aligned}$$
  3. (c)

    We have

    $$\begin{aligned} \mathcal {H}^{\infty }_{\overline{\mathcal {M}}_{\mathfrak {c}}}(\mathbb {C})\subseteq \mathcal {H}^{\infty }_{\overline{\mathcal {N}}_{\mathfrak {c}}}(\mathbb {C}). \end{aligned}$$

Thus, by combining Theorems 3.4 and 3.9, which we apply to \(N^{*}\) and \(M^{*}\), we get the following:

Theorem 1.16

Let \(M,N\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given and assume that

\((*)\):

\(1=M_0\ge M_1\) and \(1=N_0\ge N_1\),

\((*)\):

\(\lim _{p\rightarrow +\infty }(m_p)^{1/p}=\lim _{p\rightarrow +\infty }(n_p)^{1/p}=0\),

\((*)\):

both m and n are log-concave.

Then, the following are equivalent:

  1. (i)

    We have \(M{\preccurlyeq }N\).

  2. (ii)

    We have \(\mathcal {E}_{\{M\}}\subseteq \mathcal {E}_{\{N\}}\) with continuous inclusion.

  3. (iii)

    We have \(\mathcal {E}_{(M)}\subseteq \mathcal {E}_{(N)}\) with continuous inclusion.

Proof

It remains to prove (ii), (iii) \(\Rightarrow \) (i). We use the inclusion in (ii) resp. in (iii) for some compact interval I, i.e., \(\mathcal {E}_{[M]}(I)\subseteq \mathcal {E}_{[N]}(I)\). Then, the characterization shown in Theorem 3.4 yields \(\mathcal {H}^{\infty }_{\underline{\mathcal {M}^{*}}_{\mathfrak {c}}}(\mathbb {C})\subseteq \mathcal {H}^{\infty }_{\underline{\mathcal {N}^{*}}_{\mathfrak {c}}}(\mathbb {C})\) resp. \(\mathcal {H}^{\infty }_{\overline{\mathcal {M}^{*}}_{\mathfrak {c}}}(\mathbb {C})\subseteq \mathcal {H}^{\infty }_{\overline{\mathcal {N}^{*}}_{\mathfrak {c}}}(\mathbb {C})\). By the assumptions on M, N, we get \(M^{*}, N^{*}\in {\mathcal{L}\mathcal{C}}\), and then, Theorem 3.9 gives \(N^{*}{\preccurlyeq }M^{*}\) which is equivalent to \(M{\preccurlyeq }N\) (recall (ii) in Sect. 2.6) and so (i) is shown. \(\square \)

4 A criterion for boundedness of an operator on a Hilbert space

The aim of this section is to generalize results by M. Markin from [12, 13], and [14] (obtained within the so-called small Gevrey setting) to a more general weight sequence setting when considering appropriate families of small weight sequences. (In fact, Markin considers instead of \(G^{\beta }\), \(0\le \beta <1\), the sequence \(M^\beta _j:=j^{j\beta }\) but which is equivalent to \(G^{\beta }\) by Stirling’s formula. Since equivalence clearly preserves the corresponding function spaces, his results immediately transfer to \(G^{\beta }\) as well.)

Markin studies, for a Hilbert space H, and a normal (unbounded) operator A on H the associated evolution equation

$$\begin{aligned} y'(t) = A y(t) \end{aligned}$$
(4.1)

and asks whether a priori known smoothness of all solutions of (4.1) yield boundedness of the operator A.

For a detailed exposition of evolution equations on Hilbert spaces, we refer to Chapters 1 (bounded case) and 4 (unbounded case) in [15].

4.1 Solutions for bounded operators

First, let us recall quickly the situation for bounded operators A. For those, the domain is all of H. It is a classical result in this context that every solution y of (4.1) is of the form

$$\begin{aligned} y(t)=e^{tA}y_0, \end{aligned}$$

for some \(y_0 \in H\), where \(e^{tA}:= \sum _{k = 0}^{+\infty } \frac{t^k}{k!}A^k\) and which converges locally uniformly (with respect to t) in the norm topology on B(H) (the space of bounded operators on H). Moreover, y can be extended to an entire function, such that

$$\begin{aligned} \Vert y(z)\Vert \le Me^{C|z|} \end{aligned}$$

for some constants M and C and all \(z\in \mathbb {C}\). Thus, we may conclude the subsequent statement.

  1. (i)

    If A is a bounded operator on H, then each solution y of (4.1) is an entire function of exponential type.

On the other hand, we have the following:

  1. (ii)

    As outlined by M. Markin in [12, 13] and [14], there exists an unbounded normal operator A (that is actually not bounded on H), such that each (weak) solution of (4.1) is an entire function.

4.2 Motivating question

Therefore, one may ask whether one can reverse the implication in (i), and if this is possible to what extent one can weaken the assumption of exponential type. From (ii), it is clear that one cannot get completely rid of any additional growth restriction!

Markin does exactly that in [14]. Let us first recall his approach and then subsequently considerably extend it.

4.3 A generalization of Markin’s results

The main result [14, Thm. 5.1] states that if each weak solution of (4.1) is in some small Gevrey class, i.e., admitting a growth restriction expressed in germs of \(G^\alpha \) with \(\alpha <1\), then the operator A is necessarily bounded on H. This is of special interest, since, as outlined in Sect. 3.2, every small Gevrey class can be identified with a weighted class of entire functions.

Before we are able to generalize Markin’s result, we need some definitions: For a densely defined operator A on H, we first set

$$\begin{aligned} C^\infty (A):=\bigcap _{n \in \mathbb {N}} D(A^n), \end{aligned}$$

where \(D(A^n)\) is the domain of \(A^n\), the n-fold iteration of A. Then, put

$$\begin{aligned} \mathcal {E}_{\{M\}}(A):= \{f \in C^\infty (A):~ \;\exists C,h >0~ \;\forall n \in \mathbb {N}~\Vert A^nf\Vert \le C h^n M_n\}, \end{aligned}$$

and the corresponding Beurling class is defined by

$$\begin{aligned} \mathcal {E}_{(M)}(A):= \{f \in C^\infty (A):~ \forall \;h>0\;\exists \;C>0~ \;\forall n \in \mathbb {N}~\Vert A^nf\Vert \le C h^n M_n\}. \end{aligned}$$

From [4, Sect. 1.3], a different description of \(\mathcal {E}_{\{M\}}(A)\) in terms of \(E_A\), the spectral measure associated to A, can be deduced as follows:

$$\begin{aligned} \mathcal {E}_{\{M\}}(A)=\left\{ f \in H:~\;\exists t>0~ \int _\mathbb {C}e^{2 \omega _M(t|\lambda |)} \langle dE_A(\lambda ) f,f\rangle <+\infty \right\} , \end{aligned}$$

and

$$\begin{aligned} \mathcal {E}_{(M)}(A)=\left\{ f \in H:~\;\forall t>0~ \int _\mathbb {C}e^{2\omega _M(t|\lambda |)} \langle dE_A(\lambda ) f,f\rangle <+\infty \right\} . \end{aligned}$$

Now, we have the following result which generalizes [13, Thm. 3.1].

Theorem 1.17

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) be given and \(I\subseteq {\mathbb {R}}\) a closed interval. Then, a solution y of (4.1) belongs to \(\mathcal {E}_{[M]}(I,H)\) if and only if \(y(t) \in \mathcal {E}_{[M]}(A)\) for all \(t \in I\). In this case, one has \(y^{(n)}(t)=A^ny(t)\) for all \(t \in I\).

Proof

Let y be a solution of (4.1), such that \(y \in \mathcal {E}_{[M]}(I,H)\). Since \(y \in C^\infty (I,H)\), we have by [12, Prop. 4.1] that \(y^{(n)}(t)=A^ny(t)\) for all \(t \in I\) and all \(n \in \mathbb {N}\). Therefore

$$\begin{aligned} \Vert A^ny(t)\Vert = \Vert y^{(n)}(t)\Vert \le C h^n M_n, \end{aligned}$$

where h is either in the scope of an existential or universal quantifier depending on the context. This immediately gives that \(y(t) \in \mathcal {E}_{[M]}(A)\) for all t.

For the converse direction, we argue as in [13, Proof of Prop. 3.1] where it is shown that in this case for any subinterval \([a,b] \subseteq I\)

$$\begin{aligned} \max _{t \in [a,b]}\Vert y^{(n)}(t)\Vert \le \Vert y^{(n)}(a)\Vert +\Vert y^{(n)}(b)\Vert . \end{aligned}$$

Since, again, we have \(y^{(n)}(t) = A^n y(t)\), this immediately yields \(y \in \mathcal {E}_{[M]}(I,H)\). \(\square \)

We need one more result generalizing [14, Lemma 4.1] which reads as follows.

Lemma 1.18

Let \(0<\beta <+\infty \). If

$$\begin{aligned} \bigcup _{0<\beta '<\beta }\mathcal {E}_{\{G^{\beta '}\}}(A) = \mathcal {E}_{(G^{\beta })}(A), \end{aligned}$$

then the operator A is bounded.

Note that in [14], the notation \(\mathcal {E}^{[\beta ]}(A)\) is used instead of \(\mathcal {E}_{[G^\beta ]}(A)\) (i.e., the respective Gevrey class of order \(\beta \)). Since we have a generalization of [14, Thm. 5.1] as our goal, we only need a generalization of the above Lemma in the case \(\beta = 1\). Therefore, we want to conclude that an operator A on a Hilbert space H is bounded if we can write the entire functions corresponding to A (i.e., \(\beta =1\)) as an union of certain smaller Roumieu classes. Note that this statement might seem “counterintuitive” when considering the ultradifferentiable classes introduced in Sect. 2.3. However, note that the classes in Sect. 2.3 are defined using the differential operator which is unbounded.

Summarizing, our generalization of Markin’s result reads as follows.

Lemma 1.19

Let \(\mathfrak {F}\subseteq \mathcal {L}\mathcal {C}\) be a family of sequences, such that

$$\begin{aligned} \forall \;N\in \mathfrak {F}\;\exists \;M\in \mathfrak {F}:\;\;\;\omega _M(2t)=O(\omega _N(t)) \text { as } t \rightarrow +\infty , \end{aligned}$$
(4.2)

i.e., a mixed version of (3.3) (of Roumieu type, see [8, Sect. 3]).

Suppose there exists \(\textbf{a} = (a_j)\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) with the following properties:

  1. (i)

    we have \(\lim _{j \rightarrow +\infty } a_j^{1/j} = 0\),

  2. (ii)

    \(\textbf{a}\) is a uniform bound for \(\mathfrak {F}\), which means that

    $$\begin{aligned} \forall \;N \in \mathfrak {F}\;\exists \;C>0\;\forall \;j\in {\mathbb {N}}:\;\;\;(N_j/j!=)n_j \le Ca_j. \end{aligned}$$

Then

$$\begin{aligned} \bigcup _{N \in \mathfrak {F}}\mathcal {E}_{\{N\}}(A) = \mathcal {E}_{(G^1)}(A) \text { as sets } \end{aligned}$$

implies that A is bounded.

Remark 4.4

We gather some comments concerning the previous result:

\((*)\):

By choosing \(a_j = \frac{1}{\log (j)^j}\), Lemma 4.3 includes Lemma 4.2 (with \(\beta = 1\)) as a special case.

\((*)\):

Requirements (i) and (ii) in Lemma 4.3 imply that \(\lim _{j \rightarrow +\infty } n_j^{1/j} = 0\) for all \(N \in \mathfrak {F}\).

\((*)\):

If each \(N\in \mathfrak {F}\) satisfies (3.2), then (4.2) follows with \(M=N\).

\((*)\):

In [8, Thm. 3.2] condition (4.2) has been characterized for one-parameter families (weight matrices, see [8, Sect. 2.5]) in terms of the following requirement:

$$\begin{aligned} \exists \;r>1\;\forall \;N\in \mathfrak {F}\;\exists \;M\in \mathfrak {F}\;\exists \;L\in {\mathbb {N}}_{>0}:\quad \liminf _{j \rightarrow +\infty } \frac{(M_{Lj})^{1/(Lj)}}{(N_j)^{1/j}}>r, \end{aligned}$$

i.e., a mixed version of (3.2).

Actually, we show now that, if \(\mathfrak {F}\) consists of a one-parameter family of sequences having some rather mild regularity and growth properties, then it is already possible to find some sequence \(\textbf{a}\) as required in Lemma 4.3.

Proposition 1.21

Let \(\mathfrak {F}:=\{N^{(\beta )}\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}: \beta >0\}\) be a one-parameter family of sequences \(N^{(\beta )}\) which satisfies the following properties:

  1. (i)

    \(N_0^{(\beta )}=1\) for all \(\beta >0\) (normalization),

  2. (ii)

    \(N^{(\beta _1)}\le N^{(\beta _2)}\Leftrightarrow n^{(\beta _1)}\le n^{(\beta _2)}\) for all \(0<\beta _1\le \beta _2\) (point-wise order),

  3. (iii)

    \(\lim _{j\rightarrow +\infty }(n^{(\beta )}_j)^{1/j}=0\) for each \(\beta >0\),

  4. (iv)

    \(j\mapsto (n^{(\beta )}_j)^{1/j}\) is non-increasing for every \(\beta >0\),

  5. (v)

    \(\lim _{j\rightarrow +\infty }\left( \frac{N^{(\beta _2)}_j}{N^{(\beta _1)}_j}\right) ^{1/j}=\lim _{j\rightarrow +\infty }\left( \frac{n^{(\beta _2)}_j}{n^{(\beta _1)}_j}\right) ^{1/j}=+\infty \) for all \(0<\beta _1<\beta _2\) (large growth difference between the sequences).

Then, there exists \(\textbf{a}=(a_j)_j\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\), such that

\((*)\):

\(j\mapsto (a_j)^{1/j}\) is non-increasing,

\((*)\):

\(\lim _{j\rightarrow +\infty }(a_j)^{1/j}=0\), and

\((*)\):

\(\lim _{j\rightarrow +\infty }\left( \frac{a_j}{n_j^{(\beta )}}\right) ^{1/j}=+\infty \) for all \(\beta >0\).

In particular, this implies that there exists a uniform sequence/bound \(\textbf{a}\) for \(\mathfrak {F}\) as required in Lemma 4.3.

In addition, the family \(\mathfrak {F}\) satisfies (4.2).

Note:

\((*)\):

Requirement (iv) weaker than assuming log-concavity for each \(n^{(\beta )}\): Together with (i), i.e., \(n^{(\beta )}_0=1\) (for each \(\beta \)), log-concavity implies (iv); see (iv) in Sect. 2.6.

\((*)\):

Moreover, if (iv) is replaced by assuming that each \(n^{(\beta )}\) is log-concave and (i) by slightly stronger \(n^{(\beta )}_1\le n^{(\beta )}_0=1\) (for each \(\beta \)), then in view of Theorem 3.10, we see that (iii) and (v) together yield

$$\begin{aligned} \forall \;0<\beta _1<\beta _2:\;\;\;\mathcal {E}_{[N^{(\beta _1)}]}\subsetneq \mathcal {E}_{[N^{(\beta _2)}]}. \end{aligned}$$
\((*)\):

In any case, (v) implies that the sequences are pair-wise not equivalent.

\((*)\):

Finally, property (v) alone is sufficient to have (4.2) for \(\mathfrak {F}\).

Proof

Put \(j_1:=1\) and for \(k\in {\mathbb {N}}_{>0}\) set \(j_{k+1}\) to be the smallest integer \(j_{k+1}>j_k\) with

$$\begin{aligned} (n_{j_k}^{(k)})^{1/j_k}> k (n_{j_{k+1}}^{(k+1)})^{1/j_{k+1}}, \end{aligned}$$
(4.3)

and such that for all \(j\ge j_{k+1}\) and all k, we get

$$\begin{aligned} \frac{(n^{(k+1)}_{j})^{1/j}}{(n^{(k)}_j)^{1/j}}\ge k. \end{aligned}$$
(4.4)

For (4.3), we have used properties (ii), (iii), and (iv), and (4.4) holds by property (v).

Now, put \(a_0:=1\) and, for \(j_k\le j<j_{k+1}\), we set

$$\begin{aligned} (a_j)^{1/j}:=(n_{j_k}^{(k)})^{1/j_k}. \end{aligned}$$

Thus, we have by definition that \(j\mapsto (a_j)^{1/j}\) is non-increasing and tending to 0.

Finally, let \(k_0 \in \mathbb {N}_{>0}\) be given (and from now on fixed). For \(j\ge j_{k_0+1}\), we can find \(k\ge k_0\), such that \(j_{k+1}\le j<j_{k+2}\). Thus, in this situation, we can estimate as follows:

$$\begin{aligned} \frac{a_j^{1/j}}{(n^{(k_0)}_j)^{1/j}} = \frac{(n^{(k+1)}_{j_{k+1}})^{1/j_{k+1}}}{(n^{(k_0)}_j)^{1/j}}\ge \frac{(n^{(k+1)}_{j_{k+1}})^{1/j_{k+1}}}{(n^{(k)}_j)^{1/j}} \ge \frac{(n^{(k+1)}_{j})^{1/j}}{(n^{(k)}_j)^{1/j}}\ge k, \end{aligned}$$

hence \(\lim _{j\rightarrow +\infty }\frac{a_j^{1/j}}{(n^{(k_0)}_j)^{1/j}}=+\infty \). The second inequality follows from the fact that \(j\mapsto (n_j^{(k+1)})^{1/j}\) is non-increasing (property (iv)). By the point-wise order for any \(\beta >0\), we can find some \(k_0 \in \mathbb {N}_{>0}\), such that \(\frac{a_j^{1/j}}{(n^{(\beta )}_j)^{1/j}}\ge \frac{a_j^{1/j}}{(n^{(k_0)}_j)^{1/j}}\) for all \(j\ge 1\), and hence, the last desired property for \(\textbf{a}\) is verified.

Concerning (4.2), we note that by (v), we get \(2^jN^{(\beta _1)}_j\le N^{(\beta _2)}_j\) for all \(0<\beta _1<\beta _2\) and all j sufficiently large. Consequently

$$\begin{aligned} \forall \;0<\beta _1<\beta _2\;\exists \;C\ge 1\;\forall \;j\in {\mathbb {N}}:\;\;\;2^jN^{(\beta _1)}_j\le CN^{(\beta _2)}_j, \end{aligned}$$

which yields by definition of associated weights \(\omega _{N^{(\beta _2)}}(2t)\le \omega _{N^{(\beta _1)}}(t)+\log (C)\) for all \(t\ge 0\). This verifies (4.2) for \(\mathfrak {F}\). \(\square \)

Remark 4.6

The previous result shows that any family \(\mathfrak {F}\subseteq \mathcal {L}\mathcal {C}\) that can be parametrized to satisfy (ii)–(v) from Proposition 4.5 is already uniformly bounded by some sequence \(\textbf{a}\).

Consequently, in this case, the assumptions (i) and (ii) from Lemma 4.3 on the existence of \(\textbf{a}\) are superfluous, and also, assumption (4.2) for \(\mathfrak {F}\) holds true automatically.

Before we can give the proof of Lemma 4.3, we need one more technical lemma as preparation.

Lemma 1.23

Let \(\textbf{a} = (a_j)_j \in \mathbb {R}_{>0}^\mathbb {N}\) with \(\lim _{j\rightarrow +\infty }a_j^{1/j}=0\) be given. Then, there exists a function \(g = g_{\textbf{a}}: {\mathbb {R}}_{>0}\rightarrow {\mathbb {R}}_{>0}\) with the following properties:

\((*)\):

\(\lim _{t\rightarrow +\infty }g_{\textbf{a}}(t)=+\infty \).

\((*)\):

For all \(N\in {\mathcal{L}\mathcal{C}}\), such that \(n_j \le D a_j\) (for some \(D = D(N)>0\) and all \(j\in {\mathbb {N}}\)), and all \(d, s>0\), we have that

$$\begin{aligned} \lim _{t\rightarrow +\infty }s\omega _N(t/2)-dg_{\textbf{a}}(t)t=+\infty . \end{aligned}$$

Proof

Observe that

$$\begin{aligned} \omega _N(t) \ge \sup _{k \in \mathbb {N}} \log \frac{t^k}{Da_k k!} \ge \log \left( \frac{1}{2D}\sum _{k = 0}^{+\infty }\frac{(t/2)^k}{a_k k!} \right) =: h_{\textbf{a}}(t)-\log (2D). \end{aligned}$$

It is clear from the definition that \(h_{\textbf{a}}\) is non-decreasing. From the assumption \(\lim _{j\rightarrow +\infty }a_j^{1/j}=0\), it follows that for every \(R>0\), there exists \(C\in \mathbb {R}\), such that for all \(t>0\), we have

$$\begin{aligned} h_{\textbf{a}}(t) \ge C +Rt. \end{aligned}$$
(4.5)

This estimate follows, since for every (small) \(\varepsilon > 0\), there exists \(B>0\), such that \(a_k\le B \varepsilon ^k\) for all \(k \in \mathbb {N}\); and therefore

$$\begin{aligned} \log \left( \sum _{k = 0}^{+\infty }\frac{(t/2)^k}{a_k k!}\right) \ge \frac{t}{2\varepsilon }-\log (B), \end{aligned}$$

which gives (4.5).

Let us set \(f_{\textbf{a}}(t):= \frac{h_{\textbf{a}}(t/2)}{t}\), then, by (4.5), \(\lim _{t\rightarrow +\infty }f_{\textbf{a}}(t)=+\infty \). Finally, set \(g_{\textbf{a}}:=\sqrt{f_{\textbf{a}}}\), and so, \(\lim _{t\rightarrow +\infty }g_{\textbf{a}}(t)=+\infty \). Moreover, we have \(\lim _{t\rightarrow +\infty }\varepsilon f_{\textbf{a}}(t) - g_{\textbf{a}}(t)=+\infty \) for every \(\varepsilon >0\). Thus, for any arbitrary fixed \(s > 0\), we get

$$\begin{aligned} s\omega _N(t/2) - g_{\textbf{a}}(t)t \ge s h_{\textbf{a}}(t/2)-s\log (2D)-g_{\textbf{a}}(t)t=t(sf_{\textbf{a}}(t)-g_{\textbf{a}}(t))-s\log (2D); \end{aligned}$$
(4.6)

hence, \(\lim _{t\rightarrow +\infty }s\omega _N(t/2) - g_{\textbf{a}}(t)t=+\infty \). This shows the statement for \(d=1\). For \(d \ne 1\), the result simply follows by choosing s/d in (4.6). \(\square \)

Proof of Lemma 4.3

We adapt the proof of [14, Lemma 4]. Therefore, assume that the operator A is actually unbounded. Then, the spectrum \(\sigma (A)\) is unbounded as well, and so, there exists a strictly increasing sequence of natural numbers k(n), such that

  1. (i)

    \(n \le g_{\textbf{a}}(k(n))\) (and \(n \le k(n)\)) for all \(n\in {\mathbb {N}}_{>0}\),

  2. (ii)

    in each ring \(\{\lambda \in \mathbb {C}:~ k(n)< |\lambda | < k(n)+1\}\), there is a point \(\lambda _n \in \sigma (A)\),

and we can actually find a 0-sequence \(\varepsilon _n\) with \(0<\varepsilon _n<\min (1/n,\varepsilon _{n-1})\), such that \(\lambda _n\) belongs to the ring

$$\begin{aligned} r_n:= \{\lambda \in \mathbb {C}:~ k(n)-\varepsilon _n< |\lambda | < k(n)+1-\varepsilon _n\}. \end{aligned}$$

As in Markin’s proof, the subspaces \(E_A(r_n)H\) are non-trivial and pair-wise orthogonal. Thus, in each of those spaces, we may choose a non-trivial element \(e_n\), such that

$$\begin{aligned} e_n=E_A(r_n)e_n, \quad \langle e_i, e_j\rangle = \delta _{i,j}. \end{aligned}$$

Now, we define

$$\begin{aligned} f:=\sum _{n = 1}^{+\infty } g_{\textbf{a}}(k(n))^{-(k(n)+1-\varepsilon _n)}e_n. \end{aligned}$$

As in [14], the sequence of coefficients belongs to \(\ell ^2\), and

$$\begin{aligned} E_A(r_n)f=g_{\textbf{a}}(k(n))^{-(k(n)+1-\varepsilon _n)}e_n, \quad E_A\left( \bigcup _{n \in \mathbb {N}_{>0}} r_n\right) f=f. \end{aligned}$$

Moreover, for every \(t>0\), we have

$$\begin{aligned} \int _\mathbb {C}e^{2t|\lambda |}d\langle E_A(\lambda )f,f\rangle&= \int _\mathbb {C}e^{2t|\lambda |}d \left\langle E_A(\lambda )E_A\left( \bigcup _{n \in \mathbb {N}_{>0}}r_n\right) f,E_A \left( \bigcup _{n \in \mathbb {N}_{>0}}r_n\right) f\right\rangle \\&=\sum _{n = 1}^\infty \int _{r_n} e^{2t|\lambda |}d\langle E_A(\lambda )f,f\rangle \\&=\sum _{n = 1}^\infty \int _{r_n} e^{2t|\lambda |}d\langle E_A(\lambda )E_A(r_n)f,E_A(r_n)f\rangle \\&=\sum _{n=1}^\infty g_{\textbf{a}}(k(n))^{-2(k(n)+1-\varepsilon _n)} \int _{r_n} e^{2t|\lambda |}d\langle E_A(\lambda )e_n,e_n\rangle \\&\le \sum _{n=1}^\infty e^{-2\log (g_{\textbf{a}}(k(n))(k(n)+1-\varepsilon _n)}e^{2t(k(n)+1-\varepsilon _n)} \underbrace{\Vert E_A(r_n)e_n\Vert ^2}_{=1}\\&= \sum _{n = 1}^{+\infty }e^{-2(\log (g_{\textbf{a}}(k(n))-t)(k(n)+1-\varepsilon _n)}<+\infty , \end{aligned}$$

where we used in the first inequality that for \(\lambda \in r_n\), we have \(|\lambda | \le k(n)+1-\varepsilon _n\), and in the final inequality that \(g_{\textbf{a}}\) tends to infinity and that \(k(n)\ge n\). Thus, we have shown that \( f \in \mathcal {E}_{(G^1)}(A)\).

Moreover, in analogy to [14], and by a similar reasoning as above, we get for all \(N\in \mathfrak {F}\) and \(t>0\)

$$\begin{aligned} \int _\mathbb {C}e^{2\omega _N(t|\lambda |)}d\langle E_A(\lambda )f,f\rangle = \sum _{n=1}^\infty g_{\textbf{a}}(k(n))^{-2(k(n)+1-\varepsilon _n)} \int _{r_n} e^{2\omega _N(t|\lambda |)}d\langle E_A(\lambda )e_n,e_n\rangle . \end{aligned}$$
(4.7)

Next, we observe that for \(\lambda \in r_n\), we have \(\omega _N(t|\lambda |) \ge \omega _N(t(k(n)-\varepsilon _n))\ge \omega _N(t(k(n)-1))\). We continue to estimate the right-hand side of (4.7) and infer

$$\begin{aligned} \int _\mathbb {C}e^{2\omega _N(t|\lambda |)}d\langle E_A(\lambda )f,f\rangle&\ge \sum _{n=1}^\infty g_{\textbf{a}}(k(n))^{-2(k(n)+1-\varepsilon _n)} e^{2\omega _N(t(k(n)-1))} \underbrace{\int _{r_n} d\langle E_A(\lambda )e_n,e_n\rangle }_{=1}\\&\ge \sum _{n=1}^\infty e^{2(\omega _N(t(k(n)-1))-\log (g_{\textbf{a}}(k(n)))(k(n)+1))}. \end{aligned}$$

By iterating (4.2), there exist \(M\in \mathfrak {F}\), \(s>0 \) (small) and \(C > 0\) (large), such that for all \(\lambda \in \mathbb {C}\)

$$\begin{aligned} \omega _N(t|\lambda |) \ge s\omega _M(|\lambda |)-C, \end{aligned}$$

which allows us to continue the estimate and get

$$\begin{aligned} \int _\mathbb {C}e^{2\omega _N(t|\lambda |)}d\langle E_A(\lambda )f,f\rangle \ge \sum _{n=1}^\infty e^{2(s\omega _M((k(n)-1))-C-\log (g_{\textbf{a}}(k(n)))(k(n)+1))}=+\infty , \end{aligned}$$

where the last equality follows from Lemma 4.7 (applied to the sequence M and \(d=2\)). Thus, we infer that \(f \notin \mathcal {E}_{\{N\}}(A)\). Since \(N \in \mathfrak {F}\) has been arbitrary, we are done. \(\square \)

Finally, we are now in the position to prove our main theorem, a generalization of [14, Thm. 5.1] which reads as follows.

Theorem 1.24

Suppose there exists \(\textbf{a}=(a_j)_j\), such that \(\lim _{j\rightarrow +\infty }a_j^{1/j}=0\) and a family \(\mathfrak {F}\) of weight sequences as in Lemma 4.3. Assume that for any weak solution y of (4.1) on \([0,+\infty )\), there is \(N \in \mathfrak {F}\), such that \(y \in \mathcal {E}_{\{N\}}([0,+\infty ),H)\). Then, the operator A is bounded.

Proof

Let y be a weak solution of (4.1). By assumption, there exists \(N \in \mathfrak {F}\), such that \(y \in \mathcal {E}_{\{N\}}([0,+\infty ),H)\). By Theorem 4.1, we get that for every \(t \ge 0\), we have

$$\begin{aligned} y(t) \in \mathcal {E}_{\{N\}}(A), \end{aligned}$$

in particular \(y(0) \in \mathcal {E}_{\{N\}}(A)\). Via an application of [12, Thm. 3.1], we infer

$$\begin{aligned} \bigcap _{t>0}D(e^{tA})\subseteq \bigcup _{N \in \mathfrak {F}}\mathcal {E}_{\{N\}}(A). \end{aligned}$$
(4.8)

On the other hand, since

$$\begin{aligned} \bigcap _{t>0} D(e^{tA}) = \bigcap _{t>0} \left\{ f \in H:~\int _\mathbb {C}e^{2t \mathcal {R}(\lambda )} \langle d E_A(\lambda )f,f \rangle <+\infty \right\} , \end{aligned}$$

it is clear that

$$\begin{aligned} \bigcap _{t>0} D(e^{tA}) \supseteq \bigcap _{t>0} \left\{ f \in H:~\int _\mathbb {C}e^{2t |\lambda |} \langle d E_A(\lambda )f,f \rangle <+\infty \right\} = \mathcal {E}_{(G^1)}(A). \end{aligned}$$

Together with (4.8), this yields

$$\begin{aligned} \bigcup _{N \in \mathfrak {F}}\mathcal {E}_{\{N\}}(A) = \mathcal {E}_{(G^1)}(A). \end{aligned}$$

Thus, using Lemma 4.3, we conclude that A is bounded. \(\square \)

When taking \(\mathfrak {F}\) to be the family of all small Gevrey sequences, i.e., \(\mathfrak {F}=\mathfrak {G}:=\{G^\alpha :~\alpha < 1\}\), we infer [14, Thm. 5.1] (see also Remark 4.4).

4.4 An answer to the motivating question from Sect. 4.2

The final goal is now to combine the information from Theorems 3.4 and 4.8.

Therefore, suppose \(\mathfrak {F}\) is a family of weight sequences, such that:

  1. (i)

    \(N\in {\mathcal{L}\mathcal{C}}\) for all \(N\in \mathfrak {F}\) and \(1=N_0=N_1\),

  2. (ii)

    \(\mathfrak {F}\) has (4.2),

  3. (iii)

    \(\mathfrak {F}\) is uniformly bounded by some \(\textbf{a}=(a_j)_j\) with \(\lim _{j\rightarrow +\infty }a_j^{1/j}=0\), and

  4. (iv)

    for all \(N\in \mathfrak {F}\), we have that n is log-concave.

Note that (iii) gives \(\lim _{j\rightarrow +\infty }(n_j)^{1/j}=0\) for all \(N \in \mathfrak {F}\). Therefore, \(\mathfrak {F}\) is a family as required in Lemma 4.3 and by (i), (iii), and (iv) Theorem 3.4 can be applied to each \(N\in \mathfrak {F}\), and hence

$$\begin{aligned} \forall \;N\in \mathfrak {F}:\;\;\;\mathcal {E}_{\{N\}}(I,H) \cong \mathcal {H}^\infty _{\underline{\mathcal {N}^*}_\mathfrak {c}}(\mathbb {C},H). \end{aligned}$$

Summarizing, we can reformulate Theorem 4.8 as follows.

Theorem 1.25

Let \(\mathfrak {F}\) be a family of weight sequences as considered before. Suppose that for every weak solution y of (4.1), there exist \(N \in \mathfrak {F}\) and \(C,k>0\), such that y can be extended to an entire function with

$$\begin{aligned} \Vert y(z)\Vert \le Ce^{\omega _{N^*}(k|z|)}. \end{aligned}$$

Then, A is already a bounded operator.

Theorem 4.9 applies to the family \(\mathfrak {G}:=\{G^{\alpha }: 0\le \alpha <1\}\) of all small Gevrey sequences.