Ultraholomorphic extension theorems in the mixed setting

Abstract

The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.

Introduction

In the authors’ recent works [10, 14] we have shown extension theorems in the ultraholomorphic weight function framework, in the first article for spaces of Roumieu type and in the second one also for the Beurling type classes. Such results have already been known before for the weight sequence approach, see [32]. In [14] we have transferred Thilliez’s ideas to the weight function situation (by using ultradifferentiable Whitney extension results) and in [10] we have used complex methods treated by Lastra, Malek and the second author [18, 19] in the single weight sequence approach.

In the ultradifferentiable setting also Whitney extension results involving two weight sequences M and N and weight functions \(\sigma \) and \(\omega \) are known in the literature. In the weight sequence case we refer to [7] for the Whitney jet mapping and to [31] for the Borel mapping, in the weight function case see [5] for the Borel mapping and [17, 24, 25] for the general Whitney jet mapping. In our recent paper [13], which has served as motivation for this article, by involving a ramification parameter \(r\in {\mathbb {N}}_{>0}\) we have generalized the mixed setting results from [31] to r-ramification classes introduced in [30]. We have also generalized the Whitney extension results from [7] by using a parameter \(r>0\) (see [13, Theorem 5.9]).

The possibility of an extension in these mixed settings has been characterized in terms of growth properties of weight sequences and functions. We refer also to Remarks 3 and 4 where more (historical) explanations will be given.

From this theoretical point of view it seems natural to ask whether in the ultraholomorphic framework we can also prove extension results in the mixed settings and this question will be treated in this present work. We will consider Roumieu type classes in both the weight sequence and weight function setting. By inspecting the proofs of the main results in [10, 14, 32] it has turned out that, up to our ability, only the complex methods from [10] admit the possibility to generalize the result to a mixed situation, see Remark 13 below for further details.

The existence of ultraholomorphic extension results is tightly connected to the opening of the sectors where the functions are defined. In the previous results [10, 12, 14, 32] growth indices \(\gamma (M)\) and \(\gamma (\omega )\) have been introduced to measure the maximum size of these sectors, for a detailed study and comparison of these values we refer to [11]. Then a similar notion is required in the mixed setting to obtain satisfactory theorems. Therefore, motivated by the occurring mixed ramified conditions between M and N and their associated weight functions \(\omega _M\) and \(\omega _N\) appearing in [13] the definition of the mixed growth index for sequences \(\gamma (M,N)\) and for weight functions \(\gamma (\sigma ,\omega )\) has been given, see Sect. 3.1.

Under restrictions of the opening of the sectors in terms of these indices, we have stated the main extension result, Theorem 2, for a pair of two given weight functions, using the weight matrix tool described and used in [22, 28]. Then the results are transferred to the weight sequence case thanks to the associated weight functions.

Compared with the previous known extension results for weight functions (in the ultraholomorphic setting) we will also treat ”exotic” cases here, more precisely: The growth property \(\omega (2t)=O(\omega (t))\) as \(t\rightarrow +\infty \), denoted by \((\omega _1)\) in this article, will not be needed in general anymore in the mixed situation. This property is usually a very basic assumption when working with (Braun–Meise–Taylor) weight functions \(\omega \) and it is equivalent to having \(\gamma (\omega )>0\) as shown by the authors in [10]. Moreover \((\omega _1)\) has also been used to have that the class defined by \(\omega \) admits a representation by using the so-called associated weight matrix \(\varOmega \), see Sect. 2.5 for a summary. Our main extension result Theorem 2 is formulated between ultraholomorphic classes defined by weight matrices and we are able to treat such a general situation since in [10] we have worked with weight functions and their associated weight matrices also in a ”nonstandard” setting, i.e. not assuming \((\omega _1)\) necessarily. More detailed explanations will be given in Remark 12 below. In Appendix A such nonstandard examples will be constructed explicitly and underlining the different situation in our work here.

In the preceding extension results for one sequence, the opening of the sector where the functions are defined is at most \(\pi \gamma (M)\). As it will be seen in Sect. 3, for any sequences M and N satisfying standard assumptions the mixed index \(\gamma (M,N)\) is always belonging to the interval \([\gamma (N),\mu (N)]\), where \(\mu (N)\) is denoting the so-called order of quasianalyticity introduced by the second author, see [12, 26]. We know that even for strongly regular sequences N one can have \(\gamma (N)<\mu (N)\) and the gap can become as large as desired, see Remark 8. In these situations, we can provide an extension map for any opening \(\pi \gamma \) with \(\gamma (N)\le \gamma <\mu (N)\) by limiting the size of the derivatives at the origin in terms of a smaller sequence M. Furthermore, this sequence M can be chosen optimal in some sense, thanks to a modified version of the technical construction in [23, Section 4.1]. Hence we can show that the Borel map will be not surjective necessarily anymore but admitting a controlled loss of regularity, so that \(\mu (N)\), usually related to the injectivity of the Borel mapping, does have also a meaning associated with the surjectivity. For weight functions the situation is analogous by introducing the order \(\mu (\omega )\) in Sect. 3.2.

The paper is organized as follows: First, in Sect. 2 all necessary notation and conditions on weight sequences and functions used in this article will be introduced. In Sect. 3 we will define and study the new mixed growth indices \(\gamma (M,N)\) and \(\gamma (\sigma ,\omega )\) and investigate also the connection of these values to the orders \(\mu (N)\) and \(\mu (\omega )\). In Sects. 4 and 5 we will transfer the results from [10] to the mixed settings and providing only the necessary changes in the proofs, the main results will be Theorem 2 for the general mixed weight function case, Corollary 1 for mixed Braun–Meise–Taylor weight functions having \((\omega _1)\) and Theorem 4 for the mixed weight sequence case. In Sect. 6 we will prove mixed extension results fixing only the weight that defines the function space for any sector with opening smaller than \(\pi \gamma (\cdot )\), see Theorems 5 and 6. Finally, in the Appendix A, we are providing some (counter-)examples showing \(\gamma (M,N), \gamma (\sigma ,\omega )>0\), but such that all nonmixed indices \(\gamma (\cdot )\) are vanishing, see Theorem 7.

Ultradifferentiable classes defined by weight sequences and functions

General notation

Throughout this paper we will use the following notation: We will write \({\mathbb {N}}_{>0}=\{1,2,\dots \}\) and \({\mathbb {N}}={\mathbb {N}}_{>0}\cup \{0\}\), moreover we put \({\mathbb {R}}_{>0}:=\{x\in {\mathbb {R}}: x>0\}\).

Weight sequences

\(M=(M_k)_k\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) is called a weight sequence, we introduce also \(m=(m_k)_k\) defined by \(m_k:=\frac{M_k}{k!}\) and \(\mu _k:=\frac{M_k}{M_{k-1}}\), \(\mu _0:=1\). Similarly we will use this notation for sequences NSL as well. M is called normalized if \(1=M_0\le M_1\) holds true which can always be assumed without loss of generality.

For any given weight sequence M and \(r>0\) we will write \(M^{1/r}:=((M_p)^{1/r})_{p\in {\mathbb {N}}}\).

  1. (1)

    M is called log-convex, we will write \((\text {lc})\), if

    $$\begin{aligned} \forall \;j\in {\mathbb {N}}_{>0}:\;M_j^2\le M_{j-1} M_{j+1}, \end{aligned}$$

    equivalently if \((\mu _p)_p\) is nondecreasing. M is called strongly log-convex if \((\text {lc})\) holds for the sequence m. If M is log-convex and normalized, then M and the mapping \(j\mapsto (M_j)^{1/j}\) are nondecreasing, e.g. see [27, Lemma 2.0.4]. In this case we get \(M_k\ge 1\) for all \(k\ge 0\) and

    $$\begin{aligned} \forall \;k\in {\mathbb {N}}_{>0}:\;\;\;(M_k)^{1/k}\le \mu _k. \end{aligned}$$
  2. (2)

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

    $$\begin{aligned} \exists \;C\ge 1\;\forall \;j,k\in {\mathbb {N}}:\;M_{j+k}\le C^{j+k} M_j M_k. \end{aligned}$$

    We can replace in this condition M by m and by \(M^{1/r}\) (\(r>0\) arbitrary) by changing the constant C.

  3. (3)

    M is called nonquasianalytic, we write \((\text {nq})\), if

    $$\begin{aligned} \sum _{p=1}^{\infty }\frac{1}{\mu _p}<\infty . \end{aligned}$$

    More generally, for arbitrary \(r>0\) we call M to be r-nonquasianalytic, denoted by \((\text {nq}_r)\), if

    $$\begin{aligned} \sum _{p=1}^{\infty }\left( \frac{1}{\mu _p}\right) ^{1/r}<\infty , \end{aligned}$$

    and so M has \((\text {nq}_r)\) if and only if \(M^{1/r}\) has \((\text {nq})\).

  4. (4)

    M has \((\gamma _1)\), 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 the literature \((\gamma _1)\) is also called “strongly nonquasianalyticity condition”.

Due to technical reasons it is often convenient to assume several properties for M at the same time and hence we define the class

$$\begin{aligned} M\in {\mathscr {SR}},\hbox { if }M\hbox { is normalized and has }{(\text {slc})}, {(\text {mg})}\hbox { and }{(\gamma _1)}. \end{aligned}$$

Using this notation we see that \(M\in {\mathscr {SR}}\) if and only if m is a strongly regular sequence in the sense of [32, 1.1] (and this terminology has also been used by several authors so far, e.g. see [19, 26]). At this point we want to make the reader aware that here we are using the same notation as it has already been used by the authors in [10, 14], whereas in [32] and also in [11] the sequence M is precisely m in the notation in this work.

(5) For two weight sequences \(M=(M_p)_p\) and \(N=(N_p)_p\) we write \(M\le N\) if and only if \(M_p\le N_p\Leftrightarrow m_p\le n_p\) holds for all \(p\in {\mathbb {N}}\) (and similarly for the sequence of quotients \(\mu \) and \(\nu \)) and write \(M{\precsim }N\) if

$$\begin{aligned} \exists \;C_1,C_2\ge 1\;\forall \;p\in {\mathbb {N}}:\; M_p\le C_1 C_2^p N_p\Longleftrightarrow \sup _{p\in {\mathbb {N}}_{>0}}\left( \frac{M_p}{N_p}\right) ^{1/p}<+\infty \end{aligned}$$

and call them equivalent, denoted by \(M{\approx }N\), if

$$\begin{aligned} M{\precsim }N\;\text {and}\;N{\precsim }M. \end{aligned}$$

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

Some properties for weight sequences are very basic and so we introduce for convenience the following set:

$$\begin{aligned} {\mathscr {LC}}:=\{M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}:\;M\;\text {is normalized, log-convex},\;\lim _{k\rightarrow \infty }(M_k)^{1/k}=+\infty \}. \end{aligned}$$

It is well-known (e.g. see [23, Lemma 2.2]) that for any \(M\in {\mathscr {LC}}\) condition \(({\text {mg}})\) is equivalent to \(\sup _{p\in {\mathbb {N}}}\frac{\mu _{2p}}{\mu _p}<\infty \) and to \(\sup _{p\in {\mathbb {N}}_{>0}}\frac{\mu _{p+1}}{(M_p)^{1/p}}<\infty \).

A prominent example are the Gevrey sequences\(G^r:=(p!^r)_{p\in {\mathbb {N}}}\), \(r>0\), which belong to the class \({\mathscr {SR}}\) for any \(r>1\).

Weight functions

A function \(\omega :[0,+\infty )\rightarrow [0,+\infty )\) is called a weight function (in the terminology of [10, Section 2.2]), if it is continuous, nondecreasing, \(\omega (0)=0\) and \(\lim _{t\rightarrow +\infty }\omega (t)=+\infty \). If \(\omega \) satisfies in addition \(\omega (t)=0\) for all \(t\in [0,1]\), then we call \(\omega \) a normalized weight function and for convenience we will write that \(\omega \) has \({(\omega _0)}\).

Given an arbitrary function \(\omega \) we will denote by \(\omega ^{\iota }\) the function \(\omega ^{\iota }(t):=\omega (1/t)\) for any \(t>0\). Moreover, for \(r>0\), we put \(\omega ^r\) to be the function given by \(\omega ^r(t):=\omega (t^r)\).

Moreover we consider the following conditions, this list of properties has already been used in [28].

\((\omega _1)\):

\(\omega (2t)=O(\omega (t))\) as \(t\rightarrow +\infty \), i.e. \(\exists \;L\ge 1\;\forall \;t\ge 0:\;\;\;\omega (2t)\le L(\omega (t)+1)\).

\((\omega _2)\):

\(\omega (t)=O(t)\) as \(t\rightarrow +\infty \).

\((\omega _3)\):

\(\log (t)=o(\omega (t))\) as \(t\rightarrow +\infty \) (\(\Leftrightarrow \lim _{t\rightarrow +\infty }\frac{t}{\varphi _{\omega }(t)}=0\), \(\varphi _{\omega }\) being the function defined next).

\((\omega _4)\):

\(\varphi _{\omega }:t\mapsto \omega (e^t)\) is a convex function on \({\mathbb {R}}\).

\((\omega _5)\):

\(\omega (t)=o(t)\) as \(t\rightarrow +\infty \).

\((\omega _6)\):

\(\exists \;H\ge 1\;\forall \;t\ge 0:\;2\omega (t)\le \omega (H t)+H\).

\((\omega _{\text {nq}})\):

\(\int _1^{+\infty }\frac{\omega (t)}{t^2}dt<+\infty .\)

\((\omega _{\text {snq}})\):

\(\exists \;C>0\;\forall \;y>0: \int _1^{+\infty }\frac{\omega (y t)}{t^2}dt\le C\omega (y)+C\).

An interesting example is \(\sigma _s(t):=\max \{0,\log (t)^s\}\), \(s>1\), which satisfies all listed properties except \((\omega _6)\). It is well-known that the ultradifferentiable class defined by using the weight \(t\mapsto t^{1/s}\) coincides with the ultradifferentiable class given by the weight sequence \(G^s=(p!^s)_{p\in {\mathbb {N}}}\) of index \(s>1\).

For convenience we define the sets

$$\begin{aligned} {{\mathscr {W}}_0}:= & {} \{\omega :[0,\infty )\rightarrow [0,\infty ): \omega \;\text {has}\;{(\omega _0)},{(\omega _3)},{(\omega _4)}\},\\ {{\mathscr {W}}}:= & {} \{\omega \in {\mathscr {W}}_0: \omega \;\text {has}\;{(\omega _1)}\}. \end{aligned}$$

For any \(\omega \in {{\mathscr {W}}_0}\) we define the Legendre–Fenchel–Young-conjugate of \(\varphi _{\omega }\) by

$$\begin{aligned} \varphi ^{*}_{\omega }(x):=\sup \{x y-\varphi _{\omega }(y): y\ge 0\},\;\;\;x\ge 0, \end{aligned}$$

with the following properties, e.g. see [6, Remark 1.3, Lemma 1.5]: It is convex and nondecreasing, \(\varphi ^{*}_{\omega }(0)=0\), \(\varphi ^{**}_{\omega }=\varphi _{\omega }\), \(\lim _{x\rightarrow +\infty }\frac{x}{\varphi ^{*}_{\omega }(x)}=0\) and finally \(x\mapsto \frac{\varphi _{\omega }(x)}{x}\) and \(x\mapsto \frac{\varphi ^{*}_{\omega }(x)}{x}\) are nondecreasing on \([0,+\infty )\). For any \(\omega \in {{\mathscr {W}}}\) we can assume w.l.o.g. that \(\omega \) is \({\mathscr {C}}^1\) (see [6, Lemma 1.7]).

Let \(\sigma ,\tau \) be weight functions, we write \(\sigma {\preceq }\tau \) if \(\tau (t)=O(\sigma (t))\;\text {as}\;t\rightarrow +\infty \) and call them equivalent, denoted by \(\sigma {\sim }\tau \), if \(\sigma {\preceq }\tau \) and \(\tau {\preceq }\sigma \).

Motivated by the notion of a strong weight function given in [3]

$$\begin{aligned} \omega \hbox { will be called a strong weight, if } \omega \in {{\mathscr {W}}_0}\hbox { and in addition } {(\omega _{\text {snq}})}\hbox { is satisfied.} \end{aligned}$$

Note that for any weight function property \((\omega _{\text {nq}})\) implies \((\omega _5)\) because \(\int _t^{+\infty }\frac{\omega (u)}{u^2}du\ge \omega (t)\int _t^{+\infty }\frac{1}{u^2}du=\frac{\omega (t)}{t}\).

If \(\omega \) satisfies any of the properties \((\omega _1)\), \((\omega _3)\), \((\omega _4)\) or \((\omega _6)\), then the same holds for \(\omega ^r\), but \((\omega _5)\), \((\omega _{\text {nq}})\) or \((\omega _{\text {snq}})\) might not be preserved.

Concerning condition \((\omega _{\text {nq}})\) we point out that

$$\begin{aligned} \int _1^{\infty }\frac{\omega ^r(u)}{u^2}du=\int _1^{\infty }\frac{\omega (u^r)}{u^2}du=\frac{1}{r}\int _1^{\infty }\frac{\omega (v)}{u^2}\frac{dv}{u^{r-1}}=\frac{1}{r}\int _1^{\infty }\frac{\omega (v)}{v^{1+1/r}}dv, \end{aligned}$$
(1)

hence it makes sense to consider the following generalization \((\omega _{\text {nq}_r})\) (analogously to \((\text {nq}_r)\)):

$$\begin{aligned} \int _1^{\infty }\frac{\omega (u)}{u^{1+1/r}}du<+\infty . \end{aligned}$$

Then \(\omega ^r\) has \((\omega _{\text {nq}})\) if and only if \(\omega \) has \((\omega _{\text {nq}_r})\).

Weight matrices

For the following definitions see also [22, Section 4].

Let \({\mathscr {I}}={\mathbb {R}}_{>0}\) denote the index set (equipped with the natural order), a weight matrix\({\mathscr {M}}\) associated with \({\mathscr {I}}\) is a (one parameter) family of weight sequences \({\mathscr {M}}:=\{M^x\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}: x\in {\mathscr {I}}\}\), such that

$$\begin{aligned} \forall \;x\in {\mathscr {I}}:\;M^x\;\text {is normalized, nondecreasing},\;M^{x}\le M^{y}\;\text {for}\;x\le y. \end{aligned}$$

For convenience we will write \(({\mathscr {M}})\) for this basic assumption on \({\mathscr {M}}\). We call a weight matrix \({\mathscr {M}}\)standard log-convex, denoted by \(({\mathscr {M}}_{{\text {sc}}})\), if \({\mathscr {M}}\) has \(({\mathscr {M}})\) and

$$\begin{aligned} \forall \;x\in {\mathscr {I}}:\;M^x\in {\mathscr {LC}}. \end{aligned}$$

Moreover, we put \(m^x_p:=\frac{M^x_p}{p!}\) for \(p\in {\mathbb {N}}\), and \(\mu ^x_p:=\frac{M^x_p}{M^x_{p-1}}\) for \(p\in {\mathbb {N}}_{>0}\), \(\mu ^x_0:=1\).

A matrix is called constant if \(M^x {\approx }M^y\) for all \(x,y\in {\mathscr {I}}\).

Let \({\mathscr {M}}=\{M^x: x\in {\mathscr {I}}\}\) and \({\mathscr {N}}=\{N^x: x\in {\mathscr {J}}\}\) be \(({\mathscr {M}})\). We write \({\mathscr {M}}{\{\precsim \}}{\mathscr {N}}\) if

$$\begin{aligned} \forall \;x\in {\mathscr {I}}\;\exists \;y\in {\mathscr {J}}:\;M^x {\precsim }N^y, \end{aligned}$$

and call them equivalent, denoted by \({\mathscr {M}}{\{\approx \}}{\mathscr {N}}\), if \({\mathscr {M}} {\{\precsim \}}{\mathscr {N}}\) and \({\mathscr {N}} {\{\precsim \}}{\mathscr {M}}\).

Weight matrices obtained by weight functions

We summarize some facts which are shown in [22, Section 5] and will be needed in this work. All properties listed below will be valid for \(\omega \in {{\mathscr {W}}_0}\), except (3) for which \((\omega _1)\) is necessary.

(i):

The idea was that to each \(\omega \in {{\mathscr {W}}_0}\) we can associate a \(({\mathscr {M}}_{\text {sc}})\) weight matrix \(\varOmega :=\{W^l=(W^l_j)_{j\in {\mathbb {N}}}: l>0\}\) by

$$\begin{aligned} W^l_j:=\exp \left( \frac{1}{l}\varphi ^{*}_{\omega }(lj)\right) . \end{aligned}$$

In general it is not clear that \(W^x\) is strongly log-convex, i.e. \(w^x\) is log-convex, too.

(ii):

\(\varOmega \) satisfies

$$\begin{aligned} \forall \;l>0\;\forall \;j,k\in {\mathbb {N}}:\;\;\;W^l_{j+k}\le W^{2l}_jW^{2l}_k. \end{aligned}$$
(2)

In case \(\omega \) has moreover \((\omega _1)\), \(\varOmega \) has also

$$\begin{aligned} \forall \;h\ge 1\;\exists \;A\ge 1\;\forall \;l>0\;\exists \;D\ge 1\;\forall \;j\in {\mathbb {N}}:\;\;\;h^jW^l_j\le D W^{Al}_j. \end{aligned}$$
(3)
(iii):

Equivalent weight functions yield equivalent associated weight matrices.

(iv):

\((\omega _5)\) holds if and only if \(\lim _{p\rightarrow +\infty }(w^l_p)^{1/p}=+\infty \) for all \(l>0\).

Moreover we have:

Remark 1

Let \(\omega \in {{\mathscr {W}}_0}\) be given, then \(\omega \) satisfies

  1. (a)

    \((\omega _{{\text {nq}}})\) if and only if some/each \(W^l\) satisfies \(({\text {nq}})\),

  2. (b)

    \((\omega _6)\) if and only if some/each \(W^l\) satisfies \(({\text {mg}})\) if and only if \(W^l {\approx }W^n\) for each \(l,n>0\). Consequently \((\omega _6)\) is characterizing the situation when \(\varOmega \) is constant.

Associated weight functions \(\omega _M\) and \(h_M\)

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

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

For an abstract introduction of the associated function we refer to [20, Chapitre I], see also [15, 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 increasing 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 finite t which shall be considered as a basic assumption for defining \(\omega _M\).

For all \(t,r>0\) we get

$$\begin{aligned} \omega _M^r(t)=\omega _M(t^r)=\sup _{p\in {\mathbb {N}}}\log \left( \frac{t^{rp}}{M_p}\right) =\sup _{p\in {\mathbb {N}}}\log \left( \left( \frac{t^p}{(M_p)^{1/r}}\right) ^r\right) =r\omega _{M^{1/r}}(t), \end{aligned}$$
(4)

recalling \(M^{1/r}_p=(M_p)^{1/r}\).

We collect some well-known properties for \(\omega _M\) (e.g. see [10, Lem. 2.4] and [14, Lem. 3.1]).

Lemma 1

Let \(M\in {\mathscr {LC}}\) be given, then we get:

  1. (i)

    \(\omega _M\)belongs always to\({\mathscr {W}}_0\).

  2. (ii)

    IfMsatisfies\({(\gamma _1)}\), then\(\omega _M\)has\((\omega _{{\text {snq}}})\) (which implies\((\omega _1)\)).

  3. (iii)

    Mhas\(({\text {mg}})\)if and only if\(\omega _M\)has\((\omega _6)\).

Let \(M\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) (with \(M_0=1\)) and put

$$\begin{aligned} h_M(t):=\inf _{k\in {\mathbb {N}}}M_k t^k. \end{aligned}$$
(5)

The functions \(h_M\) and \(\omega _M\) are related by

$$\begin{aligned} h_M(t)=\exp (-\omega _M(1/t))=\exp (-\omega _M^{\iota }(t))\;\;\;\forall \;t>0, \end{aligned}$$
(6)

since \(\log (h_M(t))=\inf _{k\in {\mathbb {N}}}\log (t^kM_k)=-\sup _{k\in {\mathbb {N}}}-\log (t^kM_k)=-\omega _M(1/t)\) (e.g. see also [7, p. 11]).

If \(M\in {\mathscr {LC}}\), then M has \((\text {mg})\) if and only if

$$\begin{aligned} \exists \;C\ge 1\;\forall \;t>0:\;\;\;h_M(t)\le h_M(Ct)^2, \end{aligned}$$
(7)

e.g. see [23, Lemma 2.4, Remark 2.5].

Starting with a weight function we recall the following consequence of (2), see again [23, Remark 2.5].

Lemma 2

Let\(\omega \in {{\mathscr {W}}_0}\)be given and\(\varOmega =\{W^l: l>0\}\)the matrix associated with\(\omega \). Then we have

$$\begin{aligned} \exists \;A\ge 1\;\forall \;l>0\;\forall \;s>0:\;\;\;h_{W^l}(s)\le h_{W^{2l}}(As)^2. \end{aligned}$$
(8)

Classes of ultraholomorphic functions

We introduce now the classes under consideration in this paper, see also [10, Section 2.5] and [14, Section 2.5]. For the following definitions, notation and more details we refer to [26, Section 2]. Let \({\mathscr {R}}\) be the Riemann surface of the logarithm. We wish to work in general unbounded sectors in \({\mathscr {R}}\) with vertex at 0, but all our results will be unchanged under rotation, so we will only consider sectors bisected by direction 0: For \(\gamma >0\) we set

$$\begin{aligned} S_{\gamma }:=\left\{ z\in {\mathscr {R}}: |\arg (z)|<\frac{\gamma \pi }{2}\right\} , \end{aligned}$$

i.e. the unbounded sector of opening \(\gamma \pi \), bisected by direction 0.

Let M be a weight sequence, \(S\subseteq {\mathscr {R}}\) an (unbounded) sector and \(h>0\). We define

$$\begin{aligned} {\mathscr {A}}_{M,h}(S):=\{f\in {\mathscr {H}}(S): \Vert f\Vert _{M,h}:=\sup _{z\in S, p\in {\mathbb {N}}}\frac{|f^{(p)}(z)|}{h^p M_p}<+\infty \}. \end{aligned}$$

\(({\mathscr {A}}_{M,h}(S),\Vert \cdot \Vert _{M,h})\) is a Banach space and we put

$$\begin{aligned} {\mathscr {A}}_{\{M\}}(S):=\bigcup _{h>0}{\mathscr {A}}_{M,h}(S). \end{aligned}$$

\({\mathscr {A}}_{\{M\}}(S)\) is called the Denjoy–Carleman ultraholomorphic class (of Roumieu type) associated with M in the sector S (it is an (LB) space). Analogously we introduce the space of complex sequences

$$\begin{aligned} \varLambda _{M,h}:=\left\{ a=(a_p)_p\in {\mathbb {C}}^{{\mathbb {N}}}: |a|_{M,h}:=\sup _{p\in {\mathbb {N}}}\frac{|a_p|}{h^{p} M_{p}}<+\infty \right\} \end{aligned}$$

and put \(\varLambda _{\{M\}}:=\bigcup _{h>0}\varLambda _{M,h}\). The (asymptotic) Borel map\({\mathscr {B}}\) is given by

$$\begin{aligned} {\mathscr {B}}:{\mathscr {A}}_{\{M\}}(S)\longrightarrow \varLambda _{\{M\}},f\mapsto (f^{(p)}(0))_{p\in {\mathbb {N}}}, \end{aligned}$$

where \(f^{(p)}(0):=\lim _{z\in S, z\rightarrow 0}f^{(p)}(z)\).

Similarly as for the ultradifferentiable case, we now define ultraholomorphic classes associated with a normalized weight function \(\omega \) satisfying \((\omega _3)\). Given an unbounded sector S, and for every \(l>0\), we first define

$$\begin{aligned} {\mathscr {A}}_{\omega ,l}(S):=\left\{ f\in {\mathscr {H}}(S): \Vert f\Vert _{\omega ,l}:=\sup _{z\in S, p\in {\mathbb {N}}}\frac{|f^{(p)}(z)|}{\exp (\frac{1}{l}\varphi ^{*}_{\omega }(lp))}<+\infty \right\} . \end{aligned}$$

\(({\mathscr {A}}_{\omega ,l}(S),\Vert \cdot \Vert _{\omega ,l})\) is a Banach space and we put

$$\begin{aligned} {\mathscr {A}}_{\{\omega \}}(S):=\bigcup _{l>0}{\mathscr {A}}_{\omega ,l}(S). \end{aligned}$$

\({\mathscr {A}}_{\{\omega \}}(S)\) is called the Denjoy–Carleman ultraholomorphic class (of Roumieu type) associated with \(\omega \) in the sector S (it is an (LB) space). Correspondingly, we introduce the space of complex sequences

$$\begin{aligned} \varLambda _{\omega ,l}:=\left\{ a=(a_p)_p\in {\mathbb {C}}^{{\mathbb {N}}}: |a|_{\omega ,l}:=\sup _{p\in {\mathbb {N}}}\frac{|a_p|}{\exp (\frac{1}{l}\varphi ^{*}_{\omega }(lp))}<+\infty \right\} \end{aligned}$$

and put \(\varLambda _{\{\omega \}}:=\bigcup _{l>0}\varLambda _{\omega ,l}\). So in this case we get the Borel map\({\mathscr {B}}:{\mathscr {A}}_{\{\omega \}}(S)\longrightarrow \varLambda _{\{\omega \}}\).

Finally, we recall the ultraholomorphic function classes of Roumieu type defined by a weight matrix \({\mathscr {M}}\), introduced in [10, Section 2.5] and [14, Section 2.5] and inspired by the analogous ultradifferentiable classes considered in [28, Section 7] and also in [22, Section 4.2].

Given a weight matrix \({\mathscr {M}}=\{M^x\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}: x\in {\mathbb {R}}_{>0}\}\) and a sector S we may define ultraholomorphic classes \({\mathscr {A}}_{\{{\mathscr {M}}\}}(S)\) of Roumieu type as

$$\begin{aligned} {\mathscr {A}}_{\{{\mathscr {M}}\}}(S):=\bigcup _{x\in {\mathbb {R}}_{>0}}{\mathscr {A}}_{\{M^x\}}(S), \end{aligned}$$

and accordingly, \(\varLambda _{\{{\mathscr {M}}\}}:=\bigcup _{x\in {\mathbb {R}}_{>0}}\varLambda _{\{M^x\}}\).

Let now \(\omega \in {{\mathscr {W}}}\) be given and let \(\varOmega \) be the associated weight matrix defined in Sect. 2.5 (i), then

$$\begin{aligned} {\mathscr {A}}_{\{\omega \}}(S)={\mathscr {A}}_{\{\varOmega \}}(S) \end{aligned}$$
(9)

holds as locally convex vector spaces. This equality is an easy consequence of [22, Lemma 5.9 (5.10)] (see (3)) and the way how the seminorms are defined in these spaces. As one also has \(\varLambda _{\{\omega \}}=\varLambda _{\{\varOmega \}}\), the Borel map \({\mathscr {B}}\) makes sense in these last classes, \({\mathscr {B}}:{\mathscr {A}}_{\{\varOmega \}}(S)\longrightarrow \varLambda _{\{\varOmega \}}\).

In any of the considered ultraholomorphic classes, an element f is said to be flat if \(f^{(p)}(0) = 0\) for every \(p\in {\mathbb {N}}\), that is, \({\mathscr {B}}(f)\) is the null sequence.

Mixed growth indices for extension results

The indices \(\gamma (M,N)\) and \(\gamma (\sigma ,\omega )\)

First, for \(r>0\) we introduce the following condition which will be denoted by \((\gamma _r)\), see [30] for \(r\in {\mathbb {N}}_{>0}\) and [32, Lemma 2.2.1] for \(r>0\):

$$\begin{aligned} \sup _{p\in {\mathbb {N}}_{>0}}\frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p}\left( \frac{1}{\mu _k}\right) ^{1/r}<+\infty . \end{aligned}$$

It is immediate that M has \((\gamma _r)\) if and only if \(M^{1/r}\) has \((\gamma _1)\). In [32, Definition 1.3.5] the growth index \(\gamma (M)\) has been introduced (for strongly regular sequences and using a definition which is not based on property \((\gamma _r)\) directly). In [11, Thm. 3.11, Cor. 3.12] we have shown for \(M\in {\mathscr {LC}}\) that

$$\begin{aligned} \gamma (M)=\sup \{r>0: M\;\text {satisfies}\; {(\gamma _r)}\}. \end{aligned}$$

One can prove that M has \((\gamma _r)\) if and only if \(\gamma (M)>r\) (see [11, Cor. 3.12]).

Let \(M, N\in {\mathbb {R}}_{>0}^{{\mathbb {N}}}\) with \(\mu _p\le C\nu _p\) for some \(C\ge 1\) and all \(p\in {\mathbb {N}}\). For \(r>0\) we introduce the following growth property: We write \((M,N)_{\gamma _r}\) if

$$\begin{aligned} \sup _{p\in {\mathbb {N}}_{>0}}\frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p}\left( \frac{1}{\nu _k}\right) ^{1/r}<+\infty , \end{aligned}$$

and the mixed growth index is defined by

$$\begin{aligned} \gamma (M,N):=\sup \{r>0: {(M,N)_{\gamma _r}}\;\;\text {is satisfied}\}. \end{aligned}$$

If none condition \((M,N)_{\gamma _r}\) holds true, then we put \(\gamma (M,N):=0\). It is evident that \(\gamma (M,M)=\gamma (M)\) is valid.

Remark 2

Let \(M,N\in {\mathscr {LC}}\) with \(\mu _p\le C\nu _p\) (note that w.l.o.g. we can take \(C=1\), otherwise replace \(\nu _p\) by \({\widetilde{\nu }}_p:=C\nu _p)\).

  1. (i)

    Given \(r>0\), for any \(0<r'<r\) we see that \((M,N)_{\gamma _r}\) implies \((M,N)_{\gamma _{r'}}\), since we can write

    $$\begin{aligned} \frac{(\mu _p)^{1/r'}}{p}\sum _{k\ge p}\left( \frac{1}{\nu _k}\right) ^{1/r'}=\frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p}\left( \frac{1}{\nu _k}\right) ^{1/r}\left( \frac{\mu _p}{\nu _k}\right) ^{(r-r')/(r'r)}, \end{aligned}$$

    and \(\mu _p\le C\nu _p\le C\nu _k\) for all \(1\le p\le k\).

  2. (ii)

    Moreover, in \((M,N)_{\gamma _r}\) we can equivalently consider \(\frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p+1}\left( \frac{1}{\nu _k}\right) ^{1/r}\) since

    $$\begin{aligned} \frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p+1}\left( \frac{1}{\nu _k}\right) ^{1/r}\le \frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p}\left( \frac{1}{\nu _k}\right) ^{1/r}\le \frac{C^{1/r}}{p}+\frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p+1}\left( \frac{1}{\nu _k}\right) ^{1/r}. \end{aligned}$$

In order to see how these definitions have been motivated, we are describing next the appearance of such (non-)mixed relations in the literature.

Remark 3

Condition \((\gamma _1)\) has appeared as (standard) condition (M3) in [15] and in [21] where it has been used to characterize the validity of Borel’s theorem in the ultradifferentiable weight sequence setting.

Condition \((M,N)_{\gamma _1}\) has appeared in the mixed weight sequence situations in [7] (for the Whitney jet map) and in [31] for the Borel map. More precisely in [31] it has turned out that the characterizing condition is not \((M,N)_{\gamma _1}\) directly, but does coincide with this condition whenever M has \(({\text {mg}})\) (as it has been assumed in [7]), see also Remark 6 below.

In [30], condition \((\gamma _r)\) has appeared (for \(r\in {\mathbb {N}}_{>0}\)) and it has also been used by the authors in [12]. In these works \((\gamma _r)\) played a key-role proving extension theorems for ultraholomorphic classes defined by weight sequences since one is working with auxiliary ultradifferentiable-like function classes first defined in [30]. In [32, Lemma 2.2.1] this condition has been introduced for \(r>0\) arbitrary and a connection to the value \(\gamma (M)\) has been given.

Finally, condition \((M,N)_{\gamma _r}\) has appeared in the recent work by the authors [13] (mainly again for \(r\in {\mathbb {N}}_{>0}\)). There we have generalized the results from [31] to the auxiliary ultradifferentiable-like function classes, moreover in [13, Theorem 5.9], we have given a generalization of the ultradifferentiable Whitney extension results from [7] involving a ramification parameter \(r>0\).

Now we turn to the weight function situation. Let \(\omega \) be a weight function and \(r>0\), we write \((\omega _{\gamma _r})\) if

$$\begin{aligned} \exists \;C>0\;\forall \;t\ge 0:\;\;\;\int _1^{\infty }\frac{\omega (tu)}{u^{1+1/r}}du\le C\omega (t)+C \end{aligned}$$

holds true. Using this growth property, by [11, Lemma 2.10, Theorem 2.11] we have

$$\begin{aligned} \gamma (\omega )=\sup \{r>0: M\;\text {satisfies}\;{(\omega _{\gamma _r})}\}, \end{aligned}$$

with \(\gamma (\omega )\) denoting the growth index used and introduced in [10], [14] (by considering a different growth property of \(\omega \) which is not based on \((\omega _{\gamma _r})\)). Note also that \(\frac{1}{\gamma (\omega )}\) does coincide with the so-called upper Matuszewska index, see [1, p. 66]. For a more detailed study of \(\gamma (\omega )\) and its connection to the indices studied in [1] we refer to Section 2 in the authors’ recent work [11].

Let \(\omega ,\sigma \) be weight functions with \(\sigma {\preceq }\omega \) (i.e. \(\omega (t)=O(\sigma (t))\)) and \(r>0\), we write \((\sigma ,\omega )_{\gamma _r}\) if

$$\begin{aligned} \exists \;C>0\;\forall \;t\ge 0:\;\;\;\int _1^{\infty }\frac{\omega (tu)}{u^{1+1/r}}du\le C\sigma (t)+C, \end{aligned}$$

and the mixed growth index is defined by

$$\begin{aligned} \gamma (\sigma ,\omega ):=\sup \{r>0: {(\sigma ,\omega )_{\gamma _r}}\;\;\text {is satisfied}\}. \end{aligned}$$

If none condition \((\sigma ,\omega )_{\gamma _r}\) holds true, then we put \(\gamma (\sigma ,\omega ):=0\) and it is immediate that \(\gamma (\omega ,\omega )=\gamma (\omega )\). It is also clear that \((\sigma ,\omega )_{\gamma _r}\) implies \((\sigma ,\omega )_{\gamma _{r'}}\) for all \(0<r'<r\).

Similarly as before we are describing the appearance of such (non-)mixed relations in the literature.

Remark 4

\((\omega _{\gamma _1})\), which is precisely \((\omega _{{\text {snq}}})\), has appeared for \(\omega =\omega _M\) in [15], and in [3] this condition has been characterized in terms of the validity of the ultradifferentiable Whitney extension theorem in the weight function setting.

The mixed condition \((\sigma ,\omega )_{\gamma _1}\) has been treated in [5] for the Borel map and in [24] and [25] for the general Whitney jet map (see also [17] for compact convex sets). In these works, condition \((\sigma ,\omega )_{\gamma _1}\) has been identified as the characterizing property.

Finally, in [13, Theorem 5.9] we have introduced \((\omega _M,\omega _N)_{\gamma _r}\) in order to prove a generalization of the ultradifferentiable Whitney extension results from [7] (again by involving a ramification parameter \(r>0\)).

The next observation gives the connection between \(\gamma (N)\) and \(\gamma (M,N)\), resp. between \(\gamma (\omega )\) and \(\gamma (\sigma ,\omega )\).

Lemma 3

Let\(M,N\in {\mathscr {LC}}\)be given with\(\mu _p\le \nu _p\)and\(\omega ,\sigma \)be weight functions with\(\sigma {\preceq }\omega \). Then we have

$$\begin{aligned} \gamma (N)\le \gamma (M,N),\gamma (\omega )\le \gamma (\sigma ,\omega ). \end{aligned}$$

Proof

First, if \(\gamma (N)=0\), \(\gamma (\omega )=0\), then the conclusion is clear. If these values are strictly positive, then for any \(0<r<\gamma (N),\gamma (\omega )\) we get that \((\gamma _r)\) for N and \((\omega _{\gamma _r})\) for \(\omega \) hold true and so also \((M,N)_{\gamma _r}\) and \((\sigma ,\omega )_{\gamma _r}\) are valid (for any M having \(\mu _p\le \nu _p\), \(\sigma \) having \(\sigma {\preceq }\omega \)). \(\square \)

Remark 5

The motivation of defining the mixed growth indices (especially for weight functions) was arising by proving [13, Theorem 5.9].

First we wish to mention that in [7, Commentaires 32] it was observed (without giving a proof) that there is a connection between \((M,N)_{\gamma _1}\) and \((\sigma ,\omega )_{\gamma _1}\) (under suitable basic assumptions on MN): They have stated that \((M,N)_{\gamma _1}\) does imply \((\omega _M,\omega _N)_{\gamma _1}\) and so generalizing [15, Prop. 4.4] to a mixed setting. In [24, Lemma 5.7] a detailed proof of this implication is given, and in [13, Lemma 5.7] we have generalized this result by involving a ramification parameter \(r>0\).

We recall the next statement which has been shown in [13, Lemmas 5.7, 5.8] in order to see how \(\gamma (M,N)\) and \(\gamma (\omega _M,\omega _N)\) are related. This result is the generalization of [11, Corollary 4.6 (iii)] to the mixed setting.

Lemma 4

Let\(M,N\in {\mathscr {LC}}\)be given with\(\mu _p\le \nu _p\) (which is equivalent to\(\mu _p^r\le \nu _p^r\)for all\(r>0\)and implies\(M^r\le N^r\)). Assume that\((M,N)_{\gamma _r}\)holds true for\(r>0\). Then the associated weight functions are satisfying

$$\begin{aligned} \exists \;C>0\;\forall \;t\ge 0:\;\;\;\int _1^{\infty }\frac{\omega _N(tu)}{u^{1+1/r}}du\le C\omega _M(t)+C, \end{aligned}$$

i.e.\((\omega _M,\omega _N)_{\gamma _r}\)is satisfied (recall that\(M\le N\)implies\(\omega _N(t)\le \omega _M(t)\)for all\(t\ge 0\)).

Consequently, for sequencesMandNas assumed above, we always get\(\gamma (M,N)\le \gamma (\omega _M,\omega _N)\).

IfMdoes have in addition\(({\text {mg}})\), then\(\gamma (M,N)=\gamma (\omega _M,\omega _N)\)holds true.

Remark 6

For the main results about the surjectivity of the Borel map in ramified ultradifferentiable classes [13, Thm. 3.2, Thm. 4.2, Thm. 5.5] a weaker but characterizing condition has to be considered. More precisely, in [31] the following condition has been introduced (denoted by \((*)\) there):

$$\begin{aligned} \exists \;s\in {\mathbb {N}}_{>0}:\;\;\sup _{p\in {\mathbb {N}}_{>0}}\frac{\lambda _{p,s}^{M,N}}{p}\sum _{k=p}^{\infty }\frac{1}{\nu _k}<+\infty , \end{aligned}$$

with \(\lambda ^{M,N}_{p,s}:=\sup _{0\le j<p}\left( \frac{M_p}{s^p N_j}\right) ^{1/(p-j)}\).

In [13] we have generalized this condition to

$$\begin{aligned} \exists \;s\in {\mathbb {N}}_{>0}:\;\;\sup _{p\in {\mathbb {N}}_{>0}}\frac{(\lambda _{p,s}^{M,N})^{1/r}}{p}\sum _{k=p}^{\infty }\left( \frac{1}{\nu _k}\right) ^{1/r}<+\infty , \end{aligned}$$

denoted by \((M,N)_{{\text {SV}}_r}\) (and used for \(r\in {\mathbb {N}}_{>0}\) in the main results). Hence it seems to be reasonable to define

$$\begin{aligned} \gamma (M,N)_{{\text {SV}}}:=\sup \{r\in {\mathbb {R}}_{>0}: (M,N)_{{\text {SV}}_r}\;\;\text {is satisfied}\}. \end{aligned}$$

In general we only know that \((M,N)_{\gamma _r}\) implies \((M,N)_{{\text {SV}}_r}\), see [13, Lemma 2.4]. However, under the standard assumptions of the main results, that is \(M,N\in {\mathscr {LC}}\) with \(\mu _p\le \nu _p\) and such that M does have \(({\text {mg}})\), Lemma 4 combined with [13, Lemma 2.4 and (2.5)] yield

$$\begin{aligned} \gamma (\omega _M,\omega _N)=\gamma (M,N)=\gamma (M,N)_{{\text {SV}}}. \end{aligned}$$

We summarize several more properties.

Remark 7

 

  1. (i)

    By definition it is clear that \((M,N)_{\gamma _r}\) if and only if \((M^{1/r},N^{1/r})_{\gamma _1}\), recalling \(M_p^{1/r}=(M_p)^{1/r}\), \(N_p^{1/r}=(N_p)^{1/r}\).

    Similarly \((\sigma ,\omega )_{\gamma _r}\) holds true if and only if \((\sigma ^r,\omega ^r)_{\gamma _1}\) for the weights \(\omega ^r\) given by \(\omega ^r(t)=\omega (t^r)\) and \(\sigma ^r(t)=\sigma (t^r)\), because

    $$\begin{aligned} \int _1^{\infty }\frac{\omega ((tu)^r)}{u^2}du=\frac{1}{r}\int _1^{\infty }\frac{\omega (t^rv)}{u^2}\frac{dv}{u^{r-1}}=\frac{1}{r}\int _1^{\infty }\frac{\omega (t^rv)}{v^{1+1/r}}dv, \end{aligned}$$

    and then replace \(t^r\) by t (see also (1)).

    Summarizing, for all \(0<r<\gamma (M,N)\) we get \((M^{1/r},N^{1/r})_{\gamma _1}\), whereas for all \(0<r<\gamma (\sigma ,\omega )\) we have \((\sigma ^r,\omega ^r)_{\gamma _1}\) and so

    $$\begin{aligned} \forall \;r>0:\;\;\;\gamma (M^r,N^r)=r\gamma (M,N),\gamma (\sigma ^{1/r},\omega ^{1/r})=r\gamma (\sigma ,\omega ), \end{aligned}$$
    (10)

    which generalizes this fact from \(\gamma (M)\) and \(\gamma (\omega )\).

  2. (ii)

    \((\sigma ,\omega )_{\gamma _r}\) implies \(\int _1^{\infty }\frac{\omega (tu)}{u^{1+1/r}}du\ge \omega (t)\int _1^{\infty }\frac{1}{u^{1+1/r}}du=r\omega (t)\) (\(\omega \) is nondecreasing).

  3. (iii)

    The value \(p=1\) in \((M,N)_{\gamma _r}\) implies that \(N^{1/r}\) satisfies \(({\text {nq}})\), i.e. N has \(({\text {nq}}_r)\).

    The value \(t=1\) in \((\sigma ,\omega )_{\gamma _r}\) yields \(\int _1^{\infty }\frac{\omega (u)}{u^{1+1/r}}du\le C_1\) and the calculation in (i) proves that \(\omega ^r\) satisfies \((\omega _{{\text {nq}}})\), or \((\omega _{{\text {nq}}_r})\) for \(\omega \). Consequently \((\omega _5)\) follows for \(\omega ^r\), too.

Orders of quasianalyticity \(\mu (N)\) and \(\mu (\omega )\)

In the ultraholomorphic weight sequence setting another important growth index is known and related to the injectivity of the asymptotic Borel map, the so-called order of quasianalyticity. It has been introduced in [26, Def. 3.3, Thm. 3.4], see also [9, 12] and [11]. We use the notation from [11] to avoid confusion in the weight function case below and to have a unified notation (coming from [1, p. 73]).

For given \(N\in {\mathscr {LC}}\) we set

$$\begin{aligned} \mu (N):=\sup \{r\in {\mathbb {R}}_{>0}: \sum _{k\ge 1}\left( \frac{1}{\nu _k}\right) ^{1/r}<+\infty \}=\sup \{r\in {\mathbb {R}}_{>0}:\;N\;\text {has}\;{({\text {nq}}_r)}\}=\frac{1}{\lambda _{(\nu _p)_p}},\nonumber \\ \end{aligned}$$
(11)

with \(\lambda _{(\nu _p)_p}:=\inf \left\{ \alpha >0: \sum _{p\ge 1}\left( \frac{1}{\nu _p}\right) ^{\alpha }<\infty \right\} \) denoting the so-called exponent of convergence of N, see [26, Prop. 2.13, Def. 3.3, Thm. 3.4] and also [12, p. 145].

If none \(({\text {nq}}_r)\) holds true, then we put \(\mu (N):=0\). If \(M\in {\mathscr {LC}}\) and \(M{\approx }N\), then \(\mu (M)=\mu (N)\) since each \(({\text {nq}}_r)\) is stable under \(\approx \).

A first immediate consequence is the following:

Lemma 5

Let\(M,N\in {\mathscr {LC}}\)with\(\mu _p\le C\nu _p\), then

$$\begin{aligned} \gamma (M,N)\le \mu (N). \end{aligned}$$

Proof

Note that for \(r>\mu (N)\) property \((M,N)_{\gamma _r}\) cannot be valid for any choice M (see (iii) in Remark 7). \(\square \)

Remark 8

Lemmas 3 and 5 together yield

$$\begin{aligned} \gamma (N)\le \gamma (M,N)\le \mu (N). \end{aligned}$$

Hence the mixed index \(\gamma (M,N)\) can become crucial whenever \(\gamma (N)<\mu (N)\) does hold true. In general the gap between \(\gamma (N)\) and \(\mu (N)\) can be as large as desired even if \(N\in {\mathscr {SR}}\), see [11, Remark 4.13] and [8, Section 2.2.5] for more details.

According to this observation one can ask now the following question: Is it possible to get extension results for values \(\gamma >0\) with \(\gamma (N)\le \gamma <\mu (N)\)? As we will see in Sect. 5, for values \(\gamma <\gamma (M,N)\) we can prove extension results in a mixed setting between M and N but it is still not clear how large the gap between \(\gamma (M,N)\) and \(\mu (N)\) can be in general.

Given \(N\in {\mathscr {LC}}\) and \(r>0\) we consider

$$\begin{aligned} \exists \;M\in {\mathscr {LC}},\;\;\mu _p\le C\nu _p:\;\,\;\sup _{p\in {\mathbb {N}}_{>0}}\frac{(\mu _p)^{1/r}}{p}\sum _{k\ge p}\left( \frac{1}{\nu _k}\right) ^{1/r}<+\infty , \end{aligned}$$
(12)

and

$$\begin{aligned} \sup \{r\in {\mathbb {R}}_{>0}: (12)\;\;\text {is satisfied}\}. \end{aligned}$$
(13)

If for none \(r>0\) (12) holds true, then the \(\sup \) in (13) equals 0. As commented in Remark 2, given \(N\in {\mathscr {LC}}\) and \(r>0\) with having (12) for some choice \(M\in {\mathscr {LC}}\), \(\mu _p\le C\nu _p\), then this M is sufficient to guarantee (12) for all \(0<r'<r\) as well.

In [26, Theorem 3.4] (see also (11)) it has been shown that for any \(N\in {\mathscr {LC}}\) we have

$$\begin{aligned} \mu (N)=\liminf _{p\rightarrow \infty }\frac{\log (\mu _p)}{\log (p)}. \end{aligned}$$

Thus for given \(N\in {\mathscr {LC}}\) and \(0<r<\mu (N)\) we see that \(\nu _p\ge p^r\) for all \(p\in {\mathbb {N}}\) sufficiently large and \(C\nu _p\ge p^r\) for all \(p\in {\mathbb {N}}\) by choosing C large enough. Consequently, in (12) the choice \(M\equiv G^r\), i.e. the Gevrey sequence with index \(r>0\), does always make sense and the next result is becoming immediate:

Proposition 1

Let \(N\in {\mathscr {LC}}\) be given, then

$$\begin{aligned} \sup \{r\in {\mathbb {R}}_{>0}: (12)\;\;\text {is satisfied}\}=\mu (N). \end{aligned}$$

Proof

If \(\mu (N)=0\), then we have obviously equality. So let now \(\mu (N)>0\).

First, if \(0<r<\sup \{r\in {\mathbb {R}}_{>0}: (12)\;\;\text {is satisfied}\}\), then the choice \(p=1\) in (12) immediately implies \(({\text {nq}}_r)\) for N, hence \(r\le \mu (N)\) and so the first half is shown.

Conversely, let \(r<\mu (N)\) be given, then (12) is satisfied for \(\mu _p=p^r\) which can be taken as seen above. Hence \(\mu (N)\le \sup \{r\in {\mathbb {R}}_{>0}: (12)\;\;\text {is satisfied}\}\) is also shown and we are done. \(\square \)

A disadvantage of taking directly \(M\equiv G^r\) is that it is not clear that this precise choice is optimal in the sense that it is the largest sequence \(\mu _p\le C\nu _p\) admitting (12). To obtain this optimal sequence we recall the following construction: In [23, Section 4.1], which is based on an idea arising in the proof of [21, Proposition 1.1], it has been shown that to each \(N\in {\mathscr {LC}}\) satisfying \(({\text {nq}})\) we can associate a sequence \(S^N\) with good regularity properties which has been denoted by descendant.

For the reader’s convenience we recall now the construction in the following observation and are involving a ramification parameter \(r>0\) as well.

Remark 9

Let \(N\in {\mathscr {LC}}\) be given and satisfying \(({\text {nq}}_r)\), \(r>0\). Then there does exist a sequence \(S^{N,r}\), the so-called descendant of\(N^{1/r}\) defined by its quotients

$$\begin{aligned} \sigma ^{N,r}_k:=\frac{\tau ^r_1 k}{\tau ^r_k},\sigma ^{N,r}_0:=1, \end{aligned}$$

with

$$\begin{aligned} \tau ^r_k:=\frac{k}{(\nu _k)^{1/r}}+\sum _{j\ge k}\left( \frac{1}{\nu _j}\right) ^{1/r},\;\;\;k\ge 1. \end{aligned}$$

So \(S^{N,r}\) is depending on both given N and \(r>0\) and satisfies the following properties (see [23, Lemma 4.2]):

  1. (i)

    \(\sigma ^{N,r}_k\ge 1\) for all \(k\in {\mathbb {N}}\), \(s^{N,r}:=(S^{N,r}_k/k!)_{k\in {\mathbb {N}}}\in {\mathscr {LC}}\) (so \(S^{N,r}\) is strongly log-convex),

  2. (ii)

    there exists \(C>0\) such that \(\sigma ^{N,r}_k\le C(\nu _k)^{1/r}\) for all \(k\in {\mathbb {N}}\),

  3. (iii)

    \((S^{N,r},N^{1/r})_{\gamma _1}\), then \(\gamma (S^{N,r},N^{1/r})\ge 1\),

  4. (iv)

    if N enjoys \(({\text {mg}})\) (equivalently \(N^{1/r}\) does so for some/each \(r>0\)), then \(S^{N,r}\) does have \(({\text {mg}})\) too (\(r>0\) arbitrary).

  5. (v)

    \(S^{N,r}\) is optimal/maximal in the following sense: If \(M\in {\mathscr {LC}}\) is given with \(\mu _k\le C(\nu _k)^{1/r}\) and \((M,N^{1/r})_{\gamma _1}\), then \(\mu _k\le D\sigma ^{N,r}_k\) follows.

    We also have that \(C^{-1}\sigma ^{N,r}_k\le (\nu _k)^{1/r}\le C\sigma _k^{N,r}\) if and only if \(N^{1/r}\) does have \((\gamma _1)\) resp. if and only if \((N,N)_{\gamma _r}\).

We have that \(s^{N,r}\le Cs^{N,r'}\Leftrightarrow S^{N,r}\le CS^{N,r'}\) for all \(0<r'\le r\) (since \(r\mapsto \tau ^r_k\) is increasing for all \(k\in {\mathbb {N}}\) fixed).

Unfortunately this construction is in general not well-behaved under applying ramification, i.e. \(\sigma ^{N,r}\ne (\sigma ^{N,1})^{1/r}\).

Now we put

$$\begin{aligned} L^{N,r}:=(S^{N,r})^r. \end{aligned}$$
(14)

Hence \(L^{N,r}\in {\mathscr {LC}}\) and moreover

(a):

\(((L^{N,r})^{1/r},N^{1/r})_{\gamma _1}\), equivalently \((L^{N,r},N)_{\gamma _r}\) holds true and so \(\gamma (L^{N,r},N)\ge r\),

(b):

\(\lambda ^{N,r}_k=(\sigma ^{N,r}_k)^r\le C\nu _k\) for all \(k\in {\mathbb {N}}\) and so \((L^{N,r},N)_{\gamma _{r'}}\) for all \(0<r'<r\) as well (see (i) in Remark 2),

(c):

if \(M\in {\mathscr {LC}}\) is given with \(\mu _k\le C\nu _k\) and \((M,N)_{\gamma _r}\), i.e. \((M^{1/r},N^{1/r})_{\gamma _1}\), then for all \(k\in {\mathbb {N}}\), \(\mu _k\le D^r(\sigma ^{N,r}_k)^r=D^r\lambda ^{N,r}_k\) and, consequently, \(L^{N,r}\) is maximal (up to a constant) among all sequences satisfying \(\mu _k\le C\nu _k\) and \((M,N)_{\gamma _r}\),

(d):

\(S^{N,r}\) has \(({\text {mg}})\) if and only if \(L^{N,r}\) does so.

According to the maximality (c) mentioned above, we point out that we have \(\lambda ^{N,r}_k=(\sigma ^{N,r}_k)^r\ge k^r\) for all \(k\in {\mathbb {N}}\), and so \(L^{N,r}\ge G^r\). Moreover, one can show that \(\lim _{k\rightarrow \infty }\frac{\lambda ^{N,r}_k}{k^r}=\infty \) which implies \(\left( \frac{G^r_k}{L^{N,r}_k}\right) ^{1/k}\rightarrow 0\) as \(k\rightarrow +\infty \) and so \(L^{N,r}\) is strictly larger than \(G^r\).

As mentioned above, condition \(({\text {mg}})\) for N does always imply this property for the descendant \(S^{N,1}=:S\). However we can obtain a precise characterization of this growth behavior.

Lemma 6

Let \(N\in {\mathscr {LC}}\) having \(({\text {nq}})\) be given and \(S:=S^{N,1}\) shall denote its descendant. Then S does have \(({\text {mg}})\) if and only if

$$\begin{aligned} \exists \;C\ge 1\;\forall \;k\in {\mathbb {N}}_{>0}:\;\;\;\frac{\nu _{2k}}{\nu _k}\le C+C\frac{\nu _{2k}}{2k}\sum _{j\ge 2k}\frac{1}{\nu _j}. \end{aligned}$$
(15)

Proof

Since \(S\in {\mathscr {LC}}\), this sequence has \(({\text {mg}})\) if and only if \(\sup _{k\in {\mathbb {N}}}\frac{\sigma _{2k}}{\sigma _k}<\infty \), e.g. see [23, Lemma 2.2]. By the definitions given in Remark 9 we get that \(\sigma _{2k}\le D\sigma _k\Leftrightarrow \tau _k\le \frac{D}{2}\tau _{2k}\) (with \(\tau _k\equiv \tau ^1_k\)) which is equivalent to \(\frac{k}{\nu _k}+\sum _{j\ge k}\frac{1}{\nu _j}\le D\frac{k}{\nu _{2k}}+\frac{D}{2}\sum _{j\ge 2k}\frac{1}{\nu _j}\) and so to having \(\frac{k}{\nu _k}+\sum _{j=k}^{2k-1}\frac{1}{\nu _j}\le D\frac{k}{\nu _{2k}}+(\frac{D}{2}-1)\sum _{j\ge 2k}\frac{1}{\nu _j}\). So finally \(({\text {mg}})\) is equivalent to

$$\begin{aligned} \exists \;D\ge 1\;\forall \;k\in {\mathbb {N}}_{>0}:\;\;\;\frac{\nu _{2k}}{\nu _k}+\frac{\nu _{2k}}{k}\sum _{j=k}^{2k-1}\frac{1}{\nu _j}\le D+(\frac{D}{2}-1)\frac{\nu _{2k}}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}. \end{aligned}$$
(16)

The sum on the left-hand side above is estimated by below by \(\frac{k}{\nu _{2k}}\) and by above by \(\frac{k}{\nu _k}\) (since \((\nu _k)_k\) is increasing). Hence (16) implies \(\frac{\nu _{2k}}{\nu _k}+1\le D+(D-2)\frac{\nu _{2k}}{2k}\sum _{j\ge 2k}\frac{1}{\nu _j}\) and so (15) with \(C=D-1\) follows, whereas (15) implies \(\frac{\nu _{2k}}{\nu _k}+\frac{\nu _{2k}}{k}\sum _{j=k}^{2k-1}\frac{1}{\nu _j}\le \frac{\nu _{2k}}{\nu _k}+\frac{\nu _{2k}}{\nu _k}\le 2C+C\frac{\nu _{2k}}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}\) and (16) follows with \(D:=4C\). \(\square \)

Concerning the characterizing condition (15) we point out that (as expected) it is obviously satisfied if N has \(({\text {mg}})\), i.e. \(\sup _{k\in {\mathbb {N}}}\frac{\nu _{2k}}{\nu _k}<\infty \), and moreover:

(i):

The right-hand side in (15) is bounded if and only if \((\gamma _1)\) is valid for N since for all \(k\ge 1\) we get \(\frac{\nu _{2k}}{2k}\sum _{j\ge 2k}\frac{1}{\nu _j}\ge \frac{1}{2}\frac{\nu _{2k-1}}{2k-1}\sum _{j\ge 2k-1}\frac{1}{\nu _j}-\frac{1}{2}\frac{\nu _{2k-1}}{2k-1}\frac{1}{\nu _{2k-1}}\) and so \(\frac{\nu _{2k-1}}{2k-1}\sum _{j\ge 2k-1}\frac{1}{\nu _j}\le 2\frac{\nu _{2k}}{2k}\sum _{j\ge 2k}\frac{1}{\nu _j}+\frac{1}{2k-1}\).

(ii):

In this case S is equivalent to N and so S has \(({\text {mg}})\) if and only if N has this property.

(iii):

Instead of having (15) one can study the more ”compact and easy to handle” requirement

$$\begin{aligned} \liminf _{k\rightarrow \infty }\frac{\nu _{k}}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}>0. \end{aligned}$$
(17)
Since (15) is equivalent to having \(\frac{2}{C}\le 2\frac{\nu _k}{\nu _{2k}}+\frac{\nu _k}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}\) and (17) to \(\frac{\nu _{k}}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}\ge \varepsilon \) for some \(\varepsilon >0\) and all \(k\in {\mathbb {N}}_{>0}\) we immediately get that (17) implies (15).

However, the converse is not clear in general: If N has in addition \(\lim _{k\rightarrow \infty }\frac{\nu _{2k}}{\nu _k}=\infty \), i.e. \((\beta ^0_2)\) in [21], then (15) still can be valid. But, as shown in [21, Proposition 1.1.(b), \((\beta ^0_2)\Rightarrow (\gamma _2)\)], this would yield \(\lim _{k\rightarrow \infty }\frac{\nu _{k}}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}=0\) and so making (17) impossible.

We are now studying an example which shows that the characterization obtained in Lemma 6 can hold true even if N does not have \(({\text {mg}})\).

Example 1

Put \(\nu _0:=1\) and

$$\begin{aligned} \nu _k:=2^p((p+1)!-p!)=2^pp!p\text {for}p!\le k<(p+1)!,\;\;\;p\in {\mathbb {N}}_{\ge 1}. \end{aligned}$$

By definition we have \(N\in {\mathscr {LC}}\) and

$$\begin{aligned} \sum _{j\ge 1}\frac{1}{\nu _j}=\sum _{l=1}^{\infty }\sum _{j=l!}^{(l+1)!-1}\frac{1}{\nu _j}=\sum _{l=1}^{\infty }\frac{(l+1)!-l!}{2^ll!l}=\sum _{l=1}^{\infty }\frac{1}{2^l}=1, \end{aligned}$$

which shows \(({\text {nq}})\).

N does not have \(({\text {mg}})\) because for any \(k\in {\mathbb {N}}_{\ge 1}\) with \(k<(p+1)!\le 2k<(p+2)!\), \(p\ge 1\), we get \(\frac{\nu _{2k}}{\nu _k}=2\frac{(p+1)^2}{p}\) which tends to infinity as \(k\rightarrow \infty \).

Finally let us show that (17) holds (and so (15)). Let now \(k\in {\mathbb {N}}\) be given with \(p!\le 2k<(p+1)!\), \(p\ge 2\). We split the sum

$$\begin{aligned} \frac{\nu _{k}}{k}\sum _{j\ge 2k}\frac{1}{\nu _j}=\frac{\nu _k}{k}\frac{(p+1)!-2k}{2^pp!p}+\frac{\nu _k}{k}\sum _{l\ge p+1}\frac{(l+1)!-l!}{2^ll!l}=\frac{\nu _k}{k}\frac{(p+1)!-2k}{2^pp!p}+\frac{\nu _k}{k}\frac{1}{2^p}, \end{aligned}$$

and remark that both summands are nonnegative for all \(k\in {\mathbb {N}}\) under consideration.

We study the second summand and distinguish between two cases. If \(k<p!\), then we have \(k\ge p!/2\ge (p-1)!\) and so

$$\begin{aligned} \frac{\nu _k}{k}\frac{1}{2^p}=\frac{2^{p-1}(p-1)!(p-1)}{k}\frac{1}{2^p}\ge \frac{1}{2}\frac{(p-1)!(p-1)}{p!}=\frac{1}{2}\frac{p-1}{p}\ge \frac{1}{4}. \end{aligned}$$

If \(p!\le k\), then we can estimate by

$$\begin{aligned} \frac{\nu _k}{k}\frac{1}{2^p}=\frac{2^pp!p}{k}\frac{1}{2^p}\ge \frac{p!p}{(p+1)!}=\frac{p}{p+1}\ge \frac{1}{2}. \end{aligned}$$

Thus the descendant S does have \(({\text {mg}})\). But since N violates this property, N cannot be equivalent to S and so N does not satisfy \((\gamma _1)\).

Similarly, there does exist also an inverse construction concerning the descendant, called the predecessor. However, this does not provide any new insight, see Remark 17.

Analogously as in the weight sequence situation we can treat the weight function case as well. First, inspired by the weight sequence approach in (11) and condition \((\omega _{\text {nq}_r})\), we define now for any given weight function \(\omega \) (as in Sect. 2.3) the order of quasianalyticity by

$$\begin{aligned} \mu (\omega ):=\sup \left\{ r\in {\mathbb {R}}_{>0}: \int _1^{\infty }\frac{\omega (u)}{u^{1+1/r}}du<+\infty \}=\sup \{r\in {\mathbb {R}}_{>0}:\;\omega \;\text {has}\;{(\omega _{\text {nq}_r})}\right\} . \end{aligned}$$
(18)

If none condition \((\omega _{\text {nq}_r})\) holds true, then we put \(\mu (\omega ):=0\). If \(\sigma \) is another weight function with \(\omega {\sim }\sigma \Leftrightarrow \omega ^r{\sim }\sigma ^r\), then \(\mu (\omega )=\mu (\sigma )\) follows since each condition \((\omega _{\text {nq}_r})\) is stable under \(\sim \).

This growth index likely will have an interpretation for the quasianalyticity of classes of ultraholomorphic functions defined in terms of weight functions, more precisely in order to prove analogous results to [12], see also [26] and [8].

Analogously to (12) for a given weight function \(\omega \) and \(r>0\) we consider now

$$\begin{aligned} \exists \;\sigma ,\;\sigma {\preceq }\omega ,\;\exists \;C>0\;\forall \;t\ge 0:\;\,\;\int _1^{\infty }\frac{\omega (tu)}{u^{1+1/r}}du\le C\sigma (t)+C, \end{aligned}$$
(19)

and let

$$\begin{aligned} \sup \{r\in {\mathbb {R}}_{>0}: (19)\;\;\text {is satisfied}\}. \end{aligned}$$
(20)

If none (19) holds true, then the \(\sup \) in (20) equals 0.

Let \(\omega \) and \(r>0\) be given and assume that (19) holds with some \(\sigma \), then the same choice is sufficient to have (19) for all \(0<r'<r\) as well.

Lemma 5 turns now into:

Lemma 7

Let\(\sigma \)and\(\omega \)be weight functions (in the sense of Section2.3) with\(\sigma {\preceq }\omega \). Then we get

$$\begin{aligned} \gamma (\sigma ,\omega )\le \mu (\omega ). \end{aligned}$$

Proof

For any \(r>\mu (\omega )\) we see that \((\omega _{\text {nq}_r})\) is violated and so \((\sigma ,\omega )_{\gamma _r}\) cannot be valid for any choice \(\sigma \) (see (iii) in Remark 7). \(\square \)

Remark 10

Lemmas 3 and 7 together yield

$$\begin{aligned} \gamma (\omega )\le \gamma (\sigma ,\omega )\le \mu (\omega ), \end{aligned}$$

hence the mixed index \(\gamma (\sigma ,\omega )\) can become crucial whenever \(\gamma (\omega )<\mu (\omega )\) does hold true.

The next result is analogous to Proposition 1 and showing that \(\mu (\omega )\) is the upper value for our considerations.

Proposition 2

Let\(\omega \)be a weight function (in the sense of Section2.3). Then we get

$$\begin{aligned} \sup \{r\in {\mathbb {R}}_{>0}: (19)\;\;\text {is satisfied}\}=\mu (\omega ). \end{aligned}$$

Proof

If \(\mu (\omega )=0\), then we have obviously equality. So let now \(\mu (\omega )>0\).

As commented in (iii) in Remark 7, if (19) with \(r>0\) holds true, then \(\omega ^r\) has \((\omega _{\text {nq}})\) and consequently \(\sup \{r\in {\mathbb {R}}_{>0}: (19)\;\;\text {is satisfied}\}\le \mu (\omega )\).

Conversely, for each \(0<r<\mu (\omega )\), we have that there exist some \(t_0\ge 1\), \(\varepsilon >0\) and \(C\ge 1\) such that \(\omega (t)\le C t^{1/r-\varepsilon }\) for all \(t\ge t_0\). Thus for all \(t\ge 0\) we get:

$$\begin{aligned} \int _1^{\infty }\frac{\omega (tu)}{u^{1+1/r}}du=A+\int _{t_0}^{\infty }\frac{\omega (tu)}{u^{1+1/r}}du\le A+Ct^{1/r-\varepsilon }\int _{t_0}^{\infty }\frac{1}{u^{1+\varepsilon }}du\le A_1+Bt^{1/r}, \end{aligned}$$

and so (19) holds true with \(\sigma (t):=t^{1/r}\). \(\square \)

As it happens for weight sequences, the choice \(\sigma (t):=t^{1/r}\) is not optimal. For each \(0<r<\mu (\omega )\) we can define the weight \(\kappa _{\omega ^r}\), using here the notation

$$\begin{aligned} \kappa _{\omega }(t):=\int _1^{\infty }\frac{\omega (ty)}{y^2}dy=t\int _t^{\infty }\frac{\omega (y)}{y^2}dy. \end{aligned}$$
(21)

A weight \(\omega \) satisfies \((\omega _{\text {snq}})\) (or \((\omega _{\gamma _1})\)) if and only if \(\kappa _{\omega }{\sim }\omega \). \(\kappa _{\omega }\) is called a heir in [24, Defintion 3.14]. Thus \((\kappa ^{1/r}_{\omega ^r},\omega )_{\gamma _r}\) and more precisely \(\kappa ^{1/r}_{\omega ^r}\) is optimal (i.e. minimal), up to a constant, among the weights satisfying requirement (19).

Note that \(\kappa ^{1/r}_{\omega ^r}\) is a weight function (in the sense of Sect. 2.3) and moreover \((\omega _1)\), \((\omega _3)\), \((\omega _4)\) can be transferred from \(\omega ^r\) (so from \(\omega \)) to \(\kappa _{\omega ^r}\) and so to \(\kappa ^{1/r}_{\omega ^r}\) (see Sect. 2.3). The proof of [11, Theorem 2.11 \((i)\Rightarrow (ii)\)] shows that \(\kappa _{\omega ^r}\) is concave. (ii) in Remark 7 yields \(\kappa ^{1/r}_{\omega ^r}(t)\ge \omega (t)\) for all \(t\ge 0\).

We are closing this section by establishing now the connection between \(\mu (\omega )\) and \(\mu (W^l)\), with \(W^l\in \varOmega \) and \(\varOmega \) denoting the matrix associated with \(\omega \).

Lemma 8

Let\(\omega \in {{\mathscr {W}}_0}\)(i.e. a normalized weight function with\((\omega _3)\)and\((\omega _4)\)), let\(\varOmega =\{W^l: l>0\}\)be the matrix associated with\(\omega \). Then we obtain

$$\begin{aligned} \forall \;l>0:\;\;\;\mu (\omega )=\mu (\omega _{W^l})=\mu (W^l). \end{aligned}$$
(22)

In particular, if\(\omega \equiv \omega _N\)for some given sequence\(N\in {\mathscr {LC}}\), then\(\mu (\omega _N)=\mu (N)\).

The last statement yields that, if even \(n\in {\mathscr {LC}}\) holds true, then \(\mu (\omega _N)=\mu (N)=\mu (n)+1=\mu (\omega _n)+1\).

Proof

First, given \(r>0\), by the formula on p. 8 in [14] we know that the matrix associated with the weight \(\omega ^r\) coincides with the set \(\{V^{l,r}:=(W^{l/r})^{1/r}: l>0\}\), i.e taking the r-th root of the sequences belonging to \(\varOmega \) and making a re-parametrization of the index (in terms of r).

By [22, Cor. 5.8 (1)], which is based on the characterization given in [15, Lemma 4.1] (see also [29, Cor. 4.8]), applied to \(\omega ^r\) we know that \(\omega ^r\) satisfies \((\omega _{\text {nq}})\) if and only if some/each \(V^{l,r}\) does have \((\text {nq})\) and so by the definition of the growth indices given in (11) and (18) we are done.

If \(\omega \equiv \omega _N\), then this follows immediately by recalling (4) and the definitions of the growth indices. \(\square \)

Remark 11

In [26, Def. 3.3, Thm. 3.4], by using the connection between \(\mu (N)\) and the so-called exponent of convergence of a nondecreasing sequence of positive real numbers tending to infinity (see [26, Prop. 2.13] and the references therein), for any \(N\in {\mathscr {LC}}\) we obtain \(\mu ({\widehat{N}})=\mu (N)+1\), \({\widehat{N}}:=(p!N_p)_{p\in {\mathbb {N}}}\).

Optimal flat functions in the mixed setting

Construction of outer functions

The aim of this paragraph is to obtain holomorphic functions in the right half-plane of \({\mathbb {C}}\) whose growth is accurately controlled by two given weight functions \(\omega \) and \(\sigma \). Since in the forthcoming sections we want to treat the weight sequence and weight function case simultaneously we will transfer the general proofs from [10, Section 6] to a mixed setting.

First we translate \((\sigma ,\omega )_{\gamma _1}\) into a property for \(\sigma ^{\iota }\) and \(\omega ^{\iota }\) (recall \(\omega ^{\iota }(t)=\omega (1/t)\)). The next result should be compared with [10, Lemma 6.1] and [32, Lemma 2.1.1].

Lemma 9

Let \(\omega \) and \(\sigma \) be weight functions. Then, one has \((\sigma ,\omega )_{\gamma _1}\) if, and only if,

$$\begin{aligned} \exists \;C\ge 1\;\forall \;t>0:\;\;\;\int _0^1 -\omega ^{\iota }(tu)du\ge -C(\sigma ^{\iota }(t)+1). \end{aligned}$$

The next statement recalls [10, Lemma 6.2], see also [32, Lemma 2.1.2].

Lemma 10

Let\(\omega \)be a weight function. Then\(\omega \)satisfies\((\omega _{{\text {nq}}})\)if and only if

$$\begin{aligned} \int _{-\infty }^{+\infty }\frac{-\omega ^{\iota }(|t|)}{1+t^2}dt>-\infty . \end{aligned}$$
(23)

In particular (23) holds true for each\(\omega \)satisfying\((\sigma ,\omega )_{\gamma _1}\)(and\(\sigma \)some other possibly quasianalytic weight).

In the next step we are generalizing [10, Lemma 6.3] to a mixed setting, the idea of the construction in the proof is coming from [32, Lemma 2.1.3].

Lemma 11

Let\(\omega \)and\(\sigma \)be two weight functions satisfying\((\sigma ,\omega )_{\gamma _1}\). Then for all\(a>0\)there exists a function\(F_a\)which is holomorphic on the right half-plane\(H_1:=\{w\in {\mathbb {C}}: \mathfrak {R}(w)>0\}\subseteq {\mathbb {C}}\)and constants\(A,B\ge 1\)(large) depending only on the weights\(\omega \)and\(\sigma \)such that

$$\begin{aligned} \forall \;w\in H_1:\;\;\;B^{-a}\exp (-2aB\sigma ^{\iota }(B^{-1}\mathfrak {R}(w)))\le |F_a(w)|\le \exp \left( -\frac{a}{2}\omega ^{\iota }(A|w|)\right) . \end{aligned}$$
(24)

Proof

We are following the lines of the proof of [10, Lemma 6.3], see also [32, Lemma 2.1.3] for the single weight sequence case. Since \((\sigma ,\omega )_{\gamma _1}\) is valid, the weight \(\omega \) has to satisfy \((\omega _{{\text {nq}}})\). Hence for \(w\in H_1\) we can put

$$\begin{aligned} F_a(w):=\exp \left( \frac{1}{\pi }\int _{-\infty }^{+\infty } \frac{-a\omega ^{\iota }(|t|)}{1+t^2}\frac{itw-1}{it-w}dt\right) ; \end{aligned}$$

Lemma 10 implies immediately that \(F_a\) is a holomorphic function in \(H_1\). Since \(F_a(w)=(F_1(w))^a\), we need only consider in the proof \(a=1\) and put for simplicity \(F:=F_1\).

For \(w\in H_1\) write \(w=u+iv\), hence \(u>0\). We have

$$\begin{aligned} \log (|F(w)|)=\frac{1}{\pi }\int _{{\mathbb {R}}}-\omega ^{\iota }(|t|) \frac{u}{(t-v)^2+u^2}dt=-\frac{1}{\pi }f*g_u(v), \end{aligned}$$

where \(f(t):=\omega ^{\iota }(|t|)\), \(g_u(t):=u/(t^2+u^2)\). f and \(g_u\) are symmetrically nonincreasing functions, hence the convolution too, and as argued in [10, Lemma 6.3] this means that for \(w\mapsto \log (|F(w)|)\) the minimum is attained on the positive real axis and we have for all \(w\in H_1\):

$$\begin{aligned} \log (|F(w)|)\ge & {} \log (|F(\mathfrak {R}(w))|)=\log (|F(u)|)=\frac{1}{\pi }\int _{{\mathbb {R}}}-\omega ^{\iota }(|t|) \frac{u}{t^2+u^2}dt\\= & {} -\frac{1}{\pi }f*g_u(0). \end{aligned}$$

For the left-hand side in (24) consider \(K>0\) (small) and get

$$\begin{aligned} \pi \log (|F(u)|)&=\int _{{\mathbb {R}}}-\omega ^{\iota }(|t|) \frac{u}{t^2+u^2}dt\\&=\int _{\{t:|t|\ge Ku\}}-\omega ^{\iota }(|t|)\frac{u}{t^2+u^2}dt+\int _{\{t:|t|\le Ku\}}-\omega ^{\iota }(|t|)\frac{u}{t^2+u^2}dt. \end{aligned}$$

The first integral is estimated by

$$\begin{aligned}&\int _{\{|t|\ge Ku\}}-\omega ^{\iota }(|t|)\frac{u}{t^2+u^2}dt\\&\quad \ge -\omega ^{\iota }(Ku)\int _{\{|t|\ge Ku\}}\frac{u}{t^2+u^2}dt =-\omega ^{\iota }(Ku)(\pi -2\arctan (K)) \\&\quad \ge -\sigma ^{\iota }(Ku)D(\pi -2\arctan (K))-D(\pi -2\arctan (K)), \end{aligned}$$

since \(t\mapsto -\omega ^{\iota }(t)\) is nondecreasing and since by \((\sigma ,\omega )_{\gamma _1}\) we have \(\sigma {\preceq }\omega \). Thus \(-\omega ^{\iota }(t)\ge -D(\sigma ^{\iota }(t)+1)\) for some \(D\ge 1\) and all \(t>0\) follows.

For the second integral we get

$$\begin{aligned} \int _{\{t: |t|\le Ku\}}-\omega ^{\iota }(|t|)\frac{u}{t^2+u^2}dt&=\int _{\{s: |s|\le 1\}}-\omega ^{\iota }(Ku|s|)\frac{K}{K^2s^2+1}ds\\&\ge K\int _{\{s: |s|\le 1\}}-\omega ^{\iota }(Ku|s|)ds, \end{aligned}$$

since \(-\omega ^{\iota }(Ku|s|)\le 0\) holds for any \(K,u>0\) and \(|s|\le 1\). Let \(C\ge 1\) be the constant appearing in Lemma 9, then

$$\begin{aligned} K\int _{\{s:|s|\le 1\}}-\omega ^{\iota }(Ku|s|)ds\ge 2KC(-\sigma ^{\iota }(Ku)-1). \end{aligned}$$

Now follow the lines of the proof in [10, Lemma 6.3] with \(-\tau ^{\iota }\) replaced by \(-\sigma ^{\iota }\) which is also nondecreasing. Thus the first estimate in (24) is shown.

The second estimate follows without any change like in [10, Lemma 6.3]. \(\square \)

Construction of optimal sectorially flat functions in the mixed setting

Using the results from Sect. 4.1 the aim is now to transfer [32, Theorem 2.3.1] and [10, Theorem 6.7] (see also [14, Theorem 5.4]) simultaneously to a mixed setting.

Theorem 1

 

  1. (I)

    Let\(M,N\in {\mathscr {LC}}\)be given such that\(\mu _p\le \nu _p\)and\(\gamma (M,N)>0\)holds true.

    Then for any\(0<\gamma <\gamma (M,N)\)there exist constants\(K_1,K_2,K_3,K_4>0\)depending only onM, Nand\(\gamma \)such that for all\(a>0\)there exists a function\(G_a\)holomorphic on\(S_{\gamma }\)and satisfying

    $$\begin{aligned} \forall \;\xi \in S_{\gamma }:\;\;K_1^{-a}h_M(K_1|\xi |)^{2aK_2}\le |G_a(\xi )|\le h_N(K_3|\xi |)^{\frac{aK_4}{2}}. \end{aligned}$$
    (25)

    Moreover, ifNhas in addition\(({\text {mg}})\), then\(G_a\in {\mathscr {A}}_{\{{\widehat{N}}\}}(S_{\gamma })\)with\({\widehat{N}}:=(p!N_p)_{p\in {\mathbb {N}}}\) (and\(G_a\)is flat at 0).

    IfMhas in addition\(({\text {mg}})\), then there exists\(K_5>0\)depending also on given\(a>0\)such that

    $$\begin{aligned} \forall \;\xi \in S_{\gamma }:\;\;\;|G_a(\xi )|\ge K_1^{-a}h_M(K_5|\xi |). \end{aligned}$$
    (26)
  2. (II)

    Let\(\omega \)and\(\sigma \)be weight functions such that\(\gamma (\sigma ,\omega )>0\)holds true. Then for any\(0<\gamma <\gamma (\sigma ,\omega )\)there exist constants\(K_1,K_2,K_3>0\)depending only on\(\sigma \), \(\omega \)and\(\gamma \)such that for all\(a>0\)there exists a function\(G_a\)holomorphic on\(S_{\gamma }\)and satisfying

    $$\begin{aligned} \forall \;\xi \in S_{\gamma }:\;\;\;K_1^{-a}\exp (-2a\sigma ^{\iota }(K_2|\xi |))\le |G_a(\xi )|\le \exp \left( -\frac{a}{2}\omega ^{\iota }(K_3|\xi |)\right) . \end{aligned}$$
    (27)

    Moreover, if\(\omega \)is normalized and satisfies\((\omega _3)\), then\(G_a\in {\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S_{\gamma })\) (and\(G_a\)is flat at 0), more precisely

    $$\begin{aligned} \forall \;\xi \in S_{\gamma }:\;\;\;|G_a(\xi )|\le h_{W^{2/a}}(K'_2|\xi |), \end{aligned}$$
    (28)

    where\(\varOmega =\{W^x: x>0\}\)shall denote the matrix associated with\(\omega \)and\({\widehat{\varOmega }}:=\{{\widehat{W}}^x=(p!W^x_p)_{p\in {\mathbb {N}}}: x>0\}\).

    If\(\sigma \in {{\mathscr {W}}_0}\), then there exist an index\(x>0\)and a constant\(K_4>0\)depending also onasuch that

    $$\begin{aligned} \forall \;\xi \in S_{\gamma }:\;\;\;|G_a(\xi )|\ge K_4 h_{S^x}(K_2|\xi |), \end{aligned}$$
    (29)

    where\(S^x\in \varSigma \), \(\varSigma \)the matrix associated with\(\sigma \).

Proof

We will give some more details for the proof of (I) (following the lines of [32, Theorem 2.3.1]).

(I) Let \(a>0\) be arbitrary. Take \(s,\delta >0\) such that \(\gamma<\delta <\gamma (M,N)\) and \(s\delta<1<s\gamma (M,N)\).

By (10) we get \(s\gamma (M,N)>1\Leftrightarrow \gamma (M^s,N^s)>1\), hence, by Lemma 4 applied to \(M^s\) and \(N^s\), we get \(\gamma (\omega _{M^s},\omega _{N^s})>1\), too.

So we can use Lemma 11 for \(\omega _{M^s}\equiv \sigma \) and \(\omega _{N^s}\equiv \omega \) and obtain a function \(F_a\) holomorphic on the right half-plane and satisfying

$$\begin{aligned} \forall \;w\in H_1:\;\;\;B^{-a}\exp (-2aB\omega _{M^s}^{\iota }(B^{-1}\mathfrak {R}(w)))\le |F_a(w)|\le \exp \left( -\frac{a}{2}\omega _{N^s}^{\iota }(A|w|)\right) . \end{aligned}$$
(30)

Then put

$$\begin{aligned} G_a(\xi )=F_a(\xi ^s),\;\;\;\;\xi \in S_{\delta }. \end{aligned}$$

Note that, as \(s\delta <1\), the ramification \(\xi \mapsto \xi ^s\) maps holomorphically \(S_{\delta }\) into \(S_{\delta s}\subseteq S_1=H_1\), and so \(G_a\) is well-defined. We show that the restriction of \(G_a\) to \(S_{\gamma }\subseteq S_{\delta }\) satisfies the desired properties by proving that (25) holds indeed on the whole \(S_{\delta }\).

First we consider the lower estimate.

Let \(\xi \in S_{\delta }\) be given, then \(\mathfrak {R}(\xi ^s)\ge \cos (s\delta \pi /2)|\xi |^s\) (since \(s\delta \pi /2<\pi /2\)). If \(B\ge 1\) denotes the constant coming from the left-hand side in (24) applied to the weight \(\omega _{M^s}\) (or see (30)), then

$$\begin{aligned} |G_a(\xi )|&=|F_a(\xi ^s)|\ge B^{-a}\exp (-2aB\omega _{M^s}^{\iota }(B^{-1}(\mathfrak {R}(\xi ^s))))\\&\ge B^{-a}\exp (-2aB\omega _{M^s}^{\iota }(B^{-1}\cos (s\delta \pi /2)|\xi |^s))\\&=B^{-a}\exp (-2aB\omega _{M^s}^{\iota }((B_1|\xi |)^s))\\&=B^{-a}\exp (-2aBs\omega _M^{\iota }(B_1|\xi |))=B^{-a}h_M(B_1|\xi |)^{2aBs}, \end{aligned}$$

where we have put \(B_1:=(B^{-1}\cos (s\delta \pi /2))^{1/s}\). We have used that \(\omega _{M^s}^{\iota }(t^s)=\omega _{M^s}(1/t^s)=s\omega _M(1/t)=s\omega _M^{\iota }(t)\) for all \(t,s>0\), see (4), and finally (6).

Now we consider the right-hand side in (24) respectively in (30) and proceed as before. Let A be the constant coming from the right-hand side of (24) applied to \(\omega _{N^s}\), so

$$\begin{aligned} |G_a(\xi )|&=|F_a(\xi ^s)|\le \exp \left( -\frac{a}{2}\omega _{N^s}^{\iota }(A|\xi |^s)\right) =\exp \left( -\frac{a}{2}\omega _{N^s}^{\iota }((A^{1/s}|\xi |)^s)\right) \\&=\exp \left( -\frac{a}{2}s\omega _{N}^{\iota }(A^{1/s}|\xi |)\right) =h_N(A^{1/s}|\xi |)^{\frac{sa}{2}}, \end{aligned}$$

and (25) has been proved for every \(\xi \in S_{\delta }\).

In order to show \(G_a\in {\mathscr {A}}_{\{{\widehat{N}}\}}(S_{\gamma })\) we put in the estimate above \(A_1:=A^{1/s}\) and see

$$\begin{aligned} \exp \left( -\frac{a}{2}s\omega _N^{\iota }(A_1|\xi |)\right) =\exp \left( -\frac{a}{2}s\omega _N(1/(A_1|\xi |))\right) =h_N(A_1|\xi |)^{sa/2}. \end{aligned}$$

If we can show

$$\begin{aligned} \exists \;A_2\ge 1\;\forall \xi \in S_{\delta }:\ \ |G_a(\xi )|\le h_{N}(A_2|\xi |), \end{aligned}$$
(31)

then by applying [10, Lemma 6.4 (i.1)] we see that \(G_a\in {\mathscr {A}}_{\{{\widehat{N}}\}}(S_{\gamma })\) (and it is a flat function at 0). Since \(h_N\le 1\), (31) holds true whenever \(\frac{sa}{2}\ge 1\Leftrightarrow sa\ge 2\). But in general we have to use \((\text {mg})\) for N and iterate (7) (applied for N) l-times, \(l\in {\mathbb {N}}\) chosen minimal to ensure \(\frac{sa}{2}\ge \frac{1}{2^l}\).

The proof of (26) follows analogously by iterating \((\text {mg})\) for M (if necessary) in order to get rid of the exponent \(2aK_2\).

(II) Again let \(a>0\) be arbitrary but fixed, take \(s,\delta >0\) such that \(\gamma<\delta <\gamma (\sigma ,\omega )\), \(s\delta<1<s\gamma (\sigma ,\omega )=\gamma (\sigma ^{1/s},\omega ^{1/s})\) and put

$$\begin{aligned} G_a(\xi )=F_a(\xi ^s),\;\;\;\;\xi \in S_{\delta }, \end{aligned}$$

where \(F_a\) is the function constructed in Lemma 11. We apply Lemma 11 to \(\sigma ^{1/s}\equiv \sigma \), \(\omega ^{1/s}\equiv \omega \). Hence (27) holds true by following the lines in the proof of [10, Theorem 6.7] and replacing \(\tau \) by \(\omega \) for the right-hand side respectively \(\tau \) by \(\sigma \) for the left-hand side.

The remaining statements, in particular the estimate (29), follow analogously as in the proof of [10, Theorem 6.7], replacing \(\tau \) by \(\omega \) or \(\sigma \) in the arguments. \(\square \)

Right inverses for the asymptotic Borel map in ultraholomorphic classes in sectors

The aim of this section is to obtain an extension result in the ultraholomorphic classes considered in a mixed setting for both the weight sequence and the weight function approach following the proofs and techniques in [10, Section 7]. The existence of the optimal flat functions \(G_a\) obtained in Theorem 1 will be the main ingredient in the proof which is inspired by the same technique as in previous works of A. Lastra, S. Malek and the second author [18, 19] in the single weight sequence approach. Although for the general construction the weight functions \(\sigma \) and \(\omega \) need not be normalized, we are interested in working with the weight matrices associated with them, which will be standard log-convex if we ask for normalization and \((\omega _3)\) to hold.

Note that any weight function may be substituted by a normalized equivalent one (e.g. see [3, Remark 1.2 (b)]) and equivalence preserves the property \((\omega _3)\), so it is no restriction to ask for normalization from the very beginning.

Remark 12

An important difference to the complete approach in [10] is, see also the comments given in the introduction in Section 7 there, that condition \(\gamma (\omega )>0\) which amounts to \((\omega _1)\) as shown in [10, Lemma 4.2] will not be valid in general anymore in the mixed situation. In the following we are only requiring \(\gamma (\sigma ,\omega )>0\) and recall that \(\gamma (\sigma ,\omega )\ge \gamma (\omega )\) as shown above. An explicit example of this situation, having \(\gamma (\sigma ,\omega )>0\) (as large as desired) and \(\gamma (\omega )=\gamma (\sigma )=0\) will be provided in the Appendix A below. We are able to treat this situation by recognizing that in [10] we have worked in a very general framework for weight functions and the assumption \(\gamma (\omega )>0\) can be replaced by \(\gamma (\sigma ,\omega )>0\) without causing problems.

Recall that \((\omega _1)\) is standard in the ultradifferentiable setting and thus our techniques make it possible to treat ”exotic” weight function situations as well. Moreover \((\omega _1)\) has also been used to have that the class defined by \(\omega \) admits a representation by using the associated weight matrix \(\varOmega \), see Sect. 2.5. Thus the warranty that the ultraholomorphic (and also the ultradifferentiable) spaces associated with \(\omega \) and its corresponding weight matrix \(\varOmega \) coincide is not clear anymore, see the comments preceding (9).

Therefore the main and most general ultraholomorphic extension result Theorem 2 deals with a mixed situation between classes defined by (associated) weight matrices. If one imposes \((\omega _1)\) on the weights one is able to prove a mixed version of classes defined by weight functions, see Corollary 1. Finally, in Theorem 4 we will treat the mixed weight sequence case as well.

Preliminaries

We start with recalling [10, Lemma 7.1].

Lemma 12

Let\(\sigma \)and\(\omega \)be normalized weight functions with\(\gamma (\sigma ,\omega )>0\)and such that\(\omega \)satisfies\((\omega _3)\). Let\(\varOmega =\{W^x: x>0\}\)be the weight matrix associated with\(\omega \), \(0<\gamma <\gamma (\sigma ,\omega )\), and for\(a>0\)let\(G_a\)be the function constructed in Theorem 1. Let us define the function\(e_{a}:S_{\gamma }\rightarrow {\mathbb {C}}\)by

$$\begin{aligned} e_a(z):=z\,G_a(1/z),\quad z\in S_{\gamma }. \end{aligned}$$

The function\(e_{a}\)enjoys the following properties:

  1. (i)

    \(z^{-1}e_{a}(z)\)is uniformly integrable at the origin, it is to say, for any\(t_0>0\)we have

    $$\begin{aligned} \int _{0}^{t_0}t^{-1}\sup _{|\tau |<\gamma \pi /2}|e_{a}(te^{i\tau })|dt<\infty . \end{aligned}$$
  2. (ii)

    There exist constants\(K>0\), independent froma, and\(C>0\), depending ona, such that

    $$\begin{aligned} |e_{a}(z)|\le Ch_{W^{4/a}}\left( \frac{K}{|z|}\right) ,\qquad z\in S_{\gamma }. \end{aligned}$$
    (32)
  3. (iii)

    For\(\xi \in {\mathbb {R}}\), \(\xi >0\), the values of\(e_{a}(\xi )\)are positive real.

Proof

The proof is completely the same as for [10, Lemma 7.1], for (i) we apply the right-hand side in (27), for (ii) we use (28) and (8) together with the definition given in (5). \(\square \)

Analogously as in [10, Definition 7.2] we introduce now the moment function associated with \(e_a\).

Definition 1

We define the moment function associated with the function \(e_a\) (introduced in the previous Lemma) as

$$\begin{aligned} m_a(\lambda ):=\int _{0}^{\infty }t^{\lambda -1}e_{a}(t)dt= \int _{0}^{\infty }t^{\lambda }G_{a}(1/t)dt. \end{aligned}$$

From Lemma 12 and the definition of \(h_{W^x}\) in (5) we see that for every \(p\in {\mathbb {N}}\),

$$\begin{aligned} |e_{a}(z)|\le C\frac{K^pW^{4/a}_p}{|z|^p},\qquad z\in S_{\gamma }. \end{aligned}$$

So, we easily deduce that the function \(m_a\) is well defined and continuous in \(\{\lambda : \mathfrak {R}(\lambda )\ge 0\}\), and holomorphic in \(\{\lambda : \mathfrak {R}(\lambda )>0\}\). Moreover, \(m_a(\xi )\) is positive for every \(\xi \ge 0\), and the sequence \((m_a(p))_{p\in {\mathbb {N}}}\) is called the sequence of moments of \(e_{a}\).

The next result is generalizing [10, Proposition 7.3], which is similar to Proposition 3.6 in [18], to a mixed setting.

Proposition 3

Let\(\sigma \)and\(\omega \)be normalized weight functions with\(\gamma (\sigma ,\omega )>0\)and such that both weights satisfy\((\omega _3)\). Let\(\varSigma =\{S^x: x>0\}\)and\(\varOmega =\{W^x: x>0\}\)be the weight matrices associated with\(\sigma \)and\(\omega \)respectively, and for\(0<\gamma <\gamma (\sigma ,\omega )\)and\(a>0\)let\(G_a,e_a,m_a\)be the functions previously constructed. Then, there exist constants\(C_1,C_2>0\), both depending ona, such that for every\(p\in {\mathbb {N}}\)one has

$$\begin{aligned} C_1\left( \frac{K_2}{2}\right) ^pS_p^{1/(2a)}\le m_a(p)\le C_2K_3^pW_{p}^{4/a}, \end{aligned}$$
(33)

where\(K_2\)and\(K_3\)are the constants, not depending ona, appearing in (27).

Proof

The proof follows the lines as in [10, Proposition 7.3] (based on the arguments by O. Blasco in [2]). For the second estimate in (33) we use the second inequality in (27) (and here also (2) is used); the first estimate in (33) follows by applying the first inequality in (27). \(\square \)

Main extension results

Now we are able to formulate and proof the generalization of the main extension result [10, Theorem 7.4].

Theorem 2

Let\(\sigma \)and\(\omega \)be normalized weight functions with\(\gamma (\sigma ,\omega )>0\)and such that both weights satisfy\((\omega _3)\)and\(0<\gamma <\gamma (\sigma ,\omega )\). Moreover we denote by\(\varSigma =\{S^x: x>0\}\)and\(\varOmega =\{W^x: x>0\}\)the weight matrices associated with\(\sigma \)and\(\omega \)respectively and consider the matrices\({\widehat{\varSigma }}=\{{\widehat{S}}^x:x>0\}\)and\({\widehat{\varOmega }}=\{{\widehat{W}}^x:x>0\}\)where\({\widehat{S}}^x:=(p!S^x_p)_{p\in {\mathbb {N}}}\)and\({\widehat{W}}^x:=(p!W^x_p)_{p\in {\mathbb {N}}}\).

Then, there exists a constant\(k_0>0\)such that for every\(x>0\)and every\(h>0\), one can construct a linear and continuous map

$$\begin{aligned} E^{\sigma ,\omega }_h:\;\;\lambda \in \varLambda _{{\widehat{S}}^x,h}\mapsto f_{\lambda }\in {\mathscr {A}}_{{\widehat{W}}^{8x},k_0h}(S_{\gamma }) \end{aligned}$$

such that for every\(\lambda \in \varLambda _{{\widehat{S}}^x,h}\)one has\({\mathscr {B}}\circ E^{\sigma ,\omega }_h(\lambda )={\mathscr {B}}(f_{\lambda })=\lambda \). Consequently we obtain the inclusion\({\mathscr {B}}({\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S_{\gamma }))\supseteq \varLambda _{\{{\widehat{\varSigma }}\}}\).

Proof

Fix \(\delta >0\) such that \(\gamma<\delta <\gamma (\sigma ,\omega )\). Given \(\lambda =(\lambda _p)_{p\in {\mathbb {N}}}\in \varLambda _{{\widehat{S}}^x,h}\), we have

$$\begin{aligned} |\lambda _p|\le |\lambda |_{{\widehat{S}}^x,h} h^{p}p!S_p^x,\quad p\in {\mathbb {N}}_0. \end{aligned}$$
(34)

We choose \(a=1/(2x)\), and consider the function \(G_a\), defined in \(S_{\delta }\), obtained in Theorem 1 for such value of a, and the corresponding functions \(e_a\) and \(m_a\) previously defined. Next, we consider the formal power series

$$\begin{aligned} {\widehat{f}}_{\lambda }:=\sum _{p=0}^\infty \frac{\lambda _p}{p!}z^p \end{aligned}$$

and its formal (Borel-like) transform

$$\begin{aligned} \widehat{{\mathscr {B}}}_a{\widehat{f}}_{\lambda }:= \sum _{p=0}^\infty \frac{\lambda _p}{p!m_a(p)}z^p. \end{aligned}$$

By the choice of a, (34) and the first part of the inequalities in (33), we deduce that

$$\begin{aligned} \left| \frac{\lambda _p}{p!m_a(p)}\right| \le \frac{|\lambda |_{{\widehat{S}}^x,h} h^{p}p!S_p^x}{C_1\left( K_2/2\right) ^p p!S_p^x}= \frac{|\lambda |_{{\widehat{S}}^x,h}}{C_1} \left( \frac{2h}{K_2}\right) ^p, \end{aligned}$$
(35)

and so the series \(\widehat{{\mathscr {B}}}_a{\widehat{f}}_{\lambda }\) converges in the disc of center 0 and radius \(K_2/(2h)\) (not depending on \(\lambda \)), where it defines a holomorphic function \(g_\lambda \). We set \(R_0:=K_2/(4h)\), and define

$$\begin{aligned} f_{\lambda }(z):=\int _{0}^{R_0}e_{a}\left( \frac{u}{z}\right) g_{\lambda }(u)\frac{du}{u},\qquad z\in S_{\delta }. \end{aligned}$$

By virtue of Leibniz’s theorem on analyticity of parametric integrals, \(f_\lambda \) is holomorphic in \(S_{\delta }\).

Our next aim is to obtain suitable estimates for the difference between f and the partial sums of the series \({\widehat{f}}_{\lambda }\). As in the non-mixed setting, for given \(N\in {\mathbb {N}}_0\) and \(z\in S_{\delta }\) we have

$$\begin{aligned}&f_{\lambda }(z)-\sum _{p=0}^{N-1}\lambda _p\frac{z^p}{p!}= \int _{0}^{R_0}e_{a}\left( \frac{u}{z}\right) \sum _{p=0}^{\infty }\frac{\lambda _{p}}{m_a(p)}\frac{u^p}{p!} \frac{du}{u}\\&\quad -\sum _{p=0}^{N-1}\frac{\lambda _p}{m_a(p)} \int _{0}^{\infty }u^{p-1}e_{a}(u)du\frac{z^p}{p!}. \end{aligned}$$

In the second integral we make the change of variable \(v=zu\), what results in a rotation of the line of integration. By the estimate (32), one may use Cauchy’s residue theorem in order to obtain that

$$\begin{aligned} z^p\int _{0}^{\infty }u^{p-1}e_{a}(u)du= \int _{0}^{\infty }v^{p}e_{a}\left( \frac{v}{z}\right) \frac{dv}{v}, \end{aligned}$$

which allows us to write the preceding difference as

$$\begin{aligned}&\int _{0}^{R_0}e_{a}\left( \frac{u}{z}\right) \sum _{p=0}^{\infty }\frac{\lambda _{p}}{m_a(p)} \frac{u^p}{p!}\frac{du}{u} -\sum _{p=0}^{N-1}\frac{\lambda _p}{m_a(p)} \int _{0}^{\infty }u^{p}e_{a}\left( \frac{u}{z}\right) \frac{du}{u} \frac{1}{p!}\\&\quad =\int _{0}^{R_0}e_{a}\left( \frac{u}{z}\right) \sum _{p=N}^{\infty }\frac{\lambda _{p}}{m_a(p)}\frac{u^p}{p!} \frac{du}{u} -\int _{R_0}^{\infty }e_{a}\left( \frac{u}{z}\right) \sum _{p=0}^{N-1}\frac{\lambda _p}{m_a(p)} \frac{u^{p}}{p!}\frac{du}{u}. \end{aligned}$$

Then, we have

$$\begin{aligned} \left| f_{\lambda }(z)-\sum _{p=0}^{N-1}\lambda _p\frac{z^p}{p!}\right| \le |f_{1}(z)|+|f_2(z)|, \end{aligned}$$
(36)

where

$$\begin{aligned} f_{1}(z)=\int _{0}^{R_0}e_{a}\left( \frac{u}{z}\right) \sum _{p=N}^{\infty }\frac{\lambda _{p}}{m_a(p)} \frac{u^p}{p!}\frac{du}{u},\quad f_{2}(z)=\int _{R_0}^{\infty }e_{a}\left( \frac{u}{z}\right) \sum _{p=0}^{N-1}\frac{\lambda _p}{m_a(p)} \frac{u^{p}}{p!}\frac{du}{u}. \end{aligned}$$

From (35) we deduce that

$$\begin{aligned} |f_{1}(z)|&\le \frac{|\lambda |_{{\widehat{S}}^x,h}}{C_1} \int _{0}^{R_0}\left| e_{a}\left( \frac{u}{z}\right) \right| \sum _{p=N}^{\infty }\Big (\frac{2hu}{K_2}\Big )^p\frac{du}{u} =\frac{|\lambda |_{{\widehat{S}}^x,h}}{C_1} \Big (\frac{2h}{K_2}\Big )^N \int _{0}^{R_0}\left| e_{a}\left( \frac{u}{z}\right) \right| \frac{u^N}{1-\frac{2hu}{K_2}}\frac{du}{u}\nonumber \\&\le \frac{2|\lambda |_{{\widehat{S}}^x,h}}{C_1} \Big (\frac{2h}{K_2}\Big )^N \int _{0}^{R_0}\left| e_{a}\left( \frac{u}{z}\right) \right| u^{N-1}\,du, \end{aligned}$$
(37)

where in the last step we have used that \(0<u<R_0=K_2/(4h)\) we have \(1-2hu/K_2>1/2\). In order to estimate \(f_{2}(z)\), observe that for \(u\ge R_0\) and \(0\le p\le N-1\) we always have \(u^p\le R_0^pu^N/R_0^N\), and so, using again (35) and the value of \(R_0\), we may write

$$\begin{aligned} \left| \sum _{p=0}^{N-1}\frac{\lambda _pu^p}{p!m_a(p)}\right| \le \frac{|\lambda |_{{\widehat{S}}^x,h}}{C_1}\frac{u^N}{R_0^N} \sum _{p=0}^{N-1}\Big (\frac{2h}{K_2}\Big )^p R_0^p \le \frac{2|\lambda |_{{\widehat{S}}^x,h}}{C_1} \Big (\frac{4h}{K_2}\Big )^N u^N. \end{aligned}$$

Then, we deduce that

$$\begin{aligned} |f_2(z)|\le \frac{2|\lambda |_{{\widehat{S}}^x,h}}{C_1} \Big (\frac{4h}{K_2}\Big )^N \int _{R_0}^{\infty }\left| e_{a}\left( \frac{u}{z}\right) \right| u^{N-1}du. \end{aligned}$$
(38)

In order to conclude, it suffices then to obtain estimates for \(\int _{0}^{\infty }|e_{a}(u/z)|u^{N-1}du\). For this, note first that, by the estimates in (27),

$$\begin{aligned} \int _{0}^{\infty }\left| e_{a}\left( \frac{u}{z}\right) \right| u^{N-1}du&= \int _{0}^{\infty }\frac{u}{|z|} \left| G_{a}\left( \frac{z}{u}\right) \right| u^{N-1}du\\&\le \int _0^\infty \frac{u^N}{|z|} \exp \left( -\frac{a}{2}\omega \left( \frac{u}{K_3|z|}\right) \right) \,du\\&= |z|^N\int _0^\infty t^N\exp \left( -\frac{a}{2}\omega \left( \frac{t}{K_3}\right) \right) \,dt. \end{aligned}$$

Now, we can follow the first part of the proof of [10, Proposition 7.3] to obtain that

$$\begin{aligned} \int _{0}^{\infty }\left| e_{a}\left( \frac{u}{z}\right) \right| u^{N-1}du\le C_2K_3^NW_N^{4/a}|z|^N=C_2K_3^NW_N^{8x}|z|^N. \end{aligned}$$
(39)

Gathering (36), (37), (38) and (39), we get

$$\begin{aligned} \left| f_{\lambda }(z)-\sum _{p=0}^{N-1}\lambda _p\frac{z^p}{p!}\right| \le \frac{2C_2|\lambda |_{{\widehat{S}}^x,h}}{C_1} \left( \frac{4hK_3}{K_2}\right) ^NW_N^{8x}|z|^N. \end{aligned}$$
(40)

A straightforward application of Cauchy’s integral formula yields that there exists a constant r, depending only on \(\gamma \) and \(\delta \), such that whenever z is restricted to belong to \(S_{\gamma }\), one has that for every \(p\in {\mathbb {N}}\),

$$\begin{aligned} |f^{(p)}(z)|\le \frac{2C_2|\lambda |_{{\widehat{S}}^x,h}}{C_1} \left( \frac{4hK_3r}{K_2}\right) ^p p!W_p^{8x}. \end{aligned}$$

So, putting \(k_0:=\frac{4K_3r}{K_2}\) (independent from x and h), we see that \(f_{\lambda }\in {\mathscr {A}}_{{\widehat{W}}^{8x},k_0h}(S_{\gamma })\) and \(\Vert f_{\lambda }\Vert _{{\widehat{W}}^{8x},k_0h}\le \frac{2C_2}{C_1}|\lambda |_{{\widehat{S}}^x,h}\). Since the map sending \(\lambda \) to \(f_{\lambda }\) is clearly linear, this last inequality implies that the map is also continuous from \(\varLambda _{{\widehat{S}}^x,h}\) into \({\mathscr {A}}_{{\widehat{W}}^{8x},k_0h}(S_{\gamma })\). Finally, from (40) one may easily deduce that \({\mathscr {B}}(f_{\lambda })=\lambda \), and we conclude. \(\square \)

Remark 13

We point out that by checking the proofs of the main extension results [32, Theorem 3.2.1] and [14, Theorem 6.8] it seems to be not possible to transfer the techniques there to a mixed setting. In these proofs the existence of a continuous linear extension operator coming from the ultradifferentiable extension theorems is used and it is sending the sequence \(\lambda \) to the function \(g_{\lambda }\) (e.g. see [14, (6.12)]) which belongs in a mixed situation to a space defined by a different weight sequence or weight function.

Consequently, in the estimates of part (ii) in the proofs of [32, Theorem 3.2.1] and [14, Theorem 6.8], one is not able to get rid of the quotients of associated functions \(h_M(\cdot )\). This should be compared with the estimate (35) where we do not have to change the weight structure and the proofs of the extension theorems in [7] and [24].

In order to prove the generalization of [10, Corollary 7.6] first we have to recall [10, Theorem 5.3].

Theorem 3

Let\(\omega \in {{\mathscr {W}}}\), i.e. a normalized weight satisfying\((\omega _3)\), \((\omega _4)\)and\((\omega _1)\), \(\varOmega =\{W^x: x>0\}\)be the weight matrix associated with\(\omega \)and consider\({\widehat{\varOmega }}=\{{\widehat{W}}^x:x>0\}\)where\({\widehat{W}}^x=(p!W^x_p)_{p\in {\mathbb {N}}}\). Then the following identities hold as locally convex vector spaces for all sectorSand for all\(x>0\):

$$\begin{aligned} {\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S)= {\mathscr {A}}_{\{\omega _{{\widehat{W}}^x}\}}(S), \end{aligned}$$

and the same equalities are valid for the corresponding sequence classes\(\varLambda \). So, \({\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S)\)coincides with the space\({\mathscr {A}}_{\{\tau \}}(S)\)associated with\(\tau =\omega _{{\widehat{W}}^x}\in {{\mathscr {W}}}\).

Finally one gets\(\gamma (\tau )=\gamma (\omega )+1>1\).

Now we concentrate on the generalization of [10, Corollary 7.6].

Corollary 1

Let\(\sigma , \omega \in {{\mathscr {W}}}\)be given, so that\(\gamma (\sigma ,\omega )\ge \gamma (\omega )>0\), and let\(0<\gamma <\gamma (\sigma ,\omega )\)and\(\varSigma =\{S^x: x>0\}\)and\(\varOmega =\{W^x: x>0\}\)be the weight matrices associated with\(\sigma \)and\(\omega \)respectively and consider the matrices\({\widehat{\varSigma }}=\{{\widehat{S}}^x:x>0\}\)and\({\widehat{\varOmega }}=\{{\widehat{W}}^x:x>0\}\)where\({\widehat{S}}^x=(p!S^x_p)_{p\in {\mathbb {N}}}\)and\({\widehat{W}}^x=(p!W^x_p)_{p\in {\mathbb {N}}}\).

Let\(\tau _1\in {{\mathscr {W}}}\)and\(\tau _2\in {{\mathscr {W}}}\)be the weight functions coming from Theorem 3applied to\(\sigma \)and\(\omega \)respectively, so\({\mathscr {A}}_{\{{\widehat{\varSigma }}\}}(S_\gamma )= {\mathscr {A}}_{\{\tau _1\}}(S_\gamma )\)and\({\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S_\gamma )= {\mathscr {A}}_{\{\tau _2\}}(S_\gamma )\).

Then, for every\(l>0\)there exists\(l_1>0\)such that there exists a linear and continuous map

$$\begin{aligned} E^{\tau _1,\tau _2}_{l}:\;\;\;\lambda \in \varLambda _{\tau _1,l}\mapsto f_{\lambda }\in {\mathscr {A}}_{\tau _2,l_1}(S_{\gamma }) \end{aligned}$$
(41)

such that for all\(\lambda \in \varLambda _{\tau _1,l}\)one has\({\mathscr {B}}\circ E^{\tau _1,\tau _2}_l(\lambda )={\mathscr {B}}(f_{\lambda })=\lambda \). Thus we have shown that\({\mathscr {B}}({\mathscr {A}}_{\{\tau _2\}}(S_{\gamma }))\supseteq \varLambda _{\{\tau _1\}}\).

Proof

Let \({\mathscr {T}}^i:=\{T^{i,x}: x>0\}\) be the weight matrix associated with the weight function \(\tau _i\), i.e. \(T^{i,x}_p:=\exp \left( \frac{1}{x}\varphi ^{*}_{\tau _i}(xp)\right) \) for each \(x>0\) and \(p\in {\mathbb {N}}\), \(i=1,2\).

We may apply (9) in order to deduce that \({\mathscr {A}}_{\{{\widehat{\varSigma }}\}}(S_\gamma )= {\mathscr {A}}_{\{{\mathscr {T}}^1\}}(S_\gamma )\) and \({\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S_\gamma )= {\mathscr {A}}_{\{{\mathscr {T}}^2\}}(S_\gamma )\) and, as it has already been remarked in [10, Corollary 7.6], we get \({\mathscr {T}}^1{\{\approx \}}{\widehat{\varSigma }}\), \({\mathscr {T}}^2{\{\approx \}}{\widehat{\varOmega }}\).

For the rest of the proof we follow [10, Corollary 7.6] and use for the extension Theorem 2. \(\square \)

Remark 14

Of course, by the assumption on the weights in Corollary 1, we could apply [10, Corollary 7.6] directly to the weights \(\omega \) and/or \(\sigma \) because both have \((\omega _1)\) which is equivalent to have \(\gamma (\omega ),\gamma (\sigma )>0\) as shown in [11, Corollary 2.14]. Then [10, Corollary 7.6] states that for every \(l>0\) there exists \(l_1>0\) such that there exists a linear and continuous map

$$\begin{aligned} E^{\tau _i}_{l}:\;\;\;\lambda \in \varLambda _{\tau _i,l}\mapsto f_{\lambda }\in {\mathscr {A}}_{\tau _i,l_1}(S_{\gamma }), \end{aligned}$$
(42)

with \(0<\gamma <\gamma (\sigma )\) for \(i=1\) and \(0<\gamma <\gamma (\omega )\) for \(i=2\). Since in the mixed setting we are interested in the situation where \(\sigma \) is (much) larger than \(\omega \), (42) yields in particular an extension \(\varLambda _{\tau _1,l}\subseteq \varLambda _{\tau _2,l}\mapsto f_{\lambda }\in {\mathscr {A}}_{\tau _2,l_1}(S_{\gamma })\), \(0<\gamma <\gamma (\omega )\). This should be compared with (41), where we have such a mixed extension for all \(0<\gamma <\gamma (\sigma ,\omega )\) and since \(\gamma (\sigma ,\omega )\ge \gamma (\omega )\) (see Lemma 3) even here we can have a situation which is not covered by [10, Corollary 7.6] since \({\mathscr {A}}_{\tau _2,l_1}(S_{\gamma '})\subseteq {\mathscr {A}}_{\tau _2,l_1}(S_{\gamma })\) for \(\gamma \le \gamma '\).

Finally we treat the mixed weight sequence situation which is generalizing [10, Remark 7.9]. Note that condition \(({\text {mg}})\) on the smaller weight sequence has also been used in [7] and [13, Theorem 5.9].

Theorem 4

Let\(M,N\in {\mathscr {LC}}\)be given such that\(\mu _p\le C\nu _p\), Msatisfies\(({\text {mg}})\)andfinally\(\gamma (M,N)>0\). We put\({\widehat{M}}=(p!M_p)_{p\in {\mathbb {N}}}\)and\({\widehat{N}}=(p!N_p)_{p\in {\mathbb {N}}}\)and let\(0<\gamma <\gamma (M,N)(=\gamma (\omega _M,\omega _N))\).

Then, there exists a constant\(k_1>0\)such that for every\(h>0\), one can construct a linear and continuous map

$$\begin{aligned} E^{M,N}_h:\;\;\lambda \in \varLambda _{{\widehat{M}},h}\mapsto f_{\lambda }\in {\mathscr {A}}_{{\widehat{N}},k_1h}(S_{\gamma }) \end{aligned}$$

such that for every\(\lambda \in \varLambda _{{\widehat{M}},h}\)one has\({\mathscr {B}}\circ E^{M,N}_h(\lambda )={\mathscr {B}}(f_{\lambda })=\lambda \). Thus we haveshown\({\mathscr {B}}({\mathscr {A}}_{\{{\widehat{N}}\}}(S_{\gamma }))\supseteq \varLambda _{\{{\widehat{M}}\}}\).

Proof

We apply Theorem 2 to \(\sigma \equiv \omega _M\) and \(\omega \equiv \omega _N\). Recall that by the assumptions on the weight sequences we have \(0<\gamma (M,N)=\gamma (\omega _M,\omega _N)\), see Lemma 4. Moreover the matrix \(\varSigma \) associated with \(\omega _M\) is constant, see (iii) in Lemma 1 and Remark 1, and hence the matrix \({\widehat{\varSigma }}\) is constant, too. Theorem 2 provides now for all \(h_1>0\) an extension map \(E_{h_1}:\lambda \in \varLambda _{{\widehat{S}}^{1/8},h_1}\mapsto f_{\lambda }\in {\mathscr {A}}_{{\widehat{W}}^1,k_0h_1}(S_{\gamma })\), \(k_0>0\) not depending on \(h_1\).

We know that \(S^1\equiv M\) (e.g. see the proof of [29, Thm. 6.4]) and so \(S^{1/8}{\approx } M\Leftrightarrow {\widehat{S}}^{1/8}{\approx }{\widehat{M}}\). Hence \(\varLambda _{{\widehat{S}}^{1/8}}=\varLambda _{{\widehat{M}}}\), more precisely there exists some \(D\ge 1\) such that for all \(h>0\) we get \(\varLambda _{{\widehat{M}},h}\subseteq \varLambda _{{\widehat{S}}^{1/8},Dh}\) and so we have shown the desired statement with universal \(k_1=k_0D\). \(\square \)

Mixed extension results with only one fixed weight

Extension results where the weight sequence/function defining the function space is fixed

Using the properties of the index \(\mu (N)\) and the construction of the ramified descendant of Sect. 3.2 we can now prove the following variant of Theorem 4.

Theorem 5

Let\(N\in {\mathscr {LC}}\)be given with\(\mu (N)>0\)and let\(0<r<\mu (N)\). Assume that (15) holds true for\(N^{1/r}\). Then there does exist\(L\in {\mathscr {LC}}\)having\(({\text {mg}})\)such that for each\(0<\gamma <r\)we get: There exists a constant\(k_1>0\)such that for every\(h>0\), one can construct a linear and continuous map

$$\begin{aligned} E^{L,N}_h:\;\;a\in \varLambda _{{\widehat{L}},h}\mapsto f_{a}\in {\mathscr {A}}_{{\widehat{N}},k_1h}(S_{\gamma }), \end{aligned}$$
(43)

denoting\({\widehat{L}}=(p!L_p)_{p\in {\mathbb {N}}}\)and\({\widehat{N}}=(p!N_p)_{p\in {\mathbb {N}}}\). Thus we have shown that\({\mathscr {B}}({\mathscr {A}}_{\{{\widehat{N}}\}}(S_{\gamma }))\supseteq \varLambda _{\{{\widehat{L}}\}}\).

The sequenceLis maximal among those\(M\in {\mathscr {LC}}\)satisfying\(\mu _k\le C\nu _k\)and\((M,N)_{\gamma _r}\).

The important difference between Theorem 4 and this result is that, of course, L is depending here on given r.

Proof

Let \(0<\gamma<r<\mu (N)\) be given according to the requirements above. Then we consider the sequence \(L^{N,r}\) defined via the descendant \(S^{N,r}\) in (14), see Sect. 3.2.

As seen there we have that \({(L^{N,r},N)_{\gamma _r}}\) holds true which proves \(\gamma (L^{N,r},N)\ge r>\gamma \). Moreover \(\lambda ^{N,r}_k\le C\nu _k\) and since \(N^{1/r}\) has (15), Lemma 6 yields \(({\text {mg}})\) for \(S^{N,r}\) and so for \(L^{N,r}\), too.

Thus we can apply Theorem 4 to \(M\equiv L^{N,r}\) and N and \(\gamma \) unchanged to obtain: There exists a constant \(k_1>0\) such that for every \(h>0\), one can construct a linear and continuous map \(E^{L^{N,r},N}_h:\;\;a\in \varLambda _{{\widehat{L}}^{N,r},h}\mapsto f_{a}\in {\mathscr {A}}_{{\widehat{N}},k_1h}(S_{\gamma })\) with \({\widehat{L}}^{N,r}=(p!L^{N,r}_p)_{p\in {\mathbb {N}}}\) and so (43) follows by taking \(L\equiv L^{N,r}\). \(\square \)

Remark 15

Let \(N\in {\mathscr {LC}}\) be given with \(\mu (N)>0\). If N has in addition \(({\text {mg}})\), then each \(S^{N,r}\) and \(L^{N,r}\), \(0<r<\mu (N)\), share this property, see (iv) in Remark 9.

If N does not have \(({\text {mg}})\), then (15) for \(N^{1/r}\) can only hold true for values \(\gamma (N)\le r<\mu (N)\): for \(0<r<\gamma (N)\) the right-hand side in (15) is bounded by definition, whereas \(\sup _{k\in {\mathbb {N}}}\frac{(\nu _{2k})^{1/r}}{(\nu _k)^{1/r}}=\infty \) since \(N^{1/r}\) does also not have \(({\text {mg}})\).

Using \(\mu (\omega )\) we can prove Theorem 5 for the weight function setting, so we have the following variant of Theorem 2.

Theorem 6

Let\(\omega \)be a normalized weight function with\((\omega _3)\)and\(\mu (\omega )>0\). Then for all\(0<r<\mu (\omega )\)there does exist a normalized weight function\(\sigma \)satisfying\((\omega _3)\)such that for each\(0<\gamma <r\)we get: There exists a constant\(k_0>0\)such that for every\(x>0\)and every\(h>0\), one can construct a linear and continuous map

$$\begin{aligned} E^{\sigma ,\omega }_h:\;\;\lambda \in \varLambda _{{\widehat{S}}^x,h}\mapsto f_{\lambda }\in {\mathscr {A}}_{{\widehat{W}}^{8x},k_0h}(S_{\gamma }) \end{aligned}$$

such that for every\(\lambda \in \varLambda _{{\widehat{S}}^x,h}\)one has\({\mathscr {B}}\circ E^{\sigma ,\omega }_h(\lambda )={\mathscr {B}}(f_{\lambda })=\lambda \). Thus we have shown that\({\mathscr {B}}({\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S_{\gamma })\supseteq \varLambda _{\{{\widehat{\varSigma }}\}}\)(by using for\({\widehat{\varSigma }}\)and\({\widehat{\varOmega }}\)the same notation as in Theorem2).

The function\(\sigma \)is chosen minimal among those normalized weight functions\(\tau \)satisfying\((\omega _3)\), \(\tau {\preceq }\omega \) (i.e.\(\omega (t)=O(\tau (t))\)) and enjoying\((\tau ,\omega )_{\gamma _r}\).

Proof

According to this value \(r>0\) given, we consider the weight \(\kappa ^{1/r}_{\omega ^r}\) (see (21)) and so \((\kappa ^{1/r}_{\omega ^r},\omega )_{\gamma _r}\) is valid. This weight enjoys all properties like \(\omega \) except normalization (by definition). But normalization can be achieved w.l.o.g. by switching to an equivalent weight (redefining \(\kappa ^{1/r}_{\omega ^r}\) near 0, e.g. see [3, Remark 1.2 (b)]) which will be denoted by \(\sigma \).

Thus \(\gamma (\sigma ,\omega )\ge r>\gamma \) and we can apply Theorem 2 to these weights \(\sigma \) and \(\omega \) and the value \(\gamma \) and conclude. \(\square \)

Remark 16

Due to (22) one could try to restate Theorem 6 by applying Theorem 5 to \(N\equiv W^x\), \(x>0\) arbitrary. However, once chosen \(\gamma <\mu (\omega )=\mu (W^x)\) in Theorem 6 we obtain an extension for another weight function \(\sigma \) such that moving the index x we are staying in the same weight matrix associated with \(\sigma \) by the precise choice \(x\mapsto 8x\). So here we can take some uniform choice for all sequences in \(\varOmega \) (by obtaining a weight matrix not depending on given x) which is not following by applying Theorem 5.

Remark 17

Naturally one might ask what happens in the dual situation, that is, fixing the weight sequence or weight function that controls the derivatives at the origin. However, in this case the inverse construction concerning the descendant, called the predecessor, see [23, Remark 4.3], does not provide any new information, since the bounds for the opening are the same as those that are known for the one level extension theorem.

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Acknowledgements

J. Jiménez-Garrido and J. Sanz are partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under the project MTM2016-77642-C2-1-P. J. Jiménez-Garrido has been partially supported by the University of Valladolid through a Predoctoral Fellowship (2013 call) co-sponsored by the Banco de Santander. G. Schindl is supported by the Austrian research fund (FWF) Project J 3948-N35, as a part of which he has been an external researcher at the Universidad de Valladolid (Spain) for the period October 2016 - December 2018.

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Communicated by Mieczyslaw Mastylo.

An example for having \(\gamma (M,N),\gamma (\omega _M,\omega _N)>0\) but \(\gamma (\omega _M)=\gamma (M)=\gamma (N)=\gamma (\omega _N)=0\)

An example for having \(\gamma (M,N),\gamma (\omega _M,\omega _N)>0\) but \(\gamma (\omega _M)=\gamma (M)=\gamma (N)=\gamma (\omega _N)=0\)

The aim is to construct an explicit example such that Theorem 2 resp. Theorem 4 are valid but no further known extension result construct can be applied and so the situation in this present work is not covered by the approach from [10] (and also not from [14, 32]). We will construct a pair of sufficient weight sequences M and N and apply Theorem 4 to M and N directly, whereas Theorem 2 is applied to their associated weights \(\omega _M\equiv \sigma \) and \(\omega _N\equiv \omega \).

First recall that \((\gamma _1)\) can be written in a different form. We say that M has \((\beta _1)\), if

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

and \((\beta _3)\) (see [28] and [4]), if

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

By [21, Proposition 1.1] condition \((\beta _1)\) is equivalent to \((\gamma _1)\) for log-convex M and for this proof condition \((\text {nq})\), which is a general assumption in [21], was not necessary. We have that M satisfies \((\beta _1)\) if and only if m satisfies \((\beta _3)\), for more precise information concerning this relation we refer to the recent work [11].

The construction of the sequences is based on a generalization of [16, Example 3.3] and we introduce M by using the quotient sequence \((\mu _k)_{k\ge 0}\) (with \(\mu _0:=1\) and set \(M_k:=\prod _{i=0}^k\mu _i\)).

Lemma 13

Let \(\gamma >1\) be given, then there exists a sequence \(M\in {\mathscr {LC}}\) such that

  1. (i)

    Mdoes satisfy\(({\text {nq}})\), more precisely\(\mu _k\ge k^{\gamma }\)for all\(k\in {\mathbb {N}}\)and so even\(({\text {nq}}_{\gamma -\varepsilon })\)holds true for any\(\varepsilon >0\)(small),

  2. (ii)

    \(\mu _k\le k^{\gamma (2\gamma -1)}\)for all\(k\in {\mathbb {N}}\),

  3. (iii)

    M does not satisfy \((\beta _3)\) or equivalently \({\widehat{M}}=(p!M_p)_{p\in {\mathbb {N}}}\) does not satisfy \((\beta _1)\) (and consequently M is not strongly nonquasianalytic too),

  4. (iv)

    Mdoes satisfy\(({\text {mg}})\).

Proof

Let \(\gamma >1\) be given and for convenience we introduce also numbers \(\alpha \) and \(\beta \) by

$$\begin{aligned} \gamma>1,\beta :=2\gamma (>2),\alpha :=\beta -1=2\gamma -1(>\gamma >1). \end{aligned}$$

In [16, Example 3.3] the choices \(\gamma =2\), \(\beta =4\) and \(\alpha =3\) have been considered (but there (ii) has not been shown).

We define recursively two nondecreasing sequences \((c_n)_{n\ge 1}\) and \((d_n)_{n\ge 1}\) (of natural numbers) as follows and \(\lfloor a\rfloor \) shall denote the lower integer part of a given real number \(a>0\). We put

$$\begin{aligned} c_1:=1,d_n:=\lfloor c_n^{\alpha /\gamma }\rfloor +1,c_{n+1}:=\lfloor d_n^{\gamma }\rfloor +1, \end{aligned}$$

and introduce the weight sequence as follows:

$$\begin{aligned} \mu _k:= {\left\{ \begin{array}{ll} c_n^{\alpha } \quad \text {for}\;c_n\le k\le \lfloor c_n^{\alpha /\gamma }\rfloor =d_n-1\\ \frac{k^{\beta }}{d_n^{\gamma }} \quad \text {for}\;d_n\le k\le \lfloor d_n^{\gamma }\rfloor =c_{n+1}-1. \square \\ \end{array}\right. } \end{aligned}$$

(i) and (ii) together tell us that M is lying between two Gevrey sequences. By proving (i) one can verify that \(\mu (M)=\liminf _{p\rightarrow \infty }\frac{\log (\mu _p)}{\log (p)}=\gamma \) holds true.

A slight variation of Lemma 13 yields the following.

Lemma 14

Let \(\gamma >1\) be given, then there exists a sequence \(M\in {\mathscr {LC}}\) such that

  1. (i)

    Mdoes satisfy\(({\text {nq}})\), more precisely\(\mu _k\ge k^{\gamma }\)for all\(k\in {\mathbb {N}}\)and so even\(({\text {nq}}_{\gamma -\varepsilon })\)holds true for any\(\varepsilon >0\) (small),

  2. (ii)

    \(\mu _k\le k^{2\gamma ^2}\)for all\(k\in {\mathbb {N}}\),

  3. (iii)

    M does not satisfy \((\beta _3)\) or equivalently \({\widehat{M}}=(p!M_p)_{p\in {\mathbb {N}}}\) does not satisfy \((\beta _1)\) (and consequently M is not strongly nonquasianalytic too),

  4. (iv)

    Mdoes not satisfy\(({\text {mg}})\).

Proof

We use the same definition for \((\mu _k)_k\) as before. Let \(\gamma >1\) be given and set now

$$\begin{aligned} \gamma>1,\beta :=2\gamma +1(>3),\alpha :=\beta -1=2\gamma (>\gamma >1). \end{aligned}$$

In fact any choice \(\beta >2\gamma \) and \(\alpha :=\beta -1\) would be working for the following proof. \(\square \)

Again by construction \(\mu (M)=\liminf _{k\rightarrow \infty }\frac{\log (\mu _k)}{\log (k)}=\gamma \) holds true.

Using Lemma 13 we can now underline the importance of Theorem 2 and in particular of Theorem 4 as follows.

Theorem 7

There do exist sequencesMandNsatisfying all requirements from Theorem4but such that\(\gamma (M)=\gamma (N)=\gamma (\omega _M)=\gamma (\omega _N)=0\). Moreover we can achieve\(\gamma (M,N)\)to be as large as desired.

Proof

We define M and N in terms of their quotients \((\mu _p)_p\) and \((\nu _p)_p\) coming from Lemma 13 with parameters \(\gamma '\) and \(\gamma \) respectively and we require that

$$\begin{aligned} \gamma '(2\gamma '-1)\le \gamma \Leftrightarrow \frac{\gamma '(2\gamma '-1)-\gamma }{r}\le 0. \end{aligned}$$
(44)

Choosing \(1<\gamma '<\gamma \) subject to (44) it is straightforward to prove \((M,N)_{\gamma _r}\) for all \(0<r<\gamma \). This implies \(\gamma (M,N)\ge \gamma >0\) and since \(\mu (N)=\gamma \) we get equality by Lemma 5. Since \(\gamma >1\) can be chosen arbitrary large, \(\gamma (M,N)=\gamma \) can be as large as desired. Moreover one can easily verify \(\mu _k\le \nu _k\), hence Lemma 4 implies \(\gamma (\omega _M,\omega _N)=\gamma (M,N)=\gamma \).

But \(\gamma (M)=\gamma (N)=0\) holds true: \((\beta _1)\) or equivalently \((\gamma _1)\) is violated for both sequences \({\widehat{M}}=(p!M_p)_p\) and \({\widehat{N}}=(p!N_p)_p\) (by property (iii)), and so \(\gamma (M)=\gamma (N)=0\). And this is equivalent to having \(\gamma (\omega _{M})=\gamma (\omega _{N})=0\), because both sequences have \(({\text {mg}})\), for a proof see [11, Section 4]. In particular we have seen that neither \({\widehat{M}}\in {\mathscr {SR}}\) nor \({\widehat{N}}\in {\mathscr {SR}}\) and by the characterizations shown in [21], not any to \({\widehat{M}}\) or \({\widehat{N}}\) equivalent sequence L can belong to class \(\mathscr {SR}\). \(\square \)

Let M and N denote the sequences from Theorem 7 above with parameters \(\gamma '>1\) and \(\gamma >1\) subject to (44). Then by applying Theorem 4 for any given \(0<\delta <\gamma (M,N)\) there is \(k_1>0\) such that for every \(h>0\) there exists a continuous linear extension map

$$\begin{aligned} E^{M,N}_h:\;\;\lambda \in \varLambda _{{\widehat{M}},h}\mapsto f_{\lambda }\in {\mathscr {A}}_{{\widehat{N}},k_1h}(S_{\delta }). \end{aligned}$$

Thus we have shown \({\mathscr {B}}({\mathscr {A}}_{\{{\widehat{N}}\}}(S_{\delta }))\supseteq \varLambda _{\{{\widehat{M}}\}}\).

This kind of extension result is not covered by the theory developed by the authors in [10]. More precisely [10, Theorem 7.4] fails since \(\gamma (\omega _M)=\gamma (\omega _N)=0\) and also the mixed setting from [10, Section 7.1] cannot be applied, neither to M nor to N directly.

Note that both M and N have \(({\text {mg}})\), thus both matrices associated with \(\omega _M\) and \(\omega _N\) are constant, see (iii) in Lemma 1 and Remark 1. Now let M and N be the sequences constructed in Lemma 14 with parameters \(\gamma '\) and \(\gamma \) respectively and here we require that

$$\begin{aligned} 2(\gamma ')^2\le \gamma . \end{aligned}$$

Again it is straightforward to check that \((M,N)_{\gamma _r}\) holds true for all \(0<r<\gamma \) which implies \(\gamma (M,N)\ge \gamma >0\). Since \(\mu (N)=\gamma \) we again have \(\gamma (M,N)=\gamma \) and by having \(\mu _p\le \nu _p\), Lemma 4 yields \(\gamma (\omega _M,\omega _N)\ge \gamma (M,N)=\gamma \).

But here neither M nor N does satisfy \(({\text {mg}})\) and we cannot apply Theorem 4 directly. But Theorem 2 applied to \(\sigma \equiv \omega _M\) and \(\omega \equiv \omega _N\) with \(\varSigma \) denoting the matrix associated with \(\omega _M\) and \(\varOmega \) the matrix associated with \(\omega _N\), yields now the following extension result:

For any given \(0<\delta <\gamma (\omega _M,\omega _N)\) there exists a constant \(k_0>0\) such that for every \(x>0\) and every \(h>0\), one can construct a linear and continuous extension map

$$\begin{aligned} E^{\omega _M,\omega _N}_h:\;\;\lambda \in \varLambda _{{\widehat{S}}^x,h}\mapsto f_{\lambda }\in {\mathscr {A}}_{{\widehat{W}}^{8x},k_0h}(S_{\gamma }). \end{aligned}$$

Hence we have shown \({\mathscr {B}}({\mathscr {A}}_{\{{\widehat{\varOmega }}\}}(S_{\gamma }))\supseteq \varLambda _{\{{\widehat{\varSigma }}\}}\).

Note: Here, since \(({\text {mg}})\) is avoided, the associated matrices \(\varSigma \) and \(\varOmega \), and hence \({\widehat{\varSigma }}\) and \({\widehat{\varOmega }}\), are nonconstant but \(S^1\equiv M\) and \(W^1\equiv N\) (see the arguments given in the proof of Theorem 4). Also in this situation we get \(\gamma (M)=\gamma (N)=0\) (by property (iii)) but it is not clear if also \(\gamma (\omega _M)=\gamma (\omega _N)=0\) (see [11, Corollary 4.6]). In such a situation, even if \(\gamma (\omega _M)>0\) and/or \(\gamma (\omega _N)>0\), we obtain new information, see Remark 14 above.

As mentioned in the introduction and in Remark 12 we have that starting directly with a Braun–Meise–Taylor weight function \(\omega \) with \(\gamma (\omega )=0\) we do not have \((\omega _1)\) (as shown in [11, Corollary 2.14]). Hence a basic assumption in the whole theory of ultradifferentiable functions defined in terms of \(\omega \), is violated from the very beginning.

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Jiménez-Garrido, J., Sanz, J. & Schindl, G. Ultraholomorphic extension theorems in the mixed setting. Banach J. Math. Anal. (2020). https://doi.org/10.1007/s43037-020-00073-y

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Keywords

  • Spaces of ultraholomorphic functions
  • Weight sequences and weight functions
  • Growth indices
  • (non)quasianalyticity of function classes
  • Restriction and extension mappings
  • Surjectivity of the Borel map

Mathematics Subject Classification

  • 26E10
  • 30D60
  • 46A13
  • 46E10