# Nonlinear conditions for ultradifferentiability

David Nicolas Nenning, Armin Rainer, and Gerhard Schindl  Fakultät für
Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A remarkable theorem of Joris states that a function $f$ is $C^{\infty}$ if
two relatively prime powers of $f$ are $C^{\infty}$. Recently, Thilliez showed
that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We
prove that a division property, equivalent to Joris’s result, is valid in a
wide variety of ultradifferentiable classes. Generally speaking, it holds in
all dimensions for non-quasianalytic classes. In the quasianalytic case we
have general validity in dimension one, but we also get validity in all
dimensions for certain quasianalytic classes.

###### Key words and phrases:

Joris theorem, division property, ultradifferentiable classes,
(non-)quasianalytic, holomorphic approximation, almost analytic extension

###### 2020 Mathematics Subject Classification:

26E10, 30E10, 32W05, 46E10, 46E25, 58C25

AR was supported by FWF-Project P 32905-N, DNN and GS by FWF-Project P 33417-N

##  1\. Introduction

A remarkable theorem of Joris [11, Théorème 2] states: if
$f:\mathbb{R}\rightarrow\mathbb{R}$ is a function and $p,q$ are relatively
prime positive integers, then

$f^{p},f^{q}\in C^{\infty}\implies f\in C^{\infty}.$ (1)

Since smoothness can be tested along smooth curves by a theorem of Boman [3],
one immediately infers that the implication (1) holds on arbitrary open
subsets of $\mathbb{R}^{d}$, $d\geq 1$, and on smooth manifolds. (On the other
hand, the regularity of a _single_ power of a function generally says nothing
about the regularity of the function itself; e.g.
$(\mathbf{1}_{\mathbb{Q}}-\mathbf{1}_{\mathbb{R}\backslash\mathbb{Q}})^{2}=1$,
where $\mathbf{1}_{A}$ is the indicator function of a set $A$.)

It was soon realized that the statement also holds for complex valued
functions and it led to the study of so-called pseudoimmersions [7, 12, 13,
19]. A simple proof based on ring theory was given by [1].

Only recently Thilliez [30] showed that Joris’s result carries over to
Denjoy–Carleman classes of Roumieu type $\mathcal{E}^{\\{M\\}}$. These are
ultradifferentiable classes of smooth functions defined by certain growth
properties imposed upon the sequence of iterated derivatives in terms of a
weight sequence $M$ (which in view of the Cauchy estimates measures the
deviation from analyticity).

By extracting the essence of Thilliez’s proof, we show in this paper that a
broad variety of ultradifferentiable classes has a division property
equivalent to Joris’s result. Let $\mathcal{S}$ be a subring (with
multiplicative identity) of the ring of germs at $0\in\mathbb{R}^{d}$ of
complex valued $C^{\infty}$-functions. We say that $\mathcal{S}$ has the
_division property $(\mathscr{D})$_ if for any function germ $f$ at
$0\in\mathbb{R}^{d}$ we have

$\big{(}j\in\mathbb{N}_{\geq 1},~{}f^{j},f^{j+1}\in\mathcal{S}\big{)}\implies
f\in\mathcal{S}.$ (2)

If $\mathcal{S}$ has property $(\mathscr{D})$, then Joris’s theorem holds in
$\mathcal{S}$. Indeed, suppose that $p_{1},p_{2}$ are relatively prime
positive integers and $f^{p_{1}},f^{p_{2}}\in\mathcal{S}$. All integers $j\geq
p_{1}p_{2}$ can be written $j=a_{1}p_{1}+a_{2}p_{2}$ for
$a_{1},a_{2}\in\mathbb{N}$, see [11, p.270]. Hence $f^{j}\in\mathcal{S}$ for
all $j\geq p_{1}p_{2}$. Since two consecutive integers are relatively prime,
also the converse holds.111In an earlier version of the paper we considered
the division property $\big{(}j\in\mathbb{N}_{\geq
1},~{}g,q,fg\in\mathcal{S},~{}f^{j}=qg\big{)}\implies f\in\mathcal{S}$ which
resulted from our wish to prove an ultradifferentiable version of the division
theorem [13, Theorem 1]. But this property is equivalent to (2).

###  1.1. Results

Let us give an overview of our results.

The rings of germs in _one dimension_ $d=1$ of the following
ultradifferentiable classes have property $(\mathscr{D})$:

  * •

$\mathcal{E}^{[M]}$, Denjoy–Carleman class of Roumieu (Theorems 2.2 and 2.3)
and Beurling type (Theorem 2.6),

  * •

$\mathcal{E}^{[\omega]}$, Braun–Meise–Taylor classes of Roumieu and Beurling
type (Theorem 3.1),

  * •

$\mathcal{E}^{[\mathfrak{M}]}$, ultradifferentiable classes defined by weight
matrices of Roumieu and Beurling type (Theorem 4.2).

It is understood that certain minimal regularity properties of the weights are
assumed (see Table 1) which in particular guarantee that the sets of germs are
indeed rings. (Note that by convention $[\cdot]$ stands for $\\{\cdot\\}$,
i.e., Roumieu, as well as $(\cdot)$, i.e., Beurling.)

Interestingly, the proof in one dimension works for quasianalytic and non-
quasianalytic classes alike. But the tool used to reduce the multidimensional
to the one-dimensional statement is only available in the non-quasianalytic
Roumieu case ([15], [27]). The (multidimensional) Beurling case can often be
reduced to the corresponding Roumieu case. Hence we obtain the following
multidimensional non-quasianalytic results. The rings of germs in _all
dimensions_ $d$ of the following ultradifferentiable classes have property
$(\mathscr{D})$:

  * •

$\mathcal{E}^{[M]}$, non-quasianalytic Denjoy–Carleman class of Roumieu
(Theorem 2.2) and Beurling type (Theorem 2.5),

  * •

$\mathcal{E}^{[\omega]}$, non-quasianalytic Braun–Meise–Taylor classes of
Roumieu and Beurling type (Theorem 3.2),

  * •

$\mathcal{E}^{\\{\mathfrak{M}\\}}$, non-quasianalytic ultradifferentiable
classes defined by weight matrices of Roumieu type (Theorem 4.3).

For _quasianalytic_ Denjoy–Carleman classes of Roumieu type
$\mathcal{E}^{\\{M\\}}$ in _one_ dimension the implication (2) follows from
the stronger result, due to Thilliez [29], that $C^{\infty}$-solutions of a
polynomial equation

$z^{n}+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_{n}=0,$ (3)

where the coefficients $a_{j}$ are germs at $0\in\mathbb{R}$ of
$\mathcal{E}^{\\{M\\}}$-functions, are of class $\mathcal{E}^{\\{M\\}}$ (under
weak assumptions on $M$). This is false for non-quasianalytic classes. But it
seems to be unknown whether, in the presence of quasianalyticity, it holds in
higher dimensions. In fact, quasianalytic ultradifferentiability cannot be
tested on quasianalytic curves (or lower dimensional plots) even if the
function in question is known to be smooth ([10, 20]).

Hence we think that it is interesting that, combining our proof with a
description of certain quasianalytic classes $\mathcal{E}^{\\{M\\}}$ as an
intersection of suitable non-quasianalytic ones (due to [16]), we obtain that
these quasianalytic classes have property $(\mathscr{D})$ in _all_ dimensions
(see Theorem 2.7 and also Remarks 3.3 and 4.4).

Since all considered regularity classes are local, the results for germs
immediately give corresponding results for functions on open sets.

###  1.2. Summary of the results

We list in Table 1 the ultradifferentiable rings of germs known to have
property $(\mathscr{D})$, together with the needed assumptions on the weights
and the respective references. All germs are function germs at $0$ in
$\mathbb{R}^{d}$ for some dimension $d$. The dimension is added as a left
subscript, e.g., $\tensor[_{d}]{\mathcal{E}}{{}^{[M]}}$ denotes the ring of
germs at $0\in\mathbb{R}^{d}$ of $\mathcal{E}^{[M]}$-functions. All notions
will be defined below.

The Roumieu parts of the results in the first and the fifth row are due to
Thilliez [30]; see LABEL:, 2.4 and 2.5.

Table 1. Ultradifferentiable rings of germs having property $(\mathscr{D})$ Quasianalytic ring of germs | Reference
---|---
$\tensor[_{1}]{\mathcal{E}}{{}^{[M]}}$ |  | derivation closed
---
$m$ log-convex
$m_{k}^{1/k}\to\infty$ (Beurling)
Theorems 2.3 and 2.6
$\tensor[_{1}]{\mathcal{E}}{{}^{[\omega]}}$ |  |  $\omega$ concave
---
Theorem 3.1
$\tensor[_{1}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$ |  | [regular]
---
[moderate growth]
Theorem 4.2
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$ |  | intersectable
---
moderate growth
Theorem 2.7
Non-quasianalytic ring of germs | Reference
$\tensor[_{d}]{\mathcal{E}}{{}^{[M]}}$ |  | moderate growth
---
$m$ log-convex
Theorems 2.2 and 2.5
$\tensor[_{d}]{\mathcal{E}}{{}^{[\omega]}}$ |  |  $\omega$ concave
---
Theorem 3.2
$\tensor[_{1}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$ |  | [regular]
---
[moderate growth]
Theorem 4.2
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{\mathfrak{M}\\}}}$ |  | $\\{\text{regular}\\}$
---
$\\{\text{moderate growth}\\}$
Theorem 4.3

We remark that non-quasianalytic Denjoy–Carleman classes
$\mathcal{E}^{\\{M\\}}$, where the weight sequence $M$ lacks moderate growth,
do not have property $(\mathscr{D})$ in general; see [30, Remark 2.2.3]. The
moderate growth condition is rather restrictive (e.g., it implies that the
class $\mathcal{E}^{\\{M\\}}$ is contained in a Gevrey class). The
consideration of the classes $\mathcal{E}^{[\omega]}$ and
$\mathcal{E}^{[\mathfrak{M}]}$ allows to overcome this restriction in the
sense that the implication (2) holds under weaker moderate growth conditions.

###  1.3. Strategy of the proof

Thilliez’s proof of Joris’s theorem for $\mathcal{E}^{\\{M\\}}$ consists of
the following two steps:

  1. (i)

The class $\mathcal{E}^{\\{M\\}}$ admits a description by holomorphic
approximation which is based on a result of Dynkin [8] on almost analytic
extensions and a related $\overline{\partial}$-problem.

  2. (ii)

If $f^{j}$, $f^{j+1}$ are of class $\mathcal{E}^{\\{M\\}}$ and
$g_{\varepsilon}$, $h_{\varepsilon}$ are respective holomorphic
approximations, the quotient $h_{\varepsilon}/g_{\varepsilon}$ is a naive
candidate for a holomorphic approximation of $f$. In order to avoid small
divisors one considers

$u_{\varepsilon}=\varphi_{\varepsilon}\frac{\overline{g}_{\varepsilon}h_{\varepsilon}}{\max\\{|g_{\varepsilon}|,r_{\varepsilon}\\}^{2}},$

where $\varphi_{\varepsilon}$ is a suitable cutoff function and
$r_{\varepsilon}>0$. For good choices of $r_{\varepsilon}$ the function
$u_{\varepsilon}$ has uniform bounds and is close to $f$. The solution of a
$\overline{\partial}$-problem is used to modify $u_{\varepsilon}$ in order to
obtain a holomorphic approximation of $f$. By step (i) we may conclude that
$f$ belongs to $\mathcal{E}^{\\{M\\}}$.

Following the same strategy, we will work with weight matrices $\mathfrak{M}$,
since they provide a framework for ultradifferentiability (Section 4) which
encompasses Denjoy–Carleman classes (Section 2) and Braun–Meise–Taylor classes
(Section 3). In Section 5 we prove a general characterization result by
holomorphic approximation for $\mathcal{E}^{[\mathfrak{M}]}$ (Theorem 5.3)
which extends step (i); it builds on the description by almost analytic
extension presented in our recent paper [9]. Then we execute a version of step
(ii) under a quite minimal set of assumptions, see Lemma 6.1. It enables us to
easily deduce the main results in Section 6.

##  2\. Denjoy–Carleman classes have property $(\mathscr{D})$

###  2.1. Weight sequences and Denjoy–Carleman classes

Let $\mu=(\mu_{k})$ be a positive increasing (i.e. $\mu_{k}\leq\mu_{k+1}$)
sequence with $\mu_{0}=1$. We define a sequence $M$ by setting
$M_{k}:=\mu_{1}\cdots\mu_{k}$, $M_{0}:=1$, and a sequence $m$ by
$m_{k}:=\frac{M_{k}}{k!}$. Clearly, $\mu$ uniquely determines $M$ and $m$, and
vice versa. In analogy we shall use sequences $N\leftrightarrow
n\leftrightarrow\nu$, $L\leftrightarrow\ell\leftrightarrow\lambda$, etc.

That $\mu$ is increasing means that $M$ is _log-convex_ , i.e., $\log M$ is
convex or, equivalently, $M_{k}^{2}\leq M_{k-1}M_{k+1}$ for all $k$. If in
addition $M_{k}^{1/k}\rightarrow\infty$, we say that $M$ is a _weight
sequence_.

Sometimes we will make the stronger assumption that $m$ is log-convex.

For $\sigma>0$ and open $U\subseteq\mathbb{R}^{d}$, one defines the Banach
space

$\mathcal{B}^{M}_{\sigma}(U):=\Big{\\{}f\in
C^{\infty}(U):~{}\|f\|_{\sigma,U}^{M}:=\sup_{x\in
U,\,\alpha\in\mathbb{N}^{d}}\frac{|\partial^{\alpha}f(x)|}{\sigma^{|\alpha|}M_{|\alpha|}}<\infty\Big{\\}}$

and the (local) _Denjoy–Carleman classes of Roumieu type_

$\mathcal{E}^{\\{M\\}}(U):=\operatorname{proj}_{V\Subset
U}\operatorname{ind}_{\sigma>0}\mathcal{B}^{M}_{\sigma}(V).$

For later reference we also consider the global class
$\mathcal{B}^{\\{M\\}}(V):=\operatorname{ind}_{\sigma>0}\mathcal{B}^{M}_{\sigma}(V)$.
Replacing the existential quantifier for $\sigma$ by a universal quantifier,
we find the _Denjoy–Carleman classes of Beurling type_

$\mathcal{E}^{(M)}(U):=\operatorname{proj}_{V\Subset
U}\operatorname{proj}_{\sigma>0}\mathcal{B}^{M}_{\sigma}(V)$

and
$\mathcal{B}^{(M)}(V):=\operatorname{proj}_{\sigma>0}\mathcal{B}^{M}_{\sigma}(V)$.
We use the notation $\mathcal{E}^{[M]}$ for both $\mathcal{E}^{\\{M\\}}$ and
$\mathcal{E}^{(M)}$, similarly for $\mathcal{B}^{[M]}$, etc.

For positive sequences $M,N$, we write $M\preccurlyeq N$ if
$\sup_{k\in\mathbb{N}}\big{(}\frac{M_{k}}{N_{k}}\big{)}^{1/k}<\infty$ and
$M\lhd N$ if
$\lim_{k\rightarrow\infty}\big{(}\frac{M_{k}}{N_{k}}\big{)}^{1/k}=0$. We have
(cf. [23, Proposition 2.12])

$\displaystyle M\preccurlyeq N\quad$
$\displaystyle\Leftrightarrow\quad\mathcal{E}^{[M]}(U)\subseteq\mathcal{E}^{[N]}(U),$
$\displaystyle M\lhd N\quad$
$\displaystyle\Leftrightarrow\quad\mathcal{E}^{\\{M\\}}(U)\subseteq\mathcal{E}^{(N)}(U),$

where for “$\Leftarrow$” one has to assume that $M$ is a weight sequence. Note
that $\mathcal{E}^{\\{(k!)\\}}(U)$ coincides with $C^{\omega}(U)$ and so the
class of real analytic functions is contained in
$\mathcal{E}^{(M)}$($\subseteq\mathcal{E}^{\\{M\\}}$) if and only if
$m_{k}^{1/k}\to\infty$.

Log-convexity of $M$ implies that $\mathcal{E}^{[M]}(U)$ is closed under
pointwise multiplication of functions. Additional regularity properties for
$M$ endow $\mathcal{E}^{[M]}(U)$ with additional structure, e.g., log-
convexity of $m$ implies closedness under composition of functions. A crucial
assumption in [30] is _moderate growth_ of $M$, which reads as follows

$\exists C>0~{}\forall k,j\in\mathbb{N}:~{}M_{k+j}\leq C^{k+j}M_{k}M_{j}.$ (4)

It implies _derivation closedness_

$\exists C>0~{}\forall k\in\mathbb{N}:~{}M_{k+1}\leq C^{k+1}M_{k}.$ (5)

The last property we need to mention is _non-quasianalyticity_ of $M$, that is

$\sum_{k=1}^{\infty}\frac{1}{\mu_{k}}<\infty,\quad\text{ or equivalently
}\quad\sum_{k=1}^{\infty}\frac{1}{M_{k}^{1/k}}<\infty.$ (6)

By the Denjoy–Carleman theorem, this condition is equivalent to the existence
of non-trivial functions with compact support in $\mathcal{E}^{[M]}(U)$. It is
well-known that non-quasianalyticity implies $m_{k}^{1/k}\rightarrow\infty$.

Let $\tensor[_{d}]{\mathcal{E}}{{}^{[M]}}$ denote the _ring of germs_ at
$0\in\mathbb{R}^{d}$ of complex valued $\mathcal{E}^{[M]}$-functions; here we
assume that $M$ is a weight sequence in order to have a ring.

######  Remark 2.1.

There is a slight mismatch between our notation (also used in [9]) and that of
[30] (and [22]). We write $M_{j}=m_{j}j!$ for weight sequences, so our $m$
corresponds to $M$ in [30].

###  2.2. Associated functions

Let $m=(m_{k})$ be a positive sequence with $m_{0}=1$ and
$m_{k}^{1/k}\to\infty$. We define the function

$h_{m}(t):=\inf_{k\in\mathbb{N}}m_{k}t^{k},\quad\text{ for }t>0,\quad\text{
and }\quad h_{m}(0):=0,$ (7)

which is is increasing, continuous on $[0,\infty)$, and positive for $t>0$.
For large $t$ we have $h_{m}(t)=1$. Furthermore, we need

$\displaystyle\overline{\Gamma}_{m}(t)$
$\displaystyle:=\min\\{k:h_{m}(t)=m_{k}t^{k}\\},\quad t>0,$ (8)

and, provided that $m_{k+1}/m_{k}\to\infty$,

$\displaystyle\underline{\Gamma}_{m}(t)$
$\displaystyle:=\min\Big{\\{}k:\frac{m_{k+1}}{m_{k}}\geq\frac{1}{t}\Big{\\}},\quad
t>0.$ (9)

We trivially have $\underline{\Gamma}_{m}\leq\overline{\Gamma}_{m}$. If $m$ is
log-convex, then $\overline{\Gamma}_{m}=\underline{\Gamma}_{m}$.

We shall use these functions for $m_{k}=M_{k}/k!$, where $M$ is a weight
sequence satisfying $m_{k}^{1/k}\to\infty$. Then $m_{k+1}/m_{k}\to\infty$
(since $M_{k}^{1/k}\leq\mu_{k}$ for all $k$).

###  2.3. Regular weight sequences

A weight sequence $M$ is said to be _regular_ if $m_{k}^{1/k}\to\infty$, $M$
is derivation closed, and there exists a constant $C\geq 1$ such that
$\overline{\Gamma}_{m}(Ct)\leq\underline{\Gamma}_{m}(t)$ for all $t>0$.

###  2.4. Denjoy–Carleman classes of Roumieu type have property
$(\mathscr{D})$

######  Theorem 2.2 (Non-quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$).

Let $M$ be a non-quasianalytic regular weight sequence of moderate growth.
Then $\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$ has property $(\mathscr{D})$.

This is a special case of Theorem 4.3 below (cf. Section 4.5). It implies
Thilliez’s result [30, Corollary 2.2.5].

A quasianalytic one-dimensional version follows from a stronger result in
[29]:

######  Theorem 2.3 (Quasianalytic
$\tensor[_{1}]{\mathcal{E}}{{}^{\\{M\\}}}$).

Let $M$ be a quasianalytic derivation closed weight sequence such that $m$ is
log-convex. Then $\tensor[_{1}]{\mathcal{E}}{{}^{\\{M\\}}}$ has property
$(\mathscr{D})$.

###  2.5. Denjoy–Carleman classes of Beurling type have property
$(\mathscr{D})$

Let us deduce Beurling versions of Theorems 2.2 and 2.3. We use the following
lemma based on [14, Lemma 6] and [9, Lemma 7.5].

######  Lemma 2.4.

Let $L,M$ be positive sequences satisfying $L\lhd M$. Suppose that $m$ is log-
convex and satisfies $m_{k}^{1/k}\to\infty$. Then there exists a weight
sequence $S$ such that $s$ is log-convex, $s_{k}^{1/k}\to\infty$, and $L\leq
S\lhd M$. Additionally, we may assume:

  1. (i)

$S$ has moderate growth, if $M$ has moderate growth.

  2. (ii)

$S$ is derivation closed, if $M$ is derivation closed.

  3. (iii)

$S$ is non-quasianalytic, if $M$ is non-quasianalytic.

###### Proof.

Only the supplements (ii) and (iii) were not already proved in [9, Lemma 7.5].

(ii) follows from the fact that a weight sequence $M$ is derivation closed if
and only if there is a constant $C\geq 1$ such that $M_{k}\leq C^{k^{2}}$ for
all $k$, see [17, 18]. Since $S$ is a weight sequence and $S\lhd M$, also $S$
is derivation closed, by this criterion.

(iii) It suffices to show that there exists a non-quasianalytic weight
sequence $N$ such that $L\leq N\lhd M$. Then we apply the lemma to $N\lhd M$
and obtain a weight sequence $S$ with $N\leq S\lhd M$ having all desired
properties.

Let us show the existence of $N$. By $L\lhd M$, we have
$\beta_{k}:=\sup_{p\geq k}\big{(}\frac{L_{p}}{M_{p}}\big{)}^{1/p}\searrow 0$.
Applying [6, Lemme 16] (see also [28, Lemma 4.1]) to $\beta_{k}$ and
$\alpha_{k}=\gamma_{k}:=\frac{1}{\mu_{k}}$, yields an increasing sequence
$\delta=(\delta_{k})$ such that

$\displaystyle\delta_{k}\rightarrow\infty,$ (10)
$\displaystyle\delta_{k}\beta_{k}\rightarrow 0,$ (11)
$\displaystyle\frac{\mu_{k}}{\delta_{k}}\text{ is increasing},$ (12)
$\displaystyle\sum_{k=1}^{\infty}\frac{\delta_{k}}{\mu_{k}}\leq
8\delta_{1}\sum_{k=1}^{\infty}\frac{1}{\mu_{k}}<\infty.$ (13)

Then $N_{k}:=\frac{\mu_{1}\cdots\mu_{k}}{\delta_{1}\cdots\delta_{k}}$ defines
a non-quasianalytic weight sequence, by (12) and (13). (Note that
$\nu_{k}=\frac{\mu_{k}}{\delta_{k}}\to\infty$ is equivalent to
$N_{k}^{1/k}\to\infty$.) It satisfies $N\lhd M$ by (10). By (11), there is a
constant $C>0$ such that
$\delta_{k}\big{(}\frac{L_{k}}{M_{k}}\big{)}^{1/k}\leq C$ for all $k$. By the
monotonicity of $\delta$, this leads to $L_{k}\leq
C^{k}\frac{M_{k}}{\delta_{k}^{k}}\leq
C^{k}\frac{M_{k}}{\delta_{1}\cdots\delta_{k}}=C^{k}N_{k}$. After replacing
$(N_{k})$ by $(C^{k}N_{k})$ we have $L\leq N\lhd M$. ∎

######  Theorem 2.5 (Non-quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{(M)}}$).

Let $M$ be a non-quasianalytic weight sequence of moderate growth such that
$m$ is log-convex. Then $\tensor[_{d}]{\mathcal{E}}{{}^{(M)}}$ has property
$(\mathscr{D})$.

###### Proof.

Suppose that $g:=f^{j},h:=f^{j+1}\in\tensor[_{d}]{\mathcal{E}}{{}^{(M)}}$ for
some positive integer $j$. Assume that representatives of these germs are
defined in the neighborhood of the closure of some bounded $0$-neighborhood
$U$; we denote the representatives by the same symbols. Then the sequence

$L_{k}:=\max\Big{\\{}\sup_{|\alpha|=k,\,x\in
U}|g^{(\alpha)}(x)|,\sup_{|\alpha|=k,\,x\in U}|h^{(\alpha)}(x)|\Big{\\}}$ (14)

satisfies $L\lhd M$. By Lemma 2.4, there exists a weight sequence $S$
satisfying the assumptions of Theorem 2.2 and $L\leq S\lhd M$. Thus,
$f\in\tensor[_{d}]{\mathcal{E}}{{}^{\\{S\\}}}\subseteq\tensor[_{d}]{\mathcal{E}}{{}^{(M)}}$.
∎

######  Theorem 2.6 (Quasianalytic $\tensor[_{1}]{\mathcal{E}}{{}^{(M)}}$).

Let $M$ be a quasianalytic derivation closed weight sequence such that $m$ is
log-convex and $m_{k}^{1/k}\to\infty$. Then
$\tensor[_{1}]{\mathcal{E}}{{}^{(M)}}$ has property $(\mathscr{D})$.

###### Proof.

This follows from the proof of Theorem 2.3 in [29] (which also works in the
Beurling case). Alternatively, we may infer it from Theorem 2.3 by a reduction
argument based on Lemma 2.4 as in the proof of Theorem 2.5. ∎

###  2.6. A multidimensional quasianalytic result

Let $M$ be a weight sequence and consider the sequence space

$\Lambda^{\\{M\\}}:=\Big{\\{}(c_{k})\in\mathbb{C}^{\mathbb{N}}:\;\exists\rho>0:\sup_{k\in\mathbb{N}}\frac{|c_{k}|}{\rho^{k}M_{k}}<\infty\Big{\\}}.$

We call a quasianalytic weight sequence $M$ _intersectable_ if

$\Lambda^{\\{M\\}}=\bigcap_{N\in\mathcal{L}(M)}\Lambda^{\\{N\\}},$ (15)

where $\mathcal{L}(M)$ is the collection of all non-quasianalytic weight
sequences $N\geq M$ such that $n$ is log-convex. The identity (15) carries
over to respective function spaces, since
$\mathcal{B}^{\\{M\\}}(U)=\big{\\{}f\in C^{\infty}(U):(\sup_{x\in
U}\|f^{(k)}(x)\|_{L^{k}_{\operatorname{sym}}})\in\Lambda^{\\{M\\}}\big{\\}}$,
where $f^{(k)}$ denotes the $k$-th order Fréchet derivative and
$\|\cdot\|_{L^{k}_{\operatorname{sym}}}$ the operator norm.

Note that a quasianalytic intersectable weight sequence $M$ always satisfies
$m_{k}^{1/k}\to\infty$; an argument is given in Remark 2.8 below.

######  Theorem 2.7 (Quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$).

Let $M$ be a quasianalytic intersectable weight sequence of moderate growth.
Then $\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$ has property $(\mathscr{D})$.

The proof of this result is given in Section 6.

######  Remark 2.8.

In [16, Theorem 1.6] (inspired by [2]) a sufficient condition for
intersectability was given. Let $M$ be a quasianalytic weight sequence with
$1\leq M_{0}<M_{1}$. Consider the sequence $\check{M}$ defined by

$\check{M}_{k}:=M_{k}\prod_{j=1}^{k}\Big{(}1-\frac{1}{M_{j}^{1/j}}\Big{)}^{k},\quad\check{M}_{0}:=1.$

If $\check{m}$ is log-convex, then $M$ is intersectable.

Not every quasianalytic weight sequence is intersectable, for instance,

$\Lambda^{\\{(k!)\\}}\neq\bigcap_{N\in\mathcal{L}((k!))}\Lambda^{\\{N\\}}=\Lambda^{\\{Q\\}},\quad\text{
where }Q_{k}=(k\log(k+e))^{k});$

see [16, Theorem 1.8] and [26]. Every quasianalytic intersectable weight
sequence $M$ must satisfy $\Lambda^{\\{Q\\}}\subseteq\Lambda^{\\{M\\}}$, and
so $m_{k}^{1/k}$ tends to $\infty$ since clearly $q_{k}^{1/k}$ does.

A countable family $\mathbf{Q}=\\{Q^{n}\\}_{n\in\mathbb{N}_{\geq 1}}$ of
quasianalytic intersectable weight sequences of moderate growth was
constructed in [16, Theorem 1.9]:

$Q^{n}_{k}=\big{(}k\log(k)\log(\log(k))\cdots\log^{[n]}(k)\big{)}^{k},\quad\text{
for }k\geq\exp^{[n]}(1),$

where $\log^{[n]}$ denotes the $n$-fold composition of $\log$; analogously for
$\exp^{[n]}$.

See also [27, Section 11] for a generalization of this concept.

##  3\. Braun–Meise–Taylor classes have property $(\mathscr{D})$

###  3.1. Weight functions and Braun–Meise–Taylor classes

A _weight function_ is, by definition, a continuous increasing function
$\omega:[0,\infty)\rightarrow[0,\infty)$ such that

  * $(\omega_{1})$

$\omega(2t)=O(\omega(t))$ as $t\rightarrow\infty$,

  * $(\omega_{2})$

$\omega(t)=o(t)$ as $t\rightarrow\infty$,

  * $(\omega_{3})$

$\log(t)=o(\omega(t))$ as $t\rightarrow\infty$,

  * $(\omega_{4})$

$t\mapsto\omega(e^{t})=:\varphi_{\omega}(t)$ is convex on $[0,\infty)$.

One may assume that $\omega|_{[0,1]}\equiv 0$ (without changing the associated
classes $\mathcal{E}^{[\omega]}$) which we shall tacitly do if convenient.

Let $U\subseteq\mathbb{R}^{d}$ be open and $\rho>0$. We associate the Banach
space

$\mathcal{B}^{\omega}_{\rho}(U):=\Big{\\{}f\in
C^{\infty}(U):\|f\|^{\omega}_{\rho,U}:=\sup_{x\in
U,\,\alpha\in\mathbb{N}^{d}}\frac{|\partial^{\alpha}f(x)|}{e^{\varphi_{\omega}^{*}(\rho|\alpha|)/\rho}}<\infty\Big{\\}},$

where $\varphi_{\omega}^{*}(s):=\sup_{t\geq 0}\\{st-\varphi_{\omega}(t)\\}$ is
the Young conjugate of $\varphi_{\omega}$ (which is finite by $(\omega_{3})$).
Then the (local) _Braun–Meise–Taylor class of Roumieu type_ is

$\mathcal{E}^{\\{\omega\\}}(U):=\operatorname{proj}_{V\Subset
U}\operatorname{ind}_{n\in\mathbb{N}}\mathcal{B}^{\omega}_{n}(V),$

and that of _Beurling type_ is

$\mathcal{E}^{(\omega)}(U):=\operatorname{proj}_{V\Subset
U}\operatorname{proj}_{n\in\mathbb{N}}\mathcal{B}^{\omega}_{\frac{1}{n}}(V).$

Again we use $\mathcal{E}^{[\omega]}$ for $\mathcal{E}^{\\{\omega\\}}$ and
$\mathcal{E}^{(\omega)}$, similarly for $\mathcal{B}^{[\omega]}$ etc.

For two weight functions $\omega$, $\sigma$ we have (cf. [23, Corollary 5.17])

$\displaystyle\sigma(t)=O(\omega(t))\text{ as }t\to\infty\quad$
$\displaystyle\Leftrightarrow\quad\mathcal{E}^{[\omega]}(U)\subseteq\mathcal{E}^{[\sigma]}(U),$
$\displaystyle\sigma(t)=o(\omega(t))\text{ as }t\to\infty\quad$
$\displaystyle\Leftrightarrow\quad\mathcal{E}^{\\{\omega\\}}(U)\subseteq\mathcal{E}^{(\sigma)}(U).$

We say that $\omega$ and $\sigma$ are _equivalent_ if they generate the same
classes, i.e., $\sigma(t)=O(\omega(t))$ and $\omega(t)=O(\sigma(t))$ as
$t\to\infty$.

A weight function is said to be _non-quasianalytic_ if

$\int_{1}^{\infty}\frac{\omega(t)}{t^{2}}\,dt<\infty.$ (16)

This is the case if and only if $\mathcal{E}^{[\omega]}(U)$ contains non-
trivial functions of compact support (cf. [5] or [22]).

Let us emphasize that in this paper we treat condition $(\omega_{2})$ as a
general assumption for weight functions; it means that the Beurling class
$\mathcal{E}^{(\omega)}$ contains the real analytic class. It is automatically
satisfied if $\omega$ is non-quasianalytic.

Let $\tensor[_{d}]{\mathcal{E}}{{}^{[\omega]}}$ denote the _ring of germs_ at
$0\in\mathbb{R}^{d}$ of complex valued $\mathcal{E}^{[\omega]}$-functions;
note that $\mathcal{E}^{[\omega]}$ is stable by multiplication of functions
for any weight function $\omega$.

###  3.2. The associated weight matrix

Let $\omega$ be a weight function. Setting
$\Omega^{x}_{k}:=e^{\varphi_{\omega}^{*}(xk)/x}$ defines a weight sequence
$\Omega^{x}$ for every $x>0$, where $\Omega^{x}\leq\Omega^{y}$ if $x\leq y$.
Thus the collection $\mathbf{\Omega}:=\\{\Omega^{x}\\}_{x>0}$ is a weight
matrix (in the sense of Section 4). Note that $\mathbf{\Omega}$ satisfies a
_mixed_ moderate growth property, namely

$\;\forall x>0\;\forall
j,k\in\mathbb{N}:\Omega^{x}_{j+k}\leq\Omega^{2x}_{j}\Omega^{2x}_{k}.$ (17)

The importance of the associated weight matrix $\mathbf{\Omega}$ is that it
encodes an equivalent topological description of the spaces
$\mathcal{E}^{[\omega]}(U)$ as unions or intersections of Denjoy–Carleman
classes; see Section 4.5. All this can be found in [22].

###  3.3. Braun–Meise–Taylor classes have property $(\mathscr{D})$

######  Theorem 3.1 ($\tensor[_{1}]{\mathcal{E}}{{}^{[\omega]}}$).

Let $\omega$ be a concave weight function. Then
$\tensor[_{1}]{\mathcal{E}}{{}^{[\omega]}}$ has property $(\mathscr{D})$.

Evidently, it suffices to assume that $\omega$ is equivalent to a concave
weight function.

For the multidimensional analogue we additionally assume non-quasianalyticity.

######  Theorem 3.2 (Non-quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{[\omega]}}$).

Let $\omega$ be a non-quasianalytic concave weight function. Then
$\tensor[_{d}]{\mathcal{E}}{{}^{[\omega]}}$ has property $(\mathscr{D})$.

Theorems 3.1 and 3.2 are corollaries of Theorems 4.2 and 4.3 below; for the
proofs see Section 6.

######  Remark 3.3.

Every weight sequence $M$ in the family $\mathbf{Q}$ mentioned at the end of
Remark 2.8 satisfies

$\liminf_{k\to\infty}\frac{\mu_{ak}}{\mu_{k}}>1$

for some positive integer $a$. Hence there is a quasianalytic weight function
$\omega_{M}$ (take, e.g., $\omega_{M}(t):=-\log h_{M}(1/t)$) such that
$\tensor[_{d}]{\mathcal{E}}{{}^{[\omega_{M}]}}=\tensor[_{d}]{\mathcal{E}}{{}^{[M]}}$,
by [4, Theorem 14]. So for all $M\in\mathbf{Q}$, the quasianalytic ring
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{\omega_{M}\\}}}$ has property
$(\mathscr{D})$.

##  4\. The most general version of the theorem

Let us formulate the main theorems in the most general setting available. The
conditions we put on abstract weight matrices are tailored in such a way that
weight matrices associated with weight functions are contained as special
cases.

###  4.1. Weight matrices and ultradifferentiable classes

A _weight matrix_ $\mathfrak{M}$ is, by definition, a family of weight
sequences which is totally ordered with respect to the pointwise order
relation on sequences, i.e.,

  * •

$\mathfrak{M}\subseteq\mathbb{R}^{\mathbb{N}}$,

  * •

each $M\in\mathfrak{M}$ is a weight sequence in the sense of Section 2.1,

  * •

for all $M,N\in\mathfrak{M}$ we have $M\leq N$ or $M\geq N$.

Let $U\subseteq\mathbb{R}^{d}$ be open. Given a weight matrix $\mathfrak{M}$,
we define global classes

$\displaystyle\mathcal{B}^{\\{\mathfrak{M}\\}}(U)$
$\displaystyle:=\operatorname{ind}_{M\in\mathfrak{M}}\mathcal{B}^{\\{M\\}}(U),$
(18) $\displaystyle\mathcal{B}^{(\mathfrak{M})}(U)$
$\displaystyle:=\operatorname{proj}_{M\in\mathfrak{M}}\mathcal{B}^{(M)}(U).$
(19)

The limits in (18) and (19) can always be assumed countable, as is shown in
[9, Lemma 2.5]. Writing $[\cdot]$ for $\\{\cdot\\}$ and $(\cdot)$, the local
classes are defined by

$\mathcal{E}^{[\mathfrak{M}]}(U):=\operatorname{proj}_{V\Subset
U}\mathcal{B}^{[\mathfrak{M}]}(V).$

Let $\tensor[_{d}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$ denote the _ring of
germs_ at $0\in\mathbb{R}^{d}$ of complex valued
$\mathcal{E}^{[\mathfrak{M}]}$-functions; notice that
$\mathcal{E}^{[\mathfrak{M}]}$ is stable by multiplication of functions, since
each $M\in\mathfrak{M}$ is a weight sequence.

###  4.2. Regular weight matrices

A weight matrix $\mathfrak{M}$ satisfying

  * •

$m_{k}^{1/k}\to\infty$ for all $M\in\mathfrak{M}$

is called $\\{\text{\it regular}\\}$ or _R-regular_ (for Roumieu) if

  * •

$\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}\;\exists C\geq 1\;\forall
j\in\mathbb{N}:M_{j+1}\leq C^{j+1}N_{j}$,

  * •

$\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}\;\exists C\geq 1\;\forall
t>0:\overline{\Gamma}_{n}(Ct)\leq\underline{\Gamma}_{m}(t)$,

and $(\text{\it regular})$ or _B-regular_ (for Beurling) if

  * •

$\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}\;\exists C\geq 1\;\forall
j\in\mathbb{N}:N_{j+1}\leq C^{j}M_{j}$,

  * •

$\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}\;\exists C\geq 1\;\forall
t>0:\overline{\Gamma}_{m}(Ct)\leq\underline{\Gamma}_{n}(t)$.

Moreover, $\mathfrak{M}$ is called _regular_ if it is both R- and B-regular.
By our convention, $[\text{regular}]$ stands for $\\{\text{regular}\\}$ (i.e.
R-regular) in the Roumieu case and $(\text{regular})$ (i.e. B-regular) in the
Beurling case.

###  4.3. Almost analytic extensions

Let $h:(0,\infty)\to(0,1]$ be an increasing continuous function which tends to
$0$ as $t\to 0$. Let $\rho>0$ and let $U\subseteq\mathbb{R}^{d}$ be a bounded
open set. We say that a function $f:U\to\mathbb{C}$ admits an _$(h,\rho)$
-almost analytic extension_ if there is a function $F\in
C^{1}_{c}(\mathbb{C}^{d})$ and a constant $C\geq 1$ such that $F|_{U}=f$ and

$|\overline{\partial}F(z)|\leq C\,h(\rho d(z,\overline{U})),\quad\text{ for
}z\in\mathbb{C}^{d},$

where $\overline{\partial}F(z):=\sum_{j=1}^{d}\frac{\partial
F(z)}{\partial\overline{z}_{j}}d\overline{z}_{j}$ and
$d(z,\overline{U}):=\inf_{x\in\overline{U}}|z-x|$ denotes the distance of $z$
to $\overline{U}$.

Let us apply this definition to the functions $h_{m}$ from (7), where
$m_{k}=M_{k}/k!$ and $M$ belongs to a given weight matrix $\mathfrak{M}$. Let
$f:U\to\mathbb{C}$ be a function.

  * •

$f$ is called _$\\{\mathfrak{M}\\}$ -almost analytically extendable_ if it has
an $(h_{m},\rho)$-almost analytic extension for some $M\in\mathfrak{M}$ and
some $\rho>0$.

  * •

$f$ is called _$(\mathfrak{M})$ -almost analytically extendable_ if, for all
$M\in\mathfrak{M}$ and all $\rho>0$, there is an $(h_{m},\rho)$-almost
analytic extension of $f$.

######  Theorem 4.1 ([9, Corollaries 3.3, 3.5]).

Let $\mathfrak{M}$ be a [regular] weight matrix. Let
$U\subseteq\mathbb{R}^{d}$ be open. Then $f\in\mathcal{E}^{[\mathfrak{M}]}(U)$
if and only if $f|_{V}$ is $[\mathfrak{M}]$-almost analytically extendable for
each quasiconvex domain $V$ relatively compact in $U$.

In Section 5 we shall use [9, Proposition 3.12], which is a key ingredient of
the proof of Theorem 4.1.

###  4.4. Weight matrices of moderate growth

For positive sequences $M$, $N$ set

$\operatorname{mg}(M,N):=\sup_{j,k\geq 0,\,j+k\geq
1}\left(\frac{M_{j+k}}{N_{j}N_{k}}\right)^{1/(j+k)}\in(0,\infty].$ (20)

We say that a weight matrix $\mathfrak{M}$ has _R-moderate growth_ or
_$\\{\text{moderate growth}\\}$_ if

$\;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{M}:\operatorname{mg}(M,N)<\infty,$ (21)

and _B-moderate growth_ or _$(\text{moderate growth})$_ if

$\;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{M}:\operatorname{mg}(N,M)<\infty.$ (22)

Again we say that $\mathfrak{M}$ has _moderate growth_ if it has R- and
B-moderate growth, and [moderate growth] stands for $\\{\text{moderate
growth}\\}$ and $(\text{moderate growth})$, respectively.

###  4.5. Denjoy–Carleman and Braun–Meise–Taylor classes in this framework

By definition, Denjoy–Carleman classes are described by weight matrices
$\mathfrak{M}=\\{M\\}$ consisting of a single weight sequence $M$. Observe
that the weight matrix $\mathfrak{M}=\\{M\\}$ is regular if and only if the
weight sequence $M$ is regular, and it has moderate growth if and only if $M$
has moderate growth.

Let $\omega$ be a weight function and let $\mathbf{\Omega}$ be the associated
weight matrix (cf. Section 3.2). Then, by [22, Corollaries 5.8 and 5.15], as
locally convex spaces

$\mathcal{E}^{[\omega]}(U)=\mathcal{E}^{[\mathbf{\Omega}]}(U),$

and $\mathcal{E}^{[\omega]}(U)=\mathcal{E}^{[\Omega]}(U)$ for all
$\Omega\in\mathbf{\Omega}$ if and only if

$\exists H\geq 1~{}\forall t\geq 0:2\omega(t)\leq\omega(Ht)+H,$

which is in turn equivalent to the fact that some (equivalently each)
$\Omega\in\mathbf{\Omega}$ has moderate growth; see also [4].

The associated weight matrix $\mathbf{\Omega}$ always has moderate growth, by
(17). It is equivalent to a regular weight matrix $\mathfrak{S}$ (that means
$\mathcal{E}^{[\omega]}=\mathcal{E}^{[\mathfrak{S}]}$) if and only if $\omega$
is equivalent to a concave weight function. In fact, a
$\mathcal{E}^{[\omega]}$-version of the almost analytic extension theorem 4.1
holds if and only if $\omega$ is equivalent to a concave weight function; see
[9, Theorem 4.8]. The weight matrix $\mathfrak{S}=\\{S^{x}\\}_{x>0}$ has the
property that for each $x>0$ the sequence $s^{x}$ is log-convex and satisfies
$\operatorname{mg}(s^{x},s^{2x})=:H<\infty$ and thus $h_{s^{x}}(t)\leq
h_{s^{2x}}(Ht)^{2}$ for all $x,t>0$; see [25, Proposition 3].

###  4.6. General ultradifferentiable classes have property $(\mathscr{D})$

######  Theorem 4.2 ($\tensor[_{1}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$).

Let $\mathfrak{M}$ be a [regular] weight matrix of [moderate growth]. Then
$\tensor[_{1}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$ has property
$(\mathscr{D})$.

The proof is given in Section 6. It builds upon a characterization of the
class $\mathcal{E}^{[\mathfrak{M}]}$ by holomorphic approximation; see Section
5.

We may infer a multidimensional result, since non-quasianalytic
$\mathcal{E}^{\\{\mathfrak{M}\\}}$-regularity can be tested along curves; this
useful tool is available in a satisfactory manner only in the non-
quasianalytic Roumieu setting. We need two additional properties of the weight
matrix:

$\;\exists M\in\mathfrak{M}:~{}\sum_{k=0}^{\infty}\frac{1}{\mu_{k}}<\infty.$
(23)

which means that $\mathcal{E}^{\\{\mathfrak{M}\\}}$ admits non-trivial
functions of compact support, and

$\;\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}:m^{\circ}\preccurlyeq
n,$ (24)

where $m_{k}^{\circ}:=\max\\{m_{j}m_{\alpha_{1}}\cdots
m_{\alpha_{j}}:\alpha_{i}\in\mathbb{N}_{>0},\,\alpha_{1}+\dots+\alpha_{j}=k\\}$.
Condition (24) is equivalent to composition closedness of
$\mathcal{E}^{\\{\mathfrak{M}\\}}$ (which follows from the arguments in [22,
Theorem 4.9]) and is satisfied by every R-regular weight matrix. Indeed, if
$\mathfrak{M}$ is R-regular, then $\mathcal{E}^{\\{\mathfrak{M}\\}}$ has a
description by almost analytic extension, by Theorem 4.1. It is easy to see
(cf. [9, Proposition 1.1]) that the latter condition is preserved by
composition of functions.

Under these assumptions, a function $f$ defined on an open set
$U\subseteq\mathbb{R}^{d}$ is of class $\mathcal{E}^{\\{\mathfrak{M}\\}}$ if
and only if $f\circ c$ is of class $\mathcal{E}^{\\{\mathfrak{M}\\}}$ for all
$\mathcal{E}^{\\{\mathfrak{M}\\}}$-curves in $U$; see [15, 16] and [27,
Theorem 10.7.1].

######  Theorem 4.3 (Non-quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{\mathfrak{M}\\}}}$).

Let $\mathfrak{M}$ be an R-regular weight matrix of R-moderate growth
satisfying (23). Then $\tensor[_{d}]{\mathcal{E}}{{}^{\\{\mathfrak{M}\\}}}$
has property $(\mathscr{D})$.

###### Proof.

This follows immediately from Theorem 4.2 and the above observations. ∎

Note that Theorem 4.3 implies Theorem 2.2 as a special case.

######  Remark 4.4.

The family $\mathbf{Q}=\\{Q^{n}\\}_{n\in\mathbb{N}_{\geq 1}}$ of quasianalytic
intersectable weight sequences referred to at the end of Remark 2.8 actually
is a regular weight matrix of moderate growth. The Roumieu class
$\mathcal{E}^{\\{\mathbf{Q}\\}}$ is quasianalytic and, since Theorem 2.7
applies to every $M\in\mathbf{Q}$, we conclude that
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{\mathbf{Q}\\}}}$ has property
$(\mathscr{D})$.

Note that there is no weight sequence $M$ with
$\mathcal{E}^{\\{M\\}}=\mathcal{E}^{\\{\mathbf{Q}\\}}$ and no weight function
$\omega$ with $\mathcal{E}^{\\{\omega\\}}=\mathcal{E}^{\\{\mathbf{Q}\\}}$.
This follows from the fact that $Q^{n}\leq Q^{n+1}\not\preccurlyeq Q^{n}$, in
analogy to the proof given in [22, Theorem 5.22]; see also Remark 5.25 there.

##  5\. Holomorphic Approximation of functions in
$\mathcal{E}^{[\mathfrak{M}]}$

In this section we prove a characterization of the class
$\mathcal{E}^{[\mathfrak{M}]}$ (in dimension one) by holomorphic
approximation. It generalizes [30, Proposition 3.3.2].

For notational convenience, we set $\|f\|_{A}:=\sup_{z\in A}|f(z)|$ for any
complex valued function $f$, where $A$ is any set in the domain of $f$.

###  5.1. Some preparatory observations

######  Lemma 5.1.

Let $M,N$ be weight sequences satisfying $m_{k}^{1/k}\to\infty$,
$n_{k}^{1/k}\to\infty$, and $C:=\operatorname{mg}(M,N)<\infty$. Then

$\displaystyle h_{m}(t)$ $\displaystyle\leq C^{j}n_{j}t^{j}h_{n}(Ct),\quad
t>0,~{}j\in\mathbb{N},$ (25) $\displaystyle h_{m}(t)$ $\displaystyle\leq
h_{n}\Big{(}\frac{eC}{2}t\Big{)}^{2},\quad t>0.$ (26)

###### Proof.

Note that $\operatorname{mg}(m,n)\leq\operatorname{mg}(M,N)$. Thus, for all
$j\in\mathbb{N}$ and $t>0$,

$\displaystyle h_{m}(t)$ $\displaystyle\leq\inf_{k\geq
0}m_{k+j}t^{k+j}\leq\inf_{k\geq
0}n_{j}n_{k}(Ct)^{k+j}=C^{j}n_{j}t^{j}h_{n}(Ct).$

For (26) we refer to [24, Lemma 3.13]. ∎

For $\varepsilon>0$ let $\Omega_{\varepsilon}$ denote the interior of the
ellipse in $\mathbb{C}$ with vertices $\pm\cosh(\varepsilon)$ and co-vertices
$\pm i\sinh(\varepsilon)$. By $\mathcal{H}(\Omega_{\varepsilon})$ we denote
the space of holomorphic functions on $\Omega_{\varepsilon}$. The following
lemma is a simple modification of [30, Lemma 3.2.4].

######  Lemma 5.2.

Let $M,N$ be two weight sequences satisfying $m_{k}^{1/k}\to\infty$,
$n_{k}^{1/k}\to\infty$, and $C:=\operatorname{mg}(M,N)<\infty$. Let
$\varepsilon>0$. Let $g\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ and assume that there are constants
$L,a_{1},a_{2}>0$ such that

$\|g\|_{\Omega_{\varepsilon}}\leq L,\quad\|g\|_{[-1,1]}\leq
a_{1}h_{m}(a_{2}\varepsilon).$

Then with $a_{3}:=\max\\{a_{1},L\\}$ and $a_{4}:=eCa_{2}$ we have

$\|g\|_{\Omega_{\varepsilon/2}}\leq a_{3}h_{n}(a_{4}\varepsilon).$

###### Proof.

Let $f(z):=\frac{1}{a_{1}}g(\sin(\varepsilon z))$. Since
$z\mapsto\sin(\varepsilon z)$ maps the horizontal strip
$S:=\\{z\in\mathbb{C}:~{}|\operatorname{Im}(z)|<1\\}$ to
$\Omega_{\varepsilon}$, we get that $f\in\mathcal{H}(S)\cap
C^{0}(\overline{S})$ is bounded by $K:=\max\\{1,\frac{L}{a_{1}}\\}$ on the
whole of $S$ and by $h_{m}(a_{2}\varepsilon)$ on $\mathbb{R}$. Thus an
application of Hadamard’s three lines theorem gives

$|f(z)|\leq
h_{m}(a_{2}\varepsilon)^{1-|\operatorname{Im}(z)|}K^{|\operatorname{Im}(z)|},\quad
z\in S.$

Since $h_{m}\leq 1$ and every $w\in\Omega_{\varepsilon/2}$ can be written as
$w=\sin(\varepsilon z)$ for some $w\in S$ with $|\operatorname{Im}(w)|\leq
1/2$, we obtain

$|g(w)|\leq a_{1}(Kh_{m}(a_{2}\varepsilon))^{1/2}.$

The statement follows from (26). ∎

###  5.2. Condition $(\mathcal{P}_{[\mathfrak{M}]})$

Let $\mathfrak{M}$ be a weight matrix.

  1. $(\mathcal{P}_{\\{\mathfrak{M}\\}})$

We say that a function $f:[-1,1]\to\mathbb{C}$ satisfies
$(\mathcal{P}_{\\{\mathfrak{M}\\}})$ if there exist $M\in\mathfrak{M}$,
constants $K,c_{1},c_{2}>0$, and a family
$(f_{\varepsilon})_{0<\varepsilon\leq\varepsilon_{0}}$ of functions
$f_{\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that for all
$0<\varepsilon\leq\varepsilon_{0}$,

$\displaystyle\|f_{\varepsilon}\|_{\Omega_{\varepsilon}}$ $\displaystyle\leq
K,$ (27) $\displaystyle\|f-f_{\varepsilon}\|_{[-1,1]}$ $\displaystyle\leq
c_{1}h_{m}(c_{2}\varepsilon).$ (28)

  2. $(\mathcal{P}_{(\mathfrak{M})})$

We say that a function $f:[-1,1]\to\mathbb{C}$ satisfies
$(\mathcal{P}_{(\mathfrak{M})})$ if for all $M\in\mathfrak{M}$ and all
$c_{2}>0$ there exist constants $K,c_{1}>0$ and a family
$(f_{\varepsilon})_{0<\varepsilon\leq\varepsilon_{0}}$ of functions
$f_{\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that (27) and (28) hold for all
$0<\varepsilon\leq\varepsilon_{0}$.

Note that $(\mathcal{P}_{\\{\mathfrak{M}\\}})$ generalizes condition
$(\mathcal{P}_{M})$ of [30].

###  5.3. Description by holomorphic approximation

######  Theorem 5.3.

i Let $M^{(i)}$, $1\leq i\leq 3$, be weight sequences with
$(m^{(i)}_{k})^{1/k}\rightarrow\infty$ and

$\displaystyle\;\exists B_{1}\geq 1\;\forall
t>0:\overline{\Gamma}_{m^{(2)}}(B_{1}t)\leq\underline{\Gamma}_{m^{(1)}}(t),$
(29) $\displaystyle\;\exists B_{2}\geq 1\;\forall
j\in\mathbb{N}:m^{(2)}_{j+1}\leq B_{2}^{j+1}m^{(3)}_{j}.$ (30)

Then for each $f\in\mathcal{B}^{M^{(1)}}_{B_{0}}((-1,1))$ there exist positive
constants $K,c_{1},c_{2}$ and functions
$f_{\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that for all small
$\varepsilon>0$

$\|f_{\varepsilon}\|_{\Omega_{\varepsilon}}\leq
K,\quad\|f-f_{\varepsilon}\|_{[-1,1]}\leq c_{1}h_{m^{(3)}}(c_{2}\varepsilon).$
(31)

The constants $K,c_{1},c_{2}$ only depend on $B_{i}$, in particular,
$c_{2}=CB_{0}B_{1}$, where $C$ is an absolute constant.

ii Let $N^{(i)}$, $1\leq i\leq 3$, be weight sequences with
$(n^{(i)}_{k})^{1/k}\rightarrow\infty$ and
$\operatorname{mg}(N^{(i)},N^{(i+1)})=D^{(i)}<\infty$. Let
$f:[-1,1]\to\mathbb{C}$ be a function. Assume that there exist positive
constants $K,c_{1},c_{2}$ and functions
$f_{\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that for all small
$\varepsilon>0$

$\|f_{\varepsilon}\|_{\Omega_{\varepsilon}}\leq
K,\quad\|f-f_{\varepsilon}\|_{[-1,1]}\leq c_{1}h_{n^{(1)}}(c_{2}\varepsilon).$
(32)

Then $f\in\mathcal{B}^{N^{(3)}}_{\sigma}((-b,b))$ for every $b<1$, where
$\sigma:=\frac{2eD^{(1)}D^{(2)}c_{2}}{E(1-b)}$ and $E$ is an absolute
constant.

iii If $\mathfrak{M}$ is a [regular] weight matrix of [moderate growth], then

$f\in\mathcal{B}^{[\mathfrak{M}]}((-1,1))\Rightarrow
f~{}\text{satisfies}~{}(\mathcal{P}_{[\mathfrak{M}]})\Rightarrow
f\in\mathcal{E}^{[\mathfrak{M}]}((-1,1)).$ (33)

Note that [regularity] of $\mathfrak{M}$ is needed for the first implication
in (iii), [moderate growth] for the second. Item (iii) generalizes [30,
Proposition 3.3.2].

###### Proof.

We follow closely the proof of [30, Proposition 3.3.2].

(i) Let $f\in\mathcal{B}^{M^{(1)}}_{B_{0}}((-1,1))$. By [9, Proposition 3.12],
there are constants $c_{1},c_{2}>0$ and a function $F\in
C^{1}_{c}(\mathbb{C})$ extending $f$ such that

$|\overline{\partial}F(z)|\leq c_{1}h_{m^{(3)}}(c_{2}d(z,[-1,1])),\quad
z\in\mathbb{C}.$ (34)

Note that $c_{1}=c_{1}(\|f\|^{M^{(1)}}_{B_{0}},B_{0},B_{1},B_{2})$ and
$c_{2}=12B_{0}B_{1}$. Then
$w_{\varepsilon}:=\overline{\partial}F\,\mathbf{1}_{\Omega_{\varepsilon}}$
satisfies

$\|w_{\varepsilon}\|_{\mathbb{C}}\leq c_{1}h_{m^{(3)}}(Cc_{2}\varepsilon),$

where $C>0$ is an absolute constant such that $d(z,[-1,1])\leq C\varepsilon$
for $z\in\Omega_{\varepsilon}$. Moreover, the bounded continuous function

$v_{\varepsilon}(z):=\frac{1}{2\pi
i}\int_{\mathbb{C}}\frac{w_{\varepsilon}(\zeta)}{\zeta-z}\,d\zeta\wedge
d\overline{\zeta}$

satisfies $\overline{\partial}v_{\varepsilon}=w_{\varepsilon}$ in the
distributional sense, and we have

$\|v_{\varepsilon}\|_{\mathbb{C}}\leq c_{1}h_{m^{(3)}}(Cc_{2}\varepsilon).$
(35)

So $f_{\varepsilon}:=F-v_{\varepsilon}$ is holomorphic on
$\Omega_{\varepsilon}$ and continuous on $\overline{\Omega}_{\varepsilon}$.
The estimates (34) and (35) easily imply (31).

(ii) Let $f:[-1,1]\to\mathbb{C}$ satisfy (32). Consider
$g_{\varepsilon}:=f_{\varepsilon}-f_{2\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$. Then
$\|g_{\varepsilon}\|_{\Omega_{\varepsilon}}\leq 2K$ and
$\|g_{\varepsilon}\|_{[-1,1]}\leq 2c_{1}h_{n^{(1)}}(2c_{2}\varepsilon)$. By
Lemma 5.2,

$\|g_{\varepsilon}\|_{\Omega_{\varepsilon/2}}\leq\max\\{c_{1},2K\\}\,h_{n^{(2)}}(2eD^{(1)}c_{2}\varepsilon).$

There exists a (universal) constant $E>0$ such that for any $b<1$ the closed
disk with radius $E(1-b)\varepsilon$ around any $x\in[-b,b]$ is contained in
$\Omega_{\varepsilon/2}$. The Cauchy estimates and (25) yield

$\displaystyle\|g_{\varepsilon}^{(j)}\|_{[-b,b]}$
$\displaystyle\leq\frac{\max\\{c_{1},2K\\}}{(E(1-b)\varepsilon)^{j}}j!\,h_{n^{(2)}}(2eD^{(1)}c_{2}\varepsilon)$
$\displaystyle\leq\max\\{c_{1},2K\\}\Big{(}\frac{2eD^{(1)}D^{(2)}c_{2}}{E(1-b)}\Big{)}^{j}N_{j}^{(3)}\,h_{n^{(3)}}(2eD^{(1)}D^{(2)}c_{2}\varepsilon),$

which means
$\|g_{\varepsilon}\|^{N^{(3)}}_{\sigma,[-b,b]}\leq\max\\{c_{1},2K\\}\,h_{n^{(3)}}(2eD^{(1)}D^{(2)}c_{2}\varepsilon)$
for $\sigma=\frac{2eD^{(1)}D^{(2)}c_{2}}{E(1-b)}$. Thus, if
$\varepsilon_{0}>0$ is such that (32) holds for all
$0<\varepsilon\leq\varepsilon_{0}$, then

$g:=f_{\varepsilon_{0}}+\sum_{j=1}^{\infty}g_{\varepsilon_{0}2^{-j}}=f_{\varepsilon_{0}}+\sum_{j=1}^{\infty}(f_{\varepsilon_{0}2^{-j}}-f_{\varepsilon_{0}2^{-j+1)}})$

converges absolutely in the Banach space
$\mathcal{B}^{N^{(3)}}_{\sigma}([-b,b])$. Clearly, for every $k\in\mathbb{N}$,

$g=f_{\varepsilon_{0}2^{-k}}+\sum_{j=k+1}^{\infty}(f_{\varepsilon_{0}2^{-j}}-f_{\varepsilon_{0}2^{-j+1)}}),$

and $f=g$ on $[-b,b]$, since for $x\in[-b,b]$,

$|f(x)-g(x)|\leq|f(x)-f_{\varepsilon_{0}2^{-k}}(x)|+\Big{|}\sum_{j=k+1}^{\infty}(f_{\varepsilon_{0}2^{-j}}(x)-f_{\varepsilon_{0}2^{-j+1)}}(x))\Big{|}$

which tends to $0$ as $k\to\infty$, by (32) and absolute convergence of the
sum.

(iii) For the first implication in (33) in the Roumieu case, observe that for
$f\in\mathcal{B}^{\\{\mathfrak{M}\\}}((-1,1))$ we have
$f\in\mathcal{B}^{M^{(1)}}_{B_{0}}((-1,1))$ for some $B_{0}>0$ and
$M^{(1)}\in\mathfrak{M}$. Then R-regularity of $\mathfrak{M}$ implies the
existence of $M^{(2)},M^{(3)}\in\mathfrak{M}$ such that (29) and (30) are
satisfied. Thus (i) yields the desired holomorphic approximation.

In the Beurling case take any weight sequence $M^{(3)}\in\mathfrak{M}$. By
B-regularity, we find $M^{(1)},~{}M^{(2)}$ such that (29) and (30) are
satisfied. If $f\in\mathcal{B}^{(\mathfrak{M})}((-1,1))$, then
$f\in\mathcal{B}^{M^{(1)}}_{B_{0}}((-1,1))$ for any $B_{0}>0$. Again (i)
yields the desired holomorphic approximation (since $c_{2}=CB_{0}B_{1}$).

The second implication in (33) follows from (ii), since [moderate growth] of
$\mathfrak{M}$ yields weight sequences $N^{(i)}$ fulfilling the assumptions of
(ii). ∎

##  6\. Proofs

We are now ready to prove the main results. We begin with a technical lemma in
which we extract and slightly modify the essential arguments of [30, Section
4]. Its general formulation allows us to readily complete the pending proofs.

###  6.1. A technical lemma

######  Lemma 6.1.

Let $j$ be a positive integer. Let $M^{(i)}$, $1\leq
i\leq\lceil\log_{2}(j(j+1))\rceil+7=:k$, be weight sequences satisfying
$(m^{(i)}_{\ell})^{1/\ell}\rightarrow\infty$ and

$\displaystyle\;\exists B\geq 1\;\forall
t>0:\overline{\Gamma}_{m^{(2)}}(Bt)\leq\underline{\Gamma}_{m^{(1)}}(t),$
$\displaystyle\operatorname{mg}(M^{(i)},M^{(i+1)})<\infty,\quad\text{ for
}2\leq i\leq k-1.$

If $f:[-1,1]\rightarrow\mathbb{C}$ is such that
$f^{j},f^{j+1}\in\mathcal{B}^{[M^{(1)}]}((-1,1))$, then
$f\in\mathcal{E}^{[M^{(k)}]}((-1,1))$.

###### Proof.

Set $g:=f^{j}$ and $h:=f^{j+1}$.

Let us begin with the Roumieu case. By Theorem 5.3(i), there exist families of
holomorphic functions $(g_{\varepsilon})$, $(h_{\varepsilon})$ approximating
$g$, $h$, respectively. More precisely, there exist positive constants
$K,c_{1},c_{2}$ and functions
$g_{\varepsilon},h_{\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that, for all small
$\varepsilon>0$,

$\displaystyle\max\\{\|g_{\varepsilon}\|_{\Omega_{\varepsilon}},\|h_{\varepsilon}\|_{\Omega_{\varepsilon}}\\}\leq
K,$ (36)
$\displaystyle\max\\{\|g-g_{\varepsilon}\|_{[-1,1]},\|h-h_{\varepsilon}\|_{[-1,1]}\\}\leq
c_{1}h_{m^{(3)}}(c_{2}\varepsilon).$ (37)

Then
$g_{\varepsilon}^{j+1}-h_{\varepsilon}^{j}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ satisfies

$\displaystyle|g_{\varepsilon}^{j+1}-h_{\varepsilon}^{j}|$
$\displaystyle\leq|g_{\varepsilon}^{j+1}-f^{j(j+1)}|+|f^{j(j+1)}-h_{\varepsilon}^{j}|$
$\displaystyle\leq(j+1)\max\\{|g_{\varepsilon}|,|g|\\}^{j}|g_{\varepsilon}-g|+j\max\\{|h_{\varepsilon}|,|h|\\}^{j-1}|h_{\varepsilon}-h|$
$\displaystyle\leq c_{3}h_{m^{(3)}}(c_{2}\varepsilon),\quad\text{ on }[-1,1].$

Thus Lemma 5.2 implies

$\|h_{\varepsilon}^{j}-g_{\varepsilon}^{j+1}\|_{\Omega_{\varepsilon/2}}\leq
c_{4}h_{m^{(4)}}(Cec_{2}\varepsilon)=:\delta_{\varepsilon},$ (38)

where $C$ is chosen such that $C\geq\operatorname{mg}(M^{(i)},M^{(i+1)})$ for
all $2\leq i\leq k-1$. (Here and below all constants $c_{i}$ are independent
of $\varepsilon$.)

Consider the continuous function

$u_{\varepsilon}:=\varphi_{\varepsilon}\frac{\overline{g}_{\varepsilon}h_{\varepsilon}}{\max\\{|g_{\varepsilon}|,r_{\varepsilon}\\}^{2}},\quad\text{
with }r_{\varepsilon}:=\delta_{\varepsilon}^{\frac{1}{j+1}},$

where $\varphi_{\varepsilon}$ is a smooth function compactly supported in
$\Omega_{\varepsilon}$ and $1$ on $\Omega_{\varepsilon/2}$. It coincides with
$h_{\varepsilon}/g_{\varepsilon}$ in
$\Omega_{\varepsilon/2}\cap\\{|g_{\varepsilon}|>r_{\varepsilon}\\}$, but is
not holomorphic everywhere near $[-1,1]$. By taking $\varepsilon>0$
sufficiently small, we may assume that $\delta_{\varepsilon}\leq
r_{\varepsilon}\leq 1$.

Lemmas 4.2.1 to 4.2.4 in [30] (which apply without change to our situation)
lead to a holomorphic approximation $(f_{\varepsilon})$ of $f$ by solving a
suitable $\overline{\partial}$-problem. Indeed, they show (using (36), (37),
(38) and $h_{m^{(3)}}(t)\leq h_{m^{(4)}}(eCt/2)$, by (26) since
$h_{m^{(4)}}\leq 1$) that

$\displaystyle\|u_{\varepsilon}\|_{\Omega_{\varepsilon/2}}$
$\displaystyle\leq(2K)^{1/j},$ (39)
$\displaystyle\|f-u_{\varepsilon}\|_{[-1,1]}$ $\displaystyle\leq
c_{5}r_{\varepsilon}^{1/j},$ (40)

and that the bounded continuous function

$v_{\varepsilon}(z):=\frac{1}{2\pi
i}\int_{\Omega_{\varepsilon/2}}\frac{\overline{\partial}u_{\varepsilon}(\zeta)}{\zeta-z}\,d\zeta\wedge
d\overline{\zeta},$

which satisfies
$\overline{\partial}v_{\varepsilon}=\overline{\partial}u_{\varepsilon}\mathbf{1}_{\Omega_{\varepsilon/2}}$
in the distributional sense in $\mathbb{C}$, fulfills

$\|v_{\varepsilon}\|_{\Omega_{\varepsilon/2}}\leq
c_{6}\delta_{\varepsilon}^{1/s}$ (41)

where $s$ is any real number with $s>j(j+1)$ (with $c_{6}$ depending on $s$).

Then $f_{\varepsilon}:=u_{2\varepsilon}-v_{2\varepsilon}$ is holomorphic in
$\Omega_{\varepsilon}$ and continuous on $\mathbb{C}$. By (39) and (41),
$\|f_{\varepsilon}\|_{\Omega_{\varepsilon}}$ is uniformly bounded for all
small $\varepsilon$, and by (40) and (41),

$\|f-f_{\varepsilon}\|_{[-1,1]}\leq c_{7}\delta_{2\varepsilon}^{1/s}.$

Put $s:=2^{k-6}=:2^{\ell}$. A repeated application of (26) gives

$h_{m^{(4)}}(t)^{1/s}\leq h_{m^{(k-2)}}((Ce)^{\ell}t),\quad t>0.$

Thus, for all small $\varepsilon$,

$\displaystyle\|f-f_{\varepsilon}\|_{[-1,1]}\leq
c_{7}\delta_{2\varepsilon}^{1/s}$
$\displaystyle=c_{7}\big{(}c_{4}h_{m^{(4)}}(2eCc_{2}\varepsilon)\big{)}^{1/s}$
$\displaystyle\leq
c_{7}c_{4}^{1/s}h_{m^{(k-2)}}(2c_{2}(eC)^{\ell+1}\varepsilon).$ (42)

So Theorem 5.3(ii) implies that $f\in\mathcal{E}^{\\{M^{(k)}\\}}((-1,1))$.
This ends the proof in the Roumieu case.

For the Beurling we observe that, by assumption, we find for any (small)
$c_{2}>0$ approximating sequences $(g_{\varepsilon})$, $(h_{\varepsilon})$
such that (36) and (37) are satisfied. Then follow the above proof until the
end and notice that thus also in the final approximation (6.1) the constant
$2c_{2}(eC)^{\ell+1}$ gets arbitrarily small as $c_{2}$ gets small. Again an
application of Theorem 5.3 completes the proof. ∎

###  6.2. Proof of Theorem 4.2 –
$\tensor[_{1}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$

We may assume that there is a positive integer $j$ such that $g=f^{j}$,
$h=f^{j+1}$ are elements of the ring
$\tensor[_{1}]{\mathcal{E}}{{}^{[\mathfrak{M}]}}$. By composing with suitable
linear reparameterizations, we may further assume that they are represented by
elements of $\mathcal{B}^{[\mathfrak{M}]}((-1,1))$ which we denote by the same
symbols.

In the Roumieu case, there exists $M^{(1)}\in\mathfrak{M}$ such that $g$, $h$
are contained in $\mathcal{B}^{\\{M^{(1)}\\}}((-1,1))$ (by the linear order of
$\mathfrak{M}$). By R-regularity and R-moderate growth of $\mathfrak{M}$, we
find sequences $M^{(i)}\in\mathfrak{M}$ satisfying the assumptions of Lemma
6.1 which implies that $f\in\mathcal{E}^{\\{M^{(k)}\\}}((-1,1))$.

In the Beurling case, we fix an arbitrary $M\in\mathfrak{M}$ and we show that
$f\in\mathcal{E}^{(M)}((-1,1))$. By B-regularity and B-moderate growth of
$\mathfrak{M}$, we now get sequences $M^{(i)}\in\mathfrak{M}$ as required in
Lemma 6.1, where $M^{(k)}=M$. By assumption, $g$, $h$ are elements of
$\mathcal{B}^{(M^{(1)})}((-1,1))$. Thus Lemma 6.1 gives
$f\in\mathcal{E}^{(M)}((-1,1))$. ∎

###  6.3. Proof of Theorem 3.1 – $\tensor[_{1}]{\mathcal{E}}{{}^{[\omega]}}$

This is an immediate corollary of Theorem 4.2 and the discussion in Section
4.5.∎

###  6.4. Proof of Theorem 3.2 – non-quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{[\omega]}}$

We reduce the multidimensional result to the one-dimensional one.

In the Roumieu case $\tensor[_{d}]{\mathcal{E}}{{}^{\\{\omega\\}}}$, Theorem
3.2 is a simple corollary of Theorem 4.3; the weight matrix $\mathfrak{S}$
from Section 4.5 clearly satisfies (23) (since $\omega$ is non-quasianalytic).

The Beurling case $\tensor[_{d}]{\mathcal{E}}{{}^{(\omega)}}$ can be reduced
to the Roumieu case by means of the following lemma (which is an adaptation of
[21, Lemma 13]).

######  Lemma 6.2.

Let $\omega$ be a non-quasianalytic concave weight function. Suppose that
$f:[0,\infty)\to[0,\infty)$ is any function satisfying $\omega(t)=o(f(t))$ as
$t\to\infty$. Then there exists a non-quasianalytic concave weight function
$\tilde{\omega}$ satisfying $\omega(t)=o(\tilde{\omega}(t))$ and
$\tilde{\omega}(t)=o(f(t))$ as $t\to\infty$.

###### Proof.

It suffices to extract some constructions from the proof of [21, Lemma 13] (to
which we refer for details). We may assume that $\omega$ is of class $C^{1}$.
The condition $\omega(t)=o(t)$ as $t\to\infty$ implies that
$\omega^{\prime}(t)\searrow 0$ as $t\to\infty$.

Note that $\log(t)=o(\omega(t))$ and $\omega(t)=o(f(t))$ imply $f(t)\to\infty$
as $t\to\infty$. We define inductively three sequences $(x_{n})$, $(y_{n})$,
and $(z_{n})$ with $x_{1}=y_{1}=z_{1}=0$, $x_{2}>0$, and the following
properties:

$\displaystyle\int_{x_{n}}^{\infty}\frac{\omega(t)}{1+t^{2}}dt\leq\frac{1}{n^{3}},$
(43) $\displaystyle x_{n}>2y_{n-1}+n,$ (44) $\displaystyle f(t)\geq
n^{2}\omega(t),\quad\text{ for all }t\geq x_{n},$ (45)
$\displaystyle\omega(x_{n})\geq 2^{n-i}\omega(z_{i}),\quad 1\leq i\leq n-1,$
(46)
$\displaystyle\omega^{\prime}(y_{n})=\frac{n-1}{n}\omega^{\prime}(x_{n}),$
(47)
$\displaystyle\omega(z_{n})=n\omega(y_{n})-(n-1)\big{(}\omega(x_{n})+(y_{n}-x_{n})\omega^{\prime}(x_{n})\big{)}.$
(48)

Concavity of $\omega$ guarantees well-definedness of these conditions. Then

$\tilde{\omega}(t):=\begin{cases}(n-1)\big{(}\omega(x_{n})+(t-x_{n})\omega^{\prime}(x_{n})\big{)}-\sum_{i=1}^{n-2}\omega(z_{i+1})&\text{
if }x_{n}\leq t<y_{n},\\\ n\omega(t)-\sum_{i=1}^{n-1}\omega(z_{i+1})&\text{ if
}y_{n}\leq t<x_{n+1},\end{cases}$

defines a non-quasianalytic concave weight function of class $C^{1}$
satisfying

$(n-2)\omega(t)\leq\tilde{\omega}(t)\leq n\omega(t),\quad\text{ if
}t\in[x_{n},x_{n+1})\text{ and }n\geq 2.$ (49)

(Non-quasianalyticity follows from (43) and the second inequality in (49); cf.
[21, Remark 14].) Together with (45) this implies that
$\omega(t)=o(\tilde{\omega}(t))$ and $\tilde{\omega}(t)=o(f(t))$ as
$t\to\infty$. ∎

Suppose that $g=f^{j}$, $h=f^{j+1}$ are representatives (of the corresponding
germs) belonging to $\mathcal{B}^{(\omega)}(U)$ on some relatively compact
$0$-neighborhood $U$ in $\mathbb{R}^{d}$ and consider the sequence $L_{k}$
defined in (14). Then for each integer $j\geq 1$ there exists $C_{j}>1$ such
that

$L_{k}\leq C_{j}\exp(j\varphi^{*}_{\omega}(k/j)),\quad\text{ for all
}k\in\mathbb{N}.$

Defining the function $\ell:[0,\infty)\to\mathbb{R}$ by

$\ell(t):=\log\max\\{L_{k},1\\},\quad\text{ for }k\leq t<k+1,$

and performing the subsequent steps in [21, Section 5], we find that
$\ell\leq\varphi^{*}_{\tilde{\omega}}+\operatorname{const}$, where
$\tilde{\omega}$ is the weight function provided by Lemma 6.2. This means that
$g$, $h$ belong to $\mathcal{B}^{\\{\tilde{\omega}\\}}(U)$. Invoking Theorem
3.2 in the Roumieu case shows that
$f\in\tensor[_{d}]{\mathcal{E}}{{}^{\\{\tilde{\omega}\\}}}$. Since
$\omega(t)=o(\tilde{\omega}(t))$ as $t\to\infty$ we may conclude that
$f\in\tensor[_{d}]{\mathcal{E}}{{}^{(\omega)}}$. ∎

###  6.5. Proof of Theorem 2.7 – quasianalytic
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$

The following lemma is a variant of [16, Theorem 1.6(3)].

######  Lemma 6.3.

Let $M$ be a quasianalytic intersectable weight sequence. Then:

  1. (i)

$n_{k}^{1/k}\to\infty$ for all $N\in\mathcal{L}(M)$.

  2. (ii)

If $M$ has moderate growth, then for every $N\in\mathcal{L}(M)$ there exists
$N^{\prime}\in\mathcal{L}(M)$ such that
$\operatorname{mg}(N^{\prime},N)<\infty$.

###### Proof.

(i) is obvious, since $m_{k}^{1/k}\to\infty$ (cf. Remark 2.8).

(ii) If $M$ has moderate growth, then so has $m$. Set
$C:=\operatorname{mg}(m,m)<\infty$. For $N\in\mathcal{L}(M)$ we define
$N^{\prime}$ by $n^{\prime}_{k}:=C^{k}\min_{0\leq j\leq k}n_{j}n_{k-j}$ or
equivalently

$n^{\prime}_{2j}:=C^{2j}\underline{\nu}_{1}^{2}\underline{\nu}_{2}^{2}\underline{\nu}_{3}^{2}\cdots\underline{\nu}_{j}^{2},\quad
n^{\prime}_{2j+1}:=C^{2j+1}\underline{\nu}_{1}^{2}\underline{\nu}_{2}^{2}\underline{\nu}_{3}^{2}\cdots\underline{\nu}_{j}^{2}\underline{\nu}_{j+1},$

where $\underline{\nu}_{k}:=n_{k}/n_{k-1}$. Then clearly
$\operatorname{mg}(N^{\prime},N)<\infty$. Since $\underline{\nu}_{k}$ is
increasing, so is
$\underline{\nu}^{\prime}_{k}:=n^{\prime}_{k}/n^{\prime}_{k-1}$, thus
$n^{\prime}$ is log-convex. Moreover,

$n^{\prime}_{2j}=C^{2j}n_{j}^{2}\geq\frac{m_{2j}}{m_{j}^{2}}n_{j}^{2}\geq
m_{2j}$

and analogously $n^{\prime}_{2j+1}=C^{2j+1}n_{j}n_{j+1}\geq m_{2j+1}$, so that
$N^{\prime}\geq M$. It remains to check that $N^{\prime}$ is non-
quasianalytic. Since $N^{\prime}$ is log-convex, the sequence
$(N^{\prime}_{k})^{1/k}$ is increasing and so it suffices to show that
$\sum_{j}(N^{\prime}_{2j})^{-1/(2j)}<\infty$. This is clear, since

$(N^{\prime}_{2j})^{1/(2j)}=((2j)!\,C^{2j}n_{j}^{2})^{1/(2j)}\geq\frac{2C}{e}jn_{j}^{1/j}\geq\frac{2C}{e}N_{j}^{1/j}$

and $N$ is non-quasianalytic. ∎

Let $M$ be a quasianalytic intersectable weight sequence of moderate growth.
Suppose that $g=f^{j}$, $h=f^{j+1}$ are elements of
$\tensor[_{d}]{\mathcal{E}}{{}^{\\{M\\}}}$. Since $M$ is intersectable, it
suffices to show that $f\in\tensor[_{d}]{\mathcal{E}}{{}^{\\{N\\}}}$ for every
$N\in\mathcal{L}(M)$. Fix such $N$. By Lemma 6.3, there exist
$N^{(1)},\dots,N^{(k)}\in\mathcal{L}(M)$ with $N^{(k)}=N$ such that the
requirements of Lemma 6.1 are satisfied. (Note that $\mathcal{L}(M)$ is not a
weight matrix in the sense of Section 4.1, because it is not totally ordered.)

Let $U$ be an open $0$-neighborhood in $\mathbb{R}^{d}$ on which we have
$g,h\in\mathcal{E}^{\\{M\\}}(U)$ for representatives which are denoted by the
same symbols. Take any curve $c\in\mathcal{E}^{\\{N^{(1)}\\}}(\mathbb{R},U)$
with compact support. Then, by composition closedness of
$\mathcal{E}^{\\{N^{(1)}\\}}$ as $n^{(1)}$ is log-convex, we have $g\circ
c,h\circ c\in\mathcal{E}^{\\{N^{(1)}\\}}(\mathbb{R})$. After a linear change
of variables, we may assume that $g\circ c,h\circ
c\in\mathcal{B}^{\\{N^{(1)}\\}}((-1,1))$. Thus Lemma 6.1 yields that $f\circ
c\in\mathcal{E}^{\\{N\\}}((-1,1))$. This implies that
$f\in\mathcal{E}^{\\{N\\}}(U)$, by [16, Theorem 2.7].∎

### Acknowledgements

We wish to thank Prof. José Bonet Solves for drawing our attention to [30].

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