# Interpolation of derivatives and ultradifferentiable regularity

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

We review some interpolation inequalities for derivatives and results on
ultradifferentiable regularity by lacunary estimates.

###### Key words and phrases:

Interpolation inequalities, Denjoy–Carleman classes, ultradifferentiable
regularity, lacunary estimates

###### 2020 Mathematics Subject Classification:

26D10, 26E05, 26E10, 41A17, 46E10

Supported by the Austrian Science Fund (FWF), Grants DOI 10.55776/P32905 and
DOI 10.55776/P33417

##  1\. Introduction

Ultradifferentiable classes are classes of $C^{\infty}$ functions defined by
prescribed growth behavior of the infinite sequence of derivatives. The most
classical among them are the Denjoy–Carleman classes, where the growth of the
derivatives is dominated by a weight sequence $M=(M_{j})_{j\geq 0}$. The well-
known Gevrey classes are special cases thereof.

In applications, sometimes the question arises whether the ultradifferentiable
regularity can be concluded if only lacunary information on the growth of the
derivatives is available. More precisely, suppose that we know that a smooth
function satisfies certain ultradifferentiable bounds for the derivatives of
order $k_{j}$, $j\geq 0$, can we deduce that its derivatives of all orders
satisfy the ultradifferentiable bounds? An affirmative answer clearly depends
on conditions for the base sequence $(k_{j})$, the weight $M=(M_{j})$, and on
suitable interpolation inequalities.

In the recent paper [1], Albano and Mughetti gave sufficient conditions for
Denjoy–Carleman classes of Roumieu type on a compact interval of $\mathbb{R}$,
based on the Cartan–Gorny inequality (cf. Proposition 2.6). The base sequence
$(k_{j})$ is required to be such that $k_{j+1}/k_{j}$ is bounded. For the
weight sequence, the authors assume a rather strong condition (which we recall
and discuss in Remark 6.4). In fact, we prove a version under a weaker and
more natural condition in Theorem 6.1 (see also Theorem 4.1), namely,
boundedness of $m_{k_{j+1}}/m_{k_{j}}$, where $m_{j}:=M_{j}^{1/j}$.

The main purpose of this paper is to treat the interpolation problem in a
broad and comprehensive way, striving for minimal assumptions on the weights.
To this end, we review various interpolation inequalities for $L^{p}$-norms,
where $1\leq p\leq\infty$, and work in arbitrary (finite) dimensions; the
growth of the involved constants is crucial in the ultradifferentiable
setting. We consider ultradifferentiable classes of local and global type and
allow very general weight systems (so that also Braun–Meise–Taylor classes are
covered). All the ingredients seem to be well-known, even though somewhat
scattered in the literature, but we think a unified treatment can be useful.

For local Denjoy–Carleman classes of Roumieu type, the paper of Liess [14]
must be mentioned which itself refers to the work of Bolley, Camus, and
Métivier [2] in the analytic case. Note that Liess in particular showed that,
under certain assumptions on $M$, boundedness of $m_{k_{j+1}}/m_{k_{j}}$ is
also a necessary condition for the interpolation problem (see Remark 4.4). The
results on ultradifferentiable regularity by lacunary estimates have natural
applications to the case, where one has bounds for the sequence
$(P^{j}f)_{j\geq 0}$ and $P$ is an elliptic linear partial differential
operator. See for instance [17], where this approach was used to prove
Chevalley type results.

##  2\. Interpolation inequalities

###  2.1. The global setting

The Landau–Kolmogorov inequality states that a $C^{m}$ function
$f:\mathbb{R}\to\mathbb{R}$ with finite $\|f\|_{L^{\infty}(\mathbb{R})}$ and
$\|f^{(m)}\|_{L^{\infty}(\mathbb{R})}$ satisfies

$\|f^{(j)}\|_{L^{\infty}(\mathbb{R})}\leq
K_{m,j}\,\|f\|_{L^{\infty}(\mathbb{R})}^{1-j/m}\|f^{(m)}\|_{L^{\infty}(\mathbb{R})}^{j/m},\quad
j=1,\ldots,m-1.$

Due to Kolmogorov [9], the optimal constants $K_{m,j}$ are given by

$K_{m,j}=\frac{k_{m-j}}{k_{m}^{1-j/m}},$

where $k_{r}:=\frac{4}{\pi}\sum_{i=0}^{\infty}[\frac{(-1)^{i}}{2i+1}]^{r+1}$
are the Favard constants. Note that $1\leq k_{r}\leq 2$ so that $K_{m,j}\leq
2$. By a simple functional-analytic argument, Certain and Kurtz [5] inferred
that, if $(E,\|\cdot\|)$ is a real Banach space and $A$ is the generator of a
strongly continuous group of isometries, then

$\|A^{j}x\|\leq K_{m,j}\,\|x\|^{1-j/m}\|A^{m}x\|^{j/m},\quad j=1,\ldots,m-1,$

for $x$ in the domain of $A^{m}$. In particular (see also [13, Section 4.4]),
we have the following lemma.

######  Lemma 2.1.

Let $1\leq p\leq\infty$ and $m\in\mathbb{N}_{\geq 2}$. Let
$f:\mathbb{R}^{n}\to\mathbb{R}$ be a $C^{m}$ function. Then, for all
$v\in\mathbb{S}^{n-1}$,

$\|d_{v}^{j}f\|_{L^{p}(\mathbb{R}^{n})}\leq
2\,\|f\|_{L^{p}(\mathbb{R}^{n})}^{1-j/m}\|d_{v}^{m}f\|_{L^{p}(\mathbb{R}^{n})}^{j/m},\quad
j=1,\ldots,m-1,$

if the right-hand side is finite, where
$d_{v}^{j}f(x):=\partial_{t}^{j}f(x+tv)|_{t=0}$.

###### Proof.

We may assume $v=(1,0,\ldots,0)$ by choosing a suitable orthonormal system of
coordinates. Let us first consider $p=\infty$. We have

$\displaystyle\sup_{x\in\mathbb{R}^{n}}|\partial_{1}^{j}f(x)|$
$\displaystyle\leq\sup_{\begin{subarray}{c}x_{i}\in\mathbb{R}\\\ 2\leq i\leq
n\end{subarray}}\Big{[}2\,\Big{(}\sup_{x_{1}\in\mathbb{R}}|f(x)|\Big{)}^{1-j/m}\Big{(}\sup_{x_{1}\in\mathbb{R}}|\partial_{1}^{m}f(x)|\Big{)}^{j/m}\Big{]}$
$\displaystyle\leq
2\,\Big{(}\sup_{x\in\mathbb{R}^{n}}|f(x)|\Big{)}^{1-j/m}\Big{(}\sup_{x\in\mathbb{R}^{n}}|\partial_{1}^{m}f(x)|\Big{)}^{j/m}.$

For $p<\infty$, we apply the above remarks to $A=\partial_{1}$ and
$E=L^{p}(\mathbb{R}^{n})$. It generates the group of translations
$T(s)f(x_{1},\ldots,x_{n})=f(x_{1}+s,x_{2},\ldots,x_{n})$ which is strongly
continuous, by the dominated convergence theorem. ∎

######  Remark 2.2.

For $p=2$, the factor $2$ can be omitted as follows from an application of the
Fourier transform:

$\displaystyle\int|\xi_{1}^{j}\widehat{f}(\xi)|^{2}\,d\xi$
$\displaystyle=\int(|\xi_{1}|^{m}|\widehat{f}(\xi)|)^{\frac{2j}{m}}|\widehat{f}(\xi)|^{2(1-\frac{j}{m})}\,d\xi\leq\|\xi_{1}^{m}\widehat{f}(\xi)\|^{2j/m}_{L^{2}(\mathbb{R}^{n})}\|\widehat{f}(\xi)\|^{2(1-j/m)}_{L^{2}(\mathbb{R}^{n})},$

by Hölder’s inequality.

###  2.2. The local setting

The next two lemmas follow from an easy adaptation of the proof of [2, Lemmas
2.3-2.5]. We sketch the argument for the convenience of the reader.

######  Lemma 2.3.

There is a constant $C>0$ such that the following holds. Let $1\leq
p\leq\infty$, $a>0$, $m\in\mathbb{N}$, $0\leq j\leq m$, and $f\in
C^{m}([-2a,2a])$. Then

$\frac{a^{j}}{j!}\|f^{(j)}\|_{L^{p}([-a,a])}\leq
C^{m}\Big{(}\frac{a^{m}}{m!}\|f^{(m)}\|_{L^{p}([-2a,2a])}+\|f\|_{L^{p}([-2a,2a])}\Big{)}.$

###### Proof.

By rescaling, it suffices to assume $a=1$. Fix $m$. Let $\varphi$ be a
$C^{\infty}$ function with support in $(-2,2)$ that equals $1$ near $[-1,1]$
and satisfies $\|\varphi^{(k)}\|_{L^{\infty}(\mathbb{R})}\leq(C_{0}m)^{k}$ for
$0\leq k\leq m$ for a universal constant $C_{0}$ (cf. [8, Theorem 1.3.5]). By
Taylor’s theorem, for $t\in[-2,2]$ and $0\leq j\leq m-1$,

$\displaystyle(\varphi
f)^{(j)}(t)=\int_{-2}^{t}\frac{(t-s)^{m-j-1}}{(m-j-1)!}(\varphi
f)^{(m)}(s)\,ds=\sum_{i=0}^{m}\binom{m}{i}A_{i}(t),$

where

$A_{i}(t):=\frac{1}{(m-j-1)!}\int_{-2}^{t}(t-s)^{m-j-1}\varphi^{(m-i)}(s)f^{(i)}(s)\,ds.$

For $i=m$, we have

$|A_{m}(t)|\leq\frac{1}{(m-j-1)!}\int_{-2}^{2}|t-s|^{m-j-1}|\varphi(s)f^{(m)}(s)|\,ds,$

and thus, by Young’s inequality,

$\displaystyle\|A_{m}\|_{L^{p}([-1,1])}$
$\displaystyle\leq\|A_{m}\|_{L^{p}([-2,2])}$
$\displaystyle\leq\frac{1}{(m-j-1)!}\|\mathbf{1}_{[-2,2]}t^{m-j-1}\|_{L^{1}(\mathbb{R})}\|\varphi
f^{(m)}\|_{L^{p}(\mathbb{R})}$
$\displaystyle\leq\frac{2^{m-j+1}}{(m-j)!}\|f^{(m)}\|_{L^{p}([-2,2])}.$

For $0\leq i\leq m-1$ and $t\in[-1,1]$, integration by parts gives

$\displaystyle A_{i}(t)$
$\displaystyle=\frac{(-1)^{i}}{(m-j-1)!}\int_{-2}^{-1}\partial_{s}^{i}[(t-s)^{m-j-1}\varphi^{(m-i)}(s)]f(s)\,ds$
$\displaystyle=\sum_{\ell=0}^{k}\binom{i}{\ell}\frac{(-1)^{i+\ell}}{(m-j-1-\ell)!}\int_{-2}^{-1}(t-s)^{m-j-1-\ell}\varphi^{(m-\ell)}(s)f(s)\,ds,$

where $k:=\min\\{i,m-j-1\\}$. Using Young’s inequality as before, we get

$\displaystyle\|A_{i}\|_{L^{p}([-1,1])}$
$\displaystyle\leq\sum_{\ell=0}^{k}\binom{i}{\ell}\frac{2^{m-j-\ell+1}}{(m-j-\ell)!}(C_{0}m)^{m-\ell}\|f\|_{L^{p}([-2,2])}.$

Consequently,

$\displaystyle\sum_{i=0}^{m-1}\binom{m}{i}\|A_{i}\|_{L^{p}([-1,1])}$
$\displaystyle\leq\sum_{i=0}^{m-1}\sum_{\ell=0}^{k}\binom{m}{i}\binom{i}{\ell}\frac{2^{m-j-\ell+1}}{(m-j-\ell)!}(C_{0}m)^{m-\ell}\|f\|_{L^{p}([-2,2])}$

and the lemma follows if we prove that

$\sum_{i=0}^{m-1}\sum_{\ell=0}^{k}\binom{m}{i}\binom{i}{\ell}\frac{2^{m-j-\ell+1}}{(m-j-\ell)!}(C_{0}m)^{m-\ell}\leq
C^{m}j!$

for a universal constant $C$. The left-hand side equals

$\displaystyle\sum_{\ell=0}^{m-j-1}\sum_{i=\ell}^{m-1}\frac{(m-\ell)!}{(m-i)!(i-\ell)!}\frac{m!}{\ell!(m-\ell)!}\frac{2^{m-j-\ell+1}}{(m-j-\ell)!}(C_{0}m)^{m-\ell}$

which is bounded by

$\displaystyle\sum_{\ell=0}^{m-j-1}2^{m-\ell}2^{m}\frac{2^{m-j-\ell+1}}{(m-j-\ell)!}C_{0}^{m-\ell}e^{m}\frac{m!}{\ell!}$
$\displaystyle\leq
e^{m}2^{2m}2^{j+1}C_{0}^{j}\,j!\sum_{\ell=0}^{m-j}\frac{(m-j)!}{\ell!(m-j-\ell)!}(2^{2}C_{0})^{m-j-\ell}$

which is of the required form. ∎

######  Lemma 2.4.

There exist constants $C_{0},C>0$, depending only on the dimension $n$, such
that the following holds. Let $1\leq p\leq\infty$. Let $U,V$ bounded open
subsets of $\mathbb{R}^{n}$ such that $\overline{U}\subseteq V$. Let $f\in
C^{m}(\overline{V})$. For all $0<a\leq C_{0}\operatorname{dist}(U,\partial
V)$, all $0\leq j\leq m$, and all $v\in\mathbb{S}^{n-1}$,

$\frac{a^{j}}{j!}\|d^{j}_{v}f\|_{L^{p}(U)}\leq
C^{m}\Big{(}\frac{a^{m}}{m!}\|d^{m}_{v}f\|_{L^{p}(V)}+\|f\|_{L^{p}(V)}\Big{)}.$

###### Proof.

We may assume that $v=(1,0,\ldots,0)$. Let us first show the assertion for
$U=(-a,a)^{n}$ and $V=(-2a,2a)^{n}$. For
$x^{\prime}:=(x_{2},\ldots,x_{n})\in(-a,a)^{n-1}$, Lemma 2.3 gives

$\displaystyle\frac{a^{j}}{j!}\|\partial_{1}^{j}f(\cdot,x^{\prime})\|_{L^{p}((-a,a))}\leq
C^{m}\Big{(}\frac{a^{m}}{m!}\|\partial_{1}^{m}f(\cdot,x^{\prime})\|_{L^{p}((-2a,2a))}+\|f(\cdot,x^{\prime})\|_{L^{p}((-2a,2a))}\Big{)}.$

For $p=\infty$, the statement follows, by taking the supremum over all
$x^{\prime}\in(-a,a)^{n-1}$. For $p<\infty$, take the $p$-th power an
integrate over $x^{\prime}\in(-a,a)^{n-1}$.

In general, there is a constant $C_{0}>0$ depending only on $n$ such that, if
$a\leq C_{0}\operatorname{dist}(U,\partial V)$, then $U$ can be covered by a
family $\mathcal{Q}$ of cubes $Q=x+[-a,a]^{n}$ such that
$2Q=x+[-2a,2a]^{n}\subseteq V$ and such that any two cubes in $\mathcal{Q}$
have disjoint interior. Then, for each $Q\in\mathcal{Q}$,

$\displaystyle\frac{a^{j}}{j!}\|\partial_{1}^{j}f\|_{L^{p}(Q)}\leq
C^{m}\Big{(}\frac{a^{m}}{m!}\|\partial_{1}^{m}f\|_{L^{p}(2Q)}+\|f\|_{L^{p}(2Q)}\Big{)}.$

For $p=\infty$, take the maximum over all $Q\in\mathcal{Q}$, for $p<\infty$,
take the $p$-power and the sum over all $Q\in\mathcal{Q}$. ∎

Now we combine Lemma 2.4 with Lemma 2.1 (following [14, p. 193]).

######  Corollary 2.5.

Let $1\leq p\leq\infty$. Let $U,V$ bounded open subsets of $\mathbb{R}^{n}$
such that $\overline{U}\subseteq V$. Let $f\in C^{\infty}(\overline{V})$.
Then, for each integer $m\geq 2$, $j=1,\ldots,m-1$, and all
$v\in\mathbb{S}^{n-1}$,

$\|d_{v}^{j}f\|_{L^{p}(U)}\leq
C^{m}\|f\|^{1-j/m}_{L^{p}(V)}\Big{(}\|d_{v}^{m}f\|^{j/m}_{L^{p}(V)}+m^{j}\|f\|^{j/m}_{L^{p}(V)}\Big{)},$

where $C>0$ is a constant depending only on $U$, $V$, and $n$.

###### Proof.

Let $W$ be the open $\delta$-neighborhood of $\overline{U}$, where
$\delta:=\frac{1}{2}\operatorname{dist}(U,\partial V)$. Let $\varphi$ be a
$C^{\infty}$ function with support in $W$ that equals $1$ near $\overline{U}$
and satisfies $\|\partial^{\alpha}\varphi\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C_{0}^{|\alpha|+1}m^{|\alpha|}$ for $|\alpha|\leq m$, where $C_{0}$ only
depends on $U$, $V$, and $n$ (cf. [8, Theorem 1.4.2]). Then $\varphi f$ has a
$C^{\infty}$ extension by zero outside $W$ to all of $\mathbb{R}^{n}$ which we
also denote by $\varphi f$. By Lemma 2.1,

$\|d_{v}^{j}f\|_{L^{p}(U)}\leq\|d^{j}_{v}(\varphi
f)\|_{L^{p}(\mathbb{R}^{n})}\leq 2\,\|(\varphi
f)\|^{1-j/m}_{L^{p}(W)}\|d_{v}^{m}(\varphi f)\|_{L^{p}(W)}^{j/m}$

and, by Lemma 2.4 with $a=C_{1}\delta$, where $C_{1}=C_{1}(n)$,

$\displaystyle\|$ $\displaystyle d_{v}^{m}(\varphi
f)\|_{L^{p}(W)}\leq\sum_{i=0}^{m}\binom{m}{i}\|d_{v}^{m-i}\varphi\|_{L^{\infty}(\mathbb{R}^{n})}\|d_{v}^{i}f\|_{L^{p}(W)}$
$\displaystyle\leq\sum_{i=0}^{m}\binom{m}{i}C_{0}(C_{0}m)^{m-i}C^{m}\frac{i!}{(C_{1}\delta)^{i}}\Big{(}\frac{(C_{1}\delta)^{m}}{m!}\|d_{v}^{m}f\|_{L^{p}(V)}+\|f\|_{L^{p}(V)}\Big{)}$
$\displaystyle\leq\sum_{i=0}^{m}\binom{m}{i}\frac{C_{0}(eC_{0}C\max\\{C_{1}\delta,1\\})^{m}}{(C_{0}C_{1}\delta)^{i}}\big{(}\|d_{v}^{m}f\|_{L^{p}(V)}+m!\,\|f\|_{L^{p}(V)}\big{)}$
$\displaystyle\leq
C_{2}^{m}\Big{(}\|d_{v}^{m}f\|_{L^{p}(V)}+m^{m}\|f\|_{L^{p}(V)}\Big{)}.$

This implies the assertion. ∎

###  2.3. The Cartan–Gorny inequality

The following result is due to Gorny [7] and independently to Cartan [4].

######  Proposition 2.6.

Let $I\subseteq\mathbb{R}$ be a compact interval and $f\in C^{m}(I)$. Then,
for $j=1,\ldots,m-1$,

$\|f^{(j)}\|_{L^{\infty}(I)}\leq
4e^{2j}\Big{(}\frac{m}{j}\Big{)}^{j}\,\|f\|_{L^{\infty}(I)}^{1-j/m}\max\Big{\\{}\|f^{(m)}\|_{L^{\infty}(I)},\frac{m!}{|I|^{m}}\|f\|_{L^{\infty}(I)}\Big{\\}}^{j/m}.$

The factor $(\frac{m}{j})^{j}$ is bounded by $e^{m}$.

##  3\. Ultradifferentiable regularity by lacunary estimates

In this section, we prove three technical propositions on which most of the
subsequent results are based.

Let $0=:k_{0}<k_{1}<k_{2}<\cdots$ be a strictly increasing infinite sequence
of positive integers. A sequence $(k_{j})$ with these properties is called an
_base sequence_. If additionally $k_{j+1}/k_{j}$, $j\geq 1$, is bounded we say
that $(k_{j})$ is a _special base sequence_.

######  Proposition 3.1.

Let $(k_{j})$ be a base sequence. Let $(m_{j})$ and $(m^{\prime}_{j})$ be
positive increasing sequences of reals satisfying $m_{j}\leq m^{\prime}_{j}$
for all $j$. Assume that $m_{k_{j+1}}/m^{\prime}_{k_{j}}$ is bounded. Let
$1\leq p\leq\infty$ and $f\in C^{\infty}(\mathbb{R}^{n})$.

If there are $C,\rho>0$ such that

(1) $\max_{|\alpha|=k_{j}}\|\partial^{\alpha}f\|_{L^{p}(\mathbb{R}^{n})}\leq
C(\rho m_{k_{j}})^{k_{j}},\quad j\geq 0,$

then there exist $C^{\prime},\rho^{\prime}>0$ such that

(2) $\max_{|\alpha|=\ell}\|\partial^{\alpha}f\|_{L^{p}(\mathbb{R}^{n})}\leq
C^{\prime}(\rho^{\prime}m^{\prime}_{\ell})^{\ell},\quad\ell\geq 0.$

We have $\rho^{\prime}=O(\rho)$ as $\rho\to 0$.

###### Proof.

By (1), there are constants $C_{1},\rho_{1}>0$ such that

(3) $\|d_{v}^{k_{j}}f\|_{L^{p}(\mathbb{R}^{n})}\leq
C_{1}(\rho_{1}m_{k_{j}})^{k_{j}},\quad j\geq 0,\,v\in\mathbb{S}^{n-1}.$

Let $\ell\geq 0$ and let $j\geq 0$ be such that $k_{j}\leq\ell<k_{j+1}$. By
Lemma 2.1 and the assumptions on $(k_{j})$, $(m_{j})$, and $(m^{\prime}_{j})$,

(4) $\displaystyle\|d_{v}^{\ell}f\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\leq
2\,\|d_{v}^{k_{j}}f\|_{L^{p}(\mathbb{R}^{n})}^{1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}\|d_{v}^{k_{j+1}}f\|_{L^{p}(\mathbb{R}^{n})}^{\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}$ $\displaystyle\leq
2C_{1}\,(\rho_{1}m_{k_{j}})^{k_{j}\big{(}1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}\big{)}}(\rho_{1}m_{k_{j+1}})^{k_{j+1}\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}$ $\displaystyle\leq
2C_{1}\,(\rho_{1}m^{\prime}_{k_{j}})^{k_{j}\big{(}1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}\big{)}}(C_{2}\rho_{1}m^{\prime}_{k_{j}})^{k_{j+1}\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}$ $\displaystyle\leq
2C_{1}\,(C_{2}\rho_{1}m^{\prime}_{\ell})^{\ell}.$

By polarization [10, Lemma 7.13], for $|\alpha|=\ell$ and
$v_{1},\ldots,v_{\ell}$ the list of standard unit vectors

$\underbrace{e_{1},\ldots,e_{1}}_{\alpha_{1}},\underbrace{e_{2},\ldots,e_{2}}_{\alpha_{2}},\ldots,\underbrace{e_{n},\ldots,e_{n}}_{\alpha_{n}},$

we have

(5) $\displaystyle\partial^{\alpha}f$
$\displaystyle=\frac{1}{\ell!}\sum_{\epsilon_{1},\ldots,\epsilon_{\ell}=0}^{1}(-1)^{\ell-\sum\epsilon_{\ell}}\Big{(}\sum\epsilon_{i}\Big{)}^{\ell}d^{\ell}_{\sum\epsilon_{i}v_{i}/\sum\epsilon_{i}}f.$

Therefore,

$\displaystyle\|\partial^{\alpha}f\|_{L^{p}(\mathbb{R}^{n})}\leq
2C_{1}(C_{2}\rho_{1}m^{\prime}_{\ell})^{\ell}\frac{1}{\ell!}\sum_{k=0}^{\ell}\binom{\ell}{k}k^{\ell}\leq
2C_{1}(2eC_{2}\rho_{1}m^{\prime}_{\ell})^{\ell}.$

Thus (2) is proved. ∎

######  Proposition 3.2.

Let $(k_{j})$ be an special base sequence. Let $(m_{j})$ and
$(m^{\prime}_{j})$ be positive increasing sequences of reals satisfying
$m_{j}\leq m^{\prime}_{j}$ for all $j$ and such that $j/m_{j}$ is bounded.
Assume that $m_{k_{j+1}}/m^{\prime}_{k_{j}}$ is bounded. Let $1\leq
p\leq\infty$. Let $U,V$ bounded open subsets of $\mathbb{R}^{n}$ such that
$\overline{U}\subseteq V$. Let $f\in C^{\infty}(\overline{V})$.

If there are $C,\rho>0$ such that

(6) $\max_{|\alpha|=k_{j}}\|\partial^{\alpha}f\|_{L^{p}(V)}\leq C(\rho
m_{k_{j}})^{k_{j}},\quad j\geq 0,$

then there exist $C^{\prime},\rho^{\prime}>0$ such that

(7) $\max_{|\alpha|=\ell}\|\partial^{\alpha}f\|_{L^{p}(U)}\leq
C^{\prime}(\rho^{\prime}m^{\prime}_{\ell})^{\ell},\quad\ell\geq 0.$

Under the additional assumption that $j/m_{j}\to 0$, we have that
$\rho^{\prime}=O(\rho)$ as $\rho\to 0$ and $C^{\prime}$ depends on
$\rho^{\prime}$.

###### Proof.

By (6), there are constants $C_{1},\rho_{1}>0$ such that

$\|d_{v}^{k_{j}}f\|_{L^{p}(V)}\leq C_{1}(\rho_{1}m_{k_{j}})^{k_{j}},\quad
j\geq 0,\,v\in\mathbb{S}^{n-1}.$

Let $\ell\geq 0$ and let $j\geq 0$ be such that $k_{j}\leq\ell<k_{j+1}$. By
Corollary 2.5 and the assumptions on $(k_{j})$, $(m_{j})$, and
$(m^{\prime}_{j})$,

$\displaystyle\|d_{v}^{\ell}f\|_{L^{p}(U)}$ $\displaystyle\leq
C^{k_{j+1}-k_{j}}\Big{(}\|d_{v}^{k_{j}}f\|^{1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}_{L^{p}(V)}\|d_{v}^{k_{j+1}}f\|^{\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}_{L^{p}(V)}+(k_{j+1}-k_{j})^{\ell-
k_{j}}\|d_{v}^{k_{j}}f\|_{L^{p}(V)}\Big{)}$ $\displaystyle\leq
C_{1}C_{2}^{k_{j}}\Big{(}(\rho_{1}m_{k_{j}})^{k_{j}\big{(}1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}\big{)}}(\rho_{1}m_{k_{j+1}})^{k_{j+1}\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}+(C_{3}k_{j})^{\ell-
k_{j}}(\rho_{1}m_{k_{j}})^{k_{j}}\Big{)}$ $\displaystyle\leq
C_{1}C_{2}^{k_{j}}\Big{(}(\rho_{1}m^{\prime}_{k_{j}})^{k_{j}\big{(}1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}\big{)}}(C_{4}\rho_{1}m^{\prime}_{k_{j}})^{k_{j+1}\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}+(C_{5}m^{\prime}_{k_{j}})^{\ell-
k_{j}}(\rho_{1}m^{\prime}_{k_{j}})^{k_{j}}\Big{)}$ $\displaystyle\leq
C_{1}C_{2}^{k_{j}}\Big{(}(C_{4}\rho_{1}m^{\prime}_{k_{j}})^{\ell}+((C_{5}+\rho_{1})m^{\prime}_{k_{j}})^{\ell}\Big{)}$
$\displaystyle\leq C_{1}(C_{6}m^{\prime}_{\ell})^{\ell}.$

If we assume that $j/m_{j}\to 0$ (actually $k_{j}/m_{k_{j}}\to 0$ is enough),
then for each $\tau>0$ there is $j_{\tau}$ such that $k_{j}/m_{k_{j}}\leq\tau$
for all $j\geq j_{\tau}$. Thus $C_{5}=C_{3}\rho_{1}$ provided that $j\geq
j_{\rho_{1}}$ and so

$\|d_{v}^{\ell}f\|_{L^{p}(U)}\leq
C_{1}(C_{7}\rho_{1}m^{\prime}_{\ell})^{\ell},\quad\ell\geq k_{j_{\rho_{1}}}.$

For $\ell<k_{j_{\rho_{1}}}$,

$\|d_{v}^{\ell}f\|_{L^{p}(U)}\leq C_{1}(C_{6}m^{\prime}_{\ell})^{\ell}\leq
C_{1}\max\\{\rho_{1}^{-\ell}:\ell<k_{j_{\rho_{1}}}\\}\cdot(C_{6}\rho_{1}m^{\prime}_{\ell})^{\ell}.$

Since $\rho_{1}=O(\rho)$, we get

$\|d_{v}^{\ell}f\|_{L^{p}(U)}\leq
C^{\prime}(\rho^{\prime}m^{\prime}_{\ell})^{\ell},\quad\ell\geq 0,$

where $\rho^{\prime}=O(\rho)$ and $C^{\prime}=C^{\prime}(\rho^{\prime})$.

To end the proof, it suffices to apply $\|\cdot\|_{L^{p}(U)}$ to (5) and use
the estimate for $\|d_{v}^{\ell}f\|_{L^{p}(U)}$. ∎

######  Remark 3.3.

The requirement $k_{0}=0$ for the base sequence is important in Proposition
3.1. Without this assumption there is no reason why a function satisfying (1)
should also fulfill (2) for $\ell=0$. In the local setting of Proposition 3.2,
the assumption $k_{0}=0$ can be made without loss of generality, by adjusting
the constant $C$.

######  Proposition 3.4.

Let $(k_{j})$ be a base sequence. Let $(m_{j})$ and $(m^{\prime}_{j})$ be
positive increasing sequences of reals satisfying $m_{j}\leq m^{\prime}_{j}$
for all $j$. Assume that $m_{k_{j+1}}/m^{\prime}_{k_{j}}$ is bounded. Let
$f\in C^{\infty}(\mathbb{R}^{n})$.

If there are $C,\sigma>0$ such that

(8)
$\displaystyle\max_{|\alpha|=k_{j}}\|x^{\alpha}f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\leq C(\sigma m_{k_{j}})^{k_{j}},\quad j\geq 0,$

then there exist $C^{\prime},\sigma^{\prime}>0$ such that

(9)
$\displaystyle\max_{|\alpha|=\ell}\|x^{\alpha}f\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C^{\prime}(\sigma^{\prime}m^{\prime}_{\ell})^{\ell},\quad\ell\geq 0.$

We have $\sigma^{\prime}=O(\sigma)$.

###### Proof.

If $|\alpha|=k$, then, for a constant $D>0$,

$\displaystyle|x^{\alpha}|\leq|x|^{k}\leq D\sum_{i=1}^{n}|x_{i}|^{k}\leq
D\sum_{|\alpha|=k}|x^{\alpha}|\leq D\,n^{k}\max_{|\alpha|=k}|x^{\alpha}|.$

(Indeed, $\sum_{i=1}^{n}|x_{i}|^{k}$ has a positive minimum $d$ on the sphere
$\\{|x|=1\\}$, thus $\sum_{i=1}^{n}|x_{i}|^{k}\geq d|x|^{k}$, by homogeneity.)
The assumption (8) gives

$|x|^{k_{j}}|f(x)|\leq C_{1}(n\sigma\,m_{k_{j}})^{k_{j}},\quad j\geq
0,\,x\in\mathbb{R}^{n}.$

Then, for $k_{j}\leq\ell<k_{j+1}$,

$\displaystyle|x|^{\ell}|f(x)|$
$\displaystyle=(|x|^{k_{j}}|f(x)|)^{1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}(|x|^{k_{j+1}}|f(x)|)^{\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}.$

So (8) implies (9), by a computation analogous to (4). ∎

##  4\. Denjoy–Carleman classes

Let $M=(M_{k})_{k\geq 0}$ be a positive sequence of real numbers. We say that
$M$ is a _weight sequence_ if $M_{0}=1\leq
M_{1}\leq\mu_{2}\leq\mu_{3}\leq\cdots$, where $\mu_{k}:=M_{k}/M_{k-1}$. Then
also the sequence $m_{k}:=M_{k}^{1/k}$ is increasing.

Let $U$ be an open subset of $\mathbb{R}^{n}$ and $1\leq p\leq\infty$. The
_$L^{p}$ -based local Denjoy–Carleman class of Roumieu type_ on $U$ is the set
$\mathcal{E}^{\\{M\\}}_{L^{p}}(U)$ of all $f\in C^{\infty}(U)$ such that for
all open relatively compact subsets $\Omega\Subset U$ there exist constants
$C,\rho>0$ such that

(10) $\max_{|\alpha|=k}\|\partial^{\alpha}f\|_{L^{p}(\Omega)}\leq C(\rho
m_{k})^{k},\quad k\in\mathbb{N}.$

Analogously, the _$L^{p}$ -based local Denjoy–Carleman class of Beurling type_
is the set $\mathcal{E}^{(M)}_{L^{p}}(U)$ of all $f\in C^{\infty}(U)$ that
satisfy (10) for all $\Omega$ and all $\rho$ with $C=C(\Omega,\rho)$. We use
the placeholder $[\cdot]$ for either the Roumieu case $\\{\cdot\\}$ or the
Beurling case $(\cdot)$.

Replacing (10) with various global requirements we obtain natural _global
Denjoy–Carleman classes_ ,

$\displaystyle\mathcal{B}^{\\{M\\}}_{L^{p}}(\mathbb{R}^{n})$
$\displaystyle:=\\{f\in C^{\infty}(\mathbb{R}^{n}):\exists
C,\rho>0~{}\forall\alpha:\|\partial^{\alpha}f\|_{L^{p}(\mathbb{R}^{n})}\leq
C(\rho m_{|\alpha|})^{|\alpha|}\\},$
$\displaystyle\mathcal{B}^{(M)}_{L^{p}}(\mathbb{R}^{n})$
$\displaystyle:=\\{f\in C^{\infty}(\mathbb{R}^{n}):\forall\rho>0~{}\exists
C>0~{}\forall\alpha:\|\partial^{\alpha}f\|_{L^{p}(\mathbb{R}^{n})}\leq C(\rho
m_{|\alpha|})^{|\alpha|}\\},$

_Gelfand–Shilov classes_ $\mathcal{S}^{\\{M\\}}(\mathbb{R}^{n})$ (resp.
$\mathcal{S}^{(M)}(\mathbb{R}^{n})$) consisting of all $f\in
C^{\infty}(\mathbb{R}^{n})$ such that there exist $\rho,\sigma>0$ such that
(resp. for all $\rho,\sigma>0$)

$\sup_{\alpha,\beta\in\mathbb{N}^{n}}\sup_{x\in\mathbb{R}^{n}}\frac{|x^{\alpha}\partial^{\beta}f(x)|}{(\rho
m_{|\alpha|})^{|\alpha|}(\sigma m_{|\beta|})^{|\beta|}}<\infty,$

and the classes
$\mathcal{D}^{[M]}(\mathbb{R}^{n}):=\mathcal{B}^{[M]}_{L^{\infty}}(\mathbb{R}^{n})\cap
C^{\infty}_{c}(\mathbb{R}^{n})$ of all
$\mathcal{B}^{[M]}_{L^{\infty}}$-functions with compact support. (Cleary,
$\mathcal{D}^{[M]}(\mathbb{R}^{n})$ is nontrivial only in the non-
quasianalytic setting.) Note that various aspects of these global classes have
recently been studied in [11, 12, 16]. We have continuous (with respect to the
natural locally convex topologies) inclusions

$\textstyle{\mathcal{D}^{\\{M\\}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}^{\\{M\\}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}^{\\{M\\}}_{L^{p}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{\\{M\\}}_{L^{p}}(\mathbb{R}^{n})}$$\textstyle{\mathcal{D}^{(M)}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}^{(M)}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}^{(M)}_{L^{p}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{(M)}_{L^{p}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

and, for $1\leq p\leq q\leq\infty$,

(11)
$\textstyle{\mathcal{B}^{[M]}_{L^{p}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}^{[M]}_{L^{q}}(\mathbb{R}^{n})}$$\textstyle{\mathcal{E}^{[M]}_{L^{p}}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{[M]}_{L^{q}}(U),}$

provided that $M$ is _derivation closed_ , i.e.,

(12) $\sup_{k\geq 0}\Big{(}\frac{M_{k+1}}{M_{k}}\Big{)}^{1/(k+1)}<\infty.$

This follows from the Sobolev inequality, since the latter condition
guarantees stability under taking derivatives. For the local classes, we have
equality if (12) holds, because
$\|\cdot\|_{L^{p}(\Omega)}\leq|\Omega|^{1/p-1/q}\|\cdot\|_{L^{q}(\Omega)}$.

If $(k_{j})$ is a base sequence, we consider the sets
$\mathcal{E}^{[M]}_{L^{p},(k_{j})}(U)$,
$\mathcal{B}^{[M]}_{L^{p},(k_{j})}(\mathbb{R}^{n})$,
$\mathcal{D}^{[M]}_{(k_{j})}(\mathbb{R}^{n})$ of $C^{\infty}$ functions that
satisfy the defining estimates for multiindices $\alpha$ with
$|\alpha|=k_{j}$.

######  Theorem 4.1.

Let $1\leq p\leq\infty$. Let $(k_{j})$ be a base sequence and $M=(M_{j})$ a
weight sequence such that $m_{k_{j+1}}/m_{k_{j}}$ is bounded. We have:

  1. (a)

$\mathcal{B}^{[M]}_{L^{p},(k_{j})}(\mathbb{R}^{n})=\mathcal{B}^{[M]}_{L^{p}}(\mathbb{R}^{n})$.

  2. (b)

$\mathcal{D}^{[M]}_{(k_{j})}(\mathbb{R}^{n})=\mathcal{D}^{[M]}(\mathbb{R}^{n})$.

Let $(k_{j})$ be a special base sequence and $U\subseteq\mathbb{R}^{n}$ open.
Then:

  1. (c)

$\mathcal{E}^{\\{M\\}}_{L^{p},(k_{j})}(U)=\mathcal{E}^{\\{M\\}}_{L^{p}}(U)$
provided that $j/m_{j}$ is bounded.

  2. (d)

$\mathcal{E}^{(M)}_{L^{p},(k_{j})}(U)=\mathcal{E}^{(M)}_{L^{p}}(U)$ provided
that $j/m_{j}\to 0$.

###### Proof.

(a) follows from Proposition 3.1 and (b) is a consequence of (a) by
intersecting with $C^{\infty}_{c}(\mathbb{R}^{n})$. (c) and (d) follow from
Proposition 3.2. ∎

In (c) and (d), the assumption that the base sequence $(k_{j})$ is _special_
cannot be omitted; this follows from Remark 6.5 below.

That $j/m_{j}$ is bounded (resp. tends to zero) is equivalent to the fact that
$\mathcal{E}^{\\{M\\}}_{L^{\infty}}(U)$ (resp.
$\mathcal{E}^{(M)}_{L^{\infty}}(U)$) contains $C^{\omega}(U)$. Note that
$\mathcal{E}^{[M]}_{L^{p}}(U)\subseteq\mathcal{E}^{[M]}_{L^{q}}(U)$ if $1\leq
q\leq p\leq\infty$.

Before we come to a similar result in the Gelfand–Shilov case, let us recall a
result due to [6]. (In that paper only the Roumieu case is treated, but the
Beurling case can be proved analogously, see [17, Lemma 4].) To this end we
need some further conditions on the weight sequence. We say that a weight
sequence $M=(M_{j})$ has _moderate growth_ if

(13) $\sup_{j,k\geq
1}\Big{(}\frac{M_{j+k}}{M_{j}M_{k}}\Big{)}^{1/(j+k)}<\infty.$

Note that (13) implies (12).

######  Proposition 4.2 ([6]).

Let $M=(M_{j})$ be a weight sequence with moderate growth. Assume that
$j/m_{j}$ is bounded in the Roumieu case and tends to zero in the Beurling
case. Then the following are equivalent:

  1. (a)

$f\in\mathcal{S}^{[M]}(\mathbb{R}^{n})$.

  2. (b)

There are constants $C,\rho,\sigma>0$ (resp. for all $\rho,\sigma>0$ there is
$C>0$) such that

$\sup_{x}|x^{\alpha}f(x)|\leq C(\rho m_{|\alpha|})^{|\alpha|}\quad\text{ and
}\quad\sup_{x}|\partial^{\beta}f(x)|\leq C(\sigma m_{|\beta|})^{|\beta|}$

for all $\alpha,\beta\in\mathbb{N}^{n}$.

  3. (c)

There are constants $C,\rho,\sigma>0$ (resp. for all $\rho,\sigma>0$ there is
$C>0$) such that

$\sup_{x}|x^{\alpha}f(x)|\leq C(\rho m_{|\alpha|})^{|\alpha|}\quad\text{ and
}\quad\sup_{x}|x^{\beta}\widehat{f}(x)|\leq C(\sigma m_{|\beta|})^{|\beta|}$

for all $\alpha,\beta\in\mathbb{N}^{n}$.

In view of this proposition, we allow a second base sequence $(\ell_{j})$ and
define $\mathcal{S}^{[M]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})$ to be the set
of $f\in C^{\infty}(\mathbb{R}^{n})$ such that there exist $C,\rho,\sigma>0$
(resp. for all $\rho,\sigma>0$ there is $C>0$) such that

$\max_{|\alpha|=k_{j}}\sup_{x}|x^{\alpha}f(x)|\leq C(\rho
m_{|\alpha|})^{|\alpha|}\quad\text{ and
}\quad\max_{|\beta|=\ell_{j}}\sup_{x}|\partial^{\beta}f(x)|\leq C(\sigma
m_{\ell_{j}})^{\ell_{j}}$

for all $j\geq 0$.

######  Theorem 4.3.

Let $(k_{j})$ and $(\ell_{j})$ be base sequences. Let $M=(M_{j})$ be a weight
sequence with moderate growth. Assume that $m_{k_{j+1}}/m_{k_{j}}$ and
$m_{\ell_{j+1}}/m_{\ell_{j}}$ are bounded. Then:

  1. (a)

$\mathcal{S}^{\\{M\\}}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})=\mathcal{S}^{\\{M\\}}(\mathbb{R}^{n})$
provided that $j/m_{j}$ is bounded.

  2. (b)

$\mathcal{S}^{(M)}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})=\mathcal{S}^{(M)}(\mathbb{R}^{n})$
provided that $j/m_{j}\to 0$.

###### Proof.

This follows from Proposition 3.1, Proposition 3.4, and Proposition 4.2. ∎

######  Remark 4.4.

Liess [14] proved that, under a number of assumptions on the sequence
$M=(M_{j})$, the equality
$\mathcal{E}^{\\{M\\}}_{L^{2},(k_{j})}(U)=\mathcal{E}^{\\{M\\}}_{L^{2}}(U)$
implies that $m_{k_{j+1}}/m_{k_{j}}$ is bounded. His assumptions are the
following:

  1. (a)

$m_{j}\leq m_{j+1}$ for all $j$ and $j/m_{j}$ is bounded.

  2. (b)

$m_{j+1}/m_{j}$ is bounded.

  3. (c)

There exists $g:\mathbb{N}\to(0,\infty)$ such that $m_{kj}\leq g(k)m_{j}$ for
all $k$ and $j$.

  4. (d)

For every $c>0$ there is $c^{\prime}>0$ such that $m_{j}\leq c\,m_{k}$ implies
$j\leq c^{\prime}k$.

Let us assume that $M$ is a weight sequence of moderate growth such that
$j/m_{j}$ is bounded. Then it is easily seen that the conditions (a)–(c) are
satisfied. On the other hand, for a weight sequence, (c) is equivalent to (13)
(take $k=2$ and use [15, Theorem 1]).

Observe that, assuming $m_{j}\leq m_{j+1}$, (d) is equivalent to the existence
of an integer $n\geq 2$ such that $\liminf_{k\to\infty}m_{nk}/m_{k}=:a>1$.
Indeed, for all $\ell\geq 1$ and large $k$, we have $a^{\ell}m_{k}\leq
m_{n^{\ell}k}$. If $a>1$ and $m_{j}\leq c\,m_{k}$, then $c/a^{-\ell}<1$,
provided that $\ell$ is large enough, and thus $m_{j}<m_{n^{\ell}k}$. As
$m_{j}$ is increasing, we conclude that $j\leq n^{\ell}k$. It is clear that
$a=1$ violates (d).

##  5\. Other ultradifferentiable classes

Classically, besides Denjoy–Carleman classes a lot of attention was devoted to
ultradifferentiable classes, sometimes called Braun–Meise–Taylor classes due
to the foundational article [3], defined in terms of a weight function; they
have their origin in work of Beurling.

A _weight function_ is a continuous increasing function
$\omega:[0,\infty)\to[0,\infty)$ satisfying $\omega(0)=0$,
$\omega(2t)=O(\omega(t))$, $\log t=o(\omega(t))$ as $t\to\infty$ and such that
$\varphi(t):=\omega(e^{t})$ is convex. It is no restriction to assume
$\omega(t)=0$ for $0\leq t\leq 1$ (cf. [18, Section 11.1]).

The local classes $\mathcal{E}^{\\{\omega\\}}_{L^{p}}(U)$ and
$\mathcal{E}^{(\omega)}_{L^{p}}(U)$ are defined in analogy to Denjoy–Carleman
classes, where now

(14) $\max_{|\alpha|=k}\|\partial^{\alpha}f\|_{L^{p}(\Omega)}\leq
C\,\exp\big{(}\tfrac{1}{\rho}\varphi^{*}(\rho k)\big{)},\quad k\geq 0,$

with $\varphi^{*}(s):=\sup_{t\geq 0}(st-\varphi(t))$, plays the role of (10).
More precisely, we have

$\displaystyle\mathcal{E}^{\\{\omega\\}}_{L^{p}}(U)$
$\displaystyle:=\bigcap_{\Omega\Subset U}\bigcup_{\rho>0}\bigcup_{C>0}\\{f\in
C^{\infty}(U):f\text{ satisfies }\eqref{eq:DC6}\\},$
$\displaystyle\mathcal{E}^{(\omega)}_{L^{p}}(U)$
$\displaystyle:=\bigcap_{\Omega\Subset U}\bigcap_{\rho>0}\bigcup_{C>0}\\{f\in
C^{\infty}(U):f\text{ satisfies }\eqref{eq:DC6}\\}.$

The global classes $\mathcal{B}^{[\omega]}_{L^{p}}(\mathbb{R}^{n})$,
$\mathcal{S}^{[\omega]}(\mathbb{R}^{n})$, and
$\mathcal{D}^{[\omega]}(\mathbb{R}^{n})$ are defined in a straightforward way.
Furthermore, we consider $\mathcal{E}^{[\omega]}_{L^{p},(k_{j})}(U)$,
$\mathcal{B}^{[\omega]}_{L^{p},(k_{j})}(\mathbb{R}^{n})$,
$\mathcal{S}^{[\omega]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})$, and
$\mathcal{D}^{[\omega]}_{(k_{j})}(\mathbb{R}^{n})$, all of them defined in the
obvious manner. (Note that $\mathcal{D}^{[\omega]}(\mathbb{R}^{n})$ is
nontrivial if and only if $\int_{1}^{\infty}t^{-2}\omega(t)\,dt<\infty$; cf.
[3].)

There is an overarching framework for ultradifferentiable classes introduced
in [19] which goes beyond Denjoy–Carleman and Braun–Meise–Taylor classes and,
on a technical level, reduces the proofs to handling certain families of
weight sequences. See [18] for a comprehensive survey of the theory.

From now on, let $\mathfrak{M}$ be a totally ordered family of weight
sequences, i.e., if $M,M^{\prime}\in\mathfrak{M}$, then either $M_{j}\leq
M^{\prime}_{j}$ or $M^{\prime}_{j}\leq M_{j}$ for all $j$.

We define

$\displaystyle\mathcal{E}^{\\{\mathfrak{M}\\}}_{L^{p}}(U)$
$\displaystyle:=\bigcap_{\Omega\Subset
U}\bigcup_{M\in\mathfrak{M}}\bigcup_{\rho>0}\bigcup_{C>0}\\{f\in
C^{\infty}(U):f\text{ satisfies }\eqref{eq:DC3}\\},$
$\displaystyle\mathcal{E}^{(\mathfrak{M})}_{L^{p}}(U)$
$\displaystyle:=\bigcap_{\Omega\Subset
U}\bigcap_{M\in\mathfrak{M}}\bigcap_{\rho>0}\bigcup_{C>0}\\{f\in
C^{\infty}(U):f\text{ satisfies }\eqref{eq:DC3}\\},$

and in the evident analogous way also the global classes
$\mathcal{B}^{[\mathfrak{M}]}_{L^{p}}(\mathbb{R}^{n})$,
$\mathcal{S}^{[\mathfrak{M}]}(\mathbb{R}^{n})$, and
$\mathcal{D}^{[\mathfrak{M}]}(\mathbb{R}^{n})$ as well as
$\mathcal{E}^{[\mathfrak{M}]}_{L^{p},(k_{j})}(U)$,
$\mathcal{B}^{[\mathfrak{M}]}_{L^{p},(k_{j})}(\mathbb{R}^{n})$,
$\mathcal{S}^{[\mathfrak{M}]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})$, and
$\mathcal{D}^{[\mathfrak{M}]}_{(k_{j})}(\mathbb{R}^{n})$.

######  Theorem 5.1.

Let $1\leq p\leq\infty$. Let $\omega$ be a weight function. For
$\mathfrak{M}:=\\{M^{(\rho)}\\}_{\rho>0}$, where
$M^{(\rho)}_{k}:=\exp(\tfrac{1}{\rho}\varphi^{*}(\rho k))$, we have the
identities:

$\displaystyle\mathcal{E}^{[\omega]}_{L^{p}}(U)=\mathcal{E}^{[\mathfrak{M}]}_{L^{p}}(U),$
$\displaystyle\quad\mathcal{E}^{[\omega]}_{L^{p},(k_{j})}(U)=\mathcal{E}^{[\mathfrak{M}]}_{L^{p},(k_{j})}(U),$
$\displaystyle\mathcal{B}^{[\omega]}_{L^{p}}(\mathbb{R}^{n})=\mathcal{B}^{[\mathfrak{M}]}_{L^{p}}(\mathbb{R}^{n}),$
$\displaystyle\quad\mathcal{B}^{[\omega]}_{L^{p},(k_{j})}(\mathbb{R}^{n})=\mathcal{B}^{[\mathfrak{M}]}_{L^{p},(k_{j})}(\mathbb{R}^{n}),$
$\displaystyle\mathcal{S}^{[\omega]}(\mathbb{R}^{n})=\mathcal{S}^{[\mathfrak{M}]}(\mathbb{R}^{n}),$
$\displaystyle\quad\mathcal{S}^{[\omega]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})=\mathcal{S}^{[\mathfrak{M}]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n}),$
$\displaystyle\mathcal{D}^{[\omega]}(\mathbb{R}^{n})=\mathcal{D}^{[\mathfrak{M}]}(\mathbb{R}^{n}),$
$\displaystyle\quad\mathcal{D}^{[\omega]}_{(k_{j})}(\mathbb{R}^{n})=\mathcal{D}^{[\mathfrak{M}]}_{(k_{j})}(\mathbb{R}^{n}),$

where $(k_{j})$ and $(\ell_{j})$ are base sequences.

###### Proof.

This is shown by the proof of [19, Lemma 5.14] which is based on the following
property of $\mathfrak{M}$, see [19, (5.10)],

(15) $\;\forall\sigma>0\;\exists H\geq 1\;\forall\rho>0\;\exists C\geq
1\;\forall k\in\mathbb{N}:\sigma^{k}M^{(\rho)}_{k}\leq C\,M^{(H\rho)}_{k}.$

In fact, for given $\rho,\sigma>0$ there exist $C,H\geq 1$ such that

$\displaystyle\sup_{j\geq
0}\frac{\max_{|\alpha|=k_{j}}\|\partial^{\alpha}f\|_{L^{p}(\Omega)}}{M^{(H\rho)}_{k_{j}}}\leq
C\,\sup_{j\geq
0}\frac{\max_{|\alpha|=k_{j}}\|\partial^{\alpha}f\|_{L^{p}(\Omega)}}{\sigma^{k_{j}}M^{(\rho)}_{k_{j}}}$

and $M^{(H\rho)}_{k_{j}}=\exp(\tfrac{1}{H\rho}\varphi^{*}(H\rho k_{j}))$. In
the Gelfand–Shilov case, we observe that additionally

$\displaystyle\sup_{j\geq
0}\frac{\max_{|\alpha|=\ell_{j}}\sup_{x}|x^{\alpha}f(x)|}{M^{(H\rho)}_{\ell_{j}}}\leq
C\,\sup_{j\geq
0}\frac{\max_{|\alpha|=\ell_{j}}\sup_{x}|x^{\alpha}f(x)|}{\sigma^{\ell_{j}}M^{(\rho)}_{\ell_{j}}}.$

The case
$\mathcal{S}^{[\omega]}(\mathbb{R}^{n})=\mathcal{S}^{[\mathfrak{M}]}(\mathbb{R}^{n})$
follows from

$\displaystyle\sup_{\alpha,\,\beta}\frac{\sup_{x}|x^{\alpha}\partial^{\beta}f(x)|}{M^{(H\rho)}_{|\alpha|}M^{(H\rho)}_{|\beta|}}\leq
C\,\sup_{\alpha,\,\beta}\frac{\sup_{x}|x^{\alpha}\partial^{\beta}f(x)|}{\tau^{|\alpha|}\sigma^{|\beta|}M^{(\rho)}_{|\alpha|}M^{(\rho)}_{|\beta|}}.$

taking constants $C,H$ which work for both $\sigma$ and $\tau$. ∎

The family $\mathfrak{M}$ from Theorem 5.1 has an inherent moderate growth
property (see [19, (5.6)]):

(16) $M^{(\rho)}_{j+k}\leq M^{(2\rho)}_{j}M^{(2\rho)}_{k},\quad j,k\geq 0.$

In general, we say that $\mathfrak{M}$ has [_moderate growth_] if, in the
Roumieu case (i.e., $[\cdot]=\\{\cdot\\}$),

$\;\forall M\in\mathfrak{M}\;\exists M^{\prime}\in\mathfrak{M}\;\exists
C>0\;\forall j,k\geq 0:M_{j+k}\leq C^{j+k}M^{\prime}_{j}M^{\prime}_{k},$

and, in the Beurling case (i.e., $[\cdot]=(\cdot)$),

$\;\forall M\in\mathfrak{M}\;\exists M^{\prime}\in\mathfrak{M}\;\exists
C>0\;\forall j,k\geq 0:M^{\prime}_{j+k}\leq C^{j+k}M_{j}M_{k}.$

######  Lemma 5.2.

If $(k_{j})$ is a special base sequence and $\mathfrak{M}$ has [moderate
growth], then, in the Roumieu case (i.e., $[\cdot]=\\{\cdot\\}$),

(17) $\;\forall M\in\mathfrak{M}\;\exists
M^{\prime}\in\mathfrak{M}:m_{k_{j+1}}/m^{\prime}_{k_{j}}\text{ is bounded},$

and, in the Beurling case (i.e., $[\cdot]=(\cdot)$),

(18) $\;\forall M\in\mathfrak{M}\;\exists
M^{\prime}\in\mathfrak{M}:m^{\prime}_{k_{j+1}}/m_{k_{j}}\text{ is bounded}.$

###### Proof.

By assumption, there exists a positive integer $a$ such that $k_{j+1}\leq
ak_{j}$ for all $j\geq 1$. Moderate growth in the Roumieu case guarantees
that, for given $M\in\mathfrak{M}$ there exist $C>0$ and
$M^{\prime}\in\mathfrak{M}$ (depending on $a$) such that

$M_{ak_{j}}\leq C^{ak_{j}}(M^{\prime}_{k_{j}})^{a}.$

This implies $m_{k_{j+1}}\leq m_{ak_{j}}\leq Cm^{\prime}_{k_{j}}$. The
Beurling case is similar. ∎

######  Theorem 5.3.

Let $1\leq p\leq\infty$. Let $(k_{j}),(\ell_{j})$ be base sequences. Let
$\mathfrak{M}$ satisfy (17) in the Roumieu case and (18) in the Beurling case.
We have:

  1. (a)

$\mathcal{B}^{[\mathfrak{M}]}_{L^{p},(k_{j})}(\mathbb{R}^{n})=\mathcal{B}^{[\mathfrak{M}]}_{L^{p}}(\mathbb{R}^{n})$.

  2. (b)

$\mathcal{D}^{[\mathfrak{M}]}_{(k_{j})}(\mathbb{R}^{n})=\mathcal{D}^{[\mathfrak{M}]}(\mathbb{R}^{n})$.

Assume additionally that, for all $M\in\mathfrak{M}$, $j/m_{j}$ is bounded, in
the Roumieu case, and tends to zero, in the Beurling case. Then:

  1. (c)

$\mathcal{E}^{[\mathfrak{M}]}_{L^{p},(k_{j})}(U)=\mathcal{E}^{[\mathfrak{M}]}_{L^{p}}(U)$
for all open $U\subseteq\mathbb{R}^{n}$, provided that $(k_{j})$ is a special
base sequence.

  2. (d)

$\mathcal{S}^{[\mathfrak{M}]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})=\mathcal{S}^{[\mathfrak{M}]}(\mathbb{R}^{n})$,
provided that $\mathfrak{M}$ has [moderate growth].

###### Proof.

Since the sequences in $\mathfrak{M}$ are totally ordered, we may assume that
the sequences provided by (17) and (18) also satisfy $m_{j}\leq
m^{\prime}_{j}$ or $m^{\prime}_{j}\leq m_{j}$ for all $j$, respectively. Then
(a), (b), and (c) follow easily from Proposition 3.1 and Proposition 3.2.
Finally, (d) follows from Proposition 3.1, Proposition 3.4, and the
generalization [17, Lemma 4] of Proposition 4.2. ∎

To deduce a version for Braun–Meise–Taylor classes, it suffices to note that
for a weight function $\omega$ and $\mathfrak{M}=\\{M^{(\rho)}\\}_{\rho>0}$
the associated family from Theorem 5.1,

  * •

$\omega(t)=O(t)$ as $t\to\infty$ if and only if $j/m^{(\rho)}_{j}$ is bounded
for some $\rho>0$,

  * •

$\omega(t)=o(t)$ as $t\to\infty$ if and only if $j/m^{(\rho)}_{j}\to 0$ for
all $\rho>0$;

cf. [19, Lemma 5.7 and Corollary 5.15].

######  Theorem 5.4.

Let $1\leq p\leq\infty$. Let $(k_{j})$, $(\ell_{j})$ be special base
sequences. Let $\omega$ be a weight function. We have:

  1. (a)

$\mathcal{B}^{[\omega]}_{L^{p},(k_{j})}(\mathbb{R}^{n})=\mathcal{B}^{[\omega]}_{L^{p}}(\mathbb{R}^{n})$.

  2. (b)

$\mathcal{D}^{[\omega]}_{(k_{j})}(\mathbb{R}^{n})=\mathcal{D}^{[\omega]}(\mathbb{R}^{n})$.

Assume additionally that $\omega(t)=O(t)$ as $t\to\infty$, in the Roumieu
case, and $\omega(t)=o(t)$, in the Beurling case. Then:

  1. (c)

$\mathcal{E}^{[\omega]}_{L^{p},(k_{j})}(U)=\mathcal{E}^{[\omega]}_{L^{p}}(U)$
for all open $U\subseteq\mathbb{R}^{n}$.

  2. (d)

$\mathcal{S}^{[\omega]}_{(k_{j}),(\ell_{j})}(\mathbb{R}^{n})=\mathcal{S}^{[\omega]}(\mathbb{R}^{n})$.

###### Proof.

This is a corollary of Theorem 5.1, Lemma 5.2, and Theorem 5.3. ∎

######  Remark 5.5.

The family $\mathfrak{M}=\\{M^{(\rho)}\\}_{\rho>0}$ associated with a weight
function in Theorem 5.1 has the following property:

  * (a)

Let $0<\sigma\leq\rho$. If there is $C_{1}>0$ such that $m^{(\sigma)}_{j}\leq
C_{1}\,m^{(\rho)}_{k}$, then there is $C_{2}=C_{2}(\sigma,\rho,C_{1})>0$ such
that $j\leq C_{2}\,k$.

So, in this case, validity of (17) or (18) entails that $(k_{j})$ is a special
base sequence; the converse holds by (16) and Lemma 5.2.

To see (a), note first that for any positive integer $N$ we have

$m_{k}^{(N\rho)}=\exp(\tfrac{1}{N\rho k}\varphi^{*}(N\rho
k))=m_{Nk}^{(\rho)}.$

By (15), there exist $C=C(\rho)\geq 1$ and an integer $H\geq 2$ such that

$4m_{k}^{(\rho)}\leq C^{1/k}m_{k}^{(H\rho)}=C^{1/k}m_{Hk}^{(\rho)},\quad k\geq
0.$

For $k\geq k_{0}$, we have $C^{1/k}<2$ and hence
$2m_{k}^{(\rho)}<m_{Hk}^{(\rho)}$ and, by iteration,

$2^{\ell}m_{k}^{(\rho)}<m_{H^{\ell}k}^{(\rho)}.$

If $m^{(\sigma)}_{j}\leq C_{1}\,m^{(\rho)}_{k}$, then, choosing $\ell$ such
that $C_{1}2^{-\ell}<1$, we find

$m^{(\sigma)}_{j}<m^{(\rho)}_{H^{\ell}k}.$

Then the assumption $j\geq\lceil\frac{\rho}{\sigma}\rceil H^{\ell}k$ leads to

$m^{(\rho)}_{H^{\ell}k}\leq
m^{(\lceil\frac{\rho}{\sigma}\rceil\sigma)}_{H^{\ell}k}=m^{(\sigma)}_{\lceil\frac{\rho}{\sigma}\rceil
H^{\ell}k}<m^{(\rho)}_{H^{\ell}k},$

a contradiction. Thus $j<C_{2}\,k$, where
$C_{2}:=\lceil\frac{\rho}{\sigma}\rceil H^{\ell}$. For $k<k_{0}$, there are
only finitely many $j$ satisfying $m_{j}^{(\sigma)}\leq
C_{1}\,m_{k_{0}}^{(\rho)}$ (since $\lim_{j\to\infty}m^{(\sigma)}_{j}=\infty$;
see [3, Remark 1.3]). So adding this bound to $C_{2}$ gives the assertion for
all $k$.

######  Remark 5.6.

Under the [moderate growth] assumption, we have, for $1\leq p\leq
q\leq\infty$, the continuous inclusions
$\mathcal{B}^{[\mathfrak{M}]}_{L^{p}}(\mathbb{R}^{n})\subseteq\mathcal{B}^{[\mathfrak{M}]}_{L^{q}}(\mathbb{R}^{n})$
and
$\mathcal{E}^{[\mathfrak{M}]}_{L^{p}}(U)=\mathcal{E}^{[\mathfrak{M}]}_{L^{q}}(U)$,
in particular,
$\mathcal{B}^{[\omega]}_{L^{p}}(\mathbb{R}^{n})\subseteq\mathcal{B}^{[\omega]}_{L^{q}}(\mathbb{R}^{n})$
and $\mathcal{E}^{[\omega]}_{L^{p}}(U)=\mathcal{E}^{[\omega]}_{L^{q}}(U)$.

##  6\. Complements

In this section, we discuss the setting considered in [1].

Let $M=(M_{j})$ be a weight sequence. Let $I\subseteq\mathbb{R}$ be a compact
interval. We define

$\displaystyle\mathcal{B}^{\\{M\\}}_{L^{\infty}}(I)$ $\displaystyle:=\\{f\in
C^{\infty}(I):\;\exists C,\rho>0\;\forall j\geq
0:\|f^{(j)}\|_{L^{\infty}(I)}\leq C(\rho m_{j})^{j}\\},$
$\displaystyle\mathcal{B}^{(M)}_{L^{\infty}}(I)$ $\displaystyle:=\\{f\in
C^{\infty}(I):\;\forall\rho>0\;\exists C>0\;\forall j\geq
0:\|f^{(j)}\|_{L^{\infty}(I)}\leq C(\rho m_{j})^{j}\\},$

as well as $\mathcal{B}^{[M]}_{L^{\infty},(k_{j})}(I)$ in the obvious way.

######  Theorem 6.1.

Let $(k_{j})$ be a special base sequence. Let $M=(M_{j})$ be a weight sequence
such that $m_{k_{j+1}}/m_{k_{j}}$ is bounded. Then:

  1. (a)

$\mathcal{B}^{\\{M\\}}_{L^{\infty},(k_{j})}(I)=\mathcal{B}^{\\{M\\}}_{L^{\infty}}(I)$
provided that $j/m_{j}$ is bounded.

  2. (b)

$\mathcal{B}^{(M)}_{L^{\infty},(k_{j})}(I)=\mathcal{B}^{(M)}_{L^{\infty}}(I)$
provided that $j/m_{j}\to 0$.

###### Proof.

Assume that $f\in\mathcal{B}^{\\{M\\}}_{L^{\infty},(k_{j})}(I)$. Let $\ell\geq
0$ and let $j\geq 0$ be such that $k_{j}\leq\ell<k_{j+1}$. By Proposition 2.6
and the properties of $(k_{j})$ and $(m_{j})$,

$\displaystyle\|f^{(\ell)}\|_{L^{\infty}(I)}\leq 4e^{2(\ell-
k_{j})}e^{k_{j+1}-k_{j}}\,\|f^{(k_{j})}\|_{L^{\infty}(I)}^{1-\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}$ $\displaystyle\hskip
56.9055pt\cdot\max\Big{\\{}\|f^{(k_{j+1})}\|_{L^{\infty}(I)}^{\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}},\Big{(}\frac{k_{j+1}-k_{j}}{|I|}\Big{)}^{\ell-
k_{j}}\|f^{(k_{j})}\|_{L^{\infty}(I)}^{\frac{\ell-
k_{j}}{k_{j+1}-k_{j}}}\Big{\\}}$ $\displaystyle\leq 4C\,C_{1}^{\ell}\,(\rho
m_{k_{j}})^{k_{j}\big{(}1-\frac{\ell-k_{j}}{k_{j+1}-k_{j}}\big{)}}\Big{(}(\rho
m_{k_{j+1}})^{\frac{k_{j+1}(\ell-
k_{j})}{k_{j+1}-k_{j}}}+\Big{(}\frac{C_{2}k_{j}}{|I|}\Big{)}^{\ell-k_{j}}(\rho
m_{k_{j}})^{\frac{k_{j}(\ell-k_{j})}{k_{j+1}-k_{j}}}\Big{)}$
$\displaystyle\leq 4C\,C_{1}^{\ell}\,\Big{(}(C_{3}\rho
m_{k_{j}})^{\ell}+\Big{(}\frac{C_{4}m_{k_{j}}}{|I|}\Big{)}^{\ell-k_{j}}(\rho
m_{k_{j}})^{k_{j}}\Big{)}$ $\displaystyle\leq
8C\,(C_{5}\max\\{\rho,C_{4}|I|^{-1}\\})^{\ell}\,m_{\ell}^{\ell}.$

Hence $f\in\mathcal{B}^{\\{M\\}}_{L^{\infty}}(I)$.

Assume that $f\in\mathcal{B}^{(M)}_{L^{\infty},(k_{j})}(I)$. Let $\sigma>0$ be
given. If we assume that $j/m_{j}\to 0$, then we may proceed as in the proof
of Proposition 3.2 to see that choosing $\rho>0$ sufficiently small we may
find

$\|f^{(\ell)}\|_{L^{\infty}(I)}\leq D(\sigma m_{\ell})^{\ell},\quad\ell\geq
0,$

where $D=D(\sigma)$. ∎

It is now an easy exercise to deduce the following extensions.

######  Theorem 6.2.

Let $(k_{j})$ be a special base sequence. Let $\mathfrak{M}$ have [moderate
growth] and assume that, for all $M\in\mathfrak{M}$, $j/m_{j}$ is bounded, in
the Roumieu case, and tends to zero, in the Beurling case. Then
$\mathcal{B}^{[\mathfrak{M}]}_{L^{\infty},(k_{j})}(I)=\mathcal{B}^{[\mathfrak{M}]}_{L^{\infty}}(I)$.

######  Theorem 6.3.

Let $(k_{j})$ be a special base sequence. Let $\omega$ be a weight function
such that $\omega(t)=O(t)$ as $t\to\infty$, in the Roumieu case, and
$\omega(t)=o(t)$, in the Beurling case. Then
$\mathcal{B}^{[\omega]}_{L^{\infty},(k_{j})}(I)=\mathcal{B}^{[\omega]}_{L^{\infty}}(I)$.

######  Remark 6.4.

In [1], instead of assuming $m_{k_{j+1}}/m_{k_{j}}$ bounded, the authors
require that

(19) $\;\exists i_{0}\geq 0\;\forall i,j>i_{0}\text{ and }i<j<ik:m_{j}\leq
m_{i}m_{k}.$

If there is $\epsilon>0$ such that $m_{j}>\epsilon$ for all $j$, then it is
not hard to check that (19) implies

(20) $\;\exists C\geq 1\;\forall i,j\geq 1\text{ and }i<j<ik:m_{j}\leq
C\,m_{i}m_{k}.$

But (20) (for $k=3$ and $j=2i$) implies that $M$ has moderate growth (cf. [15,
Theorem 1]). And, if $(k_{j})$ is a special base sequence, then Lemma 5.2
implies that $m_{k_{j+1}}/m_{k_{j}}$ is bounded.

######  Remark 6.5.

In Theorem 6.1, the assumption that $(k_{j})$ is a _special_ base sequence
cannot be omitted. In fact, by [1, Theorem 1.6], for each positive sequence
$M=(M_{j})$ such that $\limsup_{j\to\infty}m_{j}=\infty$ there is a base
sequence $(k_{j})$ and $f\in\mathcal{B}^{\\{M\\}}_{L^{\infty},(k_{j})}(I)$ but
$f\not\in\mathcal{B}^{\\{M\\}}(I)$. More precisely, a weight sequence
$N=(N_{j})$ and sequences of positive integers $(k_{j})$ and $(\ell_{j})$ with
the following properties are constructed:

  * •

$\cdots<\ell_{j}<k_{j}<\ell_{j+1}<k_{j+1}<\cdots$,

  * •

$N_{k_{j}}=M_{k_{j}}$, and

  * •

$N_{\ell_{j}}=2^{2^{\ell_{j}}}M_{\ell_{j}}$.

It is well-known that for each weight sequence $N$ there exists
$f\in\mathcal{B}^{\\{N\\}}_{L^{\infty}}(\mathbb{R})$ with $|f^{(j)}(0)|\geq
N_{j}$ for all $j\geq 0$, from which the assertion follows easily.

Note that this result does not contradict Theorem 4.1: if also $M=(M_{j})$ is
a weight sequence, then the above conditions imply

$m_{k_{j+1}}=n_{k_{j+1}}\geq
n_{\ell_{j+1}}=2^{2^{\ell_{j+1}}/\ell_{j+1}}m_{\ell_{j+1}}\geq
2^{2^{\ell_{j+1}}/\ell_{j+1}}m_{k_{j}}$

so that $m_{k_{j+1}}/m_{k_{j}}$ is unbounded.

It is possible to adapt the proof of this result to the Beurling case as well
as to the general setting of a weight structure $\mathfrak{M}$; we omit the
details.

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