# Tameness in the Fréchet spaces of analytic functions

Aydın Aytuna  MDBF, Sabancı Üniversitesi. Orhanlı, Tuzla 34956 İstanbul
<EMAIL_ADDRESS>Dedicated to the memory of Tosun Terzioğlu

###### Abstract.

A Fréchet space, $\mathcal{X}$, with a sequence of generating semi-norms
$\\{\|\cdot\|_{k}\\}_{k=1}^{\infty}$ is called _tame_ in case there exists an
increasing function $\sigma:\ \mathbb{N}\rightarrow\mathbb{N}$, such that for
every continuous linear operator $T$ from $\mathcal{X}$ into itself, there
exists an $N_{0}$ and $C>0$ such that

$\|T(x)\|_{n}\leq C\|x\|_{\sigma(n)},\quad\forall x\in\mathcal{X}\text{ and
}n\geq N_{0}.$

This property does not depend upon the choice of fundamental system of semi-
norms for $\mathcal{X}$ and is a property of the Fréchet space $\mathcal{X}$.
In this paper we investigate tameness in the Fréchet spaces $\mathcal{O}(M)$
of analytic functions on Stein manifolds $M$ equipped with the compact open
topology. Actually we will look into tameness in the more general class of
nuclear Fréchet spaces with the properties $\underline{DN}$ and $\Omega$ of
Vogt and then specialize to analytic function spaces. We will show that for a
Stein manifold $M$, tameness of $\mathcal{O}(M)$ is equivalent to the
hyperconvexity of $M$.

###### Key words and phrases:

tameness of Fréchet spaces, analytic functions spaces, linear topological
invariants

###### 2000 Mathematics Subject Classification:

Primary 46A61, 46E10, 32A70, 46A63; Secondary 32U15

##  1\. Introduction

Tameness of Fréchet spaces is an important property frequently used in
functional analysis since it brings a kind of control on the otherwise chaotic
behaviour of continuous linear endomorphisms. This concept was used
effectively in the structure theory of nuclear Fréchet spaces, especially in
finding bases in complemented subspaces of certain infinite type power series
spaces [9]. Frequently the Fréchet spaces that arise in practical applications
(e.g. in non-linear analysis), enjoy (versions of) the tameness property ([11,
28]). In fact, an inquiry about the tameness of the analytic function spaces
was conveyed, sometime ago, in this context by D. Zarnadze [private
communication]. In this paper we answer this question in a kind of negative
way. We show that there are no tame analytic function spaces other than the
natural ones (Theorem 4.4).

The organization of the paper is as follows: After establishing the notation
and terminology, in section 1, we recall the definition of the linear
topological invariants $\underline{DN}$ and $\Omega$ and introduce the
technical tool that we will use in the later sections, namely local imbeddings
of power series spaces of finite type into Fréchet spaces. After some general
results on local imbeddings we establish a link between the approximate
diametral dimension of the space and the existence of effective local
imbeddings from certain finite type power series spaces (Theorem 2.9).

Section 2 is devoted to Theorem 3.2 and its proof which characterizes tame
nuclear Fréchet spaces having a finitely nuclear stable exponent sequence and
enjoying the properties $\underline{DN}$ and $\Omega$.

In the first subsection of section 3 we show how the local imbeddings can be
used to construct Green’s functions on complex manifolds. The second
subsection of the same section is devoted to diametral dimension
considerations of analytic function spaces and the proof of the main theorem
Theorem 4.4 of this paper.

### Notations and Terminology:

We will use the terminology of [15] and refer the reader to this book for the
undefined concepts and the standard results of functional analysis that we
will use. For the notions from complex potential theory used (especially in
section 3) we refer the reader to [12].

Power series sequence spaces play an important role in this paper. Recall that
these are Fréchet spaces

$\Lambda_{R}(\alpha)\doteq\\{(\xi_{n})^{\infty}_{n=0}:\
|(\xi_{n})|_{r}\doteq\left(\sum_{n}|\xi_{n}|^{2}e^{2r\alpha_{n}}\right)^{\frac{1}{2}}<\infty,\quad\forall\
-\infty<r<R\\}$

where $R$ is either $1$ or $\infty$, and $\alpha=(\alpha_{n})_{n}$ is a
sequence of complex numbers with $\sup\frac{\ln(n)}{\alpha_{n}}<+\infty$,
called the _exponent sequence_ of the space. The grading on these spaces will
be the hilbertian grading $\\{|\cdot|_{r}\\}_{r<R}$.

These spaces are referred to as _finite type power_ series spaces if $R=1$ and
_infinite type power_ series spaces if $R=\infty$, respectively. We will use
the notation $\Lambda_{1}[\alpha]$ to denote the Hilbert space
$\\{(\xi_{n})^{\infty}_{n=0}:\
|(\xi_{n})|_{r}\doteq\left(\sum_{n}|\xi_{n}|^{2}e^{2r\alpha_{n}}\right)^{\frac{1}{2}},$
$\forall\ 0<r<1\\}$ with the norm $|\cdot|_{r}$, $r\in\mathbb{R}$.

For a pair of Fréchet spaces $\\{\mathcal{X},\ ^{\|\cdot\|_{k}}\\}_{k}$ and
$\\{\mathcal{Y},\ ^{|\ |_{k}}\\}_{k}$ the set of all continuous linear
operators will be denoted by $\mathcal{L}(\mathcal{X},\mathcal{Y})$. We will
use the symbol

$\|T\|^{n}_{m}\doteq\sup_{\|x\|_{n\leq 1}}|Tx|_{m},\quad n,m\in\mathbb{N}$

for a $T\in\mathcal{L}(\mathcal{X},\mathcal{Y})$. Note that
$\|\cdot\|_{m}^{n}$ can take the value $+\infty$.

For a Fréchet space, $\\{\mathcal{X},\ ^{\|\cdot\|_{k}}\\}_{k}$, the local
Hilbert spaces corresponding to the norm $\|\ \|_{k}$ will be denoted by
$\mathcal{X}_{k}$, $k=1,2,\dots$. The closed unit ball in $\mathcal{X}$
corresponding to the semi-norm $\|\ \|_{k}$ will be denoted by $U_{k}$,
$k=1,2,\dots$.

Throughout the paper we will reserve the symbol $\epsilon_{n}$ to denote the
sequence $(0,0,\dots,0,1,0,\dots)$ where $1$ is in the $n^{\text{th}}$ place.

##  2\. Local Imbeddings of Power Series Spaces of Finite Type into Fréchet
Spaces

In this section we will examine local imbeddings of power series spaces of
finite type into nuclear Fréchet spaces with the properties $\underline{DN}$
and $\Omega$.

In subsection 2.1 we recall the definition of the linear topological
invariants $\underline{DN}$ and $\Omega$ of Vogt and list some properties of
Fréchet spaces with properties $\underline{DN}$ and $\Omega$, that will be
used in the sequel. We emphasize again that all the Fréchet spaces that we
will deal with in this paper will be assumed to be nuclear and satisfy the
properties $\underline{DN}$ and $\Omega$ unless otherwise specified. In
subsection 2.2 we will investigate the existence of local imbeddings. In
subsection 2.3 we consider Fréchet spaces whose diametral approximative
dimension is equal to a finite type power series space and show that this
property yields local imbeddings.

######  Definition 2.1.

A continuous linear operator $T$ from $\Lambda_{1}(\alpha)$ into a graded
Fréchet space $\\{\mathcal{X},\ ^{{\|\cdot\|}_{n}}\\}$ is called an _$(r,k)$
-local imbedding_ in case;

$\exists\ C>0:\quad{\|T(x)\|}_{k}\geq C{|x|}_{r},\quad\forall\
x\in\Lambda_{1}(\alpha).$

We will say that a Fréchet space $\mathcal{X}$ admits an _r-local imbedding_
from $\Lambda_{1}(\alpha)$, in case there exists a continuous linear operator
$T$ from $\Lambda_{1}(\alpha)$ in $\mathcal{X}$ with the property;

$\displaystyle\exists\text{ continuous semi-norm }\ ^{\|\cdot\|}\text{ of
}\mathcal{X}\text{ and }C>0$ $\displaystyle\text{ such that }\|T(x)\|\geq
C|x|_{r}\quad\forall\ x\in\Lambda_{1}(\alpha).$

###  2.1. Linear topological invariants $\underline{DN}$, $\Omega$ and
associated exponent sequences

######  Definition 2.2.

A nuclear Fréchet space $\mathcal{X}$ is said to have properties
$\underline{DN}$ and $\Omega$ ([21], [24]) in case there exists a fundamental
system of hilbertian norms $\\{\|\ \|_{k}\\}_{k}$ generating the topology of
$\mathcal{X}$ which satisfy

$\displaystyle(\underline{DN}):\qquad$ $\displaystyle\forall k\ \exists\
0<\lambda<1\text{ and }C>0\text{ such that }$
$\displaystyle\quad{\|x\|}_{k+1}\leq C\ {\|x\|}^{\lambda}_{k}\
{\|x\|}^{1-\lambda}_{k+2}\quad\forall\ x\in\mathcal{X}$
$\displaystyle(\Omega):\qquad$ $\displaystyle\forall k\ \forall p\ \exists
C>0\text{, }j\in\mathbb{N}\text{ such that }$ $\displaystyle\quad
U_{k+1}\subset Cr^{j}U_{p}+\frac{1}{r}U_{k}\quad\forall\ r>0$

For more information about these invariants and examples of Fréchet spaces
possessing these properties we refer the reader to [15].

Let $U$ and $V$, $V\subseteq U$ be 2 subsets of a Fréchet space $\mathcal{X}$.
We will use the notation $\mathcal{E}_{n}(U,V)$ to denote

$\mathcal{E}_{n}(U,V)\doteq-\ln d_{n}(V,U),\quad n=0,1,2$

where

$d_{n}(V,U)\doteq\inf_{L_{n}}\ \inf\left\\{\lambda:\ \lambda>0,\
V\subseteq\lambda U+L_{n}\right\\}$

(infimum is taken over all n-dimensional subspaces $L_{n}$ of $\mathcal{X}$)
is the $n^{th}$ Kolmogorov diameter of $V$ in $U$.

In the case of a graded Fréchet space $(\mathcal{X},\ ^{{\|\ \|}_{k}})$, we
simplify this notation by setting

$\mathcal{E}_{n}(p,k)\doteq\mathcal{E}_{n}(U_{p},U_{k}),\quad\forall\ p<k.$

Let us now fix a nuclear Fréchet space $\mathcal{X}$, with the properties
$\underline{DN}$ and $\Omega$, and choose the grading ${\left\\{{\|\
\|}_{k}\right\\}}_{k}$, derived from these properties. In [5] (c.f [4]) we
have shown that all the sequences ${\\{\mathcal{E}_{n}(k,k+1)\\}}_{n}$,
$k=0,1,2,\dots$ are equivalent and called this equivalence class the _exponent
sequence associated_ to $\mathcal{X}$. In this paper we, unless stated
otherwise, will use a concrete representation,
$\mathcal{E}_{n}\doteq\mathcal{E}_{n}(0,1),\ n=0,1,2$, of the associate
exponent sequence of $\mathcal{X}$.

We compile ([20]) some properties of these sequences that will be used in the
sequel.

######  Proposition 2.3.

Let $(\mathcal{X},\ ^{{\|\ \|}_{k}})$ be as above. An $\Omega$-type condition;

$\exists\ 0<\lambda<1,D>0:\quad U_{q}\subset
r^{\frac{1-\lambda}{\lambda}}U_{p}+\frac{D}{r}U_{k},\quad\forall\
r>C\quad(p<q<1)$

implies the inequalities:

  1. (1)

$\exists\ C>0$; $\mathcal{E}_{n}(q,k)\leq\lambda\mathcal{E}_{n}(p;k)+C$,
$\forall\ n$,

  2. (2)

$\exists\ C>0$;
$\mathcal{E}_{n}(p,k)\leq\frac{1}{1-\lambda}\mathcal{E}_{n}(p,q)+C$, $\forall\
n$,

  3. (3)

$\exists\ C>0$;
$\mathcal{E}_{n}(q,k)\leq\frac{\lambda}{1-\lambda}\mathcal{E}_{n}(p,q)+C$,
$\forall\ n$,

A $\underline{DN}$-type condition:

$\exists\ 0<\lambda<1,D>0:\quad{\|\cdot\|}_{q}\leq
D{\|\cdot\|}_{k}^{1-\lambda}{\|\cdot\|}_{p}^{\lambda}\quad(p<q<k)$

implies the inequalities:

  1. (1’)

$\exists\ C>0$;
$\mathcal{E}_{n}(p,k)\leq\frac{1}{\lambda}\mathcal{E}_{n}(q;k)+C$,
$n=0,1,\dots$,

  2. (2’)

$\exists\ C>0$; $\mathcal{E}_{n}(p,q)\leq(1-\lambda)\mathcal{E}_{n}(p;k)+C$,
$n=0,1,\dots$,

  3. (3’)

$\exists\ C>0$;
$\mathcal{E}_{n}(p,q)\leq\frac{\lambda}{1-\lambda}\mathcal{E}_{n}(q,k)+C$,
$n=0,1,\dots$,

In the presence of both $\Omega$ and $DN$-type conditions we have

  1. (4)

$\exists\ C>0,\ 0<\lambda_{1}<1,\ 0<\lambda_{2}<1;\
\mathcal{E}_{n}\leq\frac{\lambda_{1}\lambda_{2}}{(1-\lambda_{1})(1-\lambda_{2})}\mathcal{E}_{n}(p,q)+C,\
\forall\ n$

  2. (4’)

$\exists\ C>0,\ 0<\lambda_{1}<1,\ 0<\lambda_{2}<1;\
\mathcal{E}_{n}(p,q)\leq\frac{\lambda_{1}\lambda_{2}}{(1-\lambda_{1})(1-\lambda_{2})}\mathcal{E}_{n}+C,\
n=0,1,...$

###### Proof.

See [20] (c.f [5]). ∎

###  2.2. Local imbeddings of power series spaces of finite type into nuclear
Fréchet spaces with properties $\underline{DN}$ and $\Omega$

We now turn our attention to the question of existence of local imbeddings
into the graded Fréchet space $(\mathcal{X},\ ^{{\|\ \|}_{k}})$ fixed in
subsection 2.1 above. But we first consider some general considerations
concerning local imbeddings.

######  Proposition 2.4.

Let $\Lambda_{1}(\alpha)$ be a nuclear power series space of finite type. The
following assertions are equivalent;

  1. (i)

There exists an $(r,k)$-local imbedding from $\Lambda_{1}(\alpha)$ into
$\mathcal{X}$

  2. (ii)

There exists a sequence $\left\\{g_{n}\right\\}_{n=0}^{\infty}$ in
$\mathcal{X}$ that is orthogonal in $\mathcal{X}_{k}$ and satisfies;

$\overline{\lim}_{n}\frac{\ln\|g_{n}\|_{s}}{\alpha_{n}}<1-r,\quad\forall\ s$

  3. (iii)

There exists an isometry from the local Hilbert space $\Lambda_{r}[\alpha]$
into $H_{k}$ that induces a continuous linear operator from
$\Lambda_{1}(\alpha)$ into $\mathcal{X}$

  4. (iv)

There exists a closed, Hilbertian bounded disc $B$ which satisfies;

$\exists\ C>0\text{ such that }d_{n}(B,U_{k})\geq Ce^{-(1-r)\alpha_{n}}$

###### Proof.

  1. (i)$\Rightarrow$(ii):

Fix an $(r,k)$-local imbedding $T$ with $\|T(x)\|_{k}\geq C_{1}|x|_{r}$,
$\forall\ x\in\mathcal{X}$. Consider

$f_{n}\doteq\frac{\epsilon_{n}}{e^{r\alpha_{n}}},\quad n=0,1,2,\dots$

and apply the Gram-Schmidt orthogonalization procedure to the finitely
linearly independent sequence $\left\\{T(f_{n})\right\\}_{n}$ in
$\mathcal{X}_{k}$ to get a sequence,

$g_{n}\doteq\sum_{i=0}^{n}c_{i}^{n}T(f_{i}),\quad n=1,2,\dots$

in $\mathcal{X}$ that is orthogonal in $\mathcal{X}_{k}$. We estimate;

$1=\|g_{n}\|_{k}=\left\|T\left(\sum_{i=0}^{n}c_{i}^{n}g_{i}\right)\right\|_{k}\geq
C_{1}\left|\sum_{i=0}^{n}e_{i}^{n}\frac{\epsilon_{i}}{e^{r\alpha_{i}}}\right|_{r}=C_{1}\left(\sum|c_{i}^{n}|^{2}\right)^{\frac{1}{2}}.$

From continuity of $T$ we get for every $s$, a $r<\sigma(s)<1$ and $C>0$ such
that,

$\displaystyle\|g_{n}\|_{s}$
$\displaystyle=\left\|T\left(\sum_{i=0}^{n}c_{i}^{n}f_{i}\right)\right\|_{s}$
$\displaystyle\leq C\left|\sum_{i=0}^{n}c_{i}^{n}f_{i}\right|_{\sigma(s)}$
$\displaystyle\leq Ce^{(\sigma(s)-r)_{\alpha
n}}\left(\sum_{i=0}^{n}|c_{i}|^{2}\right)^{\frac{1}{2}}\leq\frac{C}{C_{1}}e^{(\sigma(s)-r)_{\alpha
n}}.$

Hence

$\overline{\lim}_{n}\frac{\ln\|g_{n}\|_{s}}{\alpha_{n}}\leq\sigma(s)-r<1-r.$

  2. (ii)$\Rightarrow$(iii):

Choose $\\{g_{n}\\}_{n}$ as in (ii) and set

$T\left(\sum_{i=0}^{\infty}c_{i}\epsilon_{i}\right)\doteq\sum_{i=0}^{\infty}e^{r\alpha_{i}}c_{i}g_{i}.$

This operator is plainly a continuous linear operator from
$\Lambda_{1}(\alpha)$ into $\mathcal{X}$ that extends to an isometry from
$\Lambda_{r}[\alpha_{n}]$ into $\mathcal{X}_{k}$.

  3. (iii)$\Rightarrow$(iv):

Choose an operator $T$ which has the properties given in (iii) and set;

$g_{n}\doteq T\left(\frac{\epsilon_{n}}{e^{r\alpha_{n}}}\right),\quad
n=0,1,\dots.$

Consider the set

$B\doteq\left\\{x\in\mathcal{X};\quad\sum_{n=0}^{\infty}|\langle
x,g_{n}\rangle_{k}|^{2}e^{2(1-r)\alpha_{n}}\leq 1\right\\},$

where $\langle,\rangle_{k}$ is the inner product corresponding to $\|\
\|_{k}$.

A direct computation will show that $B$ is the image under $T$ of the compact
set $B_{0}=\\{(\xi_{n});\quad\sum|\xi_{n}|^{2}e^{2\alpha_{n}}\leq 1\\}$ in
$\Lambda_{1}(\alpha)$. So $B$ is a compact disc that is Hilbertian and it
follows that (see Lemma 6.2.2 of [8])

$d_{n}(B,U_{k})\geq d_{n}(B\cap T(\mathcal{X}),T(\mathcal{X})\cap
U_{k})=e^{(r-1)\alpha_{n}}$

  4. (iv)$\Rightarrow$(i):

Let $H_{B}$ denote the Hilbert space generated by $B$ in $\mathcal{X}$. Since
the inclusion $H_{B}\hookrightarrow\mathcal{X}_{k}$ is a compact operator we
can choose a sequence $\\{f_{n}\\}_{n=0}^{\infty}$ of orthogonal vectors in
$H_{B}$ that are orthogonal in the Hilbert space $\mathcal{X}_{k}$ and
$\|f_{n}\|_{B}=\left(d_{n}(B,U_{k})\right)^{-1},\quad n=0,1,2,\dots$. Set

$T\big{(}(\xi_{n})_{n}\big{)}\doteq\sum_{n=0}^{\infty}\xi_{n}e^{r\alpha_{n}}f_{n},\quad\forall\
(\xi_{n})\in\Lambda_{1}(\alpha).$

We fix an $s>k$ and choose, in view of the condition $\underline{DN}$, a
$0<\lambda<1$ and $C>0$ such that $\|x\|_{s}\leq
C\|x\|_{k}^{1-\lambda}\|x\|_{B}^{\lambda},\quad\forall\ x\in H_{B}$.

Choose $\lambda^{+}$ so that, $0<r+\lambda(1-\lambda)<\lambda^{+}<1$. There
exists positive constants $C_{1},C_{2}$ such that;

$\displaystyle\left\|T\big{(}(\xi_{n})\big{)}_{n}\right\|_{s}$
$\displaystyle\leq\sum_{n}|\xi_{n}|e^{r\alpha_{n}}\|f_{n}\|_{s}$
$\displaystyle\leq
C_{1}\sum_{n}|\xi_{n}|e^{r\alpha_{n}}e^{\lambda(1-r)\alpha_{n}}$
$\displaystyle\leq
C_{2}\left(\sum_{n}|\xi_{n}|^{2}e^{2\lambda^{+}\alpha_{n}}\right)^{\frac{1}{2}}=C_{2}\left|(\xi_{n})\right|_{\lambda^{+}}.$

It follows that $T$ defines a continuous linear operator from
$\Lambda_{1}(\alpha)$ into $\mathcal{X}$. Moreover

$\|T((\xi_{n}))\|_{k}=\left(\sum_{n}|\xi_{n}|^{2}e^{2r\alpha_{n}}\right)^{\frac{1}{2}}=|(\xi_{n})|_{r}.$

Hence $T$ is a $(r,k)$-local imbedding.

∎

######  Remark.

Contrary to the infinite case ([5]), the existence of an $(r,k)$-local
imbedding from a finite type power series space into a graded Fréchet space
need not imply the existence of an $(r^{+},k)$-local imbedding for $r^{+}>r$.
In fact it is not difficult to see that for a nuclear finite type power series
space $\mathcal{X}=\Lambda_{1}(\alpha)$, there exists a $(r_{1},r_{2})$-local
imbedding from $\Lambda_{1}(\alpha)$ into $\mathcal{X}$ if and only if
$r_{1}\leq r_{2}$. Indeed if there is a bounded set $B$ in $\mathcal{X}$ such
that $d_{n}(B,U_{r_{2}})\geq Ce^{(r_{1}-1)\alpha_{n}}$ for some $C>0$, then
for any $r_{2}<s<1$, there exists a $C_{1}>0$ such that $B\subseteq
C_{1}U_{s}$ which implies that

$d_{n}(B,U_{r})\leq C_{1}d_{n}(U_{s},U_{r})=Ce^{(r_{2}-s)\alpha_{n}}$

for every $n$. Hence $r_{1}-1\leq r_{2}-1$ or $r_{1}\leq r_{2}$.

The condition in (iv) of the above Proposition can also be expressed by using
the unit balls of the grading:

######  Proposition 2.5.

There exists a $(r,p)$-local imbedding from $\Lambda_{1}(\alpha)$ into
$\mathcal{X}$ if and only if

$\sup_{k>p}\ \overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,k)}{\alpha_{n}}\leq
1-r.$

###### Proof.

  1. ($\Leftarrow$):

Let $\rho$ and $r$ satisfy the given inequality. For a given $k>p$ in view of
Proposition 2.4 (iii) we have;

$\exists\
0<\rho<1;\quad\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,k)}{\alpha_{n}}\leq\rho\
\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,k+1)}{\alpha_{n}}<1-r.$

So for each $k>p$ we can find a $N(k)\in\mathbb{N}$, strictly increasing with
respect to $k$, such that

$-\ln d_{n}(U_{k},U_{p})<(1-r)\alpha_{n}.$

Let $\delta_{1}=1/2e^{-(1-r)\alpha_{N(p+1)}}$. Since $U_{p+1}$ is precompact
in $\mathcal{X}_{p}$ there exists a finite set $Z_{1}^{1}\subseteq U_{p+1}$
such that;

$d_{n}(U_{p+1},U_{p})\leq d_{n}(Z_{1}^{1},U_{p})+\delta_{1}\quad\forall\ n$

Hence

$d_{N(p+1)}(Z_{1}^{1},U_{p})\geq
e^{-(1-r)\alpha_{n_{N(p+1)}}}-\delta_{1}=1/2e^{-(1-r)\alpha_{n_{N(p+1)}}}$

For each $n$, with $N(p+1)\leq n<N(p+2)$, we obtain, as above, a finite set
$Z^{1}_{n}\subseteq U_{p+1}$ satisfying

$d_{n}(Z_{n}^{1},U_{p})\geq\frac{e^{-(1-r)\alpha_{n}}}{2},$

and set $Z^{1}\doteq\bigcup_{N(p+1)\leq n<N(p+2)}Z_{n}^{1}$.

Continuing in this fashion we get finite sets $Z^{s},\ s=1,2,\dots$ with
$Z^{s}\subseteq U_{p+s}$ and $d_{n}(Z^{s},U_{p})\geq(e^{-(1-r)\alpha_{n}})/2$
for $N(p+s)\leq n<N(p+s+1)$. Note that $\bigcup_{s=1}^{\infty}Z^{s}$ is a
bounded set since $\forall\ k=p+s,Z^{\bar{s}}\subseteq U_{k},\bar{s}>s$. We
can find a closed Hilbertian disc $B$ in $\mathcal{X}$, containing
$\bigcup_{s=1}^{\infty}Z^{s}$ (see for example in Lemma 1.2 of [23]). For this
disc and $n>N(p+1)$(say $N(p+s)\leq n<N(p+s+1)$) we have

$d_{n}(B,U_{p})\geq d_{n}(Z^{s},U_{p})\geq\frac{e^{-(1-r)\alpha_{n}}}{2}$

So in view of Proposition 2.4 (iv) there is a (r,p)-local imbedding from
$\Lambda_{1}(\alpha)$ into $\mathcal{X}$.

  2. ($\Rightarrow$):

This implication follows immediately from Proposition 2.4 (iv). Indeed in view
of Proposition 2.4 (iv) there exists a bounded set $B$ and $C>0$ with
$d_{n}(B,U_{p})\geq Ce^{-(1-r)\alpha_{n}},\ n=1,2,\dots$. Hence $\forall\ k\
\exists\ C_{i}>0,\ i=1,2$;

$\displaystyle\mathcal{E}_{n}(p,k)$ $\displaystyle\leq-\ln
d_{n}(B,U_{p})+C_{1}$ $\displaystyle\leq(1-r)\alpha_{n}+C_{2}.$

So;

$\overline{\lim}_{n}\ \frac{\mathcal{E}_{n}(p,k)}{\alpha_{n}}\leq
1-r\quad\forall\ k=p+1,\dots$

∎

######  Remark.

Although the structural assumption $\underline{DN}$ on $\mathcal{X}$ is put
into use in the Proposition above, the implication $(\Rightarrow)$ can be
proved without any structural assumptions on $\mathcal{X}$. One can use the
argument given in Proposition 2.3 of [4] to give a proof by just using the
definition of local imbeddings.

###  2.3. Approximate diametral dimension and local imbeddings

In this subsection we investigate the relationship between approximate
diametral dimension of a nuclear Fréchet space $\mathcal{X}$ with properties
$\underline{DN}$ and $\Omega$ and the existence of local imbeddings from
$\Lambda_{1}(\mathcal{E}_{n})$, the finite type power series space
corresponding to the associated exponent sequence of $\mathcal{X}$. Since we
wish to use the information obtained in the previous subsections, we are
forced to put _an additional assumption_ on $\\{\mathcal{E}_{n}\\}_{n}$,
namely we will assume that, $\lim_{n}\ln(n)/\mathcal{E}_{n}=0$. This
assumption guaranties the nuclearity of $\Lambda_{1}(\mathcal{E}_{n})$ ([8]),
which will be needed in the sequel and also for the validity of the results of
previous subsections.

######  Definition 2.6.

[7]. _The Approximate Diametral Dimension_ of a Fréchet space $Y$,
$\delta(Y)$, is defined as;

$\displaystyle\delta(Y)$ $\displaystyle\doteq\bigcup_{U\in\text{ zero
neighborhoods of }Y}\ \bigcup_{B\in\text{ bounded subsets of
}Y}\left\\{(t_{n}):\lim_{n}\frac{t_{n}}{d_{n}(B,U)}=0\right\\}$
$\displaystyle=\\{(t_{n})_{n}:\exists\text{ a neighborhood of zero }U\text{
and a bounded set }B\text{ of }Y$ $\displaystyle\text{ such that
}\lim_{n}\frac{t_{n}}{d_{n}(B,U)}=0\\}.$

Approximate diametral dimension also admits a representation solely in terms
of basis for the zero neighborhoods, $\\{U_{n}\\}_{n}$ of the Fréchet space
$Y$ as; $\delta(Y)=\\{(t_{n}):\ \exists\ s\ \forall\ k\geq s;\
t_{n}/d_{n}(U_{k},U_{s})\rightarrow 0\\}$ (See [7]).

Now let us fix, for the rest of this subsection, a graded nuclear Fréchet
space $\left\\{\mathcal{X},\ ^{\|\ \|_{k}}\right\\}$ with the properties
$\underline{DN}$, $\Omega$, whose associated exponent sequence
$(\mathcal{E}_{n})$ satisfies
$\left(\ln(n)/\mathcal{E}_{n}\right)\rightarrow_{n}0$, the grading $\\{\|\
\|_{k}\\}_{k}$ that is coming from the definition 2.2. Given $p<p+1<q$, in
view of Proposition 2.3 (1), we have the inequality
$d_{n}(U_{q},U_{p+1})^{\frac{1}{d}}\leq Cd_{n}(U_{q},U_{p+1})$ for some $C>0$
and $0<\lambda<1$. So if a sequence $(t_{n})_{n}$ satisfies

$\frac{|t_{n}|}{d_{n}(U_{q},U_{p})}\leq C\quad\forall n,\text{ for some }C$

then

$\frac{|t_{n}|}{d_{n}(U_{q},U_{p+1})}\rightarrow 0$

Hence under our assumptions, we have

$\delta(\mathcal{X})=\\{(t_{n})_{n}:\quad\exists\ p;\forall\ q>p\
\sup_{n}|t_{n}|e^{\mathcal{E}_{n}(p,q)}<\infty\\}.$

It is not difficult to show, by direct computation, that ([7])

$\delta(\Lambda_{1}(\alpha))=\\{(t_{n})_{n}:\quad\overline{\lim}_{n}\frac{\ln|t_{n}|}{\alpha_{n}}<0\\}.$

######  Proposition 2.7.

$\delta(\mathcal{X})\supseteq\delta(\Lambda_{1}(\alpha))\Leftrightarrow\inf_{p}\sup_{q>p}\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,q)}{\alpha_{n}}=0$

###### Proof.

  1. ($\Rightarrow$):

For a fixed pair $p<q$ let

$B(p,q)\doteq\\{(t_{n}):\quad\sup_{n}|t_{n}|e^{\mathcal{E}_{n}(p,q)}<\infty\\}.$

This linear space becomes a Banach space under
$|(t_{n})|_{pq}\doteq\sup_{n}|t_{n}|e^{\mathcal{E}_{n}(p,q)}$. Since
$B(p,q^{+})\subseteq B(p,q)$ for $p<q<q^{+}$, the space
$B(p)\doteq\bigcap_{q>p}B(p,q)$ with the fundamental generating norms
$\left\\{|\cdot|_{pq}\right\\}$ becomes a Fréchet space. Moreover by means of
continuous inclusions $B(\bar{p})\hookrightarrow B(p),\bar{p}<p$, we can put
the inductive limit topology on $\delta(\mathcal{X})=\bigcup_{n}B(p)$ and thus
view $\delta(\mathcal{X})$ as a locally convex space which is an inductive
limit of Fréchet spaces. Let us fix a $r,\ 0<r<1$ and define

$S_{r}\doteq(t_{n}):\quad\|(t)\|\doteq\sup_{n}\frac{|t_{n}|}{r^{\alpha_{n}}}<\infty.$

Clearly $S_{r}$ is a Banach space with respect to the norm $\|\cdot\|$, and
$S_{r}\subset\delta(\Lambda_{1}(\alpha))$. By our assumption,
$S_{r}\subset\delta(\mathcal{X})$ and since projections onto coordinates are
continuous with respect to the inductive limit topology on
$\delta(\mathcal{X})$, the inclusion $S_{r}\hookrightarrow\delta(\mathcal{X})$
is a sequentially closed linear operator. In view of Grothendieck
Factorization theorem ([13, p. 225]), there is a $p(r)$ such that for every
$q>p(r)$

$\exists\ C>0:\quad\sup_{n}|t_{n}|e^{\mathcal{E}_{n}(p(r),q)}\leq
C\sup_{n}\frac{|t_{n}|}{r^{\alpha_{n}}}$

for any $(t_{n})\in S_{r}$. In particular for every $q>p(r)$ there exists a
$C>0:\ \mathcal{E}_{n}(p(r),q)\leq\ln C-\alpha_{n}\ln r$, which in turn
implies that

$\inf_{p}\sup_{q>p}\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,q)}{\alpha_{n}}\leq-\ln
r.$

Since this assertion holds for every $0<r<1$, we have

$\inf_{p}\sup_{q>p}\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,q)}{\alpha_{n}}=0.$

  2. ($\Leftarrow$):

Fix an $r<1$ and choose a $(t_{n})_{n}$ which satisfies

$|t_{n}|\leq Cr^{\alpha_{n}}\text{ for some }C>0\text{ and }\forall\ n.$

In view of our assumption, we choose a $p$ so that $\sup_{q\geq
p}\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,q)}{\alpha_{n}}<-\ln r$. It
follows that there exists an $n_{0}$ such that for $n\geq n_{0}$;

$e^{\mathcal{E}_{n}(p,q)}<\frac{1}{r^{\alpha_{n}}}\text{ for every }q\geq p.$

Hence, $\forall\ q>p\ \exists\ C_{q}>0$; $e^{\mathcal{E}_{n}(p,q)}\leq
C_{q}/r^{\alpha_{n}}$; which in turn implies that $\forall\ q>p$;
$\sup_{n}|t_{n}|e^{\mathcal{E}_{n}(p,q)}<+\infty$. It follows that:

$\delta\left(\Lambda_{1}(\alpha)\right)\subseteq\bigcup_{p}\bigcap_{q>p}B(p,q)=\delta(\mathcal{X}).$

∎

If we focus our attention to the associated exponent sequence
$(\mathcal{E}_{n})_{n}$ we have:

######  Corollary 2.8.

$\delta(\mathcal{X})=\delta(\Lambda_{1}(\mathcal{E}))\Leftrightarrow\inf_{p}\sup_{q>p}\overline{\lim}_{n}\frac{\mathcal{E}_{n}(p,q)}{\mathcal{E}_{n}}=0.$

###### Proof.

Let us fix a $(t_{n})\in\delta(\mathcal{X})$, and choose a $p$ such that

$\forall\ q>p;\quad\sup_{n}|t_{n}|e^{\mathcal{E}_{n}(p,q)}<+\infty.$

In view of Proposition 2.3 (4), there exists constants $C_{1}>0$, $C_{2}>0$
such that

$\mathcal{E}_{n}\leq C_{1}\mathcal{E}_{n}(p,p+1)+C_{2}\quad\forall\ n.$

It follows that;

$\sup_{n}|t_{n}|e^{\frac{1}{C_{1}}\mathcal{E}_{n}}<\infty.$

Hence $\exists\ D>0$;

$\ln|t_{n}|+\frac{\mathcal{E}_{n}}{C_{1}}<D$

or

$\overline{\lim}\frac{\ln|t_{n}|}{\mathcal{E}_{n}}\leq-\frac{1}{C_{1}}<0.$

Hence we always have
$\delta(\mathcal{X})\subseteq\delta(\Lambda_{1}(\mathcal{E}))$. Now the
corollary follows from Proposition 2.7. ∎

We conclude this section with a summary theorem:

######  Theorem 2.9.

Let $\mathcal{X}$ be a nuclear Fréchet space with the properties
$\underline{DN}$ and $\Omega$. We also assume that the associated exponent
sequence $(\mathcal{E}_{n})_{n}$ of $\mathcal{X}$, is a nuclear exponent
sequence of finite type (i.e. $\Lambda_{1}(\mathcal{E}_{n})$ is nuclear). Then
the following assertions are equivalent:

  1. (1)

$\delta(\mathcal{X})=\delta(\Lambda_{1}(\mathcal{E}))$

  2. (2)

$\inf_{1\leq
p<\infty}\sup_{q>p}\overline{\lim}_{n}\frac{\mathcal{E}_{n}(q,p)}{\mathcal{E}_{n}}=0$

  3. (3)

There exists an $r$-local imbedding from $\Lambda_{1}(\mathcal{E})$ into
$\mathcal{X}$ for every $0<r<1$.

##  3\. Tame Spaces, $\mathcal{X}$, with Properties $\underline{DN}$, $\Omega$
and $\delta(\mathcal{X})=\delta(\Lambda_{1}(\mathcal{E}))$

To every continuous linear operator $T$ between two graded F-spaces
$\\{\mathcal{X},\ ^{\|\ \|_{k}}\\}_{k}$ and $\\{\mathcal{Y},\ ^{|\
|_{k}}\\}_{k}$ one can associate a sequence of natural numbers,
$\\{\sigma_{T}(n)\\}_{n}$, called the _characteristic of continuity_ of $T$,
via

$\sigma_{T}(n)\doteq\inf\left\\{s:\quad\exists\ C>0;\ |T(x)|_{n}\leq
C\|x\|_{s}\forall\ x\in\mathcal{X}\right\\}$

In general the characteristics of continuity of operators between graded
Fréchet spaces could be very disorderly. However for certain pairs of Fréchet
spaces control over the growth of characteristics of continuity can be
obtained. For example, in the space of analytic functions on the unit disc,
$\mathcal{O}(\Delta)$, with the grading
$\|f\|_{k}\doteq\sum_{n=0}^{\infty}\left|\frac{f^{n}(0)}{n!}\right|^{2}e^{-n/k},\quad
k=1,2,\dots,\ f\in\mathcal{O}(\Delta)$, it is not difficult to see that;

$\forall\ T\in\mathcal{L}(\mathcal{O}(\Delta),\mathcal{O}(\Delta)),\ \exists\
a\in\mathbb{N}\text{ such that }\sigma_{T}(n)\leq an,\quad\forall\
n=1,2,\dots$

Following ([22], [9] c.f. [16]) we specify this property as:

######  Definition 3.1.

A pair of Fréchet spaces $\mathcal{X}$ and $\mathcal{Y}$ will be called a
_tame-pair_ in case for a given pair of generating semi-norms $\\{\|\
\|_{k}\\}_{k}$ of $\mathcal{X}$ and $\\{|\ |_{k}\\}_{k}$ of $\mathcal{Y}$
there exists an increasing function $\psi:\mathbb{N}\rightarrow\mathbb{N}$
such that

$\displaystyle\forall\ T\in\mathcal{L}(\mathcal{X},\mathcal{Y}),\quad\exists\
N\in\mathbb{N};\quad\sigma_{T}(n)\leq\psi(n),\quad\forall\ n\geq N.$

In case $\mathcal{X}=\mathcal{Y}$ we say that the Fréchet space $\mathcal{X}$
is _tame_.

We will use the symbol $(\mathcal{X},\mathcal{Y})\in\mathcal{T}$ in case
$(\mathcal{X},\mathcal{Y})$ form a tame-pair.

######  Remark.

  1. (1)

The definition does not depend on the choice of semi-norms in the spaces
$\mathcal{X}$ and $\mathcal{Y}$.

  2. (2)

Plainly the definition is equivalent to the existence of a sequence, for a
given pair of generating semi-norms $\\{\|\ \|_{k}\\}_{k}$ of $\mathcal{X}$
and $\\{|\ |_{k}\\}_{k}$ of $\mathcal{Y}$, $\\{S_{K}\\}_{k}$ of increasing
functions $S_{K}:\ \mathbb{N}\rightarrow\mathbb{N}$, such that for every
$T\in\mathcal{L}(\mathcal{X},\mathcal{Y})$, there exists a $K$ with
$\sigma_{T}(n)\leq S_{K}(n)$ for every $n=1,2,\dots$.

  3. (3)

The space $\mathcal{O}(\Delta)$ is tame. More generally every power series
space of finite type is tame, as was observed by various authors (see 2.1 on
[9] for details). A proof of this appears in 2.1 of [16].

  4. (4)

If $(\mathcal{X},\mathcal{Y})\in\mathcal{T}$, then the space of all continuous
linear operators from $\\{\mathcal{X},\ ^{\|\ \|_{k}}\\}$ into
$\\{\mathcal{Y},\ ^{|\ |_{k}}\\}$ admits a representation of the form

$\displaystyle\mathcal{L}(\mathcal{X},\mathcal{Y})=\bigcup_{K=1}^{\infty}\bigcap_{n=1}^{\infty}\\{$
$\displaystyle T\in\mathcal{L}(\mathcal{X},\mathcal{Y}):$
$\displaystyle\exists\ C>0\ \|T(x)\|_{n}\leq C|x|_{S_{K}(n)},\forall\
x\in\mathcal{X}\\}$

Using this representation one can put a linear topology on
$\mathcal{L}(\mathcal{X},\mathcal{Y})$ making it an LF-space (i.e. inductive
limit of Fréchet spaces) by considering the semi-norms (2.1, [9])
$\|T\|_{K}^{n}\doteq\sup_{|x|_{S_{K}(n)}\leq 1}\|T(x)\|_{n}$ on the space

$\\{T\in\mathcal{L}(\mathcal{X},\mathcal{Y}):\exists C>0\ \|T(x)\|_{n}\leq
C|x|_{S_{K}(n)},\forall x\in\mathcal{X}\\},K,n\in\mathbb{N}.$

therefore …..

This structure allows one to use the results of well studied LF-spaces in the
study of $\mathcal{L}(\mathcal{X},\mathcal{Y})$ for
$(\mathcal{X},\mathcal{Y})\in\mathcal{T}$. These ideas are used in the study
of nuclear Fréchet spaces $\mathcal{X}$ which form a tame pair with nuclear
stable power series spaces of finite or infinite type in [17] where a complete
characterization of such spaces in terms of the linear topological invariants
of Vogt are obtained.

In this section we will, once again, consider nuclear Fréchet spaces,
$\mathcal{X}$ with the properties $\underline{DN}$ and $\Omega$. We will
assume that the associated exponent sequence of $\mathcal{X}$,
$\mathcal{E}=(\mathcal{E}_{n})_{n}$ in addition to our usual assumption of
being finitely nuclear is also _stable_ ; i.e.
$\sup_{n}(\mathcal{E}_{2n}/\mathcal{E}_{n})<\infty$.

######  Theorem 3.2.

Let $\mathcal{X}$ be a nuclear Fréchet space with stable finitely nuclear
associated exponent sequence $\mathcal{E}=(\mathcal{E}_{n})$ and which has
properties $\underline{DN}$ and $\Omega$. Then $\mathcal{X}$ is isomorphic to
a power series space of finite type if and only if $\mathcal{X}$ is tame and
$\delta(\mathcal{X})=\delta(\Lambda_{1}(\mathcal{E}))$.

###### Proof.

  1. ($\Rightarrow$):

If $\mathcal{X}$ is isomorphic to a power series space of finite type then
$\mathcal{X}$ must be isomorphic to $\Lambda_{1}(\mathcal{E})$ by Proposition
2.3 of [5]. So $\delta(\mathcal{X})=\delta(\Lambda_{1}(\mathcal{E}))$.
Moreover $\mathcal{X}$ being a finite type power series space, is tame in view
of Remark Remark (4) above.

  2. ($\Leftarrow$):

Let us fix a tame nuclear Fréchet space with stable finitely nuclear
associated exponent sequence $\mathcal{E}=(\mathcal{E}_{n})$, and which has
properties $\underline{DN}$ and $\Omega$. In view of [22], we can imbed
$\mathcal{X}$ into $\Lambda_{1}(\mathcal{E})$ as a closed subspace. We will do
so and consider the grading on $\mathcal{X}$ induced by
$(\Lambda_{1}(\mathcal{E}),\ ^{|\ |_{r}})$. Throughout this proof we will, as
usual, use the notation $\epsilon_{n}$ to denote the sequence
$(0,\dots,0,1,0,\dots)$ where $1$ sits in the $n^{\text{th}}$ place,
$n=1,2,\dots$. In view of Theorem 2.9 above there exists an $r$-local
imbedding from $\Lambda_{1}(\mathcal{E})$ into $\mathcal{X}$ for every $0\leq
r<1$. Let us fix an $r_{0}<1$ and a corresponding $(r_{0},r_{k_{0}})$,
$r_{k_{0}}<1$, local imbedding $T$, from $\Lambda_{1}(\mathcal{E})$ into
$\mathcal{X}$. Say $|Tx|_{r_{k_{0}}}\geq C_{0}|x|_{r_{0}},\ \forall\
x\in\Lambda_{1}(\mathcal{E})$. Let
$e_{n}\doteq(\epsilon_{n}/e^{r_{0}\mathcal{E}_{n}}),\ n=1,2,\dots$ be the
canonical orthonormal basis of $\Lambda_{r_{0}}[\mathcal{E}_{n}]$, and set
$g_{n}\doteq T(e_{n}),\ n=1,2,\dots$.

We note that $\\{g_{n}\\}_{n=1}^{\infty}$ is a finitely linearly independent
sequence since $T$ is a local imbedding. We choose a sequence $\\{f_{n}\\}$ in
$\mathcal{X}\subseteq\Lambda_{1}(\mathcal{E})$ satisfying;

    1. (1)

$f_{n}\in\text{span}\\{g_{1},\dots,g_{2n}\\},\ n=1,2,\dots$.

    2. (2)

$\langle f_{n},f_{s}\rangle_{r_{k_{0}}}=0,\ s=1,\dots,n-1;\ n=1,2,\dots$.

    3. (3)

$\langle f_{n},\epsilon_{k}\rangle_{r_{k_{0}}}=0,\ k=1,\dots,n;\ n=1,2,\dots$.

    4. (4)

$\langle f_{n},f_{n}\rangle_{r_{k_{0}}}=1,\ n=1,2,\dots$.

where $\langle$, $\rangle_{r_{k_{0}}}$ is the inner product in
$\Lambda_{r_{k_{0}}}[\epsilon]$.

Such a sequence exists and can be selected by induction since for each
$n\in\mathbb{N}$, the space $\\{g_{1},\dots,g_{2n}\\}$ is $2n$-dimensional and
we impose $2n-1$ conditions for $f_{n}$.

Hence $f_{n}=\sum_{i=1}^{2n}c_{i}^{n}g_{i}$ for some scalars
$\\{c_{i}^{n}\\}_{i},\ n=1,2,\dots$ . We have;

$1=|f_{n}|_{r_{k_{0}}}\geq
C_{0}\left|\sum_{i=1}^{2n}c_{i}^{n}e_{i}\right|_{r_{0}}=C_{0}\left(\sum_{i=1}^{2n}|c_{i}^{n}|^{2}\right)^{\frac{1}{2}},\
n=1,\dots$

So;

(3.1) $\displaystyle\sum_{i=1}^{2n}|c_{i}^{n}|^{2}\leq\frac{1}{C_{0}},\
\forall\ n\in\mathbb{N}.$

Now fix an $s$, $0<s<1$, and estimate $|f_{n}|_{s},\ n\in\mathbb{N}$ as:

$\displaystyle|f_{n}|_{s}$
$\displaystyle=\left|\sum_{i=1}^{2n}c_{i}^{n}g_{i}\right|_{s}\leq\sum_{i=1}^{2n}|c_{i}^{n}||g_{i}|_{s}\leq
C_{1}\sum_{i=1}^{2n}|c_{i}||e_{n}|_{\sigma_{T}(s)}$ (3.2)
$\displaystyle=C_{1}\sum_{i=1}^{2n}|c_{i}^{n}|\left|\frac{\epsilon_{i}}{e^{r_{0}\mathcal{E}_{i}}}\right|_{\sigma_{T}(s)}=C_{1}\sum_{i=1}^{2n}|c_{i}^{n}|e^{(\sigma_{T}(s)-r_{0})\mathcal{E}_{i}}$

for some $C_{1}>0$, where $\sigma_{T}$ is the characteristic of continuity of
$T$ with respect to the connonical gradings of $\Lambda_{1}(\mathcal{E}).$

Choose a $K(s)$, with $\max\\{\sigma_{T}(s),r_{0}\\}<K(s)<1$, and proceed with
the estimate (2.2),

$\displaystyle\sum_{i=1}^{2n}|c_{i}^{n}|\ $ $\displaystyle
e^{(\sigma_{T}(s)-r_{0})\mathcal{E}_{i}+K(s)\mathcal{E}_{i}-K(s)\mathcal{E}_{i}}$
$\displaystyle\leq\ $ $\displaystyle
e^{(K(s)-r_{0})\mathcal{E}_{2n}}\left(\sum_{i=1}^{2n}|c_{i}^{n}|^{2}\right)^{\frac{1}{2}}\left(\sum_{i=1}^{2n}e^{2(\sigma_{T}(s)-K(s))\mathcal{E}_{i}}\right)^{\frac{1}{2}}$
$\displaystyle\leq\ $ $\displaystyle Ce^{(K(s)-r_{0})\mathcal{E}_{2n}},$

to obtain $\forall s<1$, $\exists C=C(S,T)$ and $K(s)<1$ such that

(3.3) $\displaystyle|f_{n}|_{s}\leq Ce^{(K(s)-r_{0})\mathcal{E}_{2n}},\quad
n=1,2,\dots$

We choose an increasing sequence $\\{K^{+}(s)\\}_{s}$ with $K(s)<K^{+}(s)<1,\
\forall s<1$.

On the other hand we also have, for each $n\in\mathbb{N}$, a representation of
$f_{n}$ as;

$f_{n}=\sum_{k=n+1}^{\infty}\beta_{k}^{n}\epsilon_{k}$

for some sequence $\\{\beta_{k}^{n}\\}_{k}$. For an $-\infty<r<r_{k_{0}}$ we
estimate;

$\displaystyle|f_{n}|_{r}^{2}$
$\displaystyle=\sum_{s=n+1}^{\infty}|\beta_{s}^{n}|^{2}e^{2r\mathcal{E}_{s}}=\sum_{s=n+1}^{\infty}|\beta_{s}^{n}|^{2}e^{2r_{k_{0}}\mathcal{E}_{s}}e^{2(r-r_{k_{0}})\mathcal{E}_{s}}$
(3.4) $\displaystyle\leq
e^{2(r-r_{k_{0}})\mathcal{E}_{n+1}}|f_{n}|^{2}_{r_{k_{0}}}=e^{2(r-r_{k_{0}})\mathcal{E}_{n+1}},\quad
n=1,2,\dots.$

In view of stability of $\\{\mathcal{E}_{n}\\}$ there exists a $C_{0}>0$ such
that $C_{0}\mathcal{E}_{2n}\leq\mathcal{E}_{n+1},\ \forall n=1,2,\dots$. Hence
proceeding with the estimate (2.4) we have;

(3.5) $\displaystyle|f_{n}|_{r}^{2}\leq
e^{2C_{0}(r-r_{k_{0}})\alpha_{n}}\quad\forall n\in\mathbb{N}\text{ and
}-\infty<r<r_{k_{0}}.$

We now choose and fix an $s_{0}$ with $-\infty<s_{0}<-2/c_{0}$. In view of
(2.5) we have;

(3.6) $\displaystyle|f_{n}|_{s_{0}}\leq
e^{(C_{0}s_{0}-C_{0}r_{k_{0}}+r_{0})\mathcal{E}_{2n}}e^{-r_{0}\mathcal{E}_{2n}}\leq
e^{-\mathcal{E}_{2n}}e^{-r_{0}\mathcal{E}_{2n}}\quad n=1,2,\dots.$

We stress that $s_{0}$ depends only on the associated exponent sequence
$\mathcal{E}$. Now for a sequence $\\{\lambda_{n}\\}_{n=1}^{\infty}$ and
$s<1$, (2.3) above gives:

$\sum_{i=1}^{\infty}|\lambda_{i}||f_{i}|_{s}e^{r_{0}\mathcal{E}_{2n}}\leq
C\left(\sum|\lambda_{i}|^{2}e^{2K^{+}(s)\mathcal{E}_{2n}}\right)^{\frac{1}{2}}$

for some $C=C(s)$.

It follows that the assignment that sends $\epsilon_{n}$ to
$f_{n}e^{r_{0}\mathcal{E}_{2n}},\ n=1,2,\dots$, defines a continuous linear
operator $\widehat{T}$ from $\Lambda_{1}(\mathcal{E}_{2n})$ into $\mathcal{X}$
that satisfies;

$\displaystyle|\widehat{T}(x)|_{r_{k_{0}}}=\left|\widehat{T}\left(\sum^{\infty}_{i=1}x_{i}\epsilon_{i}\right)\right|_{r_{k_{0}}}$
$\displaystyle=\left|\sum^{\infty}_{i=1}x_{i}f_{i}e^{r_{0}\mathcal{E}_{2i}}\right|_{r_{k_{0}}}$
$\displaystyle=\left(\sum^{\infty}_{i=1}|x_{i}|^{2}e^{2r_{0}\mathcal{E}_{2i}}\right)^{\frac{1}{2}}$
$\displaystyle=|x|_{r_{0}},\quad\forall x\in\Lambda_{1}(\mathcal{E}_{2n}).$

Hence $\widehat{T}$ is an $(r_{0},r_{k_{0}})$-local isomorphism from
$\Lambda_{1}(\mathcal{E}_{2n})$ into $\mathcal{X}$.

Moreover $\widehat{T}$ has the additional property that it extends to be
continuous from $\Lambda_{0}[(\mathcal{E}_{2n})]$ into
$\Lambda_{s_{0}}[(\mathcal{E}_{n})]$. Indeed in view of (2.6), for
$x\in\Lambda_{0}[(\mathcal{E}_{n})]$ we have;

$\displaystyle|\widehat{T}(x)|_{s_{0}}=\left|\widehat{T}\left(\sum_{i}x_{i}\epsilon_{i}\right)\right|_{s_{0}}$
$\displaystyle=\left|\sum^{\infty}_{i=1}x_{i}f_{i}e^{r_{0}\mathcal{E}_{2i}}\right|_{s_{0}}$
$\displaystyle\leq\sum^{\infty}_{i=1}|x_{i}||f_{i}|_{s_{0}}e^{r_{0}\mathcal{E}_{2i}}\leq\sum^{\infty}_{i=1}|x_{i}|e^{-\mathcal{E}_{2i}}$
$\displaystyle\leq\left(\sum^{\infty}_{i=1}e^{-2\mathcal{E}_{2i}}\right)^{\frac{1}{2}}|x|_{0}.$

We now vary $r<1$, and obtain a family $\\{\widehat{T}_{r}\\}$ of
$(r,r_{k_{r}})$-local imbeddings from
$\Lambda_{1}\big{(}(\mathcal{E}_{2n})\big{)}$ into $\mathcal{X}$ with the
additional property mentioned above.

Using the elementary inequality

$|\ |_{t_{2}}\leq|\ |_{t_{1}}^{\frac{t_{3}-t_{2}}{t_{3}-t_{1}}}|\
|_{t_{3}}^{\frac{t_{2}-t_{1}}{t_{3}-t_{1}}},$

for a given $t_{1}<t_{2}<t_{3}$, which is valid in any power series space, for
any $s_{0}<s<1$ we choose a $\rho(s)<1$ such that

(3.7) $\displaystyle\forall\ 0\leq r<1\ \exists
C>0:\quad|\widehat{T}_{r}(x)|_{s}\leq C|x|_{\rho(s)}$

After these preparations we now proceed to show that
$\Big{(}\mathcal{X},\Lambda_{1}\big{(}(\mathcal{E}_{2n})\big{)}\Big{)}\in\mathcal{T}$.

According to our assumption, there exists a sequence
$\\{S_{\alpha}\\}_{\alpha=1}^{\infty}$ of increasing functions from
$\mathbb{N}$ into $\mathbb{N}$, such that $\forall
T\in\mathcal{L}(\mathcal{X},\mathcal{X})$ and $\exists\alpha\in\mathbb{N}$
such that $\sigma(T)\leq S_{\alpha}$.

Let now $S$ be a given continuous linear operator from $\mathcal{X}$ into
$\Lambda_{1}\big{(}\mathcal{E}_{2n})\big{)}$. In view of (2.7) above, the
family, $\\{\widehat{T}_{r}\circ S\\}_{r<1}$ of continuous linear operators
from $\mathcal{X}$ into $\mathcal{X}$ satisfy;

$\sigma_{\widehat{T}_{r}\circ S}\leq\sigma_{S}\circ\rho.$

Hence this family is in
$F\doteq\\{U\in\mathcal{L}(\mathcal{X},\mathcal{X}):\quad\sigma_{U}\leq\sigma_{S}\circ\rho\\}$.
On $F$ we consider the topology coming from the semi-norms
$\\{\|\cdot\|_{s}^{\sigma_{S}\circ\rho(s)}\\}_{s=1}^{\infty}$, and on
$\mathcal{L}(\mathcal{X},\mathcal{X})=\bigcup_{\alpha=1}^{\infty}\bigcap_{s=1}^{\infty}\\{U\in\mathcal{L}(\mathcal{X},\mathcal{X}):\
\|U\|_{s}^{S_{\alpha}(s)}<\infty\\}$, we consider the LF-space structure as
explained in Remark Remark (4) above. Since evaluation at points of
$\mathcal{X}$ are continuous in both $F$ and
$\mathcal{L}(\mathcal{X},\mathcal{X})$, the inclusion
$F\subseteq\mathcal{L}(\mathcal{X},\mathcal{X})$ has a sequentially closed
graph. It follows that there exists $\alpha$ such that
$F\subseteq\bigcap_{s=1}^{\infty}\\{U\in\mathcal{L}(\mathcal{X},\mathcal{X}):\
\|U\|_{s}^{S_{\alpha(s)}}<\infty\\}$ in view of Grothendieck factorization
theorem [14, p. 68]. It follows that there exists an $\alpha\in\mathbb{N}$
such that;

$\sigma_{\widehat{T}_{r}\circ S}\leq S_{\alpha}\quad\forall r<1.$

In particular for each $r<1$ there exists a $\widehat{C}>0$ such that;

(3.8) $\displaystyle|Sx|_{r}\leq
C\left|\widehat{T}_{r}(Sx)\right|_{r_{k_{r}}}\leq\widehat{C}|x|_{S_{\alpha}(r_{k_{r}})}.$

Now if we set $\widehat{S}_{\alpha}(r)\doteq S_{\alpha}(r_{k_{r}})$, $r<1$,
$\alpha\in\mathbb{N}$, the analysis above shows that $\forall
S\in\mathcal{L}\Big{(}\mathcal{X},\Lambda_{1}\big{(}(\mathcal{E}_{2n})\big{)}\Big{)}\
\exists\ n\in\mathbb{N}$ such that $\sigma_{S}\leq\widehat{S}_{\alpha}$. Hence
it follows that
$\Big{(}\mathcal{X},\Lambda_{1}\big{(}(\mathcal{E}_{2n})\big{)}\Big{)}\in\mathcal{T}$.

Now Theorem 11 of [17] implies that $\mathcal{X}$ satisfies the strong
$\Omega$ condition, $\bar{\Omega}$, of Vogt. This together with our assumption
that, $\mathcal{X}$ has the property $\underline{DN}$ allows us to conclude
that $\mathcal{X}$ is isomorphic to a finite type power series space
(Proposition 2.9.18 of [15]). Hence $\mathcal{X}\cong\Lambda_{1}(\mathcal{E})$
(Proposition 1.1 of [5]).

∎

##  4\. Spaces of Analytic Functions

In this section we will focus our attention to a particular class of nuclear
Fréchet spaces with the properties $\underline{DN}$ and $\Omega$, namely the
spaces of analytic functions on Stein manifolds. Stein manifolds, being closed
connected submanifolds of complex euclidean spaces $\mathbb{C}^{N}$, possess a
rich supply of analytic functions. These spaces, with the usual topology of
uniform convergence on compact subsets, form an important subclass of Fréchet
spaces with the properties $\underline{DN}$ and $\Omega$. The linear
topological properties of $\mathcal{O}(M)$, the Fréchet space of analytical
functions on a Stein manifold $M$, and the complex analytic properties of $M$
that are reflected on $M$ by the type of $\mathcal{O}(M)$, have been studied
by several authors (see [3] [25] and the references therein). In this context,
we show, in subsection 4.1 that local imbedding of finite type power series
spaces into $\mathcal{O}(M)$ can be used to construct Green’s functions in
$M$. In subsection 4.2 we classify Stein manifolds $M$, for which
$\mathcal{O}(M)$ is tame. Some results and concepts from Pluripotential Theory
will be used in the sequel. For undefined terminology and background in these
matters we refer the reader to [12].

###  4.1. Local imbeddings of finite type power series spaces into
$\mathcal{O}(M)$ and Green’s functions

Let $M$ be a complex manifold and fix a point $z_{0}\in M$. We will use the
symbol $PSH(M)$ to denote the set of all plurisubharmonic functions on $M$.
Employing norms in a local chart centered at $z_{0}$ we consider

$\mathcal{L}_{z_{0}}\doteq\\{u\in PSH(M),\ u\leq 0\text{ and
}u(z)-\ln\|z-z_{0}\|\text{ is bounded near }z_{0}\\}$

and set

$g(\xi,z_{0})\doteq\sup\\{u(\xi):\ u\in\mathcal{L}_{z_{0}}\\}.$

This assignment, if not $\equiv-\infty$, defines a plurisubharmonic function
on $M$. We will call $g(\cdot z_{0})$ the _Green’s function_ of $M$ with pole
at $z_{0}$, and say that Green’s function with pole at $z_{0}$ exists in case
$g(\cdot,z_{0})$ is not identically equal to $-\infty$. If $M$ is parabolic
i.e. if it does not possess a nonconstant bounded plurisubharmonic function
then of course, no Green’s function exist. In one variable nonparabolicity
characterizes existence of Green’s functions, [19], however in several complex
variables there is no such general result. The difficulty seems in
constructing a negative plurisubharmonic function with a pole at the given
point from the class of negative plurisubharmonic functions.

######  Proposition 4.1.

Let $M$ be a Stein manifold of dimension $d$ and let $z_{0}$ be a point in
$M$. If there exists a local imbedding from $\Lambda_{1}(\alpha)$ into
$\mathcal{O}(M)$, for some finitely nuclear exponent sequence
$\\{\alpha_{n}\\}_{n}$, with $\underline{\lim}\ n^{1/d}/\alpha_{n}>0$, then
Green’s function with a pole at $z_{0}$ exists.

###### Proof.

Let us fix $M$ and the point $z_{0}$. In view of [10] there exists a local
biomorphism $\Phi:\ \Delta^{d}_{e}\twoheadrightarrow M$, from the polydisc
with centre $0$ and radius $e$ of $\mathbb{C}^{d}$ onto $M$ such that
$\Phi(0)=z_{0}$. We will use $\Phi$ to imbed $\mathcal{O}(M)$ into
$\mathcal{O}(\Delta^{d}_{e})$ via the composition operator $f\mapsto
f\circ\Phi$, $f\in\mathcal{O}(M)$ ([6]). We will choose a bijection
$\rho:\mathbb{N}\rightarrow\mathbb{N}^{d}$ such that $|\rho(n)|$ is strictly
increasing with $n$ and is ordered lexicographic on the level sets of
$|\rho(\cdot)|$, and use it to define an isomorphism between
$\Lambda_{1}(n^{1/d})$ and $\mathcal{O}(\Delta^{d}_{e})$ by the correspondence
$\epsilon_{i}\rightarrow z^{p(i)}=z_{1}^{p_{1}(i)}\dots z_{d}^{p_{d}(i)},\
i=1,2,\dots$. We note that there exists constants $\beta_{1}>0$ and
$\beta_{2}>0$ such that $\beta_{1}(\rho(n))\leq
n^{1/d}\leq\beta_{2}(\rho(n)),\ n=1,2,\dots$ (see e.g. [18]).

Let us fix an $(r_{0},k_{0})$-local imbedding, $T$, from
$\Lambda_{1}(\alpha_{n})$ into $\Lambda_{1}(n^{1/d})$,
$T(\Lambda_{1}(\alpha_{n}))\subseteq\mathcal{O}(M)\subseteq\Lambda_{1}(n^{1/d})$,
which exists in view of our assumptions. Let

$f_{n}\doteq T\left(\frac{\epsilon_{n}}{e^{r_{0}\alpha_{n}}}\right),\
n=1,2,\dots.$

The sequence $\\{f_{n}\\}_{n}$ is finitely linearly independent, so we can,
choose a sequence $\\{g_{n}\\}^{\infty}_{n=1}$ of elements of $\mathcal{O}(M)$
with

  1. (1)

$g_{n}\in\text{span}\\{f_{1},\dots,f_{n}\\},\ n=1,2,\dots$

  2. (2)

$\langle g_{n},\epsilon_{i}\rangle_{k_{0}}$, $i=1,\dots,n-1$ where $\langle$,
$\rangle_{k_{0}}$ is the inner product in $\Lambda_{k_{0}}[n^{1/d}]\ \forall\
n=1,2,\dots$

  3. (3)

$|g_{n}|_{k_{0}}=1$, $n=1,2,\dots$

Note that if $g_{n}=\sum^{n}_{k=1}c_{k}^{n}f_{k}$ then we estimate, as in the
above arguments;

$\displaystyle\sum^{n}_{k=1}|c_{k}^{n}|^{2}=\sum^{n}_{k=1}\left|\frac{c_{k}^{n}}{e^{2r_{0}\alpha_{k}}}\right|^{2}e^{2r_{0}\alpha_{k}}=\left|\sum^{n}_{k=1}c_{k}^{n}\frac{\epsilon_{k}}{e^{2r_{0}\alpha_{k}}}\right|^{2}_{r_{0}}$
$\displaystyle\leq
C\left|T\left(\sum^{n}_{k=1}c_{k}^{n}\frac{\epsilon_{k}}{e^{2r_{0}\alpha_{k}}}\right)\right|^{2}_{k_{0}}$
$\displaystyle=C|g_{n}|^{2}_{k_{0}}\leq C,$

for some $C>0$ and for all $n=1,2,\dots$.

Hence, for a given $r<1$;

$\displaystyle|g_{n}|_{r}$
$\displaystyle=\left|\sum_{i=1}^{n}c_{i}^{n}f_{i}\right|_{r}\leq
C_{1}\sum_{i=1}^{n}|c_{i}^{n}|\left|\frac{\epsilon_{i}}{e^{r_{0}\alpha_{n}}}\right|_{\sigma_{T}(r)}$
(4.1) $\displaystyle=\sum_{i=1}^{n}|c_{i}^{n}|e^{(\sigma_{T}(r)-r_{0})2n}\leq
C_{2}e^{\rho(r)\alpha_{n}},\quad n=1,2,\dots$

and for some constants $C_{1}>0$, $C_{2}>0$ where
$\sigma_{T}(r)-r_{0}\leq\rho(r)<1$ is a chosen number that depends on $r$ (and
$T$).

On the other hand in view of equation (4.4) above, each $g_{n},\ n=1,2,\dots$
has an expansion as

$g_{n}=\sum_{i\geq n}d_{i}^{n}\epsilon_{i}\leftrightarrow\sum_{i\geq
n}d_{i}^{n}z^{\rho(i)}$

in $\Lambda_{1}(n^{1/d})$ and in $\mathcal{O}(\Delta^{d}_{e})$, respectively.
By abuse of notation we will think of
$g_{n}\in\mathcal{O}(M)\subseteq\mathcal{O}(\Delta_{e}^{d}),\ n=1,2,\dots$ as
an analytic function on $\Delta_{e}^{d}$ with Taylor series;

(4.2) $\displaystyle
g_{n}(z)=\sum_{|t|\geq\frac{1}{\beta_{2}}n^{1/d}}\alpha_{t}^{n}z_{1}^{t_{1}}\dots
z_{d}^{t_{d}},\quad n=1,2,\dots.$

Choose $h_{n}\in\mathcal{O}(M)$ such that $g_{n}=h_{n}\circ\Phi,\
n=1,2,\dots$. For a given compact set $K\subseteq M$ choose $r<1$ so that
$\Phi(\Delta^{d}_{e^{r}})\supseteq K$. So in view of equation (4.1);

$\sup_{z\in K}|h_{n}(z)|\leq\sup_{\xi\in\Delta_{e^{r}}^{d}}|g_{n}(\xi)|\leq
C|g|_{r^{+}}\leq CC_{2}e^{\rho(r^{+})\alpha_{n}}$

for some constant $C>0$ and a choice of $r<r^{+}<1$.

Hence the function

$u(z)\doteq\overline{\lim_{\xi\rightarrow z}}\
\overline{\lim}_{n}\frac{\ln|h_{n}(\xi)|}{\alpha_{n}},\quad z\in M$

defines a plurisubharmonic function on $M$ that is bounded by $1$. This
plurisubharmonic function is not $\equiv-\infty$. To see this let us assume,
with the anticipation of getting a contradiction that it is identically equal
to $-\infty$. Fix large $k_{0}<r<1$ such that $|x|_{k_{0}}\leq
C\sup_{z\in\overline{\Delta}_{e^{r}}^{d}}|x(z)|,\ \forall
x\in\mathcal{O}(\Delta^{d}_{e})$ and choose a compact set $K\subset M$ that
contains $\Phi(\overline{\Delta}_{e^{r}}^{d})$.

Our assumption and Hartogs theorem ([12, p. 70]) gives for each
$N\in\mathbb{N}$, an $n_{0}\in\mathbb{N}$ such that:

$\sup_{z\in K}|h_{n}(z)|\leq e^{-N\alpha_{n}},\quad n\geq n_{0}.$

This in turn gives that for some $C>0$

$1=|g_{n}|_{r_{0}}\leq Ce^{-N\alpha_{n}},\quad\text{ for }n\geq n_{0}.$

So $u$ is not identically $-\infty$.

Now consider a $z\in M$ near $z_{0}$, say with $\|z-z_{0}\|=e^{r}$ for some
very large and negative $r$. Since $\Phi$ is a local biholomorphism, there
exists a $C_{0}>0$, independent of $r$, and $\xi\in\Delta_{e}^{d}$ with
$\|\xi\|<C_{0}e^{r}$ such that $\phi(\xi)=Z$. Using (4.2) we estimate, with,
$k_{0}^{-}<k_{0}$ and $C_{1}>0$,

$\displaystyle|g_{n}(\xi)|$ $\displaystyle\leq\sum_{|t|\geq
Cn^{1/d}}|\alpha_{t}^{n}|C_{0}^{|t|}e^{r|t|}e^{-k_{0}^{-}}|t|e^{k_{0}^{-}|t|}$
$\displaystyle\leq
C_{0}e^{C(r-k_{0}^{-})n^{1/d}}\left(\sum_{|t|}|\alpha_{t}^{n}|^{2}e^{2k_{0}|t|}\right)^{1/2}\leq
C_{1}e^{C(r-k_{0}^{-})n^{1/d}},\quad n=1,2,\dots$

Hence our assumption on $(\alpha_{n})$ implies $\exists\ C_{2}>0$, $C_{3}>0$:

$\overline{\lim}_{n}\frac{\ln|h_{n}(z)|}{\alpha_{n}}\leq\overline{\lim}_{n}C(r-k_{0}^{-})\frac{n^{1/d}}{\alpha_{n}}\leq
C_{2}\ln\|z-z_{0}\|+C_{3}.$

So $\frac{u}{C_{2}}$ has a logarithmic singularity at $z_{0}$, and is a
bounded plurisubharmonic function. It follows that $g_{m}(\cdot,z_{0})$ is not
identically $-\infty$. ∎

###  4.2. Diametral dimensions of analytic function spaces

In this subsection we will investigate the diametral dimension of the spaces
of analytic functions on Stein manifolds. These invariants for Fréchet spaces
are in a sense dual to approximate diametral dimensions, however they are more
extensively studied.

For a nuclear Fréchet space $\mathcal{X}$ with a neighborhood basis of $0$,
$\\{U_{p}\\}_{p=1}^{\infty}$, consisting of discs, $\Delta(\mathcal{X})$, is
defined by (in the notation of Section 1):

$\displaystyle\Delta(\mathcal{X})$ $\displaystyle=\\{(t_{n});\ \forall p\
\exists q:\quad t_{n}d_{n}(U_{q},U_{p})\rightarrow 0\\}.$
$\displaystyle=\\{(t_{n});\ \forall p\ \exists q:\quad
t_{n}e^{-\mathcal{E}_{n}(p,q)}\rightarrow 0\\}.$

As the notation suggests, it is easy to see that, this sequence space does not
depend upon the neighborhood basis chosen and is an invariant of the Fréchet
space $\mathcal{X}$.

In this subsection we will use a spacial generating norm-system for the
analytic function spaces, unless stated otherwise. To describe these norms,
suppose a Stein manifold $M$ of dimension $d$ is given. We choose a strictly
plurisubharmonic $C^{\infty}$ exhaustion function, $\rho$, of $M$ and consider
a sequence $r_{p}\uparrow\infty$ such that the sub-level sets
$D_{p}\doteq\\{z\in M;\ \rho(z)<r_{p}\\},\ p=1,2,\dots$, are strictly
pseudoconvex. We set $K_{p}\doteq\bar{D}_{p},\ p=1,2,\dots$. Following ([27],
c.f. [25]) we choose for each $p$, a Hilbert space, $H_{p}$, that satisfies;

$A(K_{p})\hookrightarrow H_{p}\hookrightarrow
AC(K_{p})\hookrightarrow\mathcal{O}(D_{p})$

where $A(K_{p})$ denotes the germs of analytic functions on the compact set
$K_{p}$ with the inductive limit topology, $AC(K_{p})$ denotes the Banach
space that is the closure of $A(K_{p})$ in $C(K_{p})$, the Banach space of
continuous functions on $K_{p}$ with the sup-norm, and the $\hookrightarrow$’s
are imbeddings with dense range, $p=1,2,\dots$. The norms that we will use to
generate the topology $\mathcal{O}(M)$ will be the hilbertian norms of
$H_{p}$’s, $\\{\|\ \|_{p}\\}_{p}$. Their corresponding unit balls in
$\mathcal{O}(M)$ will be denoted by $U_{p}$, $p=1,2,\dots$. The primary reason
of our usage of these semi-norms is because of the beautiful formula of
Nivoche, Poletsky, Zaharyuta:

(4.3) $\displaystyle\forall
p<q\quad\lim_{n}\frac{\mathcal{E}_{n}(p,q)}{\left(\frac{2\pi
d!n}{\tilde{C}(K_{p},D_{q})}\right)^{\frac{1}{d}}}=1$

where $\tilde{C}(K_{p},D_{q})=\sup\\{\int_{K_{p}}(dd^{c}u)^{n}:\ u\in
PSH(D_{q})-1\leq u\leq 0\\}$ (Proposition 4.6, 4.8, 4.12 on [27]). We refer
the reader to [27] and [25] for a proof and a discussion about the history of
this formula.

An immediate consequence of equation (4.3) is;

######  Proposition 4.2.

Let $M$ be a Stein manifold of dimension $d$. Then

$\Delta(\mathcal{O}(M))\equiv\\{(t_{n}):\ \forall p\ \exists q\text{ such that
}\sup_{n}|t_{n}|e^{-\alpha_{n}/c(p,q)}<\infty\\}$

where for $p<q$, $c(p,q)\doteq\tilde{C}(K_{p},D_{q})^{\frac{1}{d}}$, and
$\alpha_{n}\doteq(2\pi d!)^{\frac{1}{d}}n^{\frac{1}{d}},\ n=1,2,\dots$.

###### Proof.

  1. ($\subseteq$):

Choose a $(t_{n})_{n}\in\Delta(\mathcal{O}(M))$. Fix $p$ and choose $q$ such
that

$\sup_{n}|t_{n}|e^{-\mathcal{E}_{n}(p,q)}<+\infty.$

Now in view of the (2’) of Proposition 2.3 there exists a $q^{+}$, $C>0$ and
$0<\lambda<1$ such that;

$\mathcal{E}_{n}(p,q)\leq(1-\lambda)\mathcal{E}_{n}(p,q^{+})+C.$

For $\varepsilon>0$ such that $(1+\varepsilon)<\frac{1}{1-\lambda}$ in view of
(4.3) there exists an $N$ such that

$(1-\varepsilon)\frac{\alpha_{n}}{C(p,q^{+})}\leq\mathcal{E}_{n}(p,q^{+})\leq(1+\varepsilon)\frac{\alpha_{n}}{C(p,q^{+})},\quad
n\geq N.$

Hence we have;

$\displaystyle|t_{n}|\leq C_{1}e^{\mathcal{E}_{n}(p,q)}$ $\displaystyle\leq
C_{2}e^{(1-\lambda)\mathcal{E}_{n}(p,q^{+})}$ $\displaystyle\leq
e^{(1-\lambda)(1+\varepsilon)\frac{\alpha_{n}}{C(p,q^{+})}}\leq
e^{\frac{\alpha_{n}}{C(p,p^{+})}},\quad n\geq N.$

  2. ($\supseteq$):

Choose a sequence $(t_{n})$ from the right hand side. For a fixed $p$, choose
a $q$ such that $\sup_{n}|t_{n}|e^{-\alpha_{n}/c(p,q)}<\infty$. By (2’) of
Proposition 2.3 choose $k$, $0<\rho<1$ and $C>0$ so that
$\mathcal{E}_{n}(p,q)\leq(1-\rho)\mathcal{E}_{n}(p,k)+C$. Let $\varepsilon>0$
be so that $1-\rho<1-\varepsilon$ and choose an $N$ (from (4.3))

$\frac{-(1+\varepsilon)\alpha_{n}}{C(p,q)}\leq-\mathcal{E}_{n}(p,q)\leq-\frac{(1-\varepsilon)\alpha_{n}}{C(p,q)},\quad
n\geq N$

Hence

$|t_{n}|^{(1-\varepsilon)}e^{-\mathcal{E}_{n}(p,q)}\leq\left(|t_{n}|e^{\frac{-\alpha_{n}}{C(p,q)}}\right)^{1-\varepsilon}\leq
Q<\infty.$

for some $Q>0$. Moreover since
$-\mathcal{E}_{n}(p,q)\geq-(1-\varepsilon)\mathcal{E}_{n}(p,k)-C$ we have;

$\sup_{n}\left(|t_{n}|e^{-\mathcal{E}_{n}(p,k)}\right)^{1-\varepsilon}\leq
Qe^{C}\Rightarrow\sup_{n}|t_{n}|d_{n}(U_{k},U_{p})<\infty.$

It follows that $(t_{n})\in\Delta(\mathcal{O}(M))$.

∎

######  Corollary 4.3.

Let $M$ be a Stein manifold of dimension $d$. Then

$\Delta(\mathcal{O}(M))=\Delta(\Lambda_{1}(n^{\frac{1}{d}}))\Leftrightarrow\inf_{p}\sup_{q\geq
p}\frac{1}{C(p,q)}=0\Leftrightarrow\delta(\mathcal{O}(M))=\delta(\Lambda_{1}(n^{\frac{1}{d}})).$

###### Proof.

Suppose that $\Delta(\mathcal{O}(M))=\Delta(\Lambda_{1}(n^{\frac{1}{d}}))$.
Suppose that there is an $\varepsilon>0$ such that $\inf_{p}\sup_{q\geq
p}\frac{1}{c(p,q)}>\varepsilon>0$. In view of equation (4.3) this implies,

$\inf_{p}\sup_{q\geq
p}\lim_{n}\frac{\mathcal{E}_{n}(p,q)}{n^{\frac{1}{d}}}>\varepsilon(d!2\pi)^{\frac{1}{d}}$

Setting $\kappa\doteq\varepsilon(d!2\pi)^{\frac{1}{d}}$, we have;

$\forall p\ \exists q\text{ and }N\text{ such that
}\frac{1}{d_{n}(U_{q},U_{p})}\geq e^{\kappa n^{\frac{1}{d}}},\quad n\geq N.$

Hence,

$e^{\kappa
n^{\frac{1}{d}}}\in\Delta(\mathcal{O}(M))=\Delta(\Lambda_{1}(n^{\frac{1}{d}}))=\\{(\xi_{n}):\
\forall\ r<1,\quad\lim_{n}|\xi_{n}|r^{n^{1/d}}=0\\}.$

This contradiction shows that $\inf_{p}\sup_{q\geq p}1/c(p,q)=0$.

On the other hand if $\inf_{p}\sup_{q\geq p}1/c(p,q)=0$, and $r<1$ is given,
choose a $p$ such that, in the notation of Proposition 4.1,

$\sup_{q\geq p}\frac{\alpha_{n}}{c(p,q)n^{\frac{1}{d}}}\leq-\ln r\quad\forall
n.$

In view of Proposition 4.1, for $p$ there is a $q>>p$ such that
$\sup_{n}|t_{n}|e^{\frac{-\alpha_{n}}{c(p,q)}}<\infty$. Hence
$\sup_{n}|t_{n}|r^{n^{\frac{1}{d}}}<\infty$. This shows that

$(t_{n})\in\\{(\xi_{n}):\ \forall r<1,\
\lim_{n}|\xi_{n}|r^{n^{\frac{1}{d}}}=0\\}=\Delta(\Lambda_{1}(n^{\frac{1}{d}})).$

So $\Delta(\mathcal{O}(M))\subseteq\Delta(\Lambda_{1}(n^{\frac{1}{d}}))$.

On the other hand, $(n^{\frac{1}{d}})_{n}$, being the associated exponent
sequence of $\mathcal{O}(M)$, always satisfies
$\Delta(\Lambda_{1}(n^{\frac{1}{d}}))\subseteq\Delta(\mathcal{O}(M))$
(Proposition 1.1, [5]). So we have
$\Delta(\Lambda_{1}(n^{\frac{1}{d}}))=\Delta(\mathcal{O}(M))$. Since
$\mathcal{O}(M)$ is isomorphic to a closed subspace of $\Lambda_{1}(n^{1/d})$,
the other equivalence directly follows from Proposition 2.7 and the fact that
approximate diametral dimension of a nuclear Fréchet space is greater than the
approximate diametral dimension of its subspaces ([7]). ∎

######  Theorem 4.4.

Let $M$ be a Stein manifold of dimension $d$. Then either
$\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\Delta^{d}))$ or
$\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\mathbb{C}^{d}))$.

###### Proof.

Let us fix a Stein manifold $M$ with dimension $d$. We will use, in what
follows, the grading on $\mathcal{O}(M)$ described in the beginning of the
section. Using the notation above, for a given $p\geq 1$, we set,
$c(p)\doteq\lim_{q>p}c(p,q)=\inf_{q>p}c(p,q)$. Since $c(\cdot,\cdot)$ is
increasing in the first variable, the sequence $\\{c(p)\\}$ is an increasing
sequence. We have two cases:

  1. 1.

$c(p)$ is zero for all $p\geq 1$. Suppose this is the case. Let us choose a
$(t_{n})_{n}\in\Delta(\mathcal{O}(\mathbb{C}^{d}))=\Delta(\Lambda_{\infty}(n^{\frac{1}{d}}))$,
and choose an $R>1$ and $C>0$, so that $|t_{n}|\leq CR^{n^{\frac{1}{d}}},\
\forall n$. For a given $p$ let us choose a $q$ such that
$c(p,q)\leq\frac{1}{\ln(\tilde{R})}$ for some $\tilde{R}$ with $R^{1/(2\pi
d!)^{\frac{1}{d}}}<<\tilde{R}$. Then

$|t_{n}|e^{-\frac{\alpha_{n}}{c(p,q)}}\leq|t_{n}|e^{-\alpha_{n}\ln\tilde{R}+n^{1/d}\ln
R-n^{1/d}\ln R}\leq C.$

So $(t_{n})_{n}\in\Delta(\mathcal{O}(M))$ in view of Proposition 4.1. Hence

$\Delta(\Lambda_{\infty}(n^{1/d}))\subseteq\Delta(\mathcal{O}(M)).$

However, $\\{n^{\frac{1}{d}}\\}_{n}$, being the associated exponent sequence
of $\mathcal{O}(M)$, the inclusion
$\Delta(\mathcal{O}(M))\subseteq\Delta(\Lambda_{\infty}(n^{1/d}))$ is always
true. It follows that in this case
$\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\mathbb{C}^{d}))$.

  2. 2.

The sequence $\\{c(p)\\}$ increases to a non-zero number $c$. Suppose this is
the case. We have two possibilities either $c\in\mathbb{R}$ or $c=\infty$.

Let us suppose that $c>0$ is a real number. Fix natural numbers
$\alpha<\gamma<\beta$. For given natural numbers $n$ and $m$, plainly
$d_{n+m}(U_{\beta},U_{\alpha})\leq
d_{n}(U_{\beta},U_{\gamma})d_{m}(U_{\gamma},U_{\alpha})$, where $U_{t}$’s,
$1\leq t<\infty$, are the unit balls corresponding to our grading. In
particular we have

(4.4)
$\displaystyle\mathcal{E}_{n+m}(\alpha,\beta)\geq\mathcal{E}_{n}(\gamma,\beta)+\mathcal{E}_{m}(\alpha,\gamma);\quad
n,m\geq 1$

Fix an $\varepsilon>0$ and using (4.3), choose $N=N(\alpha,\gamma,\beta)$ such
that

$\displaystyle(1-\varepsilon)\frac{\alpha_{n+m}}{c(\alpha,\beta)}$
$\displaystyle\leq\mathcal{E}_{n+m}(\alpha,\beta)\leq(1+\varepsilon)\frac{\alpha_{n+m}}{c(\alpha,\beta)}$
$\displaystyle(1-\varepsilon)\frac{\alpha_{n}}{c(\gamma,\beta)}$
$\displaystyle\leq\mathcal{E}_{n}(\gamma,\beta)\leq(1+\varepsilon)\frac{\alpha_{n}}{c(\gamma,\beta)}$
$\displaystyle(1-\varepsilon)\frac{\alpha_{m}}{c(\alpha,\gamma)}$
$\displaystyle\leq\mathcal{E}_{m}(\alpha,\gamma)\leq(1+\varepsilon)\frac{\alpha_{m}}{c(\alpha,\gamma)},\text{
for }n,m\geq N$

Hence we have

$(1-\varepsilon)\frac{m^{\frac{1}{d}}}{c(\alpha,\gamma)}+(1-\varepsilon)\frac{n^{\frac{1}{d}}}{c(\gamma,\beta)}\leq(1+\varepsilon)\frac{(n+m)^{\frac{1}{d}}}{c(\alpha,\beta)}$

for $n,m\geq N$, taking $m=n$, and $n>N$, we get, after cancellation:

$\frac{(1-\mathcal{E})}{c(\alpha,\gamma)}+\frac{(1-\mathcal{E})}{c(\gamma,\beta)}\leq\frac{(1+\mathcal{E})2^{1/d}}{c(\alpha,\beta)}$

which upon letting first $\beta$ then $\gamma$ and $\alpha$ to go to infinity,
respectively, gives $2\leq 2^{1/d}$. Hence for $d>1$, $c$ must be $\infty$.

For $d=1$, we will use the exhaustion given in [1, p. 145] and an associated
fundamental hilbertian norm system as explained in the beginning of the
section. In this context we will use the _modulus inequality_ as given in [19,
p. 14], which in our notation states that for $1\leq p<q<s<\infty$,

(4.5) $\displaystyle\frac{1}{c(p,s)}\geq\frac{1}{c(p,q)}+\frac{1}{c(q,s)}$

First letting $s\rightarrow\infty$ and then $q\rightarrow\infty$ we see that
$1/(c(q))$ approaches to $0$. Hence in this case we conclude that $c=\infty$,
as well. So the first possibility does not occur and we conclude that

$\lim_{p}\frac{1}{c(p)}=\inf_{p}\sup_{q\geq p}\frac{1}{c(p,q)}=0.$

The theorem now follows from Corollary 4.3.

∎

Now we turn our attention to tameness in the spaces of analytic functions.
Recall that a Stein manifold $M$ is called _hyperconvex_ in case it possesses
a bounded plurisubharmonic exhaustion function. We refer the reader to [12]
and the references therein for an account of hyperconvex manifolds. From a
functional analysis point of view, hyperconvex Stein manifolds $M$ are
precisely those Stein manifolds that satisfy,
$\mathcal{O}(M)\approx\mathcal{O}(\Delta^{d})$, $d=\text{dim}M$, ([26, 2]).
Hence for hyperconvex manifold $M$, $\mathcal{O}(M)$ is a tame Fréchet space.
Our next and final result tells us that they are the only ones having this
property:

######  Main Theorem 4.5.

Let $M$ be a Stein manifold. Then $\mathcal{O}(M)$ is tame if and only if $M$
is hyperconvex.

###### Proof.

Let us fix a Stein manifold $M$ with dimension $d$. In view of the remarks
preceding the theorem, it suffices to show that if $\mathcal{O}(M)$ is tame
then $\mathcal{O}(M)\approx\mathcal{O}(\Delta^{d})$. In view of Theorem 4.4
either $\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\mathbb{C}^{d}))$ or
$\Delta(\mathcal{O}(M))=\Delta(\mathcal{O}(\Delta^{d}))$. The first case
cannot occur. To see this first observe that the assumption on the diametral
dimension implies that $\mathcal{O}(M)$ contains a complemented copy of
$\mathcal{O}(\mathbb{C}^{d})$ by Theorem 1.3 of [4]. Since plainly tameness
passes to complemented subspaces and
$\mathcal{O}(\mathbb{C}^{d})=\Lambda_{\infty}(n^{1/d})$ is not tame [9],
indeed this case cannot occur. Now the theorem follows from Corollary 4.3 and
Theorem 3.2. ∎

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# Time-reversibility and nonvanishing Lévy area

Georg Gottwald School of Mathematics and Statistics, University of Sydney,
Sydney 2006 NSW, Australia Ian Melbourne Mathematics Institute, University of
Warwick, Coventry, CV4 7AL, UK

(20 October, 2022)

###### Abstract

We give a complete description and clarification of the structure of the Lévy
area correction to Itô/Stratonovich stochastic integrals arising as limits of
time-reversible deterministic dynamical systems. In particular, we show that
time-reversibility forces the Lévy area to vanish only in very specific
situations that are easily classified.

##  1 Introduction

The classical Wong-Zakai question [21] from 1965 is concerned with weak
convergence of smooth processes $W_{n}$ to Brownian motion $W$ and the
consequences for the interpretation of the corresponding stochastic integral
$\int W\,dW$.

In simple situations [21, 9], the limiting stochastic integrals are
Stratonovich, denoted $\int W\circ dW$, but numerous counterexamples exist in
higher dimensions [17, 20]. In general, there is a correction, reminiscent of
the Itô-Stratonovich correction, given by a deterministic quantity known as
the Lévy area $E$, whereby $\int_{0}^{t}W_{n}\,dW_{n}$ converges weakly to
$\int_{0}^{t}W\circ dW+Et$ as $n\to\infty$. Many references give closed-form
formulas for the Lévy area $E$, see [1, 4, 5, 8, 11, 13, 14, 15, 16].

It is by now well-understood (if not always well-known) that the Lévy area is
an important and nontrivial correction to the Wong-Zakai question. However,
there are few investigations of whether this formula for $E$ as displayed in
Section 2 (which looks like it may be nonzero) is actually nonzero in the
presence of additional constraints such as time-reversibility.

The Lévy area is skew-symmetric ($E^{T}=-E$) and hence vanishes in the scalar
case. In higher dimensions, $E=0$ is a consequence of exactness, or
commutativity of the defining vector fields, but such conditions are atypical
outside the scalar case. On the other hand, an example where it is proved that
$E\neq 0$ is given by Hairer, Pavliotis & Stuart, see [19, Section 11.7.7].

From time to time, we have been asked whether time-reversibility can force
${E=0}$. (The example of Hairer et al. in [19] is not time-reversible.) In
dispersing billiard examples considered by Chernov & Dolgopyat [2], the Lévy
area is indeed zero as a consequence of time-reversibility and the structure
of the equations. For Markov processes, there is a related condition,
_detailed balance_ , which forces $E=0$, see for example [10, Section 3.3.2]
or [18, Section 5.1]. See also [15, Section 1.4.2] for further comments on $E$
being zero in certain time-reversible situations.

The recent preprint [6] gives numerical evidence that $E\neq 0$ in certain
examples, but it is written (at the end of Section 4 of that paper) that $E$
could be negligible in many situations.

In this paper, we offer a complete description and clarification of the
structure of the Lévy area. In particular, we classify the constraints on $E$
imposed by time-reversibility. The cases where $E$ is forced to vanish are
easily described. Outside of these rare situations, we find that $E$ is
typically far from negligible.

###### Remark 1.1

Since our conclusions are contrary to those for systems satisfying detailed
balance, our results can be viewed as advising caution on when to invoke
detailed balance as a modelling assumption.

The remainder of this paper is organised as follows. Section 2 contains the
setup in this paper and gives a rough description of our main result, namely
Theorem 4.3. In Section 3, we introduce time-reversibility and determine the
constraints imposed by the time-reversibility on the Lévy area. In Section 4,
we state and prove Theorem 4.3, showing that there are no further constraints
on the Lévy area beyond those determined in Section 3.

##  2 The setup

We consider fast-slow ordinary differential equations (ODEs) on
${\mathbb{R}}^{d}\times{\mathbb{R}}^{m}$ of the form

$\displaystyle\dot{x}$ $\displaystyle=a(x)+\epsilon^{-1}b(x)v(y),\quad
x(0)=\xi\in{\mathbb{R}}^{d}$ $\displaystyle\dot{y}$
$\displaystyle=\epsilon^{-2}g(y),\qquad\qquad\qquad y(0)\in(\Lambda,\mu)$
(2.1)

where $\Lambda\subset{\mathbb{R}}^{m}$ is a compact subset and $\mu$ is a
Borel probability measure on $\Lambda$. The functions

$a:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},\quad
b:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d},\quad
v:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d},\quad
g:{\mathbb{R}}^{m}\to{\mathbb{R}}^{m},$

are assumed to be $C^{r}$ (for some $r\geq 1$) with $\int_{\Lambda}v\,d\mu=0$.
Let $g_{t}:\Lambda\to\Lambda$ denote the flow on $\Lambda$ generated by the
ODE $\dot{y}=g(y)$. We assume that $\mu$ is $g_{t}$-invariant, ergodic and
mixing.

The aim of homogenisation is to establish, as $\epsilon\to 0$ in (2), a
limiting stochastic differential equation (SDE) of the form

$dX=\tilde{a}(X)\,dt+b(X)\circ dW,\quad X(0)=\xi,$ (2.2)

such that $x=x^{(\epsilon)}$ converges weakly to $X$. Here, $W$ is
$d$-dimensional Brownian motion, the stochastic integral $b(X)\circ dW$ has
the Stratonovich interpretation, and $\tilde{a}$ is a modified drift term
incorporating the correction (if any) to the Stratonovich integral.

In the deterministic setting of (2), convergence to an SDE of the form (2.2)
was obtained in [11] under suitable chaoticity assumptions (subsequently
optimised in [4, 7, 12]) on the fast dynamics $\dot{y}=g(y)$. Under these
assumptions, we have convergent series of Green-Kubo-type:

Covariance

$\Sigma=\int_{0}^{\infty}\int_{\Lambda}\big{\\{}v\otimes(v\circ g_{t})+(v\circ
g_{t})\otimes v\big{\\}}\,d\mu\,dt$,

Lévy area

$E=\int_{0}^{\infty}\int_{\Lambda}\big{\\{}v\otimes(v\circ g_{t})-(v\circ
g_{t})\otimes v\big{\\}}\,d\mu\,dt$,

where $u\otimes v=uv^{T}\in{\mathbb{R}}^{d\times d}$ for
$u,v\in{\mathbb{R}}^{d}$. These series define a positive semi-definite
symmetric matrix $\Sigma\in{\mathbb{R}}^{d\times d}$ and a skew-symmetric
matrix $E\in{\mathbb{R}}^{d\times d}$. Moreover (under smoothness assumptions
on $a$ and $b$ which play no further role in this paper), the solutions
$x^{(\epsilon)}$ converge weakly to solutions of the SDE (2.2) where the
Brownian motion $W$ has covariance matrix $\Sigma$ and the modified drift term
is given by

$\tilde{a}(X)=a(X)+\tfrac{1}{2}\sum_{\alpha,\beta,\gamma=1}^{d}E^{\gamma\beta}\partial_{\alpha}b^{\beta}(X)b^{\alpha\gamma}(X).$

Generally, $\Sigma$ is positive definite in the setting of [11]. Indeed, the
case $\det\Sigma=0$ is infinitely unlikely in a sense that can be made
precise, see for example [3, Section 2.3]. Given its antisymmetry, a natural
question is to ask whether the Lévy area $E$ may be forced to vanish in
certain circumstances. Clearly, if $d=1$ then $E=0$. More generally, if
$b:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d}$ satisfies an exactness
condition, namely that $b^{-1}=dh$ for some
$h:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$, then $E=0$ as shown in [9].

If $v$ transforms as $v\circ R=v$ or $v\circ R=-v$, where $R$ is a time-
reversal symmetry for the fast dynamics, then again it is easily verified (see
Remark 3.3) that $E=0$. This situation can occur in simple physical situations
where $v$ represents position or velocity, such as in the dispersing billiards
examples in [2].

In this paper we show that the cases $v\circ R=v$ and $v\circ R=-v$ are the
only situations where time-reversal symmetry forces the Lévy area to vanish,
and typically $E\neq 0$ for the remaining time-reversible systems. In
particular, we show that $v$ transforms as $v\circ R=Av$ where
$A=I_{d^{+}}\oplus(-I_{d^{-}})$ in appropriate coordinates (with
$d^{+}+d^{-}=d$), and that $E=\left(\begin{array}[]{cc}0&E_{0}\\\
-E_{0}^{T}&0\end{array}\right)$ where $E_{0}$ is a general $d^{+}\times d^{-}$
matrix.

##  3 Time-reversal symmetry

In this section, we introduce time-reversal symmetry into the fast-slow ODE
(2) and derive simplified formulas for the covariance $\Sigma$ and Lévy area
$E$. Our only assumption in this section is that the Green-Kubo-type formulas
for $\Sigma$ and $E$ converge.

We assume that there is a time-reversal symmetry $(x,y)\mapsto(Sx,Ry)$ where
$S\in{\mathbb{R}}^{d\times d}$ and $R:\Lambda\to\Lambda$ satisfy $S^{2}=I$ and
$R\circ R=I$. As usual, this means that $(Sx(-t),Ry(-t))$ is a solution of (2)
whenever $(x(t),y(t))$ is a solution. We suppose also that $\mu$ is
$R$-invariant.

###### Proposition 3.1

For the fast-slow system (2), the constraints arising from time-reversibility
read as follows:

  * (a)

$g(Ry)=-Rg(y)$ for all $y\in\Lambda$;

  * (b)

$a(Sx)=-Sa(x)$ for all $x\in{\mathbb{R}}^{d}$;

  * (c)

$b(Sx)v(Ry)=-Sb(x)v(y)$ for all $x\in{\mathbb{R}}^{d},\,y\in\Lambda$.

  * (d)

$g_{t}(Ry)=Rg_{-t}(y)$ for all $y\in\Lambda,\,t\in{\mathbb{R}}$.

###### Proof.

(a)–(c) are equivalent to $(S,R)$ defining a time-reversal symmetry, while (d)
follows from (a). ∎

###### Corollary 3.2

$\begin{split}\Sigma&=\int_{0}^{\infty}\int_{\Lambda}\big{\\{}v\otimes(v\circ
g_{t})+(v\circ R)\otimes(v\circ R\circ g_{t})\big{\\}}\,d\mu\,dt,\\\
E&=\int_{0}^{\infty}\int_{\Lambda}\big{\\{}v\otimes(v\circ g_{t})-(v\circ
R)\otimes(v\circ R\circ g_{t})\big{\\}}\,d\mu\,dt.\end{split}$

###### Proof.

By invariance of $\mu$ under the fast flow $g_{t}$ and $R$, and Proposition
3.1(d),

$\begin{split}\int_{\Lambda}(v\circ g_{t})\otimes
v\,d\mu&=\int_{\Lambda}(v\circ R)\otimes(v\circ g_{-t}\circ R)\,d\mu\\\
&=\int_{\Lambda}(v\circ R)\otimes(v\circ R\circ g_{t})\,d\mu.\end{split}$

Substituting this into the covariance and Lévy area formulas from Section 2
yields the result. ∎

###### Remark 3.3

It can be seen already from Corollary 3.2 that $E=0$ if either $v\circ R=v$ or
$v\circ R=-v$. (This was the case in [2].) Our main result, Theorem 4.3 below,
implies that typically $E\neq 0$ outside of these cases.

To avoid pathologies, from now on we suppose that
$b:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d}$ defines a nonsingular matrix
in ${\mathbb{R}}^{d\times d}$ on a dense subset of ${\mathbb{R}}^{d}$ and that
$\\{v(y):y\in\Lambda\\}$ spans ${\mathbb{R}}^{d}$. Then condition (c) of
Proposition 3.1 simplifies as follows.

###### Lemma 3.4

Condition (c) holds if and only if there exists $A\in{\mathbb{R}}^{d\times d}$
with $A^{2}=I$ such that

$b(Sx)=-Sb(x)A\quad\text{for all $x\in{\mathbb{R}}^{d}$}$ (3.1)

and

$v(Ry)=Av(y)\quad\text{for all $y\in\Lambda$.}$ (3.2)

###### Proof.

It is immediate that the existence of $A$ with $A^{2}=I$ satisfying (3.1) and
(3.2) implies that condition (c) holds. Conversely, let
$X=\\{x\in{\mathbb{R}}^{d}:\det b(x)\neq 0\\}$. Define

$A:X\to{\mathbb{R}}^{d\times d},\qquad A(x)=-b(x)^{-1}Sb(Sx).$

For $x_{1},x_{2}\in X$, it follows from condition (c) that

$v(y)=A(x_{1})v(Ry),\qquad v(Ry)=A(x_{2})v(y)\quad\text{for all
$y\in\Lambda$}.$

Hence $v(y)=A(x_{1})A(x_{2})v(y)$ for all $y$ and it follows from the
condition on $v$ that

$A(x_{1})A(x_{2})=I\quad\text{for all $x_{1},x_{2}\in X$}.$

Taking $x_{1}=x_{2}$, we obtain $A(x)=A(x)^{-1}$ for all $x\in X$ and hence
that $A(x)\equiv A$ is constant on $X$. We immediately obtain (3.2). Also
(3.1) holds on the dense set $X\subset{\mathbb{R}}^{d}$ and hence on the whole
of ${\mathbb{R}}^{d}$ by continuity of $b$. ∎

Let $\pi^{\pm}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be the projections onto
the $\pm 1$ eigenspaces of $A$. Then we can write $v$ in (2) uniquely as
$v=v^{+}+v^{-}$ where $v^{\pm}=\pi^{\pm}v$. Note that
$\int_{\Lambda}v^{\pm}\,d\mu=0$.

###### Corollary 3.5

In the $(\pi^{+},\pi^{-})$ coordinates,

$\begin{split}\Sigma=\left(\begin{array}[]{cc}\Sigma^{+}&0\\\
0&\Sigma^{-}\end{array}\right),\qquad E=\left(\begin{array}[]{cc}0&E_{0}\\\
-E_{0}^{T}&0\end{array}\right),\end{split}$

where

$\Sigma^{\pm}=2\int_{0}^{\infty}\int_{\Lambda}v^{\pm}\otimes(v^{\pm}\circ
g_{t})\,d\mu\,dt,\qquad
E_{0}=2\int_{0}^{\infty}\int_{\Lambda}v^{+}\otimes(v^{-}\circ
g_{t})\,d\mu\,dt.$

###### Proof.

Since $v\circ R=Av=v^{+}-v^{-}$, we obtain

$\begin{split}\Sigma&=2\int_{0}^{\infty}\int_{\Lambda}\big{\\{}v^{+}\otimes(v^{+}\circ
g_{t})+v^{-}\otimes(v^{-}\circ g_{t})\big{\\}}\,d\mu\,dt,\\\
E&=2\int_{0}^{\infty}\int_{\Lambda}\big{\\{}v^{+}\otimes(v^{-}\circ
g_{t})+v^{-}\otimes(v^{+}\circ g_{t})\big{\\}}\,d\mu\,dt.\end{split}$

The result follows. ∎

##  4 Generality of $E_{0}$

In this section, we show that there are no further restrictions on the Lévy
area $E$ beyond those in Corollary 3.5. Recall that the vector field $g$ in
(2) is assumed to be $C^{r}$ for some $r\geq 1$. We suppose that the fast
dynamics defined by $g$ is mixing sufficiently quickly, so that the series for
$E$ in Section 2 and hence $E_{0}$ in Corollary 3.5 converge for all $C^{r}$
functions $v:\Lambda\to{\mathbb{R}}^{d}$ with $\int_{\Lambda}v\,d\mu=0$. We
exclude the uninteresting case where $\Lambda$ is a fixed point. (For then
$v|_{\Lambda}\equiv 0$, so $\Sigma=E=0$ and there is no stochasticity in the
limit.)

###### Remark 4.1

In the setting of [4, 7, 11, 12], the matrices $\Sigma$ and $E$ depend
continuously on $v\in C^{r}$.

###### Proposition 4.2

For any $f\in C^{r}({\mathbb{R}}^{m},{\mathbb{R}}^{d^{+}})$, $h\in
C^{r+1}({\mathbb{R}}^{m},{\mathbb{R}}^{d^{-}})$ with
$\int_{\Lambda}f\,d\mu=0$, there exists $v\in
C^{r}({\mathbb{R}}^{m},{\mathbb{R}}^{d})$ with $\int_{\Lambda}v\,d\mu=0$ such
that $E_{0}=\int_{\Lambda}f\otimes h\,d\mu$.

###### Proof.

Recall that $g_{t}$ denotes the fast flow generated by the vector field $g$ in
(2). We define $v=v^{+}+v^{-}:\Lambda\to{\mathbb{R}}^{d}$ by setting

$v^{+}=-f,\qquad v^{-}=w-\int_{\Lambda}w\,d\mu$

where

$w=\nabla h\cdot g=\sum_{j}\frac{\partial h}{\partial y_{j}}g_{j}.$

Along solutions $y(t)$ to $\dot{y}=g(y)$,

$w(y(t))=\sum_{j}\frac{\partial h}{\partial
y_{j}}(y(t))\,\dot{y}_{j}(t)=\frac{d}{dt}h(y(t)).$

Hence $\int_{0}^{T}w(y(t))\,dt=h(y(T))-h(y(0)).$ In other words,

$\int_{0}^{T}w\circ g_{t}\,dt=h\circ g_{T}-h.$

Since the flow is mixing and $\int_{\Lambda}v^{+}\,d\mu=0$,

$\begin{split}\int_{0}^{T}\int_{\Lambda}&v^{+}\otimes(v^{-}\circ
g_{t})\,d\mu\,dt=\int_{\Lambda}\int_{0}^{T}v^{+}\otimes(w\circ
g_{t})\,dt\,d\mu\\\ &=\int_{\Lambda}v^{+}\otimes\Big{(}h\circ
g_{T}-h\Big{)}\,d\mu\to-\int_{\Lambda}v^{+}\otimes
h\,d\mu=\int_{\Lambda}f\otimes h\,d\mu\end{split}$

as $T\to\infty$. By the definition of $E_{0}$ in Corollary 3.5 (up to a factor
of $2$ which can be incorporated into $f$) we have proved the result. ∎

###### Theorem 4.3

For any $d^{+}\times d^{-}$ matrix $F$, there exists a $C^{r}$ function
$v:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d}$ with $\int_{\Lambda}v\,d\mu=0$ such
that $E_{0}=F$.

###### Proof.

We claim that it is possible to choose $v$ so that $E_{0}$ has full rank,
namely $\min\\{d_{+},d_{-}\\}$. Assuming this is the case, let
$L^{\pm}\in{\mathbb{R}}^{d^{\pm}\times d^{\pm}}$. By the definition of $E_{0}$
in Corollary 3.5, transforming $v$ to $(L^{+}v^{+},L^{-}v^{-})$ changes
$E_{0}$ to $L^{+}E_{0}(L^{-})^{T}$. By standard linear algebra, this results
in any desired matrix in ${\mathbb{R}}^{d^{+}\times d^{-}}$.

It remains to prove the claim. The first step is to construct suitable
functions $f_{1}\in L^{2}(\Lambda,{\mathbb{R}}^{d^{+}})$ and $h_{1}\in
L^{2}(\Lambda,{\mathbb{R}}^{d^{-}})$ such that $\int_{\Lambda}f_{1}\otimes
h_{1}\,d\mu$ has full rank. The second step is to approximate $f_{1}$ and
$h_{1}$ by smooth functions $f:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d^{+}}$ and
$h:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d^{-}}$ so that $\int_{\Lambda}f\otimes
h\,d\mu$ has full rank. The third step is to apply Proposition 4.2.

Step 1 Since $\Lambda$ is not a fixed point, we can choose infinitely many
orthonormal functions $\phi_{j}\in L^{2}(\Lambda)$ with
$\int_{\Lambda}\phi_{j}\,d\mu=0$ and
$\int_{\Lambda}\phi_{i}\phi_{j}\,d\mu=\delta_{ij}$. Let
$f_{1}=\sum_{i=1}^{d^{-}}\alpha_{i}\phi_{i}$ and
$h_{1}=\sum_{j=1}^{d^{-}}\beta_{j}\phi_{j}$ where
$\alpha_{i}\in{\mathbb{R}}^{d^{+}}$, $\beta_{j}\in{\mathbb{R}}^{d^{-}}$. Then

$\int_{\Lambda}f_{1}\otimes
h_{1}\,d\mu=\sum_{i,j}(\alpha_{i}\otimes\beta_{j})\int_{\Lambda}\phi_{i}\phi_{j}\,d\mu=\sum_{j=1}^{d^{-}}\alpha_{j}\otimes\beta_{j}.$

Let $F_{1}=\int_{\Lambda}f_{1}\otimes h_{1}\,d\mu$. Taking $\beta_{j}$ to be
the $j$’th canonical unit vector, $F_{1}$ is the $d^{+}\times d^{-}$ matrix
with columns $\alpha_{1},\dots,\alpha_{d^{-}}$. In particular, $F_{1}$ is
arbitrary and we can choose the $\alpha_{i}$ so that $F_{1}$ is a matrix of
full rank.

Step 2 Now choose functions $\tilde{\phi}_{j}\in C^{\infty}({\mathbb{R}}^{m})$
with $\int_{\Lambda}|\phi_{j}-\tilde{\phi}_{j}|^{2}\,d\mu$ small111For
instance, to approximate $\phi_{1}$ by a $C^{\infty}$ function, first
approximate $\phi_{1}$ in $L^{2}$ by a simple function
$\sum_{k=1}^{\ell}c_{k}1_{E_{k}}$ (with $c_{k}\in{\mathbb{R}}$ and
$E_{k}\subset\Lambda$ measurable). By outer regularity of the Borel
probability measure $\mu$, there exist open neighbourhoods
$U_{k}\subset{\mathbb{R}}^{m}$ of $E_{k}$ with $\mu(U_{k}\setminus E_{k})$
small. Hence $\sum_{k=1}^{\ell}c_{k}1_{U_{k}}$ is $L^{2}$-close to $\phi_{1}$.
Choose $V_{k}\subset{\mathbb{R}}^{m}$ closed such that
$\overline{U_{k}}\subset\operatorname{Int}V_{k}$ with $\mu(V_{k}-U_{k})$
small. By Urysohn’s Lemma, there exists a continuous function
$\psi_{k}:{\mathbb{R}}^{m}\to[0,1]$ supported on $V_{k}$ with
$\psi_{k}|_{U_{k}}{\equiv 1}$. In this way we obtain a continuous function
$\sum_{k=1}^{\ell}c_{k}\psi_{k}$ that is $L^{2}$-close to $\phi_{1}$. Finally,
each $\psi_{k}$ can be uniformly approximated by a $C^{\infty}$ function
$\zeta_{k}$ resulting in a $C^{\infty}$ function
$\tilde{\phi}_{1}=\sum_{k=1}^{\ell}c_{k}\zeta_{k}$ that is $L^{2}$-close to
$\phi_{1}$. and define $f$, $h$ using $\tilde{\phi}_{j}$ in place of
$\phi_{j}$ with $\alpha_{i}$, $\beta_{j}$ unchanged. This results in a matrix
$F=\int_{\Lambda}f\otimes h\,d\mu$ close to $F_{1}$. In particular, taking the
approximation close enough ensures that $F$ is still of full rank.

Step 3 It follows from Proposition 4.2 that we can choose $v\in
C^{r}({\mathbb{R}}^{m},{\mathbb{R}}^{d})$ with $\int_{\Lambda}v\,d\mu=0$ so
that $E_{0}=\int_{\Lambda}f\otimes h\,d\mu$. In particular, such $E_{0}$ has
full rank, proving the claim. ∎

### Acknowledgements

This research was supported in part by the visitor program at the Sydney
Mathematical Research Institute (SMRI) in early 2020. IM is very grateful for
the hospitality of the University of Sydney, and in particular SMRI, at both
ends of the pandemic.

We would like to thank Greg Pavliotis for his comments on detailed balance.

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