On the Siegel-Sternberg Linearization Theorem

https://doi.org/10.1007/s10884-021-09947-7

On the Siegel-Sternberg Linearization Theorem

Jürgen Pöschel¹

Received: 3 December 2020 / Revised: 4 January 2021 / Accepted: 11 January 2021 / Published online: 8 March 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021

Abstract

We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffen- tiable maps which includes the analytic case, the smooth case and the Gevrey case. It may be regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode’s and pde’s.

We consider the problem of linearizing a smooth mapgin the neighbourhood of a fixed point. Placing this fixed point at the origin we writeg=Λ+ ˆg, whereΛdenotes its linear part andgˆcomprises its nonlinear terms. We then look for a diffeomorphismϕ =id+ ˆϕ around the origin such that

ϕ⁻¹◦g◦ϕ=Λ.

It is well known that any solution to this problem depends on the eigenvalues of its lineariza- tion. Letgbe a map ins-dimensional space, and letλ1, . ., λsbe the eigenvalues ofΛ. Within the category of formal power series there is always a formal solution to this problem, if the infinitely manynonresonance conditions

λ^k−λi =0, |k| ≥2, 1≤i ≤s, are satisfied, where fork∈ {0,1, . .}^s,

λ^k=λ^k₁¹· ·λ^ks^s, |k| =k₁+. .+k_s.

Convergence of these formal solutions for analyticg, however, can be established only if the map is analytic and certainsmall divisor conditionsare satisfied, such as

|λ^k−λi| ≥ γ

|k|^τ, |k| ≥2, 1≤i≤s,

with someγ >0 and largeτ. This is the celebrated result of Siegel [33,34], who was the first to overcome those small divisor difficulties. Later, those small divisor conditions were considerably relaxed by Bruno and Rüssmann [5,6,31]. On the other hand, it is also well known that bad small divisors destroy analyticity.

B

1 IADM, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

A few years later, Sternberg [35] established the same result within the category of smooth maps without any small divisor conditions. More precisely, ifgis real, smooth and nonres- onant, thengcan be linearized by a smooth diffeomorphism around the fixed point. So it appears that small divisors are invisible in the smooth category [36].

The purpose of this paper is to prove that this is not the case. Rather, the smooth category by itself is simply too indifferent to discern small divisiors. But looking more closely in terms of classes of ultradifferentiable functions one can clearly quantify the effect of near resonances no matter how small they get how fast. The results of Siegel and Sternberg are then two instances of one and the same general theorem.

Ultradifferentiable functions form subclasses of smooth functions that are described by growth restrictions on their derivatives. Those restrictions are given in terms of a sequence of positive real numbers that serve as weights for those derivatives. More specifically, if m = (m_n)n≥1 is a sequence of positive numbers, then a smooth function f on somes- dimensional domain is said to be of classE^m, if for any point in the domain of f there is a neighborhoodUand a positive numberr>0 such that

x∈Usupsup

n≥1

1 n!

Dⁿf(x) mn

rⁿ <∞.

Those classes are also called local Denjoy-Carleman classes of Roumieu type and are often denoted byE^{m} orC{m}. Best known among those are the Gevrey classesG^s defined in terms of the weights

m=(n!^s−1), s>1.

Within these classes, a version of the Sternberg theorem was recently proven by Stolovitch [36].

To state the general result we measure the size of near-resonances in terms of theirnon- resonance functionΩdefined by

Ω(q)= max

2≤|k|≤q max

1≤i≤s|λ^k−λi|⁻¹, q≥2.

We say that a weightwdominates a nonresonance functionΩ, if there exists a constanta such that

ν≤log2|k|

logΩ(2^ν+1)

2^ν ≤a+logwk

|k| , |k| ≥2.

The point is thatanynonresonance function can be dominated by an appropriate weight.

Hence, the following theorem may be regarded as a small divisior theorem without explicit small divisior conditions.

Siegel-Sternberg Linearization TheoremConsider a smooth map g of class E^min a neigh- bourhood of a fixed point. If its linearization at the fixed point is nonresonant, then the map can be linearized by a local diffeomorphismϕof class E^{m w},wherewis any weight dom- inating the associated nonresonance functionΩsuch that m wis log-convex and strongly non-analytic.

Here,m w=(wkm_k)k≥0is log-convex, if the logs of the weights are convex with respect tok. Strongly non-analytic weights are defined in Sect.7. Both conditions essentially amount to weak growth conditions.

Note that the reality and nonresonance conditions imply that the fixed point is hyperbolic.

So this need not be stipulated explicitly.

This theorem comprises all versions of the Sternberg linearization theorem established so far. We discuss the most relevant cases.

No small divisors:In this case,Ω is bounded. This amounts to the classical theorem of Poincaré [25] and is a particularly simple instance of the next case.

Good small divisors:If the eigenvalues ofΛsatisfy small divisor conditions of Siegel or Bruno-Rüssmann type, then

ν≥0

logΩ(2^ν+1) 2^ν <∞.

Indeed, this is the generaldefinitionof admissible small divisors for convergent majorant techniques as introduced by Bruno [5,6]. In this case, we simply choose the constant weight w =(1). So the normalizing transformationϕis of thesame class E^m as the mapg. This applies to the analytic categoryC^ω– which amounts to the classical results of Siegel [33,34]

–, the Gevrey categoryG^s– see Stolovitch [36] –, andany otherfdbspaceE^mas defined in Lemma3.

Gevrey small divisors:This amounts to the existence of aδ >0, so that

1≤ν≤log₂|k|

logΩ(2^ν+1)

2^ν ≤a+δlog|k|

for almost allk. In this case, we can choosew=(k!^δ). So ifgis of Gevrey classG^s, thenϕ is of Gevrey classG^s+δ– see again [36]. But the same loss of regularity is observed in any otherfdbspaceE^m.

Arbitrarily small divisorThe theorem also applies to the case where no small divisor estimate and no smoothness class are given at all.Anysmooth mapgis ofsomeclassE^m, since we only need to choose an appropriate weightm in dependence on the growth of the deriva- tives ofg. Moreover, there always issomeweightwdominating the associated resonance functionΩ. Increasingwif necessary,m wis log-convex and strongly non-analytic. Hence, the theorem applies also in this case and amounts to the general Sternberg theorem with additional quantitative information.

OutlineAn indispensable prerequisite for doing analysis within spaces of ultradifferentiable functions is their stability with respect to composition. Partial results are well known and rely on the Faà di Bruno formula for higher derivatives of composite functions and the assumption that derivation is well behaved. The latter, however, amounts to a severe growth restriction on the weightsw. The essential step is to completely remove the latter restriction and to give an exact description of those spaces. The proof is also much simpler and works by considering formal power series only. Along this way we reprove regularity results for solutions of ode’s and pde’s without employing tedious estimates.

The proof of the Siegel-Sternberg theorem then consists of two parts. First, a small divisor problem is solved to linearize the mapgup to a flat term. But working within the category of ultradifferentiable functions it is not necessary to use any bounds on those divsisors. It suffices to keep control of their effect and proliferation. Subsequently, the flat term is removed using hyperbolicity, which is a consequence of nonresonance and reality. Here, we transfer Steinberg’s classical approach to the properE^mclasses using heavily their closedness under composition und flows and also using a version of the Whitney extension theorem.

1 Ultradifferentiable Functions and Maps

First consider complex valued functions. With any smooth complex valued function fdefined in a neighborhood of a pointainR^swe associate its formal Taylor series expansion ata,

Taf :=

k∈Λ

fk(a)x^k, where as usual

fk:= 1

k!∂^kf := 1

k1! · ·ks!∂_x^k₁¹· ·∂_x^k_s^sf, x^k=x₁^k1· ·x^k_s^s

fork=(k₁, . .,k_s)inΛ= {0,1, . .}^s. As constant terms won’t play a role in our considera- tions, we also let

T˙af :=Taf − f0=

k=0

fk(a)x^k.

Aweightis any mapm: Λ→(0,∞). Theweighted Taylor seriesof fwith and without constant term are then defined as

M_a^mf :=

k∈Λ

|f_k(a)|

m_k x^k, M˚_a^mf :=M_a^mf − f₀, respectively, and we set

f^m_a,r :=

k=0

|fk(a)|

mk

r^|k|=M˚_a^mf(r, . .,r).

Obviously,f is of classE^mif and only if any point in the domain of f has a neighborhoodU and a positive numberr>0 such that

f^m_U,r :=sup

a∈Uf^m_a,r <∞.

Note that these are only semi-norms, as we omit the constant term. In addition, if they are finite for somer > 0, then they can be made arbitrarily small by choosingr sufficiently small.

Here are some standard examples. In the one-dimensional case the constant sequence m=(1)n≥0defines the class of analytic functions on open subsets of the real line, denoted byC^ω=E⁽¹⁾. More generally, as shown in Appendix A,

C^ω=E^m⇔0<infm^1/n_n ≤supm^1/n_n <∞.

Form=(n!^s−1)_n≥1we obtain the Gevrey spaces [12]

G^s=E^(n!s−1), s>1, well known in pde theory.

These examples naturally extend to the multi-dimensional case. Here, one usually con- siders weights as functions of|k| =k1+. .+ksrather thank. For instance,

G^s=E^(k!s−1)=E^(|k|!s−1)

by standard inequalities for the factorial. But with true multi-dimensional weights one may also consider functions with anisotropic differentiability properties – see for example [7] and Sect.6.

We also need to consider smooth maps fromR^sinto some spaceR^t. For f =(f₁, . .,f_t) we set

f^m_a,r := max

1≤i≤tf_i^m_a,r. The Taylor coefficients f_kof f aret-vectors. Defining

M˚_a^mf :=

k=0

f_k(a)

m_k x^k, f_k :=(|f_k,1|, . .,|f_k,t|), and denoting the usual sup-norm by·∞we have

f^m_a,r = M˚_a^mf(r, . .,r)∞.

In either case, fis of classE^mif and only if ˚M_a^mf is analytic on as-dimensional polydisc with a radius that can be chosen locally constant.

2 Basic Properties and Assumptions

For the time being we focus on one-dimensional weights. Multi-dimensional weights will be considered again in Sect.4.

Certain properties of the spacesE^mare more directly connected with the sequenceM= (Mn)n≥1of the associated weights

Mn :=n!mn, n≥1,

controlling the derivatives f⁽ⁿ⁾rather than the Taylor coefficients fn of a function f. If A:=lim infM_n^1/n<∞,

thenE^mis a proper subspace ofC^ωand not closed under composition of maps – see Appendix A. Hence we will assume thatA= ∞. In this case one always has

E^m=E^m˘,

wherem˘is characerized by the fact thatM˘ is the largest log-convex minorant ofM. That is, we have

M˘_n²≤ ˘M_n−1M˘_n+1,

which is tantamount toM˘_n/M˘_n−1forming an increasing sequence [1,22]. — From now on we therefore make the following

General AssumptionThe weightsm=(mn)n≥1are ‘weakly log-convex’:

m_n= M_n n!

with a log-convex sequence M = (Mn)n≥1 so that one can write Mn = μ1 · ·μn with an increasing sequence0 < μ1 ≤ μ2 ≤ . ..As a consequence,M_n^1/n is increasing and M_n^1/n ∞if and only ifμn ∞.

Under this assumption a weightmis always ‘weakly submultiplicative’: we haveMkMl ≤ Mk+land thus, by the binomial formula,

mkml = M_kM_l

k!l! ≤ (k+l)!

k!l! M_k+l

(k+l)! ≤2^k+lmk+l, k,l≥1.

As a consequence, E^m is always an algebra. But note that m is not necessarily almost submultiplicative as defined in Lemma2.

Another important consequence of this assumption is the existence of so called charac- teristicE^m-functions. The following lemma is well known, as is its proof [1,17]. We state it for functions on an interval.

Lemma 1 Under the general assumption the space E^mcontains for any given point o in the considered interval a functionηsuch thatη(o)=0and

ηn(o)=iⁿ⁻¹sn, sn ≥mn, n≥1.

Proof SetTn=μⁿ_n/Mn. We then have T_n =sup

k>0

μ^k_n M_k,

asμn/μk≥1 fork≤nandμn/μk ≤1 fork ≥n. Now assume for simplicity thatois the origin on the real line and define the functionρ_νby

ρ_ν(x)= e^iμνx

T_ν , ν≥1.

Itsn-th derivative is uniformly bounded by ρ_ν⁽ⁿ⁾ ≤μⁿ_ν

T_ν ≤M_n, n≥1. Hence, if we defineϕby

ϕ(x)=

ν≥1

2^−νρ_ν(x),

thenϕ⁽ⁿ⁾ ≤M_nandϕn ≤M_n/n! =m_nfor alln≥1. Henceϕis inE^m, and its Taylor coefficients at zero are

ϕn=iⁿs_n, n≥1, with

sn= 1 n!

ν≥1

1 2^ν

μⁿ_ν T_ν ≥ 1

n!

1 2ⁿ

μⁿ_n T_n = 1

2ⁿ M_n

n! = m_n 2ⁿ.

So the rescaled functionη = −iϕ◦2−η0 with a suitable constantη0 has all the required

properties.

We note that the proof does not make use of the assumption thatμn → ∞. But if not, thenE^mconsists of analytic functions, and the result is trivial.

From the existence of characteristic functions it follows that

E^m˜ ⊂E^m ⇔ m˜ m ⇔ m˜n ≤λⁿmn, n≥1, the latter holding with someλ≥1. Consequently,

E^m˜ =E^m ⇔ m˜ m :⇔ ˜mm ∧mm˜.

Obviously,is an equivalence relation among weights, identifying all weights which define the sameE-space.

3 Properties of Weights

All of the following properties pertain to the equivalence class of a weight, thus are properties of the associated spacesE^m. The corresponding analytical properties will be discussed later.

We will use Stirling’s inequality in the form n

e ≤n!^1/n≤ 2n

e , n≥2.

For instance, asMn^1/nis increasing by the general assumption it follows that m_k^1/k

m_l^1/l = l!^1/l k!^1/k

M_k^1/k Ml1/l ≤ l!^1/l

k!^1/k ≤2l

k , 1≤k ≤l. (1)

We will use this estimate in the next proof.

Lemma 2 The following two properties are equivalent.

(i) m is ‘almost increasing’: there is aλ≥1such that m^1/k_k ≤λm^1/l_l , 1≤k≤l.

(ii) m is ‘almost submultiplicative’ or ‘asm’: there is aλ≥1such that m_k1· ·mkr ≤λ^km_k, k=k₁+. .+k_r, for any choice of r ≥1and k1, . .,kr ≥1.

Proof Ifmis almost increasing andk=k₁+. .+k_r, then mk_i ≤λ^kim^k_k^i/k, 1≤i≤r.

Taking the product over 1≤i ≤rgives the second property. Conversely, ifmisasm, then in particularmⁿ_k ≤λ^nkmnkand thus

m^1/k_k ≤λm^1/nk_nk , n,k≥1.

Given 1≤k≤land choosingn≥1 so thatnk≤l≤nk+kwe further conclude with (1) that

m^1/nk_nk ≤4m_l^1/l.

These two estimates together show thatmis almost increasing.

In the next lemma, ‘fdb’ is short for ‘Faà di Bruno’. The property thus named is motivated by the composition rule for formal power series – see the Main Lemma8– and reflects the higher order chain rule named after Faà di Bruno [10]. The term was coined in [27]. But note that this chain rule itself is never used in this paper.

Lemma 3 Each of the following properties implies the next one.

(i) m is ‘log-convex’: m²_n≤m_n−1m_n+1for all n≥2, or equivalently, m_n =α1α2· ·αn, 0< α1≤α2≤. ..

(ii) m is ‘block-convex’:

maxn≤2^ναn ≤min

n>2^ναn, ν≥0.

(iii) m is ‘strongly submultiplicative’: there is aλ≥1such that m_km_l≤λm_k+l−1, k,l≥1.

(iv) m is ‘strictlyfdb’: there is aλ≥1so that for all r≥1and k₁, . .,k_r =0, m_rm_k1· ·mkr ≤λ^rm_{k1+. .+kr}.

(v) m is ‘fdb’: there is aλ≥1so that for all r≥1and k₁, . .,k_r =0, m_rm_k1· ·mkr ≤λ^km_{k1+. .+kr}, k =k₁+. .+k_r. Moreover,(iii)and(iv)hold withλ=1by passing to an equivalent weight.

We remark that in (v) the factorλcan not be normalized to 1. We do not know whether (iv) and (v) are equivalent or not. — The block-convex property seems to be a new concept and offers a more flexible way to construct strictlyfdbweights – see Example2.

Proof (i)⇒(ii) This is obvious.

(ii)⇒(iii) Dividing allm_n bym₁we may assume thatm₁=1. Now, for given 1≤l≤k we fixν≥0 so that 2^ν ≤k<2^ν+1. Ifl≤2^ν, then by hypnosis

α2· ·αl ≤αk+1· ·αk+l−1. Otherwise,k≥l>2^ν, and we argue that

α2· ·αl =(α2· ·α2^ν)(α2^ν+1· ·αl)

≤(α_k+1· ·α_k+2^ν₋₁)(α2^ν+1· ·αl)

≤(αk+1· ·αk+2^ν−1)(αk+2^ν· ·αk+l−1)

=αk+1· ·αk+l−1.

This is equivalent tom_km_l ≤m_k+l−1. So we indeed obtain (iii) withλ=1.

(iii)⇒(iv) Ifmis strongly submultiplicative, then

mrmk1· ·mkr ≤λmk1+r−1mk2· ·mkr

≤λ²mk1+k2+r−2· ·mkr ≤ · ·

≤λ^rm_{k1+. .+kr}.

(iv)⇒(v) This again is obvious, ask≥r.

The properties of beingfdbandasmare closely related but not equivalent. Examples to this effect are given in Appendix B. But ifmisfdb, then the situation is clear cut.

Lemma 4 Suppose m isfdb. Then m isasmif and only if m is ‘analytic’:

α:= inf

n≥1m^1/n_n >0.

Proof Supposemisfdb. Ifα >0, then

mk₁· ·mkr ≤α^−rmrmk₁· ·mkr ≤α^−rλ^kmk

withk = k1+. .+kr ≥r. Hencemis alsoasm. Conversely, ifmisasm, thenmis also almost increasing by Lemma2. The latter property includes that λm^1/nn ≥ m1 for alln,

whenceα >0.

The conditionα >0 amounts toC^ω⊂E^m– see Lemma9. So wheneverE^mcontains all the analytic functions,fdbimpliesasm. The converse, however, is not true without further assumptions. To this end, the following property is usually required, wherem´ denotes the

‘left shift’ ofmdefined bym´n=m_n+1forn≥1.

Lemma 5 The following properties are equivalent.

(i) m´ m.

(ii) m is ‘diff-stable’:δ:=sup_n≥2 mn

m_n−1

_1/n

<∞.

(iii) sup_n≥1μ^1/nn <∞.

Proof On one hand,

´

mm⇔m_n+1≤λⁿm_n, n≥1.

On the other hand,

mn

mn−1 = (n−1)!

n!

Mn

Mn−1 = μn

n .

From this the equivalence of all three statements follows.

Lemma 6 If m isasmand diff-stable, then m isfdb.

Proof Givenm_rm_k1· ·m_krwe arrange the factors so thatk₁=. .=k_s =1 andk_s+1, . .,k_r >

1 for some 0≤s≤n. Thenk_s+1+. .+kr =k−s, and makingδ≥m1we get m_rm_k1· ·m_kr ≤m_rm^s₁δm_ks+1−1· ·δm_kr−1

≤δ^rm_rm_ks+1−1· ·mkr−1

≤δ^rλ^km_k.

Hence,mis alsofdb.

The assumption of diff-stability, however, represents a severe growth restriction and is certainly not necessary, and it will never be required in the sequel. There is, however, an interesting relation betweenfdbandasmweights, which has not been noticed before and will be used in the proof of Theorem3.

Lemma 7 The weight m isfdbif and only if its left shiftm is´ asm. Proof First supposem´ isasm. Withr≥2 andm´₀:=m₁we get

mrmk₁· ·mk_r = ´mr−1m´k₁−1· · ´mk_r−1≤λ^k−1m´k−1

withk=r+k1−1+. .+kr−1=k1+. .+kr as usual. Asm´k−1 =mk, the weightm isfdb.

Conversely, supposemisfdbandm₁≥1 for simplicity. Letl_i =k_i+1 for 1≤i≤r.

Ifl₁≥r−1, say, we write

´

m_k1· · ´m_kr =m_l1m_l2· ·m_lr ≤m_l1m_l2· ·m_lrm^l₁^1−r+1.

We then apply thefdbproperty to the last term withm_l1as the ‘leading factor’ andl₁trailing factors to get

´

m_k1· · ´m_kr ≤λ^l−r+1m_l−r+1, l=l₁+. .+l_r.

Asl−r=kandm_k+1 = ´m_kwe get theasmproperty form´ in this case.

Otherwise, we havel₁, . .,l_r <r−1. We then proceed by induction to get ml₁ml₂· ·ml_r ≤λ^nsmn_s−n_s−1ml_a+n_s−1+1· ·ml_r

witha =l₁ andn_s =l₂+. .+l_a+ns−1 fors ≥1 withn₀ =0. After finitely many steps, we getn_s−n_s−1 ≥r−a−1−n_s−1, orn_s ≥r−1−l₁. Now we apply the immediate estimate to obtain

m_l1m_l2· ·m_lr ≤λ^ns+nm_n with

n=l_a+ns−1+1+. .+l_r+l₁+n_s−r+1

=l1+. .+lr −r+1

=k+1

as before. So also in this case we obtain theasmproperty form.´ We note that for Gevrey weights we have

Mn =n!^s, mn =n!^s−1.

Hence, fors>0 they are weakly log-convex. Fors≥1 they are log-convex and thusasm and striclyfdb. They are also diff-stable, as

n→∞lim mn

mn−1

_1/n

= lim

n→∞n^s/n =1.

4 Composition

To study the composition ofE-maps we first consider formal power series, which avoids the cumbersome Faà di Bruno formula. We employ the standard notation

k∈Λ

f_kx^k

k∈Λ

g_kx^k

for two formal power series inR^t, when

f_kg_k ⇔ |f_k,i| ≤g_k,i, 1≤i≤t, holds for all coefficients.

To simplify notation we consider a s-dimensional weight as a weight on any lower- dimensional index space as well by identifying(k₁, . .,k_t)with(k₁, . .,k_t,0, . .,0). In other words, we add dummy coordinates to make the dimensions equal.

Main Lemma 8 Let g and h be two formal power series without constant terms. If m andw are two weights such that, with someλ >0,

wlm_k1· ·m_kr ≤λ^km_k

for all l=0and k1, . .,kr =0such that r= |l|and k1+. .+kr =k, then M˚₀^m(g◦h)M˚₀^wg◦M˚₀^mh◦λ.

Proof Write

g=

l=0

glz^l, h=

k=0

hkx^k, wherel=(l1, . .,lt)andk=(k1, . .,ks). Then

g◦h=

l=0

gl

⎛

⎝

k=0

hkx^k

⎞

⎠

l

=

k=0

⎛

⎝

l=0

k₁,. .,k_r

glk1· ·kr

⎞

⎠x^k,

where the last sum is taken over alll=0 andk₁, . .,k_r =0 such thatr= |l|andk₁+. .+kr = k, and wherekstands for a certain component of thet-vectorh_kwhich we do not need to make explicit.

By hypotheses,mk≥λ^−kwlmk1· ·mkr in all cases. Therefore, M˚₀^m(g◦h)

k=0

⎛

⎝

l=0

k₁,. .,k_r

gl|k1| · ·|kr|

⎞

⎠x^k mk

k=0

⎛

⎝

l=0

k1,. .,kr

gl wl

|k1| mk1

· ·|kr| mkr

⎞

⎠(λx)^k.

This is tantamount to

M˚₀^m(g◦h)

l=0

g_l wl

⎛

⎝

k=0

h_k m_k (λx)^k

⎞

⎠

l

=M˚₀^wg◦M˚₀^mh◦λ.

Theorem 1 Let g∈ E^wand h∈ E^mand suppose g◦h is well defined. Ifwand m satisfy the assumption of the preceding lemma, then g◦h∈E^m. In particular,

g◦h^m_a,r ≤ g^w_h(a),ρ, ρ= h^m_a,λr.

Proof Lethbe of classE^mnearaandgbe of classE^wnearb=h(a). Theng◦his well defined neara. Moreover, there is a neighborhoodV ofband aρ0such thatg^w_V,ρ0 <∞.

There is another neighborhoodU ofaand anrsuch that h(U) ⊂ V andh^m_U,r < ∞.

As this semi-norm does not include a constant term, we can make it as small as we like by makingrsmall. So in particular we can chooserso that

h^m_U,λr < ρ0.

Applying the preceding lemma to the formal Taylor series expansions ofgatbandhata we obtain

M˚_a^m(g◦h)M˚_b^wg◦M˚_a^mh◦λ.

Considering each component ofg◦hseparately we conclude that g◦h^m_a,r =M˚_a^m(g◦h)(r, . .,r)

≤M˚_b^wg◦M˚_a^mh(λr, . ., λr)

≤M˚_b^wg(h₁^m_a,λr, . .,h_t^m_a,λr).

With max_1≤i≤th_i^m_a,λr = h^m_a,λr =:ρ < ρ0we get

g◦h^m_a,r ≤M˚_b^wg(ρ, . ., ρ)= g^w_b,ρ <∞.

As this holds locally around any point in the domain ofhand hence ofg◦h, the latter is also

of classE^m.

5 Characterizations ofE^m

We now characterize thoseE^m-spaces, which are stable under composition. There are two distinct cases, the holomorphically orC^ω-closed and theE^m-closed spaces. Neither of these characterizations require the property of being closed under derivation. These results are obviously optimal and more general than those in [27,28].

The first theorem already appeared in [32] but apparently did not receive much attention.

Its roots go back to [3,30].

Theorem 2 The following statements are equivalent.

(i) m isasm.

(ii) E^mis ‘holomorphically closed’: for g∈C^ωand h∈E^m, also g◦h∈E^m. (iii) E^mis ‘inverse closed’: for h∈E^m, also1/h∈E^mwherever defined.

Proof (i)⇒(ii) Recall thatC^ω = E^wwithw =(1)n≥1. Ifmisasm, thenwandmsatisfy the hypothesis of the Main Lemma8. Thus, ifg∈C^ωandh∈E^m, then alsog◦h∈E^mby Lemma1.

(ii)⇒(iii) This is obvious, asz→1/zis holomorphic forz=0.

(iii)⇒(i) Assume for simplicity thatE^mconsists of functions on an interval around 0. Let ηbe the characteristic function of Lemma1andρ: z→(1−z)⁻¹. The Taylor coefficients ofρareρr =1 for allr≥1, so

T˙0(ρ◦η)=

n>0

r>0

n1+. .+nr=n

ρrηn₁· ·ηn_r

xⁿ

=

n>0

i^n−r

r>0

n1+. .+nr=n

sn1· ·snr

xⁿ and

ρ◦η^m_0,r ≥

n>0

r>0

n₁+. .+n_r=n

m_n1· ·m_nr m_n rⁿ.

Hence, forρ◦ηto be inE^m, the weightmhas to beasm. The complementary theorem forE^m-closedness is

Theorem 3 The following statements are equivalent.

(i) m isfdb.

(ii) E^mis ‘composition closed’: with g,h∈E^m, also g◦h∈E^m.

(iii) E^m is ‘inversion closed’: if g is a local diffeomorphism in E^m, then its local inverse g⁻¹is also in E^m.

The first results of this kind are apparently due to Gevrey [12] and Cartan [8]. For log- convex weights this was shown by Roumieu [29], and for more details see [11,14,27,28]

among many others. The necessity of thefdbcondition generalizes results in [28] and is new, as is the proof of (i)⇒(iii).

Note also that whenE^mis inversion closed, the corresponding implicit function theorem holds as well.

Proof (i)⇒(ii) Ifmisfdb, then we can apply Lemma1withw=mand conclude thatE^m is stable under composition.

(i)⇒(iii) We may reduce the problem to a local diffeomorphism g = id+ ˆg in the neighbourhood of its fixed point 0, where the hat denotes higher order terms. Its local inverse may be written asρ = id− ˆρ. Fromg◦ρ = idwe obtainρˆ = ˆg◦ρand henceDρˆ = (Dgˆ◦ρ)(I +Dρ)ˆ . With g ∈ E^m we have Dgˆ ∈ E^m´. Asm´ isasmby Lemma7,E^m´ is holomorphically closed, so in particular an algebra, and we may normalizem´ so that uv^m_a,r^´ ≤ u^m_a,r^´ v^m_a,r^´ . Furthermore, there exists a neighbourhoodU of 0 and anr >0 such that

Dgˆ^m_U,r^´ ≤1/2, and another neighboorhoudVso thatρ(V)⊂U.

As we do not know yet whether Dρˆ ^m_V^´_,r is finite, we first consider its N-th Taylor polynomialT^NDρˆ. As in the above equation its terms do not depend on higher oder terms we conclude that

T^NDρˆ ^m_V,r^´ ≤ Dgˆ^m_U,r^´ (1+ T^NDρˆ ^m_V,r^´ ).

Hence,

T^NDρˆ ^m_V^´_,r ≤2Dgˆ ^m_U^´_,r.

As this hold for allN, this implies thatDρˆ∈E^m´ and consequently thatρ∈E^m.

(ii)⇒(i) Assume again that 0 is in the domain under consideration. For the characteristic E^m-functionηof Lemma1we obtain

T˙0(η◦η)=

n>0

r>0

n1+. .+nr=n

ηrηn₁· ·ηn_r

xⁿ

=

n>0

iⁿ⁻¹

r>0

n1+. .+nr=n

srsn1· ·snr

xⁿ,

as the ‘signs’ of all terms forxⁿcombine to i^r−1iⁿ¹⁻¹· ·i^nr−1=i^r−1i^n−r =iⁿ⁻¹. It follows that

η◦η^m_0,r ≥

n>0

r>0

n1+. .+nr=n

mrmn₁· ·mnr

mn

rⁿ. Hence, forη◦ηto be inE^m, the weightmhas to befdb.

(iii)⇒(i) We may modifiy the linear term of the characteristic functionηso thatη=id+ð.

This is a local diffeomorphism at 0 with an inverse function of the formρ=id− ˆρfor which we make the ansatz

ˆ

ρ=

n≥2

ρnxⁿ =

n≥2

iⁿ⁻¹r_nxⁿ.

Thenη◦ρ=idis equivalent toρˆ=ð◦ρor

n≥2

ρnxⁿ=

n≥2

r>0

n1+. .+nr=n

ηrρn₁· ·ρn_r

xⁿ. Comparison of coefficients leads to

rn=sn+

1<r<n

n1+. .+nr=n

srrn1· ·rnr, n≥2, wherer1=1. It follows thatrn ≥sn ≥mnfor alln≥1, and finally that

r_n≥m_rm_n1· ·m_nr.

Forρto be inE^m, the weightmthus has to befdb.

For the sake of completeness and comparison we mention Theorem 4 The following statements are equivalent.

(i) m is diff-stable.

(ii) E^mis ‘derivation closed’: with f ∈E^malso f∈E^m.

Proof It follows from the definitions that with f ∈E^mwe have f∈E^m´. Ifmis diff-stable, thenm´ mandE^m´ ⊂ E^mand thus f ∈E^m. The converse is proven with characteristic

functions as usual.

Corollary 1 If E^m is holomorphically and derivation closed, then E^m is also composition closed.

Proof IfE^mis holomorphically closed, thanmis almost submultiplicative by Theorem2. If E^mis also derivation closed, thenmis also diff-stable and thusfdbby Lemma6. SoE^mis

composition closed by Theorem3.

GeneralizationsThe preceding results hold forlocalDenjoy-Carleman classesE^m, where f belongs toE^m, if any point in its domain has aneighbourhood Usuch that

sup

n≥1

1 n!

DⁿfU

m_n rⁿ<∞

for somer>0. But as noted in [32] they also hold for Denjoy-Carleman classesE^m(I)for anarbitraryintervalI, where f belongs toE^m(I), if

supn≥1

1 n!

DⁿfI

m_n rⁿ <∞

for somer>0, with·I denoting the usual sup-norm onI. IfIis compact, these classes coincide with the local classes, and there is nothing new. Otherwise, it is no longer true that

E^m(I)=E^m˘(I)

with aweakly log-convexsequencem. Instead, the regularized weight˘ m˘ has to be defined slightly differently, depending on the nature of the intervalI – see [32] and [22]. But the crucial fact is that also in this case characteristicE^m-functions exist, and the proofs remain essentially the same.