Nonlinear conditions for ultradifferentiability
aa r X i v : . [ m a t h . C A ] F e b NONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY
DAVID NICOLAS NENNING, ARMIN RAINER, AND GERHARD SCHINDL
Abstract.
A remarkable theorem of Joris states that a function f is C ∞ if two relatively prime powers of f are C ∞ . Recently, Thilliez showed thatan analogous theorem holds in Denjoy–Carleman classes of Roumieu type.We prove that a division property, equivalent to Joris’s result, is valid in awide variety of ultradifferentiable classes. Generally speaking, it holds in alldimensions for non-quasianalytic classes. In the quasianalytic case we havegeneral validity in dimension one, but we also get validity in all dimensions forcertain quasianalytic classes. Introduction
A remarkable theorem of Joris [11, Th´eor`eme 2] states: if f : R → R is a functionand p, q are relatively prime positive integers, then f p , f q ∈ C ∞ = ⇒ f ∈ C ∞ . (1)Since smoothness can be tested along smooth curves by a theorem of Boman [3],one immediately infers that the implication (1) holds on arbitrary open subsets of R d , d ≥
1, and on smooth manifolds. (On the other hand, the regularity of a single power of a function generally says nothing about the regularity of the functionitself; e.g. ( Q − R \ Q ) = 1, where A is the indicator function of a set A .)It was soon realized that the statement also holds for complex valued functionsand it led to the study of so-called pseudoimmersions [7, 12, 13, 19]. A simple proofbased on ring theory was given by [1].Only recently Thilliez [30] showed that Joris’s result carries over to Denjoy–Carleman classes of Roumieu type E { M } . These are ultradifferentiable classes ofsmooth functions defined by certain growth properties imposed upon the sequenceof iterated derivatives in terms of a weight sequence M (which in view of the Cauchyestimates measures the deviation from analyticity).By extracting the essence of Thilliez’s proof, we show in this paper that a broadvariety of ultradifferentiable classes has a division property equivalent to Joris’sresult. Let S be a subring (with multiplicative identity) of the ring of germs at0 ∈ R d of complex valued C ∞ -functions. We say that S has the division property ( D ) if for any function germ f at 0 ∈ R d we have (cid:0) j ∈ N ≥ , f j , f j +1 ∈ S (cid:1) = ⇒ f ∈ S . (2)If S has property ( D ), then Joris’s theorem holds in S . Indeed, suppose that p , p are relatively prime positive integers and f p , f p ∈ S . All integers j ≥ p p can Mathematics Subject Classification.
Key words and phrases.
Joris theorem, division property, ultradifferentiable classes, (non-)quasianalytic, holomorphic approximation, almost analytic extension.AR was supported by FWF-Project P 32905-N, DNN and GS by FWF-Project P 33417-N. be written j = a p + a p for a , a ∈ N , see [11, p.270]. Hence f j ∈ S for all j ≥ p p . Since two consecutive integers are relatively prime, also the converseholds. Results.
Let us give an overview of our results.The rings of germs in one dimension d = 1 of the following ultradifferentiableclasses have property ( D ): • E [ M ] , Denjoy–Carleman class of Roumieu (Theorems 2.2 and 2.3) andBeurling type (Theorem 2.6), • E [ ω ] , Braun–Meise–Taylor classes of Roumieu and Beurling type (Theo-rem 3.1), • E [ M ] , ultradifferentiable classes defined by weight matrices of Roumieu andBeurling type (Theorem 4.2).It is understood that certain minimal regularity properties of the weights are as-sumed (see Table 1) which in particular guarantee that the sets of germs are indeedrings. (Note that by convention [ · ] stands for {·} , i.e., Roumieu, as well as ( · ), i.e.,Beurling.)Interestingly, the proof in one dimension works for quasianalytic and non-quasianalytic classes alike. But the tool used to reduce the multidimensional tothe one-dimensional statement is only available in the non-quasianalytic Roumieucase ([15], [27]). The (multidimensional) Beurling case can often be reduced tothe corresponding Roumieu case. Hence we obtain the following multidimensionalnon-quasianalytic results. The rings of germs in all dimensions d of the followingultradifferentiable classes have property ( D ): • E [ M ] , non-quasianalytic Denjoy–Carleman class of Roumieu (Theorem 2.2)and Beurling type (Theorem 2.5), • E [ ω ] , non-quasianalytic Braun–Meise–Taylor classes of Roumieu and Beurl-ing type (Theorem 3.2), • E { M } , non-quasianalytic ultradifferentiable classes defined by weight ma-trices of Roumieu type (Theorem 4.3).For quasianalytic Denjoy–Carleman classes of Roumieu type E { M } in one di-mension the implication (2) follows from the stronger result, due to Thilliez [29],that C ∞ -solutions of a polynomial equation z n + a z n − + · · · + a n − z + a n = 0 , (3)where the coefficients a j are germs at 0 ∈ R of E { M } -functions, are of class E { M } (under weak assumptions on M ). This is false for non-quasianalytic classes. Butit seems to be unknown whether, in the presence of quasianalyticity, it holds inhigher dimensions. In fact, quasianalytic ultradifferentiability cannot be tested onquasianalytic curves (or lower dimensional plots) even if the function in question isknown to be smooth ([10, 20]).Hence we think that it is interesting that, combining our proof with a descriptionof certain quasianalytic classes E { M } as an intersection of suitable non-quasianalyticones (due to [16]), we obtain that these quasianalytic classes have property ( D ) in all dimensions (see Theorem 2.7 and also Remarks 3.3 and 4.4). In an earlier version of the paper we considered the division property (cid:0) j ∈ N ≥ , g, q, fg ∈S , f j = qg (cid:1) = ⇒ f ∈ S which resulted from our wish to prove an ultradifferentiable version ofthe division theorem [13, Theorem 1]. But this property is equivalent to (2). ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 3
Since all considered regularity classes are local, the results for germs immediatelygive corresponding results for functions on open sets.1.2.
Summary of the results.
We list in Table 1 the ultradifferentiable rings ofgerms known to have property ( D ), together with the needed assumptions on theweights and the respective references. All germs are function germs at 0 in R d forsome dimension d . The dimension is added as a left subscript, e.g., E [ M ] d denotesthe ring of germs at 0 ∈ R d of E [ M ] -functions. All notions will be defined below.The Roumieu parts of the results in the first and the fifth row are due to Thilliez[30]; see Sections 2.4 and 2.5. Table 1.
Ultradifferentiable rings of germs having property ( D ) Quasianalytic ring of germs Reference E [ M ]1 derivation closed m log-convex m /kk → ∞ (Beurling) Theorems 2.3 and 2.6 E [ ω ]1 ω concave Theorem 3.1 E [ M ]1 [regular][moderate growth] Theorem 4.2 E { M } d intersectablemoderate growth Theorem 2.7 Non-quasianalytic ring of germs Reference E [ M ] d moderate growth m log-convex Theorems 2.2 and 2.5 E [ ω ] d ω concave Theorem 3.2 E [ M ]1 [regular][moderate growth] Theorem 4.2 E { M } d { regular }{ moderate growth } Theorem 4.3We remark that non-quasianalytic Denjoy–Carleman classes E { M } , where theweight sequence M lacks moderate growth, do not have property ( D ) in general;see [30, Remark 2.2.3]. The moderate growth condition is rather restrictive (e.g.,it implies that the class E { M } is contained in a Gevrey class). The consideration ofthe classes E [ ω ] and E [ M ] allows to overcome this restriction in the sense that theimplication (2) holds under weaker moderate growth conditions.1.3. Strategy of the proof.
Thilliez’s proof of Joris’s theorem for E { M } consistsof the following two steps:(i) The class E { M } admits a description by holomorphic approximation whichis based on a result of Dynkin [8] on almost analytic extensions and a related ∂ -problem. D.N. NENNING, A. RAINER, AND G. SCHINDL (ii) If f j , f j +1 are of class E { M } and g ε , h ε are respective holomorphic ap-proximations, the quotient h ε /g ε is a naive candidate for a holomorphicapproximation of f . In order to avoid small divisors one considers u ε = ϕ ε g ε h ε max {| g ε | , r ε } , where ϕ ε is a suitable cutoff function and r ε >
0. For good choices of r ε the function u ε has uniform bounds and is close to f . The solution of a ∂ -problem is used to modify u ε in order to obtain a holomorphic approxi-mation of f . By step (i) we may conclude that f belongs to E { M } .Following the same strategy, we will work with weight matrices M , since they pro-vide a framework for ultradifferentiability (Section 4) which encompasses Denjoy–Carleman classes (Section 2) and Braun–Meise–Taylor classes (Section 3). In Sec-tion 5 we prove a general characterization result by holomorphic approximation for E [ M ] (Theorem 5.3) which extends step (i); it builds on the description by almostanalytic extension presented in our recent paper [9]. Then we execute a version ofstep (ii) under a quite minimal set of assumptions, see Lemma 6.1. It enables usto easily deduce the main results in Section 6.2. Denjoy–Carleman classes have property ( D )2.1. Weight sequences and Denjoy–Carleman classes.
Let µ = ( µ k ) be apositive increasing (i.e. µ k ≤ µ k +1 ) sequence with µ = 1. We define a sequence M by setting M k := µ · · · µ k , M := 1, and a sequence m by m k := M k k ! . Clearly, µ uniquely determines M and m , and vice versa. In analogy we shall use sequences N ↔ n ↔ ν , L ↔ ℓ ↔ λ , etc.That µ is increasing means that M is log-convex , i.e., log M is convex or, equiv-alently, M k ≤ M k − M k +1 for all k . If in addition M /kk → ∞ , we say that M is a weight sequence .Sometimes we will make the stronger assumption that m is log-convex.For σ > U ⊆ R d , one defines the Banach space B Mσ ( U ) := n f ∈ C ∞ ( U ) : k f k Mσ,U := sup x ∈ U, α ∈ N d | ∂ α f ( x ) | σ | α | M | α | < ∞ o and the (local) Denjoy–Carleman classes of Roumieu type E { M } ( U ) := proj V ⋐ U ind σ> B Mσ ( V ) . For later reference we also consider the global class B { M } ( V ) := ind σ> B Mσ ( V ).Replacing the existential quantifier for σ by a universal quantifier, we find the Denjoy–Carleman classes of Beurling type E ( M ) ( U ) := proj V ⋐ U proj σ> B Mσ ( V )and B ( M ) ( V ) := proj σ> B Mσ ( V ). We use the notation E [ M ] for both E { M } and E ( M ) , similarly for B [ M ] , etc.For positive sequences M, N , we write M N if sup k ∈ N (cid:0) M k N k (cid:1) /k < ∞ and M ✁ N if lim k →∞ (cid:0) M k N k (cid:1) /k = 0. We have (cf. [23, Proposition 2.12]) M N ⇔ E [ M ] ( U ) ⊆ E [ N ] ( U ) ,M ✁ N ⇔ E { M } ( U ) ⊆ E ( N ) ( U ) , ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 5 where for “ ⇐ ” one has to assume that M is a weight sequence. Note that E { ( k !) } ( U )coincides with C ω ( U ) and so the class of real analytic functions is contained in E ( M ) ( ⊆ E { M } ) if and only if m /kk → ∞ .Log-convexity of M implies that E [ M ] ( U ) is closed under pointwise multiplicationof functions. Additional regularity properties for M endow E [ M ] ( U ) with additionalstructure, e.g., log-convexity of m implies closedness under composition of functions.A crucial assumption in [30] is moderate growth of M , which reads as follows ∃ C > ∀ k, j ∈ N : M k + j ≤ C k + j M k M j . (4)It implies derivation closedness ∃ C > ∀ k ∈ N : M k +1 ≤ C k +1 M k . (5)The last property we need to mention is non-quasianalyticity of M , that is ∞ X k =1 µ k < ∞ , or equivalently ∞ X k =1 M /kk < ∞ . (6)By the Denjoy–Carleman theorem, this condition is equivalent to the existenceof non-trivial functions with compact support in E [ M ] ( U ). It is well-known thatnon-quasianalyticity implies m /kk → ∞ .Let E [ M ] d denote the ring of germs at 0 ∈ R d of complex valued E [ M ] -functions;here we assume that M is a weight sequence in order to have a ring. Remark 2.1.
There is a slight mismatch between our notation (also used in [9])and that of [30] (and [22]). We write M j = m j j ! for weight sequences, so our m corresponds to M in [30].2.2. Associated functions.
Let m = ( m k ) be a positive sequence with m = 1and m /kk → ∞ . We define the function h m ( t ) := inf k ∈ N m k t k , for t > , and h m (0) := 0 , (7)which is is increasing, continuous on [0 , ∞ ), and positive for t >
0. For large t wehave h m ( t ) = 1. Furthermore, we needΓ m ( t ) := min { k : h m ( t ) = m k t k } , t > , (8)and, provided that m k +1 /m k → ∞ ,Γ m ( t ) := min n k : m k +1 m k ≥ t o , t > . (9)We trivially have Γ m ≤ Γ m . If m is log-convex, then Γ m = Γ m .We shall use these functions for m k = M k /k !, where M is a weight sequencesatisfying m /kk → ∞ . Then m k +1 /m k → ∞ (since M /kk ≤ µ k for all k ).2.3. Regular weight sequences.
A weight sequence M is said to be regular if m /kk → ∞ , M is derivation closed, and there exists a constant C ≥ m ( Ct ) ≤ Γ m ( t ) for all t > D.N. NENNING, A. RAINER, AND G. SCHINDL
Denjoy–Carleman classes of Roumieu type have property ( D ) .Theorem 2.2 (Non-quasianalytic E { M } d ) . Let M be a non-quasianalytic regularweight sequence of moderate growth. Then E { M } d has property ( D ) . This is a special case of Theorem 4.3 below (cf. Section 4.5). It implies Thilliez’sresult [30, Corollary 2.2.5].A quasianalytic one-dimensional version follows from a stronger result in [29]:
Theorem 2.3 (Quasianalytic E { M } ) . Let M be a quasianalytic derivation closedweight sequence such that m is log-convex. Then E { M } has property ( D ) . Denjoy–Carleman classes of Beurling type have property ( D ) . Let usdeduce Beurling versions of Theorems 2.2 and 2.3. We use the following lemmabased on [14, Lemma 6] and [9, Lemma 7.5].
Lemma 2.4.
Let
L, M be positive sequences satisfying L ✁ M . Suppose that m islog-convex and satisfies m /kk → ∞ . Then there exists a weight sequence S suchthat s is log-convex, s /kk → ∞ , and L ≤ S ✁ M . Additionally, we may assume: (i) S has moderate growth, if M has moderate growth. (ii) S is derivation closed, if M is derivation closed. (iii) S is non-quasianalytic, if M is non-quasianalytic.Proof. Only the supplements (ii) and (iii) were not already proved in [9, Lemma7.5].(ii) follows from the fact that a weight sequence M is derivation closed if andonly if there is a constant C ≥ M k ≤ C k for all k , see [17, 18]. Since S is a weight sequence and S ✁ M , also S is derivation closed, by this criterion.(iii) It suffices to show that there exists a non-quasianalytic weight sequence N such that L ≤ N ✁ M . Then we apply the lemma to N ✁ M and obtain a weightsequence S with N ≤ S ✁ M having all desired properties.Let us show the existence of N . By L ✁ M , we have β k := sup p ≥ k (cid:0) L p M p (cid:1) /p ց β k and α k = γ k := µ k , yieldsan increasing sequence δ = ( δ k ) such that δ k → ∞ , (10) δ k β k → , (11) µ k δ k is increasing , (12) ∞ X k =1 δ k µ k ≤ δ ∞ X k =1 µ k < ∞ . (13)Then N k := µ ··· µ k δ ··· δ k defines a non-quasianalytic weight sequence, by (12) and (13).(Note that ν k = µ k δ k → ∞ is equivalent to N /kk → ∞ .) It satisfies N ✁ M by (10).By (11), there is a constant C > δ k (cid:0) L k M k (cid:1) /k ≤ C for all k . By themonotonicity of δ , this leads to L k ≤ C k M k δ kk ≤ C k M k δ ··· δ k = C k N k . After replacing( N k ) by ( C k N k ) we have L ≤ N ✁ M . (cid:3) ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 7
Theorem 2.5 (Non-quasianalytic E ( M ) d ) . Let M be a non-quasianalytic weightsequence of moderate growth such that m is log-convex. Then E ( M ) d has property ( D ) .Proof. Suppose that g := f j , h := f j +1 ∈ E ( M ) d for some positive integer j . Assumethat representatives of these germs are defined in the neighborhood of the closureof some bounded 0-neighborhood U ; we denote the representatives by the samesymbols. Then the sequence L k := max n sup | α | = k, x ∈ U | g ( α ) ( x ) | , sup | α | = k, x ∈ U | h ( α ) ( x ) | o (14)satisfies L ✁ M . By Lemma 2.4, there exists a weight sequence S satisfying theassumptions of Theorem 2.2 and L ≤ S ✁ M . Thus, f ∈ E { S } d ⊆ E ( M ) d . (cid:3) Theorem 2.6 (Quasianalytic E ( M )1 ) . Let M be a quasianalytic derivation closedweight sequence such that m is log-convex and m /kk → ∞ . Then E ( M )1 has property ( D ) .Proof. This follows from the proof of Theorem 2.3 in [29] (which also works in theBeurling case). Alternatively, we may infer it from Theorem 2.3 by a reductionargument based on Lemma 2.4 as in the proof of Theorem 2.5. (cid:3)
A multidimensional quasianalytic result.
Let M be a weight sequenceand consider the sequence spaceΛ { M } := n ( c k ) ∈ C N : ∃ ρ > k ∈ N | c k | ρ k M k < ∞ o . We call a quasianalytic weight sequence M intersectable ifΛ { M } = \ N ∈L ( M ) Λ { N } , (15)where L ( M ) is the collection of all non-quasianalytic weight sequences N ≥ M such that n is log-convex. The identity (15) carries over to respective functionspaces, since B { M } ( U ) = (cid:8) f ∈ C ∞ ( U ) : (sup x ∈ U k f ( k ) ( x ) k L k sym ) ∈ Λ { M } (cid:9) , where f ( k ) denotes the k -th order Fr´echet derivative and k · k L k sym the operator norm.Note that a quasianalytic intersectable weight sequence M always satisfies m /kk → ∞ ; an argument is given in Remark 2.8 below. Theorem 2.7 (Quasianalytic E { M } d ) . Let M be a quasianalytic intersectableweight sequence of moderate growth. Then E { M } d has property ( D ) . The proof of this result is given in Section 6.
Remark 2.8.
In [16, Theorem 1.6] (inspired by [2]) a sufficient condition for inter-sectability was given. Let M be a quasianalytic weight sequence with 1 ≤ M < M .Consider the sequence ˇ M defined byˇ M k := M k k Y j =1 (cid:16) − M /jj (cid:17) k , ˇ M := 1 . If ˇ m is log-convex, then M is intersectable. D.N. NENNING, A. RAINER, AND G. SCHINDL
Not every quasianalytic weight sequence is intersectable, for instance,Λ { ( k !) } = \ N ∈L (( k !)) Λ { N } = Λ { Q } , where Q k = ( k log( k + e )) k );see [16, Theorem 1.8] and [26]. Every quasianalytic intersectable weight sequence M must satisfy Λ { Q } ⊆ Λ { M } , and so m /kk tends to ∞ since clearly q /kk does.A countable family Q = { Q n } n ∈ N ≥ of quasianalytic intersectable weight se-quences of moderate growth was constructed in [16, Theorem 1.9]: Q nk = (cid:0) k log( k ) log(log( k )) · · · log [ n ] ( k ) (cid:1) k , for k ≥ exp [ n ] (1) , where log [ n ] denotes the n -fold composition of log; analogously for exp [ n ] .See also [27, Section 11] for a generalization of this concept.3. Braun–Meise–Taylor classes have property ( D )3.1. Weight functions and Braun–Meise–Taylor classes. A weight function is, by definition, a continuous increasing function ω : [0 , ∞ ) → [0 , ∞ ) such that( ω ) ω (2 t ) = O ( ω ( t )) as t → ∞ ,( ω ) ω ( t ) = o ( t ) as t → ∞ ,( ω ) log( t ) = o ( ω ( t )) as t → ∞ ,( ω ) t ω ( e t ) =: ϕ ω ( t ) is convex on [0 , ∞ ).One may assume that ω | [0 , ≡ E [ ω ] )which we shall tacitly do if convenient.Let U ⊆ R d be open and ρ >
0. We associate the Banach space B ωρ ( U ) := n f ∈ C ∞ ( U ) : k f k ωρ,U := sup x ∈ U, α ∈ N d | ∂ α f ( x ) | e ϕ ∗ ω ( ρ | α | ) /ρ < ∞ o , where ϕ ∗ ω ( s ) := sup t ≥ { st − ϕ ω ( t ) } is the Young conjugate of ϕ ω (which is finite by( ω )). Then the (local) Braun–Meise–Taylor class of Roumieu type is E { ω } ( U ) := proj V ⋐ U ind n ∈ N B ωn ( V ) , and that of Beurling type is E ( ω ) ( U ) := proj V ⋐ U proj n ∈ N B ω n ( V ) . Again we use E [ ω ] for E { ω } and E ( ω ) , similarly for B [ ω ] etc.For two weight functions ω , σ we have (cf. [23, Corollary 5.17]) σ ( t ) = O ( ω ( t )) as t → ∞ ⇔ E [ ω ] ( U ) ⊆ E [ σ ] ( U ) ,σ ( t ) = o ( ω ( t )) as t → ∞ ⇔ E { ω } ( U ) ⊆ E ( σ ) ( U ) . We say that ω and σ are equivalent if they generate the same classes, i.e., σ ( t ) = O ( ω ( t )) and ω ( t ) = O ( σ ( t )) as t → ∞ .A weight function is said to be non-quasianalytic if Z ∞ ω ( t ) t dt < ∞ . (16)This is the case if and only if E [ ω ] ( U ) contains non-trivial functions of compactsupport (cf. [5] or [22]). ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 9
Let us emphasize that in this paper we treat condition ( ω ) as a general assump-tion for weight functions; it means that the Beurling class E ( ω ) contains the realanalytic class. It is automatically satisfied if ω is non-quasianalytic.Let E [ ω ] d denote the ring of germs at 0 ∈ R d of complex valued E [ ω ] -functions;note that E [ ω ] is stable by multiplication of functions for any weight function ω .3.2. The associated weight matrix.
Let ω be a weight function. Setting Ω xk := e ϕ ∗ ω ( xk ) /x defines a weight sequence Ω x for every x >
0, where Ω x ≤ Ω y if x ≤ y .Thus the collection Ω := { Ω x } x> is a weight matrix (in the sense of Section 4).Note that Ω satisfies a mixed moderate growth property, namely ∀ x > ∀ j, k ∈ N : Ω xj + k ≤ Ω xj Ω xk . (17)The importance of the associated weight matrix Ω is that it encodes an equivalenttopological description of the spaces E [ ω ] ( U ) as unions or intersections of Denjoy–Carleman classes; see Section 4.5. All this can be found in [22].3.3. Braun–Meise–Taylor classes have property ( D ) .Theorem 3.1 ( E [ ω ]1 ) . Let ω be a concave weight function. Then E [ ω ]1 has property ( D ) . Evidently, it suffices to assume that ω is equivalent to a concave weight function.For the multidimensional analogue we additionally assume non-quasianalyticity. Theorem 3.2 (Non-quasianalytic E [ ω ] d ) . Let ω be a non-quasianalytic concaveweight function. Then E [ ω ] d has property ( D ) . Theorems 3.1 and 3.2 are corollaries of Theorems 4.2 and 4.3 below; for theproofs see Section 6.
Remark 3.3.
Every weight sequence M in the family Q mentioned at the end ofRemark 2.8 satisfies lim inf k →∞ µ ak µ k > a . Hence there is a quasianalytic weight function ω M (take, e.g., ω M ( t ) := − log h M (1 /t )) such that E [ ω M ] d = E [ M ] d , by [4, Theorem 14].So for all M ∈ Q , the quasianalytic ring E { ω M } d has property ( D ).4. The most general version of the theorem
Let us formulate the main theorems in the most general setting available. Theconditions we put on abstract weight matrices are tailored in such a way that weightmatrices associated with weight functions are contained as special cases.4.1.
Weight matrices and ultradifferentiable classes. A weight matrix M is,by definition, a family of weight sequences which is totally ordered with respect tothe pointwise order relation on sequences, i.e., • M ⊆ R N , • each M ∈ M is a weight sequence in the sense of Section 2.1, • for all M, N ∈ M we have M ≤ N or M ≥ N . Let U ⊆ R d be open. Given a weight matrix M , we define global classes B { M } ( U ) := ind M ∈ M B { M } ( U ) , (18) B ( M ) ( U ) := proj M ∈ M B ( M ) ( U ) . (19)The limits in (18) and (19) can always be assumed countable, as is shown in [9,Lemma 2.5]. Writing [ · ] for {·} and ( · ), the local classes are defined by E [ M ] ( U ) := proj V ⋐ U B [ M ] ( V ) . Let E [ M ] d denote the ring of germs at 0 ∈ R d of complex valued E [ M ] -functions;notice that E [ M ] is stable by multiplication of functions, since each M ∈ M is aweight sequence.4.2. Regular weight matrices.
A weight matrix M satisfying • m /kk → ∞ for all M ∈ M is called { regular } or R-regular (for Roumieu) if • ∀ M ∈ M ∃ N ∈ M ∃ C ≥ ∀ j ∈ N : M j +1 ≤ C j +1 N j , • ∀ M ∈ M ∃ N ∈ M ∃ C ≥ ∀ t > n ( Ct ) ≤ Γ m ( t ),and ( regular ) or B-regular (for Beurling) if • ∀ M ∈ M ∃ N ∈ M ∃ C ≥ ∀ j ∈ N : N j +1 ≤ C j M j , • ∀ M ∈ M ∃ N ∈ M ∃ C ≥ ∀ t > m ( Ct ) ≤ Γ n ( t ).Moreover, M is called regular if it is both R- and B-regular. By our convention,[regular] stands for { regular } (i.e. R-regular) in the Roumieu case and (regular)(i.e. B-regular) in the Beurling case.4.3. Almost analytic extensions.
Let h : (0 , ∞ ) → (0 ,
1] be an increasing con-tinuous function which tends to 0 as t →
0. Let ρ > U ⊆ R d be a boundedopen set. We say that a function f : U → C admits an ( h, ρ ) -almost analytic ex-tension if there is a function F ∈ C c ( C d ) and a constant C ≥ F | U = f and | ∂F ( z ) | ≤ C h ( ρd ( z, U )) , for z ∈ C d , where ∂F ( z ) := P dj =1 ∂F ( z ) ∂z j dz j and d ( z, U ) := inf x ∈ U | z − x | denotes the distanceof z to U .Let us apply this definition to the functions h m from (7), where m k = M k /k !and M belongs to a given weight matrix M . Let f : U → C be a function. • f is called { M } -almost analytically extendable if it has an ( h m , ρ )-almostanalytic extension for some M ∈ M and some ρ > • f is called ( M ) -almost analytically extendable if, for all M ∈ M and all ρ >
0, there is an ( h m , ρ )-almost analytic extension of f . Theorem 4.1 ([9, Corollaries 3.3, 3.5]) . Let M be a [regular] weight matrix. Let U ⊆ R d be open. Then f ∈ E [ M ] ( U ) if and only if f | V is [ M ] -almost analyticallyextendable for each quasiconvex domain V relatively compact in U . In Section 5 we shall use [9, Proposition 3.12], which is a key ingredient of theproof of Theorem 4.1.
ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 11
Weight matrices of moderate growth.
For positive sequences M , N setmg( M, N ) := sup j,k ≥ , j + k ≥ (cid:18) M j + k N j N k (cid:19) / ( j + k ) ∈ (0 , ∞ ] . (20)We say that a weight matrix M has R-moderate growth or { moderate growth } if ∀ M ∈ M ∃ N ∈ M : mg( M, N ) < ∞ , (21)and B-moderate growth or ( moderate growth ) if ∀ M ∈ M ∃ N ∈ M : mg( N, M ) < ∞ . (22)Again we say that M has moderate growth if it has R- and B-moderate growth, and[moderate growth] stands for { moderate growth } and (moderate growth), respec-tively.4.5. Denjoy–Carleman and Braun–Meise–Taylor classes in this frame-work.
By definition, Denjoy–Carleman classes are described by weight matrices M = { M } consisting of a single weight sequence M . Observe that the weight ma-trix M = { M } is regular if and only if the weight sequence M is regular, and it hasmoderate growth if and only if M has moderate growth.Let ω be a weight function and let Ω be the associated weight matrix (cf. Sec-tion 3.2). Then, by [22, Corollaries 5.8 and 5.15], as locally convex spaces E [ ω ] ( U ) = E [ Ω ] ( U ) , and E [ ω ] ( U ) = E [Ω] ( U ) for all Ω ∈ Ω if and only if ∃ H ≥ ∀ t ≥ ω ( t ) ≤ ω ( Ht ) + H, which is in turn equivalent to the fact that some (equivalently each) Ω ∈ Ω hasmoderate growth; see also [4].The associated weight matrix Ω always has moderate growth, by (17). It isequivalent to a regular weight matrix S (that means E [ ω ] = E [ S ] ) if and only if ω is equivalent to a concave weight function. In fact, a E [ ω ] -version of the almostanalytic extension theorem 4.1 holds if and only if ω is equivalent to a concave weightfunction; see [9, Theorem 4.8]. The weight matrix S = { S x } x> has the propertythat for each x > s x is log-convex and satisfies mg( s x , s x ) =: H < ∞ and thus h s x ( t ) ≤ h s x ( Ht ) for all x, t >
0; see [25, Proposition 3].4.6.
General ultradifferentiable classes have property ( D ) .Theorem 4.2 ( E [ M ]1 ) . Let M be a [regular] weight matrix of [moderate growth].Then E [ M ]1 has property ( D ) . The proof is given in Section 6. It builds upon a characterization of the class E [ M ] by holomorphic approximation; see Section 5.We may infer a multidimensional result, since non-quasianalytic E { M } -regularitycan be tested along curves; this useful tool is available in a satisfactory manner onlyin the non-quasianalytic Roumieu setting. We need two additional properties of theweight matrix: ∃ M ∈ M : ∞ X k =0 µ k < ∞ . (23) which means that E { M } admits non-trivial functions of compact support, and ∀ M ∈ M ∃ N ∈ M : m ◦ n, (24)where m ◦ k := max { m j m α · · · m α j : α i ∈ N > , α + · · · + α j = k } . Condition (24)is equivalent to composition closedness of E { M } (which follows from the argumentsin [22, Theorem 4.9]) and is satisfied by every R-regular weight matrix. Indeed,if M is R-regular, then E { M } has a description by almost analytic extension, byTheorem 4.1. It is easy to see (cf. [9, Proposition 1.1]) that the latter condition ispreserved by composition of functions.Under these assumptions, a function f defined on an open set U ⊆ R d is of class E { M } if and only if f ◦ c is of class E { M } for all E { M } -curves in U ; see [15, 16] and[27, Theorem 10.7.1]. Theorem 4.3 (Non-quasianalytic E { M } d ) . Let M be an R-regular weight matrixof R-moderate growth satisfying (23) . Then E { M } d has property ( D ) .Proof. This follows immediately from Theorem 4.2 and the above observations. (cid:3)
Note that Theorem 4.3 implies Theorem 2.2 as a special case.
Remark 4.4.
The family Q = { Q n } n ∈ N ≥ of quasianalytic intersectable weightsequences referred to at the end of Remark 2.8 actually is a regular weight matrix ofmoderate growth. The Roumieu class E { Q } is quasianalytic and, since Theorem 2.7applies to every M ∈ Q , we conclude that E { Q } d has property ( D ).Note that there is no weight sequence M with E { M } = E { Q } and no weightfunction ω with E { ω } = E { Q } . This follows from the fact that Q n ≤ Q n +1 Q n ,in analogy to the proof given in [22, Theorem 5.22]; see also Remark 5.25 there.5. Holomorphic Approximation of functions in E [ M ] In this section we prove a characterization of the class E [ M ] (in dimension one)by holomorphic approximation. It generalizes [30, Proposition 3.3.2].For notational convenience, we set k f k A := sup z ∈ A | f ( z ) | for any complex valuedfunction f , where A is any set in the domain of f .5.1. Some preparatory observations.Lemma 5.1.
Let
M, N be weight sequences satisfying m /kk → ∞ , n /kk → ∞ , and C := mg( M, N ) < ∞ . Then h m ( t ) ≤ C j n j t j h n ( Ct ) , t > , j ∈ N , (25) h m ( t ) ≤ h n (cid:16) eC t (cid:17) , t > . (26) Proof.
Note that mg( m, n ) ≤ mg( M, N ). Thus, for all j ∈ N and t > h m ( t ) ≤ inf k ≥ m k + j t k + j ≤ inf k ≥ n j n k ( Ct ) k + j = C j n j t j h n ( Ct ) . For (26) we refer to [24, Lemma 3.13]. (cid:3)
For ε > ε denote the interior of the ellipse in C with vertices ± cosh( ε )and co-vertices ± i sinh( ε ). By H (Ω ε ) we denote the space of holomorphic functionson Ω ε . The following lemma is a simple modification of [30, Lemma 3.2.4]. ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 13
Lemma 5.2.
Let
M, N be two weight sequences satisfying m /kk → ∞ , n /kk → ∞ ,and C := mg( M, N ) < ∞ . Let ε > . Let g ∈ H (Ω ε ) ∩ C (Ω ε ) and assume thatthere are constants L, a , a > such that k g k Ω ε ≤ L, k g k [ − , ≤ a h m ( a ε ) . Then with a := max { a , L } and a := eCa we have k g k Ω ε/ ≤ a h n ( a ε ) . Proof.
Let f ( z ) := a g (sin( εz )). Since z sin( εz ) maps the horizontal strip S := { z ∈ C : | Im( z ) | < } to Ω ε , we get that f ∈ H ( S ) ∩ C ( S ) is bounded by K := max { , La } on the whole of S and by h m ( a ε ) on R . Thus an application ofHadamard’s three lines theorem gives | f ( z ) | ≤ h m ( a ε ) −| Im( z ) | K | Im( z ) | , z ∈ S. Since h m ≤ w ∈ Ω ε/ can be written as w = sin( εz ) for some w ∈ S with | Im( w ) | ≤ /
2, we obtain | g ( w ) | ≤ a ( Kh m ( a ε )) / . The statement follows from (26). (cid:3)
Condition ( P [ M ] ) . Let M be a weight matrix.( P { M } ) We say that a function f : [ − , → C satisfies ( P { M } ) if there exist M ∈ M , constants K, c , c >
0, and a family ( f ε ) <ε ≤ ε of functions f ε ∈ H (Ω ε ) ∩ C (Ω ε ) such that for all 0 < ε ≤ ε , k f ε k Ω ε ≤ K, (27) k f − f ε k [ − , ≤ c h m ( c ε ) . (28)( P ( M ) ) We say that a function f : [ − , → C satisfies ( P ( M ) ) if for all M ∈ M and all c > K, c > f ε ) <ε ≤ ε of functions f ε ∈ H (Ω ε ) ∩ C (Ω ε ) such that (27) and (28) hold for all0 < ε ≤ ε .Note that ( P { M } ) generalizes condition ( P M ) of [30].5.3. Description by holomorphic approximation.Theorem 5.3. (i)
Let M ( i ) , ≤ i ≤ , be weight sequences with ( m ( i ) k ) /k → ∞ and ∃ B ≥ ∀ t > m (2) ( B t ) ≤ Γ m (1) ( t ) , (29) ∃ B ≥ ∀ j ∈ N : m (2) j +1 ≤ B j +12 m (3) j . (30) Then for each f ∈ B M (1) B (( − , there exist positive constants K, c , c and func-tions f ε ∈ H (Ω ε ) ∩ C (Ω ε ) such that for all small ε > k f ε k Ω ε ≤ K, k f − f ε k [ − , ≤ c h m (3) ( c ε ) . (31) The constants
K, c , c only depend on B i , in particular, c = CB B , where C isan absolute constant. (ii) Let N ( i ) , ≤ i ≤ , be weight sequences with ( n ( i ) k ) /k → ∞ and mg( N ( i ) , N ( i +1) ) = D ( i ) < ∞ . Let f : [ − , → C be a function. Assume that there exist positive constants K, c , c and functions f ε ∈ H (Ω ε ) ∩ C (Ω ε ) such thatfor all small ε > k f ε k Ω ε ≤ K, k f − f ε k [ − , ≤ c h n (1) ( c ε ) . (32) Then f ∈ B N (3) σ (( − b, b )) for every b < , where σ := eD (1) D (2) c E (1 − b ) and E is anabsolute constant. (iii) If M is a [regular] weight matrix of [moderate growth], then f ∈ B [ M ] (( − , ⇒ f satisfies ( P [ M ] ) ⇒ f ∈ E [ M ] (( − , . (33)Note that [regularity] of M is needed for the first implication in (iii), [moderategrowth] for the second. Item (iii) generalizes [30, Proposition 3.3.2]. Proof.
We follow closely the proof of [30, Proposition 3.3.2].(i) Let f ∈ B M (1) B (( − , c , c > F ∈ C c ( C ) extending f such that | ∂F ( z ) | ≤ c h m (3) ( c d ( z, [ − , , z ∈ C . (34)Note that c = c ( k f k M (1) B , B , B , B ) and c = 12 B B . Then w ε := ∂F Ω ε satisfies k w ε k C ≤ c h m (3) ( Cc ε ) , where C > d ( z, [ − , ≤ Cε for z ∈ Ω ε .Moreover, the bounded continuous function v ε ( z ) := 12 πi Z C w ε ( ζ ) ζ − z dζ ∧ dζ satisfies ∂v ε = w ε in the distributional sense, and we have k v ε k C ≤ c h m (3) ( Cc ε ) . (35)So f ε := F − v ε is holomorphic on Ω ε and continuous on Ω ε . The estimates (34)and (35) easily imply (31).(ii) Let f : [ − , → C satisfy (32). Consider g ε := f ε − f ε ∈ H (Ω ε ) ∩ C (Ω ε ).Then k g ε k Ω ε ≤ K and k g ε k [ − , ≤ c h n (1) (2 c ε ). By Lemma 5.2, k g ε k Ω ε/ ≤ max { c , K } h n (2) (2 eD (1) c ε ) . There exists a (universal) constant
E > b < E (1 − b ) ε around any x ∈ [ − b, b ] is contained in Ω ε/ . The Cauchyestimates and (25) yield k g ( j ) ε k [ − b,b ] ≤ max { c , K } ( E (1 − b ) ε ) j j ! h n (2) (2 eD (1) c ε ) ≤ max { c , K } (cid:16) eD (1) D (2) c E (1 − b ) (cid:17) j N (3) j h n (3) (2 eD (1) D (2) c ε ) , which means k g ε k N (3) σ, [ − b,b ] ≤ max { c , K } h n (3) (2 eD (1) D (2) c ε ) for σ = eD (1) D (2) c E (1 − b ) .Thus, if ε > < ε ≤ ε , then g := f ε + ∞ X j =1 g ε − j = f ε + ∞ X j =1 ( f ε − j − f ε − j +1) ) ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 15 converges absolutely in the Banach space B N (3) σ ([ − b, b ]). Clearly, for every k ∈ N , g = f ε − k + ∞ X j = k +1 ( f ε − j − f ε − j +1) ) , and f = g on [ − b, b ], since for x ∈ [ − b, b ], | f ( x ) − g ( x ) | ≤ | f ( x ) − f ε − k ( x ) | + (cid:12)(cid:12)(cid:12) ∞ X j = k +1 ( f ε − j ( x ) − f ε − j +1) ( x )) (cid:12)(cid:12)(cid:12) which tends to 0 as k → ∞ , by (32) and absolute convergence of the sum.(iii) For the first implication in (33) in the Roumieu case, observe that for f ∈B { M } (( − , f ∈ B M (1) B (( − , B > M (1) ∈ M . ThenR-regularity of M implies the existence of M (2) , M (3) ∈ M such that (29) and (30)are satisfied. Thus (i) yields the desired holomorphic approximation.In the Beurling case take any weight sequence M (3) ∈ M . By B-regularity,we find M (1) , M (2) such that (29) and (30) are satisfied. If f ∈ B ( M ) (( − , f ∈ B M (1) B (( − , B >
0. Again (i) yields the desired holomorphicapproximation (since c = CB B ).The second implication in (33) follows from (ii), since [moderate growth] of M yields weight sequences N ( i ) fulfilling the assumptions of (ii). (cid:3) Proofs
We are now ready to prove the main results. We begin with a technical lemmain which we extract and slightly modify the essential arguments of [30, Section 4].Its general formulation allows us to readily complete the pending proofs.6.1.
A technical lemma.Lemma 6.1.
Let j be a positive integer. Let M ( i ) , ≤ i ≤ ⌈ log ( j ( j +1)) ⌉ +7 =: k ,be weight sequences satisfying ( m ( i ) ℓ ) /ℓ → ∞ and ∃ B ≥ ∀ t > m (2) ( Bt ) ≤ Γ m (1) ( t ) , mg( M ( i ) , M ( i +1) ) < ∞ , for ≤ i ≤ k − . If f : [ − , → C is such that f j , f j +1 ∈ B [ M (1) ] (( − , , then f ∈ E [ M ( k ) ] (( − , .Proof. Set g := f j and h := f j +1 .Let us begin with the Roumieu case. By Theorem 5.3(i), there exist families ofholomorphic functions ( g ε ), ( h ε ) approximating g , h , respectively. More precisely,there exist positive constants K, c , c and functions g ε , h ε ∈ H (Ω ε ) ∩ C (Ω ε ) suchthat, for all small ε >
0, max {k g ε k Ω ε , k h ε k Ω ε } ≤ K, (36)max {k g − g ε k [ − , , k h − h ε k [ − , } ≤ c h m (3) ( c ε ) . (37)Then g j +1 ε − h jε ∈ H (Ω ε ) ∩ C (Ω ε ) satisfies | g j +1 ε − h jε | ≤ | g j +1 ε − f j ( j +1) | + | f j ( j +1) − h jε |≤ ( j + 1) max {| g ε | , | g |} j | g ε − g | + j max {| h ε | , | h |} j − | h ε − h |≤ c h m (3) ( c ε ) , on [ − , . Thus Lemma 5.2 implies k h jε − g j +1 ε k Ω ε/ ≤ c h m (4) ( Cec ε ) =: δ ε , (38)where C is chosen such that C ≥ mg( M ( i ) , M ( i +1) ) for all 2 ≤ i ≤ k −
1. (Here andbelow all constants c i are independent of ε .)Consider the continuous function u ε := ϕ ε g ε h ε max {| g ε | , r ε } , with r ε := δ j +1 ε , where ϕ ε is a smooth function compactly supported in Ω ε and 1 on Ω ε/ . Itcoincides with h ε /g ε in Ω ε/ ∩ {| g ε | > r ε } , but is not holomorphic everywhere near[ − , ε > δ ε ≤ r ε ≤ f ε ) of f by solving a suitable ∂ -problem.Indeed, they show (using (36), (37), (38) and h m (3) ( t ) ≤ h m (4) ( eCt/ h m (4) ≤
1) that k u ε k Ω ε/ ≤ (2 K ) /j , (39) k f − u ε k [ − , ≤ c r /jε , (40)and that the bounded continuous function v ε ( z ) := 12 πi Z Ω ε/ ∂u ε ( ζ ) ζ − z dζ ∧ dζ, which satisfies ∂v ε = ∂u ε Ω ε/ in the distributional sense in C , fulfills k v ε k Ω ε/ ≤ c δ /sε (41)where s is any real number with s > j ( j + 1) (with c depending on s ).Then f ε := u ε − v ε is holomorphic in Ω ε and continuous on C . By (39) and(41), k f ε k Ω ε is uniformly bounded for all small ε , and by (40) and (41), k f − f ε k [ − , ≤ c δ /s ε . Put s := 2 k − =: 2 ℓ . A repeated application of (26) gives h m (4) ( t ) /s ≤ h m ( k − (( Ce ) ℓ t ) , t > . Thus, for all small ε , k f − f ε k [ − , ≤ c δ /s ε = c (cid:0) c h m (4) (2 eCc ε ) (cid:1) /s ≤ c c /s h m ( k − (2 c ( eC ) ℓ +1 ε ) . (42)So Theorem 5.3(ii) implies that f ∈ E { M ( k ) } (( − , c > g ε ), ( h ε ) such that (36) and (37) are satisfied. Thenfollow the above proof until the end and notice that thus also in the final approxi-mation (42) the constant 2 c ( eC ) ℓ +1 gets arbitrarily small as c gets small. Againan application of Theorem 5.3 completes the proof. (cid:3) ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 17
Proof of Theorem 4.2 – E [ M ]1 . We may assume that there is a positiveinteger j such that g = f j , h = f j +1 are elements of the ring E [ M ]1 . By compos-ing with suitable linear reparameterizations, we may further assume that they arerepresented by elements of B [ M ] (( − , M (1) ∈ M such that g , h are contained in B { M (1) } (( − , M ). By R-regularity and R-moderategrowth of M , we find sequences M ( i ) ∈ M satisfying the assumptions of Lemma 6.1which implies that f ∈ E { M ( k ) } (( − , M ∈ M and we show that f ∈E ( M ) (( − , M , we now get sequences M ( i ) ∈ M as required in Lemma 6.1, where M ( k ) = M . By assumption, g , h areelements of B ( M (1) ) (( − , f ∈ E ( M ) (( − , (cid:3) Proof of Theorem 3.1 – E [ ω ]1 . This is an immediate corollary of Theo-rem 4.2 and the discussion in Section 4.5. (cid:3)
Proof of Theorem 3.2 – non-quasianalytic E [ ω ] d . We reduce the multi-dimensional result to the one-dimensional one.In the Roumieu case E { ω } d , Theorem 3.2 is a simple corollary of Theorem 4.3; theweight matrix S from Section 4.5 clearly satisfies (23) (since ω is non-quasianalytic).The Beurling case E ( ω ) d can be reduced to the Roumieu case by means of thefollowing lemma (which is an adaptation of [21, Lemma 13]). Lemma 6.2.
Let ω be a non-quasianalytic concave weight function. Suppose that f : [0 , ∞ ) → [0 , ∞ ) is any function satisfying ω ( t ) = o ( f ( t )) as t → ∞ . Then thereexists a non-quasianalytic concave weight function ˜ ω satisfying ω ( t ) = o (˜ ω ( t )) and ˜ ω ( t ) = o ( f ( t )) as t → ∞ .Proof. It suffices to extract some constructions from the proof of [21, Lemma 13](to which we refer for details). We may assume that ω is of class C . The condition ω ( t ) = o ( t ) as t → ∞ implies that ω ′ ( t ) ց t → ∞ .Note that log( t ) = o ( ω ( t )) and ω ( t ) = o ( f ( t )) imply f ( t ) → ∞ as t → ∞ . Wedefine inductively three sequences ( x n ), ( y n ), and ( z n ) with x = y = z = 0, x >
0, and the following properties: Z ∞ x n ω ( t )1 + t dt ≤ n , (43) x n > y n − + n, (44) f ( t ) ≥ n ω ( t ) , for all t ≥ x n , (45) ω ( x n ) ≥ n − i ω ( z i ) , ≤ i ≤ n − , (46) ω ′ ( y n ) = n − n ω ′ ( x n ) , (47) ω ( z n ) = nω ( y n ) − ( n − (cid:0) ω ( x n ) + ( y n − x n ) ω ′ ( x n ) (cid:1) . (48)Concavity of ω guarantees well-definedness of these conditions. Then˜ ω ( t ) := ( ( n − (cid:0) ω ( x n ) + ( t − x n ) ω ′ ( x n ) (cid:1) − P n − i =1 ω ( z i +1 ) if x n ≤ t < y n ,nω ( t ) − P n − i =1 ω ( z i +1 ) if y n ≤ t < x n +1 , defines a non-quasianalytic concave weight function of class C satisfying( n − ω ( t ) ≤ ˜ ω ( t ) ≤ nω ( t ) , if t ∈ [ x n , x n +1 ) and n ≥ . (49)(Non-quasianalyticity follows from (43) and the second inequality in (49); cf. [21,Remark 14].) Together with (45) this implies that ω ( t ) = o (˜ ω ( t )) and ˜ ω ( t ) = o ( f ( t ))as t → ∞ . (cid:3) Suppose that g = f j , h = f j +1 are representatives (of the corresponding germs)belonging to B ( ω ) ( U ) on some relatively compact 0-neighborhood U in R d andconsider the sequence L k defined in (14). Then for each integer j ≥ C j > L k ≤ C j exp( jϕ ∗ ω ( k/j )) , for all k ∈ N . Defining the function ℓ : [0 , ∞ ) → R by ℓ ( t ) := log max { L k , } , for k ≤ t < k + 1 , and performing the subsequent steps in [21, Section 5], we find that ℓ ≤ ϕ ∗ ˜ ω + const,where ˜ ω is the weight function provided by Lemma 6.2. This means that g , h belongto B { ˜ ω } ( U ). Invoking Theorem 3.2 in the Roumieu case shows that f ∈ E { ˜ ω } d . Since ω ( t ) = o (˜ ω ( t )) as t → ∞ we may conclude that f ∈ E ( ω ) d . (cid:3) Proof of Theorem 2.7 – quasianalytic E { M } d . The following lemma is avariant of [16, Theorem 1.6(3)].
Lemma 6.3.
Let M be a quasianalytic intersectable weight sequence. Then: (i) n /kk → ∞ for all N ∈ L ( M ) . (ii) If M has moderate growth, then for every N ∈ L ( M ) there exists N ′ ∈L ( M ) such that mg( N ′ , N ) < ∞ .Proof. (i) is obvious, since m /kk → ∞ (cf. Remark 2.8).(ii) If M has moderate growth, then so has m . Set C := mg( m, m ) < ∞ . For N ∈ L ( M ) we define N ′ by n ′ k := C k min ≤ j ≤ k n j n k − j or equivalently n ′ j := C j ν ν ν · · · ν j , n ′ j +1 := C j +1 ν ν ν · · · ν j ν j +1 , where ν k := n k /n k − . Then clearly mg( N ′ , N ) < ∞ . Since ν k is increasing, so is ν ′ k := n ′ k /n ′ k − , thus n ′ is log-convex. Moreover, n ′ j = C j n j ≥ m j m j n j ≥ m j and analogously n ′ j +1 = C j +1 n j n j +1 ≥ m j +1 , so that N ′ ≥ M . It remains tocheck that N ′ is non-quasianalytic. Since N ′ is log-convex, the sequence ( N ′ k ) /k isincreasing and so it suffices to show that P j ( N ′ j ) − / (2 j ) < ∞ . This is clear, since( N ′ j ) / (2 j ) = ((2 j )! C j n j ) / (2 j ) ≥ Ce jn /jj ≥ Ce N /jj and N is non-quasianalytic. (cid:3) Let M be a quasianalytic intersectable weight sequence of moderate growth.Suppose that g = f j , h = f j +1 are elements of E { M } d . Since M is intersectable, itsuffices to show that f ∈ E { N } d for every N ∈ L ( M ). Fix such N . By Lemma 6.3,there exist N (1) , . . . , N ( k ) ∈ L ( M ) with N ( k ) = N such that the requirements of ONLINEAR CONDITIONS FOR ULTRADIFFERENTIABILITY 19
Lemma 6.1 are satisfied. (Note that L ( M ) is not a weight matrix in the sense ofSection 4.1, because it is not totally ordered.)Let U be an open 0-neighborhood in R d on which we have g, h ∈ E { M } ( U )for representatives which are denoted by the same symbols. Take any curve c ∈E { N (1) } ( R , U ) with compact support. Then, by composition closedness of E { N (1) } as n (1) is log-convex, we have g ◦ c, h ◦ c ∈ E { N (1) } ( R ). After a linear change ofvariables, we may assume that g ◦ c, h ◦ c ∈ B { N (1) } (( − , f ◦ c ∈ E { N } (( − , f ∈ E { N } ( U ), by [16, Theorem2.7]. (cid:3) Acknowledgements.
We wish to thank Prof. Jos´e Bonet Solves for drawing ourattention to [30].
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