Nonlinear Conditions for Ultradifferentiability

A remarkable theorem of Joris states that a function f is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}C∞ if two relatively prime powers of f are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}C∞. Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

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 function itself; e.g. (1 Q − 1 R\Q ) 2 = 1, where 1 A is the indicator function of a set A. ) It was soon realized that the statement also holds for complex valued functions and it led to the study of so-called pseudoimmersions [7,12,13,19]. A simple proof based on ring theory was given by [1].
Only recently Thilliez [30] showed that Joris's result carries over to Denjoy-Carleman classes of Roumieu type E {M} . These are ultradifferentiable classes of smooth functions defined by certain growth properties imposed upon the sequence of iterated derivatives in terms of a weight sequence M (which in view of the Cauchy estimates measures the deviation from analyticity).
By extracting the essence of Thilliez's proof, we show in this paper that a broad variety of ultradifferentiable classes has a division property equivalent to Joris's result. Let S be a subring (with multiplicative identity) of the ring of germs at 0 ∈ 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 If S has property (D), then Joris's theorem holds in S. Indeed, suppose that p 1 , p 2 are relatively prime positive integers and f p 1 , f p 2 ∈ S. All integers j ≥ p 1 p 2 can be written j = a 1 p 1 + a 2 p 2 for a 1 , a 2 ∈ N, see [11, p.270]. Hence, f j ∈ S for all j ≥ p 1 p 2 . Since two consecutive integers are relatively prime, also the converse holds. 1

Results
Let us give an overview of our results. It is understood that certain minimal regularity properties of the weights are assumed (see Table 1) which in particular guarantee that the sets of germs are indeed rings. (Note that by convention [·] stands for {·}, i.e., Roumieu, as well as (·), i.e., Beurling.) Interestingly, the proof in one dimension works for quasianalytic and nonquasianalytic classes alike. But the tool used to reduce the multidimensional to the one-dimensional statement is only available in the non-quasianalytic Roumieu 1 In an earlier version of the paper we considered the division property j ∈ N ≥1 , g, q, f g ∈ S, f j = qg ⇒ f ∈ S which resulted from our wish to prove an ultradifferentiable version of the division theorem [13,Theorem 1]. But this property is equivalent to (2). case ( [15], [27]). The (multidimensional) Beurling case can often be reduced to the corresponding Roumieu case. Hence we obtain the following multidimensional non-quasianalytic results. The rings of germs in all dimensions d of the following ultradifferentiable classes have property (D): • E [M] , non-quasianalytic Denjoy-Carleman classes of Roumieu (Theorem 2.2) and Beurling type (Theorem 2.5), • E [ω] , non-quasianalytic Braun-Meise-Taylor classes of Roumieu and Beurling type (Theorem 3.2), • E {M} , non-quasianalytic ultradifferentiable classes defined by weight matrices of Roumieu type (Theorem 4.3).
For quasianalytic Denjoy-Carleman classes of Roumieu type E {M} in one dimension the implication (2) follows from the stronger result, due to Thilliez [29], that C ∞ -solutions of a polynomial equation z n + a 1 z n−1 + · · · + a n−1 z + a n = 0, 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. But it seems to be unknown whether, in the presence of quasianalyticity, it holds in higher dimensions. In fact, quasianalytic ultradifferentiability cannot be tested on quasianalytic curves (or lower dimensional plots) even if the function in question is known to be smooth ( [10,20]). Hence, we think that it is interesting that, combining our proof with a description of certain quasianalytic classes E {M} as an intersection of suitable non-quasianalytic ones (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).
Since all considered regularity classes are local, the results for germs immediately give corresponding results for functions on open sets.

Summary of the Results
We list in Table 1 the ultradifferentiable rings of germs known to have property (D), together with the needed assumptions on the weights and the respective references. All germs are function germs at 0 in R d for some dimension d. The dimension is added as a left subscript, e.g., d E [M] denotes the 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 Sects. 2.4, 2.5.
We remark that non-quasianalytic Denjoy-Carleman classes E {M} , where the weight 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 of the classes E [ω] and E [M] allows to overcome this restriction in the sense that the implication (2) holds under weaker moderate growth conditions.

Strategy of the Proof
Thilliez's proof of Joris's theorem for E {M} consists of the following two steps: (i) The class E {M} admits a description by holomorphic approximation which is based on a result of Dynkin [8] on almost analytic extensions and a related ∂-problem. (ii) If f j , f j+1 are of class E {M} and g ε , h ε are respective holomorphic approximations, the quotient h ε /g ε is a naive candidate for a holomorphic approximation of f . In order to avoid small divisors one considers 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 approximation 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 provide a framework for ultradifferentiability (Sect. 4) which encompasses Denjoy-Carleman classes (Sect. 2) and Braun-Meise-Taylor classes (Sect. 3). In Sect. 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 almost analytic extension presented in our recent paper [9]. Then we execute a version of step (ii) under a quite minimal set of assumptions, see Lemma 6.1. It enables us to easily deduce the main results in Sect. 6.
That μ is increasing means that M is log-convex, i.e., log M is convex or, equivalently, M 2 k ≤ M k−1 M k+1 for all k. If in addition M 1/k k → ∞, we say that M is a weight sequence. Sometimes we will make the stronger assumption that m is log-convex.
For σ > 0 and open U ⊆ R d , one defines the Banach space

and the (local) Denjoy-Carleman classes of Roumieu type
For later reference we also consider the global class Replacing the existential quantifier for σ by a universal quantifier, we find the Denjoy-Carleman classes of Beurling type and where for "⇐" one has to assume that M is a weight sequence. Note that A crucial assumption in [30] is moderate growth of M, which reads as follows It implies derivation closedness The last property we need to mention is non-quasianalyticity of M, that is By the Denjoy-Carleman theorem, this condition is equivalent to the existence of non-trivial functions with compact support in E [M] (U ). It is well-known that nonquasianalyticity implies m 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].

Associated Functions
which is is increasing, continuous on [0, ∞), and positive for t > 0. For large t we have h m (t) = 1. Furthermore, we need and, provided that m k+1 /m k → ∞, We trivially have m ≤ m . If m is log-convex, then m = m . We shall use these functions for

Regular Weight Sequences
A weight sequence M is said to be regular if m 1/k k → ∞, M is derivation closed, and there exists a constant C ≥ 1 such that m (Ct) ≤ m (t) for all t > 0.

Theorem 2.2 (Non-quasianalytic d E {M} ) Let M be a non-quasianalytic regular weight sequence of moderate growth. Then, d E {M} has property (D).
This is a special case of Theorem 4.3 below (cf. Sect. 4.5). It implies Thilliez's result [30,Corollary 2.2.5].
A quasianalytic one-dimensional version follows from a stronger result in [29]:

Denjoy-Carleman Classes of Beurling Type Have Property (D)
Proof Only the supplements (ii) and (iii) were not already proved in [9, Lemma 7.5].
(ii) follows from the fact that a weight sequence M is derivation closed if and only if there is a constant C ≥ 1 such that M k ≤ C k 2 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 weight sequence 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 Applying [6, Lemme 16] (see also [28,Lemma 4.1]) to β k and α k = γ k := 1 μ k , yields an increasing sequence δ = (δ k ) such that Then N k := μ 1 ···μ k δ 1 ···δ k defines a non-quasianalytic weight sequence, by (12) and (13). (10). By (11), there is a constant C > 0 such that δ k for some positive integer j. Assume that representatives of these germs are defined in the neighborhood of the closure of some bounded 0-neighborhood U ; we denote the representatives by the same symbols. Then, the sequence satisfies L M. By Lemma 2.4, there exists a weight sequence S satisfying the assumptions of Theorem 2.2 and L ≤ S M.

Theorem 2.6 (Quasianalytic 1 E (M) ) Let M be a quasianalytic derivation closed weight sequence such that m is log-convex and m
Proof This follows from the proof of Theorem 2.3 in [29] (which also works in the Beurling case). Alternatively, we may infer it from Theorem 2.3 by a reduction argument based on Lemma 2.4 as in the proof of Theorem 2.5.

A Multidimensional Quasianalytic Result
Let M be a weight sequence and consider the sequence space We call a quasianalytic weight sequence M intersectable if 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 function spaces, since denotes the k-th order Fréchet derivative and · L k sym the operator norm. Note that a quasianalytic intersectable weight sequence M always satisfies m 1/k k → ∞; an argument is given in Remark 2.8 below.

Theorem 2.7 (Quasianalytic d E {M} ) Let M be a quasianalytic intersectable weight sequence of moderate growth. Then d E {M} has property (D).
The proof of this result is given in Sect. 6.

Remark 2.8
In [16, Theorem 1.6] (inspired by [2]) a sufficient condition for intersectability was given. Let M be a quasianalytic weight sequence with 1 ≤ M 0 < M 1 . Consider the sequenceM defined by Ifm is log-convex, then M is intersectable. Not every quasianalytic weight sequence is intersectable, for instance, where log [n] denotes the n-fold composition of log; analogously for exp [n] . See also [27,Sect. 11] for a generalization of this concept.

Weight Functions and Braun-Meise-Taylor Classes
A weight function is, by definition, a continuous increasing function ω One may assume that ω| [

. Then the (local) Braun-Meise-Taylor class of Roumieu type is
and that of Beurling type is We say that ω and σ are equivalent if they generate the same classes, i.e., This is the case if and only if E [ω] (U ) contains non-trivial functions of compact support (cf. [5] or [22]). Let us emphasize that in this paper we treat condition (ω 2 ) as a general assumption for weight functions; it means that the Beurling class E (ω) contains the real analytic class. It is automatically satisfied if ω is non-quasianalytic.
Let d E [ω] 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.

The Associated Weight Matrix
Let ω be a weight function. Setting x k := e ϕ * ω (xk)/x defines a weight sequence x for every x > 0, where x ≤ y if x ≤ y. Thus the collection := { x } x>0 is a weight matrix (in the sense of Sect. 4). Note that satisfies a mixed moderate growth property, namely The importance of the associated weight matrix is that it encodes an equivalent topological description of the spaces E [ω] (U ) as unions or intersections of Denjoy-Carleman classes; see Sect. 4.5. All this can be found in [22]. Evidently, it suffices to assume that ω is equivalent to a concave weight function.

Braun-Meise-Taylor Classes Have Property (D)
For the multidimensional analogue we additionally assume non-quasianalyticity.

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

Weight Matrices and Ultradifferentiable Classes
A weight matrix M is, by definition, a family of weight sequences which is totally ordered with respect to the pointwise order relation on sequences, i.e., The limits in (18) and (19)

Regular Weight Matrices
and (regular) or B-regular (for Beurling) if 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.

Almost Analytic Extensions
Let h : (0, ∞) → (0, 1] be an increasing continuous function which tends to 0 as t → 0. Let ρ > 0 and let U ⊆ R d be a bounded open set. We say that a function f : U → C admits an (h, ρ)-almost analytic extension if there is a function F ∈ C 1 c (C d ) and a constant C ≥ 1 such that F| U = f and ∂z j dz j and d(z, U ) := inf x∈U |z − x| denotes the distance of 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.

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 analytically extendable for each quasiconvex domain V relatively compact in U .
In Sect. 5 we shall use [9, Proposition 3.12], which is a key ingredient of the proof of Theorem 4.1.

Weight Matrices of Moderate Growth
For positive sequences M, N set We say that a weight matrix M has R-moderate growth or {moderate growth} if and B-moderate growth or (moderate growth) if 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), respectively.

Denjoy-Carleman and Braun-Meise-Taylor classes in this Framework
By definition, Denjoy- which is in turn equivalent to the fact that some (equivalently each) ∈ has moderate growth; see also [4].
The associated weight matrix always has moderate growth, by (17). It is equivalent to a regular weight matrix S (that means E

E [M] has property (D).
The proof is given in Sect. 6. It builds upon a characterization of the class E [M] by holomorphic approximation; see Sect. 5.
We may infer a multidimensional result, since non-quasianalytic E {M} -regularity can be tested along curves; this useful tool is available in a satisfactory manner only in the non-quasianalytic Roumieu setting. We need two additional properties of the weight matrix: which means that E {M} admits non-trivial functions of compact support, and where m • k := max{m j m α 1 · · · m α j : α i ∈ N >0 , α 1 + · · · + α j = k}. Condition (24) is equivalent to composition closedness of E {M} (which follows from the arguments in [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, by

Theorem 4.3 (Non-quasianalytic d E {M} ) Let M be an R-regular weight matrix of R-moderate growth satisfying (23). Then d E {M} has property (D).
Proof This follows immediately from Theorem 4.2 and the above observations. Note that Theorem 4.3 implies Theorem 2.2 as a special case.

Remark 4.4
The family Q = {Q n } n∈N ≥1 of quasianalytic intersectable weight sequences referred to at the end of Remark 2.8 actually is a regular weight matrix of moderate growth. The Roumieu class E {Q} is quasianalytic and, since Theorem 2.7 applies to every M ∈ Q, we conclude that d E {Q} has property (D).
Note that there is no weight sequence M with E {M} = E {Q} and no weight function 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.

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 f A := sup z∈A | f (z)| for any complex valued function f , where A is any set in the domain of f .
For ε > 0 let ε denote the interior of the ellipse in C with vertices ± cosh(ε) and co-vertices ±i sinh(ε). By H( ε ) we denote the space of holomorphic functions on ε . The following lemma is a simple modification of [30, Lemma 3.2.4].
The statement follows from (26).

Description by Holomorphic Approximation
Then for each f ∈ B M (1) B 0 ((−1, 1)) there exist positive constants K , c 1 , c 2 and func- The constants K , c 1 , c 2 only depend on B i , in particular, Assume that there exist positive constants K , c 1 , c 2 and functions f ε ∈ H( ε ) ∩ C 0 ( ε ) such that for all small ε > 0 and E is an absolute constant.

iii) If M is a [regular] weight matrix of [moderate growth], then
satisfies ∂v ε = w ε in the distributional sense, and we have So f ε := F − v ε is holomorphic on ε and continuous on ε . The estimates (34) and (35) easily imply (31).
The second implication in (33) follows from (ii), since [moderate growth] of M yields weight sequences N (i) fulfilling the assumptions of (ii).

Proofs
We are now ready to prove the main results. We begin with a technical lemma in which we extract and slightly modify the essential arguments of [30,Sect. 4]. Its general formulation allows us to readily complete the pending proofs.
For the Beurling we observe that, by assumption, we find for any (small) c 2 > 0 approximating sequences (g ε ), (h ε ) such that (36) and (37) are satisfied. Then follow the above proof until the end and notice that thus also in the final approximation (42) the constant 2c 2 (eC) +1 gets arbitrarily small as c 2 gets small. Again an application of Theorem 5.3 completes the proof.

Proof of Theorem 4.2-1 E [M]
We may assume that there is a positive integer j such that g = f j , h = f j+1 are elements of the ring 1 E [M] . By composing with suitable linear reparameterizations, we may further assume that they are represented by elements of B [M] ((−1, 1)) which we denote by the same symbols.
In the Roumieu case, there exists M (1) ∈ M such that g, h are contained in B {M (1) } ((−1, 1)) (by the linear order of M). By R-regularity and R-moderate growth of M, we find sequences M (i) ∈ M satisfying the assumptions of Lemma 6.1 which implies that f ∈ E {M (k) } ((−1, 1)).

Proof of Theorem 3.1-1 E [!]
This is an immediate corollary of Theorem 4.2 and the discussion in Sect. 4.5.

Proof of Theorem 3.2-Non-quasianalytic d E [!]
We reduce the multidimensional result to the one-dimensional one.
In the Roumieu case d E {ω} , Theorem 3.2 is a simple corollary of Theorem 4.3; the weight matrix S from Sect. 4.5 clearly satisfies (23) (since ω is non-quasianalytic).
The Beurling case d E (ω) can be reduced to the Roumieu case by means of the following lemma (which is an adaptation of [21,Lemma 13]). 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 1 . The condition We define inductively three sequences (x n ), (y n ), and (z n ) with x 1 = y 1 = z 1 = 0, x 2 > 0, and the following properties: x n > 2y n−1 + n, ω(z n ) = nω(y n ) − (n − 1) ω(x n ) + (y n − x n )ω (x n ) .
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 and consider the sequence L k defined in (14). Then for each integer j ≥ 1 there exists C j > 1 such that L k ≤ C j exp( jϕ * ω (k/ j)), for all k ∈ N.