The Kotake–Narasimhan theorem in general ultradifferentiable classes

We prove a Kotake–Narasimhan type theorem in general ultradifferentiable classes given by weight matrices. In doing so we simultaneously recover and generalize significantly the known results for classes given by weight sequences and weight functions. In particular, we obtain a sharp Kotake–Narasimhan theorem for Beurling classes.


Historical background
In this paper we consider the problem of iterates for elliptic operators with coefficients in general ultradifferentiable structures. 1 In its general form the problem of iterates for an ultradifferentiable structure U can be stated in the following way: Let u be a smooth function which satisfies the defining estimates of U with respect to the iterates of a differential operator P. Can we conclude that u is already an element of U?
If the answer to the question above is affirmative for an ultradifferentiable structure U and an operator P then we say that the theorem of iterates holds for the operator P with respect to U.
Recently, there has been a resurged interest in the problem of iterates in various different settings, see e.g.[6,14,17,20,21].For surveys on the problem of iterates we refer the reader to [9,16].In this paper we are going to revisit the classical case of elliptic operators in open sets of R n in view of the recently expanded theory on general ultradifferentiable classes.
The main starting point of the problem of iterates was when in 1962 Kotake-Narasimhan [27] and Komatsu [24] separately proved the following statement: If P is an elliptic operator of order d with analytic coefficients on an open set ⊆ R n , then a smooth function u ∈ E( ) is analytic in if and only if for each relatively compact set U ⊆ there are constants C, h > 0 such that Nelson [33] proved an analogous statement for an elliptic system of analytic vector fields.
The next natural step is then to ask if a similar result holds if one considers instead of the analytic class more general ultradifferentiable classes.We are going to say that the Kotake-Narasimhan theorem holds for an ultradifferentiable structure U if the theorem of iterates with respect to U holds for every elliptic operator P with coefficients in U.
For example Bolley-Camus [7] proved the Kotake-Narasimhan theorem for Gevrey classes.If we want to consider more general families of ultradifferentiable classes then the commonly used spaces are the Denjoy-Carleman classes which are defined by weight sequences, for an introduction see e.g.[25], and the Braun-Meise-Taylor classes which are given by weight functions, introduced in their modern form by [13].Both of these classes are generalizations of the Gevrey classes, however, they do not in general coincide, see [11].In this paper we take a step further and consider ultradifferentiable classes given by weight matrices, i.e. countable families of weight sequences, which were introduced in [34,35].These classes include both Denjoy-Carleman and Braun-Meise-Taylor classes.In [20] we showed that the theorem of iterates with respect to classes given by weight matrices holds for elliptic operators with analytic coefficients using a microlocal analytic approach, generalizing results of [8] in the case of weight sequences and of [5] for weight functions.
More generally, the Kotake-Narasimhan theorem for Denjoy-Carleman classes was proven in [29] and in a more general form in [15].In the case of Braun-Meise-Taylor classes the Kotake-Narasimhan theorem was shown by [6].All these instances followed generally the lines of the proofs of [24,27] and [7], which used a technique involving apriori L 2 -estimates and nested neighborhoods first introduced by Morrey-Nirenberg [32].In this paper we prove the Kotake-Narasimhan theorem for ultradifferentiable classes given by weight matrices by adapting the proof in [7] resp.[14].In doing so we not only recover the known statements in Denjoy-Carleman classes and Braun-Meise-Taylor classes but also partially generalizing them.In particular, in the case of Denjoy-Carleman classes Lions and Magenes [29] asked, what the optimal conditions on the weight sequences are in order for the Kotake-Narasimhan Theorem to hold.As we will see, our main result implies especially that the Kotake-Narasimhan Theorem holds for the Denjoy-Carleman classes determined by the sequences where q > 1 is a real parameter.These sequences have not been covered by the previous works on the Kotake-Narasimhan Theorem for Denjoy-Carleman classes, cf.Remark 1.5.

Statement of main results
In order to formulate our main results for weight sequences and weight functions we need to fix some notations: will always be an open set of R n and we set D j = −i∂ j = −i∂ x j where ∂ x j is the j-th partial derivative, j = 1, . . ., n.We denote the set of positive integers by N whereas N 0 = N ∪ {0}.Furthermore we say that a sequence for all k ∈ N. Next we define the Denjoy-Carleman class associated to M over an open set ⊆ R n .To a weight sequence M we can in fact associate two different ultradifferentiable classes.First the Roumieu class associated to M is given by A basic question in the theory of ultradifferentiable classes is that of quasianalyticity.We recall that an algebra E of smooth functions is called non-quasianalytic if the only flat function in E is the zero function.The Denjoy-Carleman theorem, see [23], says that a Denjoy-Carleman class We call a weight sequence M non-quasianalytic if (1.2) holds and otherwise quasianalytic.
In order to formulate the Kotake-Narasimhan Theorem for Denjoy-Carleman classes we have to specify additional conditions on the weight sequence M. It is often easier to formulate these conditions not in terms of the sequence (M k ) k directly but to use other sequences associated to M, like We say that M is weakly regular if the following conditions hold: The Gevrey sequence G s , which is given by G s k = (k!) s , is weakly regular.More generally the weight sequences B s,σ , defined by B s,σ k = (k!) s log(k + e) σ k , are weakly regular for all s ≥ 1 and σ > 0. The sequences B s,σ are quasianalytic if and only if s = 1 and 0 < σ ≤ 1.
Another examples of weakly regular weight sequences are the following: For q > 1 and 1 < r let L q,r be given by L q,r k = q k r .The weight sequences L q,r are weakly regular for all q > 1 and 1 < r ≤ 2. In particular the case r = 2 are the q-Gevrey sequences N q given by N q = q k 2 .
If P = {P 1 , . . ., P } is a system of partial differential operators with smooth coefficients a jα ∈ E( ), then we recall that the system P is elliptic in if for every x ∈ the principal symbols have no common nontrivial real zero in ξ .
Our main result in the case of Denjoy-Carleman classes is the following theorem.

Theorem 1.3
Let M be a weakly regular weight sequence and P = {P 1 , . . ., P } be an elliptic system of differential operators.Then the following statements hold: (1) Assume that the coefficients of the operators P j , j = 1, . . ., , are all elements of E {M} ( ).Then u ∈ E {M} ( ) if and only if u ∈ E( ) and for all U there are constants C, h > 0 such that for all k ∈ N 0 we have that for all α ∈ {1, . . ., } k where P α = P α 1 . . .P α k and d α = d α 1 + • • • + d α k with d j denoting the order of the operator P j , j = 1, . . ., .
(2) Assume that the coefficients of the operators P j , j = 1, . . ., , are in E (M) ( ).Then u ∈ E (M) ( ) if and only if u ∈ E( ) and for all U and every h > 0 there exists a constant C > 0 such that 4 The notion of a weakly regular weight sequence is inspired by [18].In that article a weight sequence M is called regular if M satisfies (1.3), (1.5) and instead of (1.4) the sequence M is strongly logarithmic convex, i.e.
Remark 1. 5 We need to point out that Theorem 1.3 is a considerably more general statement then the previous known results, like [15,29] or [14] in the Roumieu case and [12] in the Beurling case.In these papers varying conditions on the weight sequence are assumed, but these conditions always include (1.2), and To compare these conditions with those of our result we may note that it is easy to see that (1.7) is another consequence of (1.6).However, if M is a weight sequence we can see that (1.4) implies also (1.7):For k = 0 we have that m 0 m = m for all ∈ N 0 and the same is true for = 0 and all k ∈ N 0 .So we have to show (1.7) for k, ∈ N, but then we have by (1.4).Thence in that regard our conditions are formally more restrictive, but this is compensated by the fact that we replaced the other conditions noted above by far weaker conditions.The most interesting property above in that regard is the last one.It is clear that (1.8) implies (1.5).However, (1.8) is far more restrictive than (1.5), see e.g.[30].In fact, if a weight sequence M satisfies (1.8) then there is some s > 1 such that E [M] ( ) ⊆ G s ( ).For example, the sequences L q,r cannot satisfy (1.8) for any choice of q, r > 1.
Moreover, we have replaced (1.2) by the non-analyticity condition (1.3).We will see in the next section, cf.Remark 2.4, that (1.3) is still nearly superfluous but we use it in order to allow for a unified formulation of Theorem 1.3.This only excludes formally the analytic case, which is well-known to hold.
We may also observe that in the Beurling case Theorem 1.3 is also a strict statement in the sense that we allow that the coefficients are in the class given by the same weight sequence as the space of vectors considered.In contrast, for example [12] requires that the coefficients of the operators are in a strictly smaller class than the vectors considered.We are able to remove this restriction by applying an argument given in [26], which essentially allows us to reduce the Beurling case to the Roumieu case.
(1) Assume that all coefficients of the operators P j , j = 1, . . ., , are in E {ω} ( ).Then u ∈ E {ω} ( ) if and only if u ∈ E( ) and for all U there are constants C, h > 0 such that for all α ∈ {1, . . ., } k and every k ∈ N 0 .( 2) Assume that all coefficients of the P j , j = 1, . . ., are in if and only if u ∈ E( ) and for all U and every h > 0 there is a constant C > 0 such that for all α ∈ {1, . . ., } k and every k ∈ N 0 .
Remark 1.7 Theorem 1.6 generalizes the main theorem in [6] from a single elliptic operator to elliptic systems of operators.Furthermore, in the Beurling case the Kotake-Narasimhan theorem was proven in [6], again, only for operators whose coefficients are in a strictly smaller class As we have announced above we are going to prove our main Theorem 2.14 in the far more general setting of weight matrices, that is countable families of weight sequences.We utilize the approach of [7], which in particular allows us to apply the aforementioned technique of [26] in order to prove the Beurling version of Theorem 2.14 by effectively reducing it to the Roumieu case even in the case of weight matrices.However, we need to point out that in general it is not possible to use this technique in the case of non-trivial weight matrices, see for example [19,Section 7].
The structure of the paper is the following: In Sect. 2 we recall basic definitions and facts about weight sequences, weight matrices and the ultradifferentiable classes of functions and vectors generated by them.Having sufficiently developed the theory of ultradifferentiable structures given by weight matrices we can formulate our main result Theorem 2.14 at the end of Sect. 2. In Sect. 3 we prove the fundamental L 2 -estimate which is used in Sect. 4 to prove Theorem 2.14.We conclude the paper by stating some remarks in Sect. 5.

Weight sequences
The following properties of a weight sequence are well known.

Lemma 2.1 Let M be a weight sequence and set
For later convenience we shall take a closer look at the structure of Denjoy-Carleman classes, for more details see [25].Here we do not to require that M is a weight sequence.
So let U ⊆ R n be an open set, M be an increasing sequence of positive numbers and h > 0 be a parameter.Then E(U ) is the space of smooth functions f on U such that ∂ α f extends continuously to U and we define a seminorm on E(U ) by setting is the Roumieu resp.Beurling class of global ultradifferentiable functions associated to the weight sequence M over U .The local ultradifferentiable classes E [M] (U ) associated to M over U can thus be described as We recall that to a weight sequence M (or an abritary sequence) we associate another sequence m k = M k /k!.It follows from above that the condition We are also discussing the other conditions appearing in the beginning of this article, e.g.
If M is a weight sequence satisfying (2.1) and (2.2) then clearly m = (m k ) k is also a weight sequence.In particular Furthermore, we have the following property: Therefore the sequence k is strictly increasing.(2) The estimate (2.3) is equivalent to For later use we note the following Lemma.

Lemma 2.3
Let M be a sequence with M 0 = 1 ≤ M 1 satisfying (2.1).Then the following statements hold: (2) Assume that the sequence k √ m k is increasing, i.e. (2.5) holds.If we consider the sequence Proof Recall from Remark 2.2 that m is a weight sequence and hence we can apply Lemma 2.1.By Lemma 2.1(3) we thus have that the sequence μ k /k is increasing and therefore This gives instantly (1) since μ k /k ≥ 1.
For (2) we need only to observe that Remark 2. 4 We recall that condition (2.1) implies that A(U ) . However, it is nearly superfluous in view of (2.2): Iterating the estimate in Lemma 2.3(1) we obtain that for any strongly logarithmically convex weight sequence M we have that k ≤ μ k and therefore k! ≤ M k .In particular (k!) M k which in turn implies that A(U ) ⊆ E {M} (U ).In fact, for that relation to hold it is enough to assume that k √ m k is increasing, since this gives also that lim inf k→∞ k √ m k > 0 (recall that m 1 ≥ 1 by assumption).In the Beurling situation, we know that (2.1) holds if and only if A(R) ⊆ E (M) (R).Thus we only formally exclude the analytic case by our assumptions, which of course has been well investigated.
In order to deal later with the Beurling case we need the following statement, which will allow us to reduce the key argument in the Beurling case to the Roumieu case.Its proof is based on the proof of [26,Lemma 6], cf. also [20,Lemma 2.2].

Proposition 2.5
Let M be a weight sequence satisfying (2.1) and (2.5).If L = (L k ) k is a sequence of positive numbers such that L M then there is a sequence N with N 0 = 1 ≤ N 1 satisfying (2.1) and (2.5) For h > 0 we take C • h to be the smallest number such that (2.7) holds.We set and define a auxillary sequence by ¯ k = ˜ k / ˜ 0 .Since the infimum in (2.8) is assumed for some h > 0 we have that Hence the sequence (m k / ¯ k ) k is logarithmic convex and by definition ¯ 0 = 1 and therefore the sequence Thence the sequence N satisfies the conditions (2.1) and (2.5).
Finally we observe that Thence L ≤ N M.

Weight matrices
Now we are in the position to introduce the concept of weight matrices.

Definition 2.6
A weight matrix M is a family of weight sequences such that for all M, N ∈ M we have either If M is a weight matrix then the Roumieu class of global ultradifferentiable functions associated to M is whereas the Beurling class is Then the local ultradifferentiable classes associated to M are given by If M and N are two weight matrices then we set Definition 2.7 Let M be a weight matrix.
(1) We say that M is R-semiregular if the following conditions hold: (2.10) (2) M is B-semiregular if (2.9) and are satisfied.
Remark 2.8 Let d ∈ N be fixed and M be a weight matrix.
Let U be an open set in R n .We say that U has Lipschitz boundary if for all x 0 ∈ ∂U there are some r > 0, local coordinates (x 1 , . . ., x n ) and a Lipschitz function where B(x 0 , r ) is the ball of radius r in R n centered at x 0 .Remark 2.9 For completeness we give an alternative characterization of E [M] ( ) when M is a [semiregular] weight matrix.
(1) Let U ⊆ R n be a bounded open set with Lipschitzian boundary.The Sobolev Theorem [1] implies that a smooth function f ∈ E(U ) is an element of B {M} (U ) (of B (M) (U )) if and only if there are constants C, h > 0 and some M ∈ M (for every h > 0 and M ∈ M there is a constant C > 0) such that We consider the more difficult Beurling case and leave the Roumieu case to the reader.Hence let f ∈ E(U ) and suppose that for all M ∈ M and h > 0 there is some C > 0 such that (2.14) is satisfied for all α ∈ N n 0 .By the Sobolev Imbedding Theorem we have that for integers σ > n/2 the following estimate holds for all u ∈ E(U ) where the constant A depends only on U , n and σ .We fix now σ > n/2.If we set u = D α f then we obtain Thence, using (2.13) we conclude that for all M ∈ M and h > 0 there is a constant C > 0 such that Thus applying also (2.15) gives that f ∈ B (M) (U ).The other direction follows from the trivial estimate (2) On the other hand, it is easy to see that, if ⊆ R n is an arbitrary open set and f ∈ E( ) (3) Combining these two statements we have the following characterizations: Let f ∈ E( ).

and only if for all
x ∈ there is a neighborhood U of x such that • If M is a B-semiregular weight matrix then f ∈ E (M) ( ) if and only if for all x ∈ there is a neighborhood U of x such that Definition 2.10 Let M be a weight matrix.

Remark 2.11
If ω is a weight function then the weight matrix W associated to ω consists of the weight sequences W λ , λ > 0, which are given by It is easy to see that W is R-and B-semiregular.Furthermore, (2.10) and (2.11) hold both for W, cf.[34].
When ω is concave then there exists a weight matrix T such that T{≈}W and T(≈)W and moreover T satisfies (2.16); in fact, every weight sequence T ∈ T satisfies (2.2), see [36,Proposition 3].Moreover, T satisfies (2.9), since condition (2.9) is clearly invariant under ( ) and T is weakly R-and weakly B-regular since the conditions (2.10) and (2.11) are also invariant under the equivalence relations {≈} and (≈), respectively.

Ultradifferentiable vectors associated to weight matrices
Definition 2.12 Let P = {P 1 , . . ., P } be a system of smooth differential operators in U , d j be the order of P j for j = 1, . . ., , u ∈ E(U ) and M be a weight matrix.
(1) Assume that a jλ ∈ B {M} (U ) for all |λ| ≤ d j and every j ∈ {1, . . ., }.Then we say that u is a global vector of class {M} if there are M ∈ M and C, h > 0 such that for all τ ∈ {1, . . ., } k , d τ = k j=1 d τ j and all k ∈ N 0 .(2) When a jλ ∈ B (M) (U ) for all |λ| ≤ d j and every j ∈ {1, . . ., } then u is a global vector of class (M) if for all M ∈ M and all h > 0 there is a constant C > 0 such that (2.17) holds for all τ ∈ {1, . . ., } k and every k ∈ N 0 .(3) Assume that a jλ ∈ E {M} (U ) for all |λ| ≤ d j and every j ∈ {1, . . ., }.Then we say that u is a (local) vector of class {M} if for all V U there are M ∈ M and C, h > 0 such that for α ∈ {1, . . ., } k and k ∈ N 0 .(4) When a jλ ∈ E (M) (U ) for all |λ| ≤ d j and every j ∈ {1, . . ., } then u is a local vector of class (M) if for all V U , all M ∈ M and all h > 0 there is a constant C > 0 such that (2.18) holds for all α ∈ {1, . . ., } k and every k ∈ N 0 .
We denote the space of global vectors of class [M] in U by B [M] (U ; P) and E [M] (U ; P) is the space of all local vectors of class [M].

Proposition 2.13
Let M be a weight matrix and P = {P 1 , . . ., P } be a system of smooth differential operators in an open set U ⊆ R n .Then the following holds: (1) If the coefficients of the operators P j are all in E [M] (U ) then E [M] (U ) ⊆ E [M] (U ; P).
(2) If U is bounded and the coefficients of the operators P j are all in B [M] (U ) then Then we can assume that there exist some h > 0 and for some constants C > 0 and h > 0. From this estimate we can conclude that B {M} (U ) ⊆ B {M} (U ; P).
In the Beurling case we have that b α , f ∈ B (M) (U ) = M∈M B (M) (U ).We define a sequence L by If M ∈ M is arbitrary then L M. According to the proof of [20,Lemma 2.2] there is a weight sequence N such that L ≤ N M. Thus b α , f ∈ B N,h (U ) for some h.The estimate above gives for some C , h > 0. Hence for all h > 0 there is a constant C > 0 such that Since M ∈ M was chosen arbitrarily we conclude that B (M) (U ) ⊆ B (M) (U ; P).
The local case follows analogously.
We can now state our main theorem: Theorem 2.14 Let M be a [weakly regular] weight matrix and P = {P 1 , . . ., P } be an elliptic system of differential operators of class [M] in .Then Remark 2. 15 Note that we can assume that all the operators P j ∈ P in Theorem 2.14 have the same order.Indeed, if d j denotes the order of P j for all j ∈ {1, . . ., } then we set Then all Q j are of order d = j d j and it is easy to see that the system Q = {Q 1 , . . ., Q } is elliptic if and only if P is elliptic.Furthermore E [M] ( ; P) ⊆ E [M] ( ; Q) for all weight matrices M. Hence, Theorem 2.14 will be proven if we show

The fundamental estimate
By Remark 2.15 we will assume in this and the next section that all operators P j ∈ P are of the same order d ∈ N. We will also denote the open ball of radius R centered at a point x ∈ R n by B(x, R).
We follow closely the structure of the proof given in [7] (see also [9]).
Beginning with a well-known a-priori estimate for elliptic systems of smooth operators of equal order d (see e.g.[2]), i.e. for all U there is a constant C > 0 such that we can deduce two other estimates following the arguments in [7]:

(cf. [7, Proposition I-2]) Let P be an elliptic system of differential operators with equal order d and smooth coefficients in and let W W be open sets. Then there exists a constant C
for all u ∈ E( ).

Proposition 3.2 (cf. [7, Proposition I-3]) Let P be as in Proposition
Then there exists a constant C > 0 such that for all u ∈ E( ), for all α ∈ N n 0 with |α| ≤ d and for all ρ, ρ > 0 with ρ + ρ < R we have We are now in the position to formulate and prove the main estimate which will be used in the proof of Theorem 2.14.Note that if Moreover, we recall also that for a weight sequence M we have defined the auxiliary sequence and assume P = {P 1 , . . ., P } is an elliptic system of smooth differential operators P j = |λ|≤d a jλ D λ on such that a jλ | W ∈ B {M} (W ).Then there exists a constant A > 0 such that for all 0 < ρ < R, all u ∈ E( ) and all k ∈ N we have that where |α| ≤ dk and Proof We begin by observing that S k (u) ≤ S k+1 (u) and for all k ∈ N since ρ ≤ 1 by assumption.Furthermore there exists a constant H > 0 such that for all γ ∈ N n 0 and λ ∈ N n 0 with |λ| ≤ d and every j ∈ {1, . . ., }.We are going to prove (3.3) by induction in k.To begin with, (3.1) implies that In the following we use notation β < α if β ≤ α and β = α for α, β ∈ N n 0 .We observe that can be estimated by ) for all < k.Now, the induction hypothesis implies the following estimates: Hence we have obtained that we are able to choose A large enough and independent of α and ρ so that the bracket on the right-hand side of (3.7) is ≤ 1.
4 Proof of Theorem 2.14

The Roumieu case
Let M be an R-regular weight matrix and u ∈ E {M} ( ; P).We have to prove that for all x ∈ there is a neighborhood U of x such that u| U ∈ B {M} (U ).Therefore we fix x 0 ∈ and choose . Then by assumptation there are a weight sequence M ∈ M, satisfying (2.1) and (2.2), and constants C, h > 0 such that We conclude that Hence by (3.3) we have that Thus, we have for d(k − 1) ≤ |α| ≤ dk that dk ≤ |α| + d and therefore Now, note that the sequence k is strictly increasing and the Stirling formula implies that and therefore Thus we can, if we enlarge h when necessary, estimate that for every k ∈ N and all α ∈ N n 0 with d(k − 1) < |α| ≤ dk.Since d does not depend on α or k we have by (2.12) that there is a weight sequence M and constants C 1 , h 1 such that Thus u ∈ E {M} ( ) by Remark 2.9(3).

The Beurling case
Now we assume that M is weakly B-regular and u ∈ E (M) (U ; P).Note that Thus we consider first the case where u ∈ E (M) ( ; P) and a jλ ∈ E (M) ( ), 1 ≤ j ≤ , |λ| ≤ d, with M being a weight sequence for which (2.1) and (2.5) hold.We fix x ∈ and let 0 < R < R 1 ≤ 1 be such that W = B(x; R 1 ) .We define a sequence L by setting According to Lemma 2.5 there is a sequence N with N 0 = 1 ≤ N 1 satisfying (2.1) and (2.5) such that L N M. Hence u ∈ B {N} (W ; P) and a jλ ∈ B {N} (W ).It follows that we can apply Proposition 3.3 and obtain that there is a constant A such that for all k ∈ N and every α with d(k − 1) < |α| ≤ dk we have where S k (u) is as in Proposition 3.3 with M replaced by N and 0 < ρ < R is chosen arbitrarily but fixed.Since (2.5) still implies that N and k √ N k are increasing, the arguments in the previous subsection yield that there are a neighborhood U of x 0 and constants C 1 , h 1 > 0 such that for all ∈ N and all d(k − 1) < |α| ≤ dk.Since N M we conclude that for all h > 0 there is a constant C > 0 such that But M ∈ M has been chosen arbitrarily and therefore we obtain the above estimate for all M ∈ M if u ∈ E (M) ( ; P).Now we can employ (2.13) to conclude that for all weight sequences M and h 1 > 0 there is a constant C 1 > 0 such that Applying Remark 2.9(3) we observe that u ∈ E (M) ( ).

Elliptic regularity in ultradifferentiable classes
Let M be a weight matrix and P be an elliptic system of differential operators with E [M] ( )coefficients.We note that in that case instead of u ∈ E( ) we can just assume that u ∈ D ( ) in Definition 2.12 by the subellipticity of the elliptic system P, cf.[9] or [37]. 3This allows us to deduce results on ultradifferentiable hypoellipticity from Theorem 2.14.Definition 5.1 Let M be a weight matrix and P = {P 1 , . . ., P } be a system of differential operators with E [M] ( )-coefficients.We say that P is It is worthwile to compare the conditions on the weight matrix in Theorem 5.2 with the hypothesis needed in the microlocal regularity results given in [19,Section 7].For simplicity we restrict our discussion to Denjoy-Carleman classes.In [19] we proved the following result: Theorem 5.3 ([19,Theorem 7.1 & Theorem 7.4]) Let M be a weight sequence satisfying (1.3), (1.4) and (1.8).Then for any differential operator with coefficients in E [M] ( ) we have that Here WF [M] u denotes the ultradifferentiable wavefront set with respect to the weight sequence as defined by [22] for Roumieu classes (for the Beurling case see [19]).Moreover, Char P is the characteristic set of the linear differential operator P, cf.e.g.[23].Since j=1 Char P j = ∅ for any elliptic system of differential operators and under our assumptations it holds that WF [M] Pu ⊆ WF [M] u for all u ∈ D ( ) and any differential operator P of class [M], cf.[19, Proposition 5.4(7)], we obtain the following corollary from Theorem 5.3.

Theorem 5.6 Let
⊆ R n be a bounded open set with Lipschitzian P = {P 1 , . . ., P } be a system of operators with constant coefficients and M be a [weakly regular] weight matrix.Then the following statements are equivalent: The system P satisfies (5.2).
Proof First, assume that (1) holds.We set On the other hand, Y ( ) ⊆ E( ) is a closed subspace of E( ) with the usual topology.The Sobolev embedding theorem implies moreover that the topology of E( ) is generated by the system of seminorms { .H k ( ) | k ∈ N 0 }.Thus on Y ( ) the topologies coming from E( ) and H k ( ), k ∈ N 0 agree.Furthermore, has only finitely many connected components j since is bounded with Lipschitz boundary and for each two points x, y in a connected components j there is a continuous path γ connecting x with y such that the length of γ is smaller than C j |x − y| where C j is a constant only depending on j , see [3].It follows that E( ) is nuclear and therefore also Y ( ) according to [31].Thence Y ( ) is a nuclear Banach space and thus Y ( ) has to be finite dimensional according to [38].
But if ξ 0 ∈ C n satisfies P j (ξ 0 ) = 0 for all 1 ≤ j ≤ then the function u(x) = e i xξ 0 is a solution of P j u = 0 for all 1 ≤ j ≤ .Thence the set of all common complex zeros of the polynomials P j (ξ ), 1 ≤ j ≤ , has to be finite.
On the other hand, suppose that (2) is true.Let ξ 1 , . . ., ξ ν be the common complex zeros of the polynomials P j (ξ ), 1 ≤ j ≤ .For each 1 ≤ j ≤ n we consider the polynomial that means that the polynomials Q j (ξ ) vanish on the set of common complex zeros of the polynomials P j , 1 ≤ j ≤ .The Nullstellensatz, cf.[28, Theorem IX.1.5],implies that there exists an integer ρ ≥ 1 such that the polynomials Q ρ j , 1 ≤ j ≤ n, belong to the ideal spanned by the polynomials P r , 1 ≤ r ≤ .Thence, there exist polynomials A jr such that A jr (ξ )P r (ξ ), 1 ≤ j ≤ n.
The polynomials Q ρ j (ξ ) are of order νρ whose principal part is equal to (ξ j ) νρ .Thus 0 is the only complex common zero of these principal parts and therefore Q = {Q ρ 1 , . . ., Q ρ n } is globally elliptic in .Furthermore, if u ∈ D ( ) and P j u ∈ B [M] ( ) for 1 ≤ j ≤ then Q ρ j (D)u ∈ B [M] ( ) for 1 ≤ j ≤ n.From [10] it follows that u ∈ E( ) and therefore u ∈ B [M] ( ; Q).Finally, according to Theorem 5.5 we have u ∈ B [M] ( ).

Final Remarks
We can ask if the conditions, we have imposed on the data of the ultradifferentiable class E [M] for the Kotake-Narasimhan Theorem to hold, can be further loosened.For this, we recall that we proved the Theorem of Iterates for E [M] in the case of elliptic operators with analytic coefficients when M is [semiregular], cf.[20].But if M is [semiregular] then E [M] is closed under derivation and invariant under composition with analytic mappings by [19].On the other hand, if the weight matrix M is [weakly regular], then E [M] is closed under derivation and invariant under composition with ultradifferentiable mappings of class [M].Although we must note that the assumption of weakly regularity is not a priori optimal for this fact to hold, see [34].However, in the case of Braun-Meise-Taylor classes we have that the space E [ω] is invariant under composition with maps of class [ω] if and only if ω is equivalent to a concave weight function.We observe also that E [ω] is closed under derivation by definition.
All these arguments motivate the following conjecture: Conjecture Let U be an ultradifferentiable structure which is closed under derivation and is invariant under composition with mappings of class U. Then the Kotake-Narashiman Theorem holds in the class U.
As we have stated, the conjecture is verified for Braun-Meise-Taylor classes, but we claim moreover that the same is true for Denjoy-Carleman classes.Recall from [34] that a Denjoy-Carleman class E [M] ( ) which contains A( ) (cf.Remark 2.4) and is closed under derivation, i.e. satisfies (1.5), is closed under composition with mappings of class [M] if and only if the sequene k √ m k is almost increasing, that is (5.3) In order to prove our claim let M be a weight sequence such that (5.3) and, if we exclude the analytic case, (2.1) hold.We define a new sequence M by the following procedure. 4For k ∈ N we set where C > 0 is the constant from (5.3).Thence the sequence ν k is increasing and k √ m k ≤ ν k ≤ C k √ m k .We define M by M 0 = 1 and M k = k!ν k k for k ∈ N, in particular M 1 ≥ 1.It follows that M satisfies (2.1) and (2.5).We need to point out, that we cannot conclude that M satisfies (1.1) and therefore cannot assume that M is a weigth sequence.But a close inspection of the proof of Theorem 2.14 in both the Roumieu and Beurling case shows that we still obtain the assertion of Theorem 2.14 for the sequence M.But since M ≈ M, i.e.E [M] ( ) = E [ M] ( ) and E [M] ( ; P) = E [ M] ( ; P), we have in fact shown the assertion of Theorem 2.14 for M. Therefore the conjecture is also true for Denjoy-Carleman classes.Remark 5. 7 In view of the proof of Proposition 2.5 and the argument above, it would make sense to adapt the definition of a weight sequence by replacing (1.1) by the following condition: the sequence k M k is increasing.However, (1.1) is a standard assumption for weight sequences in context of Denjoy-Carleman classes, see e.g.[25] or [34] for classes given by weight matrices.Moreover, we have needed the concept of logarithmic convexity for the proof of Proposition 2.5.

Theorem 5 . 2
Let M be a [weakly regular] weight matrix and P be an elliptic system of operator of class [M] in .Then P is [M]-hypoelliptic in .
( ) and denote by Y k ( ) the space Y ( ) equipped with the H k ( )-norm, for k ∈ N 0 .It is easy to see that the Y k ( ) are all Banach spaces.Moreover, the identity mapping from Y k+1 ( ) into Y k ( ) is clearly continuous and therefore an isomorphism.Thus all H k ( )-norms are pairwise equivalent to each other on Y ( ).