The Metivier inequality and ultradifferentiable hypoellipticity

In 1980 M{\'e}tivier characterized the analytic (and Gevrey) hypoellipticity of $L^2$-solvable partial linear differential operators by a-priori estimates. In this note we extend this characterization to ultradifferentiable hypoellipticity with respect to Denjoy-Carleman classes given by suitable weight sequences. We also discuss the case when the solutions can be taken as hyperfunctions and present some applications.


Introduction
In this work we study the regularity of solutions of linear partial differential equations within the framework of Denjoy-Carleman classes defined by appropriate weight sequences.In fact, our principal concern here is to what extend a result due to G. Métivier (cf.[8]), proved in the study of analytic and Gevrey regularity, could still be valid in this more general set up.
More precisely, in [8] a characterization of analytic hypoellipticity is presented for L 2solvable linear partial differential equations in terms of a very precise a priori inequality.The author also mentions that a similar characterization for Gevrey hypoellipticity is also valid, and in [1] the result is extended for pseudodifferential operators.
In the present work we are able to extend this Métivier result for what we call here admissible weight sequences (see Definition 1 below).The corresponding Denjoy-Carleman classes contain the Gevrey classes of order s ≥ 1 properly.It is important to note that it is irrelevant in our presentation if the classes are either quasi-analytic or non quasi-analytic.
One of the main points in Métivier's argument is to give a positive answer to a question related to the concept of solvability: assume that P = P (x, D) is a real analytic, linear partial differential operator in an open set Ω ⊂ R n , which is L 2 -solvable and Gevrey hypoelliptic of some order s ≥ 1.Take f ∈ G s (Ω), an open set U ⋐ Ω and let u ∈ L 2 (U) solve P u = f in U with minimum L 2 -norm.Automatically u ∈ G s (U).Is it possible to bound the Sobolev norms of u| V , where V ⋐ U is another open set, in terms of the Gevrey norms of f | U ?
This is what is achieved by Métivier when s = 1.The extension to the Gevrey case of arbitrary order is straightforward.Indeed, Métivier's argument in the analytic case is based on an interpolation method for which it is needed the choice of suitable subsequences of (j!) 1/j ∼ j (in this case the subsequences are explicitly described).The main difference in the Gevrey case is that now it is needed subsequences of (j! s ) 1/j ∼ j s , which can be obtained from the ones of the real analytic case after applying the uniform deformation j → j s , cf. [1].
This deformation argument is no longer possible in the Denjoy-Carleman case.The situation is now much more delicate, and the determination of the class of weight sequences for which the result is valid is indeed one of the key points in our work (see Definition 1 below).Moreover the proof of our main result requires several new insights which we believe justifies its publication.
After we discuss several results on weight sequences and Denjoy-Carleman classes is Section 1, we state our main result in Section 2 (Theorem 1) and prove some consequences of it.Section 3 is devoted to the proof of Theorem 1. Finally, in Section 4, we extend our result to the hyperfunction set up and discuss the case of Hörmander's sum of squares operators.

Preliminaries on weight sequences and the corresponding Denjoy-Carleman classes
We say that a sequence of positive numbers and lim If M is a weight sequence the Denjoy-Carleman class E {M} (Ω), Ω ⊆ R n open, of ultradifferentiable functions associated to M, consists of all functions f ∈ E(Ω), for which the following holds: for every compact set K ⊆ Ω there are constants C, h > 0 such that It follows from ( 1) that E {M} (Ω) is an algebra with respect to the pointwise addition and multiplication of functions.If M = (k!) k then E {M} (Ω) = A(Ω) is the space of analytic functions on Ω, More generally, the Gevrey classes G s (Ω), s ≥ 1, are the Denjoy-Carleman classes associated to the weight sequences In order to be able to impose further properties on the classes, we need to discuss some additional conditions on the weight sequences we shall deal with.On the set of weight sequences we can establish the following relation: if M and N are two weight sequences we set The relation is both reflexive and transitive.When we also consider the equivalence relation ≈ given by M ≈ N :⇐⇒ M N ∧ N M and identify any pair M, N with M ≈ N then is also antisymmetric, i.e. is a partial order.We may write M N if M N and N M.
We finally introduce a condition taken from [4]: Condition (4) implies in particular that E {M} (Ω) is closed under differentiation and that M G s for some s > 1, see Matsumoto [7].
If M is a weight sequence then we are also going to use the following sequences We note that (1) gives that Λ k ≤ µ k for all k ∈ N and therefore by (2) it follows that (5) lim Furthermore, (3) is equivalent to the existence of δ > 0 such that From (4) it follows that there is σ > 0 such that It is easy to see that (7) is the condition (M2') of Komatsu [4] written in terms of the sequence (Λ k ) k .This condition is sufficient to guarantee that E {M} is closed under differentiation.
Definition 1.A weight sequence satisfying properties (3) and (4) will be referred to as an admissible weight sequence.
Clearly the Gevrey sequences G s are admissible weight sequences for any s ≥ 1.
A more general family of admissible weight sequences is defined as follows: Let s ≥ 1 and σ ≥ 0. The weight sequence N s,σ is given by N s,σ k = (k!) s (log(k + e)) σk .It is easy to see that N s,σ is admissible for any choice of s ≥ 1 and σ ≥ 0. Furthermore N s,0 = G s for all s ≥ 1 and we have for s ≥ 1 fixed that for all σ > 0 and every s ′ > s.
In order to present a weight sequence which is not admissible let q > 1 be a parameter.The sequence L q given by L q k = q k 2 is a weight sequence which satisfies (3) and (7).However, since G s L q for all s, q > 1, we conclude that L q cannot satisfy (4) for any q > 1.
Lemma 1. 1 Let M be an admissible weight sequence.Then there is a constant σ > 1 such that the following holds: For each k ∈ N there is a sequence (k j ) j such that Λ k 0 ≤ Λ k and Proof.We note that due to (1) the sequence (Λ m ) m is increasing and that Λ m → ∞ for m → ∞ by (2).Furthermore we can assume that σ > 1 in (7).Now fix k ∈ N. We construct the sequence k j iteratively.First set We choose k 1 to be the greatest element of T k 1 .Now assume that we have chosen k j as the greatest number in T k j = {m ∈ N : is non-empty and we choose k j+1 to be the greatest element in T k j+1 .Definition 2. Let M be a weight sequence.The weight function associated to M is given by ( 8) We recall that the associated weight function ω M is a continuous and increasing function on the positive real line, cf.[6].
Lemma 2. Let M be an admissible weight sequence.Then the associated weight functions satisfies the following estimate Proof.Since ω M is a continuous and increasing function we have that ω Hence it is enough to show that ω M (µ k ) ≤ Hk.By Mandelbrojt [6] we know the following fact: According to Matsumoto [7] condition ( 4) is equivalent to Here we can choose D = 2A where A is the constant from (4).It follows that where H = log D. Since we can assume without loss of generality that D ≥ e we have that H ≥ 1.
We need also to dwell a little bit on the functional analytic structure of Denjoy-Carleman classes, for more details see [4].We shall use the following notation: if U ⊆ R n is an open subset then B(U) denotes the space of all bounded, smooth functions on U which have all its derivatives also bounded.For each weight sequence M and all open sets U ⊆ R n we can define a norm on B(U) by The resulting Banach space is Clearly we have thus there is a continuous and injective map We can then introduce the classes of global ultradifferentiable functions on U: We observe that B {M} (U) is a (DFS)-space and thus, in particular, a webbed space [5, p. 63, (8)].Notice also that (10) Next we localize the preceding concepts.The (local) Denjoy-Carleman class associated to the weight sequence M is defined as As before we have It is easy to see that B {M} (U) and E {M} (U) are algebras with respect to the pointwise operations.We are going to occasionally refer to f to be of class Métivier [8] gave a criterion for analytic (and Gevrey) hypoellipticity at a point in the case of differential operators P with analytic coefficients in Ω which satisfy the following condition: There is a continuous operator R: L 2 (Ω) → L 2 (Ω) such that P R = Id.
Our aim is to generalize Métivier's result to {M}-hypoellipticity.In order to do so we need to introduce a weighted Sobolev norm: For an open set U ⊆ R n , a weight sequence M and k ∈ N we set Our main result is the following theorem: Theorem 1.Let M be an admissible weight sequence, P be a differential operator with ultradifferentiable coefficients of class {M} in Ω which satisfies (H).
Then P is {M}-hypoelliptic at a point x 0 ∈ Ω if and only if there is a neighborhood U 0 ⊆ Ω of x 0 such that for all open sets V ⋐ U ⊆ U 0 there are constants C, L > 0 such that for all D ′ (U) and all k ∈ Z + we have It follows immediately from the condition (11) that if a differential operator P is {M}hypoelliptic at x 0 for some admissible weight sequence M then P is smooth hypoelliptic at x 0 .Furthermore we have the following corollaries.
If P is {M}-hypoelliptic at x 0 then there is a neighborhood U 0 of x 0 such that for all V ⋐ U ⊆ U 0 the condition (29) holds and (30) is satisfied for M and for some constants C 2 , L independent of k and u ∈ D ′ (U).Hence As a special case we obtain Corollary 2. Let P be a differential operator with analytic coefficients in Ω ⊆ R n satisfying (H).If P is analytic hypoelliptic at some point x 0 ∈ Ω then P is {M}-hypoelliptic at x 0 for all admissible weight sequences M.

Proof of Theorem 1
We start by introducing the Ehrenpreis-Hörmander cut-off functions: For every open sets V ⋐ U ⊆ R n there exists a sequence We will call such a sequence an Ehrenpreis-Hörmander cut-off sequence which is supported in U and centered in V .Ehrenpreis-Hörmander cut-off sequences have been heavily used in local and microlocal regularity theory in the analytic and ultradifferentiable category.
Proof.Note first if u ∈ H k (U) then χ k u can be extended to an element of H k (R n ) by setting 0 outside U. We note also that (6) gives that there is some δ > 0 such that k ≤ δΛ k .The Leibniz rule gives Thus we obtain It follows that and we have proven the Lemma since there is a constant γ > 0 such that Proposition 1.Let M be an admissible weight sequence, Ω be a neighborhood of x 0 and let E be a Banach space continuously injected in L 2 (Ω).If we suppose that there is an open set U 0 ⊆ Ω such that u| U 0 ∈ E {M} (U 0 ) for all u ∈ E then for any V ⋐ U ⋐ U 0 and all Ehrenpreis-Hörmander cut-off sequence (χ k ) k supported in U and centered on V there exist constants C, γ > 0 such that for all k ∈ N and u ∈ E.
Proof.We have that the restriction map | for a constant only depending on U. On the other hand, f j L 2 (Ω) → 0 since E is continuously injected in L 2 (Ω).It follows that g = 0 and thus the graph of T U is closed.By the version of the closed graph theorem given in [5, p.56, (1)] we have that T U is continuous.If we denote the unit ball in E by B 1 then we deduce that T U (B 1 ) is bounded and thence there is some h > 0 such that T U (B 1 ) ⊆ B M,h (U) which gives that T U (B) ⊆ B M,h (U).The closed graph theorem for Banach spaces implies now that T U is continuous from E to B M,h (U).We denote the norm of this map by C T and obtain for |α| = k that since Λ k is increasing and by (6) there is some δ > 0 such that k ≤ δΛ k .It follows that where If M is a weight sequence it is easy to see that for every u ∈ H k (R n ) and k ∈ Z + .We are also going to use the space Then G {M} ⊆ E {M} (R n ) is a Hilbert space with respect to the topology inherited by L 2 (R n ).
Lemma 4. Let M be an admissible weight sequence and k ∈ N. Then every u ∈ H k (R n ) can be written in the form u ∞ j=0 u j with the u j ∈ G {M} satisfying: where the constants C, γ > 0 are independent of k, j and (k j ) j is the sequence from Lemma 1.
Proof.We may set k −1 = 0 and For |ξ| ≤ Λ k j we conclude that On the other hand, in the case Λ k j−1 ≤ |ξ| we note first that for j ≥ 2 we have Hence, due to (15), for some γ > 0.
Lemma 5. Assume that M is an admissible weight sequence and that E is a Banach space which is continuously injected in L 2 (Ω).If there is an open set U 0 ⊆ Ω such that u| U 0 ∈ E {M} (U 0 ) for every u ∈ E then for all V ⋐ U 0 there exists a constant C such that for all k ∈ N and every sequence Proof.For k = 1 the condition (17) implies that u converges absolutely in L 2 (Ω).By Proposition 1 we have that for all open sets V ⋐ U ⋐ Ω and for every Ehrenpreis-Hörmander cut-off sequence χ k ∈ D(U) centered in V there are constants C 0 , γ > 0 such that for all k ∈ Z + and all u ∈ B: We introduce the following functions: By (18) we have that It is clear that the function v = ∞ j=0 χ k j u j coincides with u on V .It suffices to show that v ∈ H k (R n ) satisfies the estimate: and thus it will be enough to show that holds, where C > 0 is independent of k.We write and conclude that where Hence (21) (and thus the Lemma) is proven by ( 19) and the estimate where C is some constant independent of k.In order to establish (22) we set If e 2H γΛ k j ≤ |ξ|, where H is the constant from (9), then we have that by Lemma 2. This gives On the other hand, if e 2H γΛ k j ≥ |ξ| then we estimate If we set j 0 = min{j ∈ Z + | γe 2H Λ k j ≥ |ξ|} for a fixed ξ then we have that and conclude that If P is a differential operator with smooth coefficients then we set for an open set U. Clearly P 0 (U) is a closed subspace of L 2 (U).
Proposition 2. Let Ω ⊆ R n be open, M be an admissible weight sequence.Furthermore assume that P is a linear differential operator with E {M} (Ω)-coefficients.
If P is {M}-hypoelliptic at some point x 0 ∈ Ω then there is a neighborhood U 0 of x 0 such that for all open sets V ⋐ U ⊆ U 0 there are constants C, h > 0 such that for all k ∈ Z + and all u ∈ L 2 (U) with P u = 0 we have Proof.If V ⋐ U are open sets and h > 0 then we set We define where h 1 = 2(1 + h 2 ).Thus we obtain that there is a continuous embedding and consequently Since P is {M}-hypoelliptic at x 0 , we know that there is a neighborhood U 0 ⊆ Ω of x 0 such that u| V ∈ B {M} (V ) for all u ∈ P 0 (U) and all pairs V ⋐ U ⊆ U 0 .Then similarly to the proof of Proposition 1 we observe that the graph of the restriction map is closed.Hence the Closed Graph Theorem of De Wilde implies that T V is continuous and therefore the map V ) is continuous.Again as before in the proof of Proposition 1 we thus can conclude that for all V ⋐ U there exists h > 0 such that the map u → u| V is continuous from P 0 (U) to H {M} (V ) which is equivalent to the existence of some constant C > 0 such that Proof of Theorem 1.If we assume that u ∈ D ′ (U), U being an open subset of U 0 , is such that P u ∈ E {M} (U) then (1) implies that u| V ∈ E(V ) for any V ⋐ U.In particular, u ∈ H k (V ) for all k ∈ N 0 .We observe also that we have for a weight sequence M that On the other hand assume now that P is a differential operator which is {M}-hypoelliptic in Ω and satisfies (H), i.e. there is a continuous map R : L 2 (Ω) → L 2 (Ω) such that P R = Id.Furthermore it is easy to see that (28) M k P u L 2 (U ) ≤ |||P u||| U,M,k .
We are now able to finish the proof: by ( 24), ( 25) and (26).Applying ( 27) and (28) we see that Hence we have shown that there are constants C, h > 0 such that for all k ∈ Z + and every u ∈ D ′ (U) with P u ∈ H k (U) the following estimate holds:

The hyperfunction case -Final remarks
Denote by B the sheaf of (germs of) hyperfunctions in R n .We strength the definition of hypoellipticity in the following way: Furthermore P is {M}-hypoelliptic at x 0 ∈ U in the hyperfunction sense if there is a neighborhood U 0 of x 0 such that P is {M}-hypoelliptic in U 0 in the hyperfuntion sense.
A close look at the proof of Theorem 1 leads to the following result: Theorem 2. Let M be an admissible weight sequence and P be a differential operator with real analytic coefficients in Ω satisfying (H).
Then P is {M}-hypoelliptic at a point x 0 ∈ Ω in the hyperfunction sense if and only if there is a neighborhood U 0 ⊆ Ω of x 0 such that for all open sets V ⋐ U ⊆ U 0 there are constants C, L > 0 such that for all B(U) and all k ∈ Z + we have An operator P = P (x, D) in an open set W ⊆ R n will be said to belong to the Hörmander class H(W ) if it can be written in the form

2 .
The concept of {M}-hypoellipticity.Statement of the main results.Definition 3. Let U ⊆ R n be an open set and M be a weight sequence.If P

Corollary 1 .
Let M and M ′ be two admissible weight sequences such that M M ′ .Moreover, let Ω ⊆ R n be an open set and P be a differential operator with E {M} (Ω)coefficients of class {M} such that (H) holds.If P is {M}-hypoelliptic at x 0 ∈ Ω then P is {M ′ }-hypoelliptic at x 0 .Proof of Corollary 1.If we set Λ

Lemma 3 .
Let (χ k ) k be an Ehrenpreis-Hörmander cut-off sequence supported in an open set U ⊆ R n .If M is a weight sequence satisfying (3) then there is a constant γ > 0 such that for all k ∈ N and u ∈ H k (U) we have

Definition 4 .
Let U ⊆ R n be an open set and M be a weight sequence.If P = P (x, D) is a linear differential operator with real-analytic coefficients then P is {M}-hypoelliptic in U in the hyperfunction sense if given u ∈ B(U) the following holds: if V ⊆ U is open and if P u| V ∈ E {M} (V ) then u| V ∈ E {M} (V ).