A NON-LINEAR THEORY OF INFRAHYPERFUNCTIONS ANDREAS DEBROUWERE, HANS VERNAEVE, AND JASSON VINDAS 7 Abstract. We develop a non-linear theory for infrahyperfunctions (also known as 1 quasianalytic (ultra)distributions [25]). In the hyperfunction case our work can be 0 summarizedas follows: We constructa differentialalgebrathat containsthe spaceof 2 hyperfunctionsasalineardifferentialsubspaceandinwhichthemultiplicationofreal n analytic functions coincides with their ordinary product. Moreover, by proving an a analogueofSchwartz’simpossibility resultfor hyperfunctions, we show that this em- J beddingisoptimal. OurresultsfullysolveaquestionraisedbyM.Oberguggenberger 4 [34, p. 286, Prob. 27.2]. 2 ] A F 1. Introduction . h t The non-linear theory of generalized functions was initiated by J.-F. Colombeau a m [5, 6] who constructed a differential algebra containing the space of distributions as a linear differential subspace and the space of smooth functions as a subalgebra. The [ algebrasofgeneralized functions provide aframework forlinear equations withstrongly 1 v singular data and non-linear equations [19, 20]. We refer to the monograph [34] for 6 many interesting applications to the theory of partial differential equations. 9 On the other hand, for several natural linear problems the space of distributions is 9 6 not the suitable setting, e.g. Cauchy problems for weakly hyperbolic equations – even 0 with smooth coefficients – are in general not well-posed in the space of distributions . 1 [7, 9]. Such considerations motivated the search for and study of new spaces of lin- 0 ear generalized functions like ultradistributions [23] and hyperfunctions [39, 33]. For 7 1 instance, under suitable conditions the above Cauchy problems become well-posed in : v spaces of ultradistributions [7, 8, 15], while the space of hyperfunctions is the con- Xi venient setting for the treatment of partial differential equations with real analytic coefficients [3, 21]. r a In [11] we presented a non-linear theory of non-quasianalytic ultradistributions (cf. [13, 18]). The goal of this paper is to further extend these results and develop a non-linear theory for hyperfunctions. In fact, we consider general spaces of infrahyper- functions of class M [25] for a quasianalytic weight sequence M [23] and construct p p { } a differential algebra that contains the space of infrahyperfunctions of class M as a p { } differential subspace and in which the multiplication of quasianalytic functions of class 2010 Mathematics Subject Classification. Primary 46F30. Secondary 46F15. Keywordsandphrases. Generalizedfunctions;hyperfunctions;Colombeaualgebras;multiplication of infrahyperfunctions; sheaves of infrahyperfunctions. A.Debrouweregratefullyacknowledgessupportby GhentUniversity,througha BOFPh.D.-grant. H. Vernaeve was supported by grant 1.5.138.13Nof the Research Foundation Flanders FWO. The work of J. Vindas was supported by Ghent University, through the BOF-grant01N01014. 1 2 A.DEBROUWERE,H.VERNAEVE,ANDJ. VINDAS M coincides with their pointwise product. Moreover, by establishing a Schwartz p { } impossibility type result for infrahyperfunctions, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. The case M = p! corresponds to the hyperfunction case, thereby fully p settling a question of M. Oberguggenberger [34, p. 286, Prob. 27.2]. We also believe that our work puts the notion of ‘very weak solution’ introduced by C. Garetto and M. Ruzhansky in their study on well-posedness of weakly hyperbolic equations with time dependent nonregular coefficients [16] in a broader perspective and could possibly lead to a natural framework for extensions of their results. This paper is organized as follows. In the preliminary Section 2 we collect some useful properties of the spaces of quasianalytic functions and their duals (quasianalytic functionals) that will be used later on in the article. Sheaves of infrahyperfunctions are introduced in Section 3. They were first constructed by L. H¨ormander1 [25] but we give a short proof of their existence based on H¨ormander’s support theorem for quasianalytic functionals and a general method for the construction of flabby sheaves due to K. Junker and Y. Ito [27, 28]. In Section 4 we define our algebras of generalized functions, provide a null-characterization for the space of negligible sequences, and show that the totality of our algebras forms a sheaf on Rd. Due to the absence of non- trivial compactly supported quasianalytic functions the latter is much more difficult to establish than for Colombeau algebras of non-quasianalytic type. The proof is based on some of our recent results on the solvability of the Cousin problem for vector-valued quasianalytic functions [12]. We mention that the motivation for [12] was precisely this problem. In Section 5 we embed the infrahyperfunctions of class M into our p { } algebras and show that the multiplication of ultradifferentiable functions of class M p { } is preserved under the embedding we shall construct. Finally, we present a Schwartz impossibility type result for infrahyperfunctions which shows that our embeddings are optimal. 2. Spaces of quasianalytic functions and their duals In this section we fix the notation and introduce spaces of quasianalytic functions and their duals, the so called spaces of quasianalytic functionals. We also discuss the notion of support for quasianalytic functionals and state a result about the solvability of the Cousin problem for vector-valued quasianalytic functions [12] that will be used in Section 4. Set N = N 0 . Let (M ) be a weight sequence of positive real numbers and 0 ∪{ } p p∈N0 set m := M /M , p N. We shall always assume that M = 1 and that m p p p−1 0 p ∈ → ∞ as p . Furthermore, we will make use of various of the following conditions: → ∞ (M.1) M2 M M , p N, p ≤ p−1 p+1 ∈ (M.2)′ M AHp+1M , p N , for some A,H 1, p+1 p 0 ≤ ∈ ≥ (M.2) M AHp+qM M , p,q N , for some A,H 1, p+q p q 0 ≤ ∈ ≥ (M.2)∗ 2m Cm , p N, for some Q N, C > 0. p pQ ≤ ∈ ∈ 1He uses the name quasianalytic distributions in [25]; the terminology infrahyperfunctions comes from [14, 35]. NON-LINEAR THEORY OF INFRAHYPERFUNCTIONS 3 ∞ 1 (QA) = , m ∞ p p=1 X (NE) p! ChpM , p N , for some C,h > 0. p 0 ≤ ∈ We write M = M for α Nd. The associated function of M is defined as α |α| ∈ 0 p tp M(t) := sup log , t > 0, M p∈N0 p and M(0) = 0. We define M on Rd as the radial function M(x) = M( x ), x Rd. | | ∈ As usual, the relation M N between two weight sequences means that there are p p ⊂ C,h > 0 such that M ChpN , p N . The stronger relation M N means p p 0 p p ≤ ∈ ≺ that the latter inequality remains valid for every h > 0 and a suitable C = C > 0. h Condition (NE) can therefore be formulated as p! M . We refer to [23, 2] for the p ⊂ meaning of these conditions and their translation in terms of the associated function. In particular, under (M.1), the assumption (M.2) holds [23, Prop. 3.6] if and only if 2M(t) M(Ht)+logA, t 0. ≤ ≥ Suppose K is a regular compact subset of Rd, that is, intK = K. For h > 0 we write Mp,h(K) for the Banach space of all ϕ C∞(K) such that E ∈ ϕ(α)(x) ϕ := sup sup | | < . K,h k k h|α|M ∞ α∈Ndx∈K α 0 For an open set Ω in Rd, we define {Mp}(Ω) = lim lim Mp,h(K). E E K←⋐−Ωh−→→∞ The elements of {Mp}(Ω) are called ultradifferentiable functions of class M (or p E { } Roumieu type) in Ω. When M = p! the space {Mp}(Ω) coincides with the space p E (Ω) of real analytic functions in Ω. By the Denjoy-Carleman theorem, (QA) means A that there are no non-trivial compactly supported ultradifferentiable functions of class M , or equivalently, that a function ϕ {Mp}(Ω) which vanishes on an open subset p { } ∈ E Ω′ that meets every connected component of Ω, is necessarily identically zero. It should be mentioned that, under (M.1), (M.2) and (NE), a weight sequence M p satisfies (M.2)∗ [2, Thm. 14] if and only if ω = M is a Braun-Meise-Taylor type weight function [1] (as defined in [2, p. 426]). In such a case, we have {Mp}(Ω) = (Ω) as {ω} E E locally convex spaces. We write for the family of all positive real sequences (r ) with r = 1 that R j j∈N0 0 increase (not necessarily strictly) to infinity. This set is partially ordered and directed by the relation r s , which means that there is an j N such that r s for j j 0 0 j j (cid:22) ∈ ≤ all j j . Following Komatsu [24], we removed the non-quasianalyticity assumption 0 ≥ in his projective description of {Mp}(Ω) and we found in [12] an explicit system of E semi-norms generating the topology of {Mp}(Ω) also in the quasianalytic case. E 4 A.DEBROUWERE,H.VERNAEVE,ANDJ. VINDAS Lemma 2.1. ([12, Prop. 4.8]) Let M be a weight function satisfying (M.1), (M.2)′, p and (NE). A function ϕ C∞(Ω) belongs to {Mp}(Ω) if and only if ∈ E ϕ(α)(x) ϕ := sup sup | | < , k kK,rj α∈Nd0x∈K Mα j|α=|0rj ∞ for all K ⋐ Ω and r . Moreover, the topQology of {Mp}(Ω) is generated by the j ∈ R E system of semi-norms : K ⋐ Ω,r . {kkK,rj j ∈ R} Suppose that the weight sequence M satisfies (M.1), (M.2)′, (QA), and (NE). The p elements of the dual space ′{Mp}(Ω) are called quasianalytic functionals of class M p E { } (or Roumieu type) in Ω, while those of ′(Ω) are called analytic functionals in Ω. The A properties (M.1), (M.2)′, and (NE) imply that the space of entire functions is dense in {Mp}(Ω) [25, Prop 3.2]. Hence, for any Ω′ Ω and any other weight sequence N with p E ⊆ p! N M we may identify ′{Mp}(Ω′) with a subspace of ′{Np}(Ω). In particular, p p ⊂ ⊂ E E we always have ′{Mp}(Ω′) ′(Ω). E ⊆ A An ultradifferential operator of class M is an infinite order differential operator p { } P(D) = a Dα, a C, α α ∈ αX∈Nd0 (Dα = ( i∂)α) where the coefficients satisfy the estimate − CL|α| a α | | ≤ M |α| for every L > 0 and some C = C > 0. If M satisfies (M.2), then P(D) acts L p continuously on {Mp}(Ω) and hence it can be defined by duality on ′{Mp}(Ω). E E Next, we discuss the notion of support for quasianalytic functionals. For a compact K in Rd, we define the space of germs of ultradifferentiable functions on K as {Mp}[K] = lim {Mp}(Ω), E E K−→⋐Ω a (DFS)-space. Since {Mp}(Rd) = lim {Mp}[K] ∼ E E K←⋐−Rd as locally convex spaces, and {Mp}(Rd) is dense in each {Mp}[K] we have the following E E isomorphism of vector spaces ′{Mp}(Rd) = lim ′{Mp}[K]. ∼ E E K−⋐→Rd Let f ′{Mp}(Rd). A compact K ⋐ Rd is said to be a M -carrier of f if p ∈ E { } f ′{Mp}[K]. It is well known that for every f ′(Ω) there is a smallest com- pa∈ct EK ⋐ Rd among the p! -carriers of f, called ∈theAsupport of f and denoted by { } supp f. This essentially follows from the cohomology of the sheaf of germs of an- A′ alytic functions (see e.g. [33]). An elementary proof based on the properties of the Poisson transform of analytic functionals is given in [26, Sect 9.1]. For a proof based on the heat kernel method see [31]. H¨ormander noticed that a similar result holds for NON-LINEAR THEORY OF INFRAHYPERFUNCTIONS 5 quasianalytic functionals of Roumieu type. More precisely, he showed the following important result: Proposition 2.2. ([25, Cor.3.5]) Let M be a weightsequence satisfying (M.1), (M.2)′, p (QA) and (NE). For every f ′{Mp}(Ω) there is a smallest compact set among the ∈ E M -carriers of f and that set coincides with supp f. We simply denote this set by { p} A′ suppf. Finally, we introduce vector-valued quasianalytic functions and state a sufficient condition on the target space for the solvability of the Cousin problem. Let F be a locally convex space. We write {Mp}(Ω;F) for the space of all ϕ C∞(Ω;F) such that for each continuous semi-noErm q on F, K ⋐ Ω, and r it h∈olds that j ∈ R q(ϕ(α)(x)) q (ϕ) := sup sup < . K,rj α∈Nd0x∈K Mα j|α=|0rj ∞ We endow it with the locally convex topology geQnerated by the system of semi-norms q : q continuous semi-norm on F,K ⋐ Ω,r . { K,rj j ∈ R} A Fr´echet space E with a generating system of semi-norms : k N has the k {kk ∈ } property (DN) [32, p. 368] if ( m N)( k N)( j N)( τ (0,1))( C > 0) ∃ ∈ ∀ ∈ ∃ ∈ ∃ ∈ ∃ y C y 1−τ y τ, y E. k kk ≤ k km k kj ∈ We use the notation F′ for F′ endowed with the strong topology. β Proposition 2.3. ([12, Thm. 6.7]) Let M be a weight sequence satisfying (M.1), p (M.2)′, (QA), and (NE). Let Ω Rd be open, let = Ω : i I be an open i ⊆ M { ∈ } coveringof Ω, andletF be a(DFS)-spacesuch thatF′ hasthe property(DN). Suppose β ϕ {Mp}(Ω Ω ;F), i,j I, are given F-valued functions such that i,j i j ∈ E ∩ ∈ ϕ +ϕ +ϕ = 0 on Ω Ω Ω , i,j j,k k,i i j k ∩ ∩ for all i,j,k I.Then, there are ϕ {Mp}(Ω ;F), i I, such that i i ∈ ∈ E ∈ ϕ = ϕ ϕ on Ω Ω , i,j i j i j − ∩ for all i,j I. ∈ Naturally, rephrased in the language of cohomology [33] Proposition 2.3 says that the first cohomology group of the open covering with coefficients in the sheaf of F- M valued quasianalytic functions {Mp}( ;F) vanishes, that is, H1( , {Mp}( ;F)) = 0. E · M E · 3. Sheaves of infrahyperfunctions In [25] H¨ormander constructed a flabby sheaf {Mp} with the property that its set B of sections with support in K, K ⋐ Rd, coincides with the space of quasianalytic functionals of class M supported in K. In this section, we give a short proof of p { } H¨ormander’s result and discuss some of the basic properties of the sheaf {Mp}. B 6 A.DEBROUWERE,H.VERNAEVE,ANDJ. VINDAS Let X be a topological space and let be a sheaf on X. For U X open and F ⊆ A U we write Γ (U, ) for the set of sections over U with support in A. Define A ⊆ F Γ (U, ) = Γ (U, ). c K F F K⋐U [ Proposition 3.1. ([25, Sect. 6]) Let M be a weight sequence satisfying (M.1), (M.2)′, p (QA), and (NE). Then, there exists an (up to sheaf isomorphism) unique flabby sheaf {Mp} over Rd such that B Γ (Rd, {Mp}) = ′{Mp}[K], K ⋐ Rd. K B E Moreover, for every relatively compact open subset Ω of Rd, one has {Mp}(Ω) = ′{Mp}[Ω]/ ′{Mp}[∂Ω]. B E E We call {Mp} the sheaf of infrahyperfunctions of class M (or Roumieu type). p B { } For M = p! this is exactly the sheaf of hyperfunctions . Our proof of Proposition p B 3.1 relies on the following general method for the construction of flabby sheaves with prescribed compactly supported sections, due to Junker and Ito [27, 28] (they used it to construct sheaves of vector-valued (Fourier) hyperfunctions). The idea goes back to Martineau’s duality approach to hyperfunctions [30, 40]. Lemma 3.2. ([27, Thm. 1.2]) Let X be a second countable, locally compact topological space. Assume that for each compact K ⋐ X a Fr´echet space F is given and that K for each two compacts K ,K ⋐ X, K K , there is an injective linear continuous 1 2 1 2 ⊆ mapping ι : F F such that for K K K , ι = ι ι and K2,K1 K1 → K2 1 ⊆ 2 ⊆ 3 K3,K1 K3,K2 ◦ K2,K1 ι = id. We shall identify F with its image under the mapping ι . Suppose K1,K1 K1 K1,K2 that the following conditions are satisfied: (FS1) If K ,K ⋐ X, K K , have the property that every connected component of 1 2 1 2 ⊆ K meets K , then F is dense in F . 2 1 K1 K2 (FS2) For K ,K ⋐ X the mapping 1 2 F F F : (f ,f ) f +f K1 × K2 → K1∪K2 1 2 → 1 2 is surjective. (FS3) (i) For K ,K ⋐ X it holds that 1 2 F = F F . K1∩K2 K1 ∩ K2 (ii) Let K K ... be a decreasing sequence of compacts in X and set 1 2 ⊇ ⊇ K = K . Then, n n ∩ F = F . K Kn n∈N \ (FS4) F = 0 . ∅ { } Then, there exists an (up to sheaf isomorphism) unique flabby sheaf over X such F that Γ (X, ) = F , K ⋐ X. K K F Moreover, for every relatively compact open subset U of X it holds that (U) = F /F . F U ∂U NON-LINEAR THEORY OF INFRAHYPERFUNCTIONS 7 We shall often use the following extension principle. Lemma 3.3. ([22, Lemma 2.3, p. 226]) Let X be a second countable topological space and let and be soft sheaves on X. Let ρ : Γ (X, ) Γ (X, ) be a linear c c c F G F → G mapping such that suppρ (T) suppT, T Γ (X, ). c c ⊆ ∈ F Then, there is a unique sheaf morphism ρ : such that, for every open set U in F → G X, we have ρ (T) = ρ (T) for all T Γ (U, ). If, moreover, U c c ∈ F suppρ (T) = suppT, T Γ (X, ), c c ∈ F then ρ is injective. Proof of Proposition 3.1. The uniqueness of the sheaf {Mp} follows from Lemma 3.32. B For its existence we use Lemma 3.2. Set X = Rd, F = ′{Mp}[K], and ι = tr K Eβ K2,K1 K2,K1 with r the canonical restriction mapping K2,K1 {Mp}[K ] {Mp}[K ]. 2 1 E → E Condition(FS1)isaconsequence of(QA)while (FS3)and(FS4)aresatisfied because of Proposition 2.2. By the well known criterion for surjectivity of continuous linear mappings between Fr´echet spaces [42, Thm. 37.2], (FS2) follows from the fact that the mapping {Mp}[K K ] {Mp}[K ] {Mp}[K ] : ϕ (r (ϕ),r (ϕ)) E 1 ∪ 2 → E 1 ×E 2 → K1∪K2,K1 K1∪K2,K2 is injective and has closed range. (cid:3) Proposition 3.4. Let M be a weight sequence satisfying (M.1), (M.2)′, (QA) and p (NE). (i) Let N be another weight sequence satisfying (M.1), (M.2)′, (QA) and (NE). p If N M , then {Mp} is a subsheaf of {Np}. In particular, {Mp} is always p p ⊂ B B B a subsheaf of the sheaf of hyperfunctions . B (ii) The sheaf of distributions ′ is a subsheaf of {Mp} and the following diagram D B commutes ′ D B {Mp} B (iii) For every ultradifferential operator P(D) of class M there is a unique sheaf p { } morphism P(D) : {Mp} {Mp} that coincides on Γ (Rd, {Mp}) = ′{Mp}(Rd) c B → B B E with the usual action of P(D) on quasianalytic functionals of class M . p { } Proof. In view of Lemma 3.3, (i) follows from Proposition 2.2, (ii) from the fact that the distributional and hyperfunctional support of a distribution coincide, and (iii) holds because suppP(D)f suppf, f ′{Mp}(Rd). ⊆ ∈ E 2A flabby sheaf on a paracompactspace is soft. 8 A.DEBROUWERE,H.VERNAEVE,ANDJ. VINDAS (cid:3) 4. Algebras of generalized functions of class M p { } We now introduce differential algebras {Mp}(Ω) of generalized functions of class G M as quotients of algebras consisting of sequences of ultradifferentiable functions of p { } class M and satisfying an appropriate growth condition. Furthermore, we provide p { } a null characterization of the negligible sequences and discuss the sheaf-theoretic prop- erties of the functor Ω {Mp}(Ω). Unless otherwise stated, throughout this section → G M stands for a weight sequence satisfying (M.1), (M.2), (QA), and (NE). p We define the space of M -moderate sequences as p { } {Mp}(Ω) = (f ) {Mp}(Ω)N :( K ⋐ Ω)( λ > 0)( h > 0) EM { n n ∈ E ∀ ∀ ∃ sup f e−M(λn) < , n K,h k k ∞} n∈N and the space of M -negligible sequences as p { } {Mp}(Ω) = (f ) {Mp}(Ω)N :( K ⋐ Ω)( λ > 0)( h > 0) EN { n n ∈ E ∀ ∃ ∃ sup f eM(λn) < . n K,h k k ∞} n∈N Notice that {Mp}(Ω) is an algebra under pointwise multiplication of sequences, as EM follows from (M.1) and (M.2), and that {Mp}(Ω) is an ideal of {Mp}(Ω). Hence, we EN EM can define the algebra {Mp}(Ω) of generalized functions of class M as the factor p G { } algebra {Mp}(Ω) = {Mp}(Ω)/ {Mp}(Ω). G EM EN We denote by [(f ) ] the equivalence class of (f ) {Mp}(Ω). Note that {Mp}(Ω) n n n n ∈ EM E can be regarded as a subalgebra of {Mp}(Ω) via the constant embedding G (4.1) σ (f) := [(f) ], f {Mp}(Ω). Ω n ∈ E We also remark that {Mp}(Ω) can be endowed with a canonical action of ultradifferen- G tial operators P(D) of class M . In fact, since P(D) acts continuously on {Mp}(Ω), p { } E we have that {Mp}(Ω) and {Mp}(Ω) are closed under P(D) if we define its action EM EN as P(D)((f ) ) := (P(D)f ) . Consequently, every ultradifferential operator P(D) of n n n n class M induces a well defined linear operator p { } P(D) : {Mp}(Ω) {Mp}(Ω). G → G We now provide a null characterization of the ideal {Mp}(Ω). EN Lemma 4.1. Let (f ) {Mp}(Ω). Then, (f ) {Mp}(Ω) if and only if for every K ⋐ Ω there is λ > 0n snuc∈hEtMhat n n ∈ EN supsup f (x) eM(λn) < . n | | ∞ n∈Nx∈K NON-LINEAR THEORY OF INFRAHYPERFUNCTIONS 9 Proof. The proof is based on the following multivariable version of Gorny’s inequality: Let K and K′ be regular compact subsets of Rd such that K′ ⋐ K. Set d(K′,Kc) = δ > 0. Then, m k max f(α) 4e2k f 1−k/m |α|=kk kL∞(K′) ≤ k k kL∞(K)× (cid:16) (cid:17) f m! k/m (4.2) max dm max f(α) , k kL∞(K) |α|=mk kL∞(K) δm (cid:18) (cid:26) (cid:27)(cid:19) for all f Cm(K) and 0 < k < m. We prove (4.2) below, but let us assume it for the ∈ moment and show how the result follows from it. Suppose (f ) satisfies the 0-th order estimate. Let K′ ⋐ Ω be an arbitrary regular n n compact set. Choose a regular compact K such that K′ ⋐ K ⋐ Ω. For every λ > 0 1 there are h ,C > 0 such that 1 1 f(α) C h|α|M eM(λ1n), α Nd,n N, k n kL∞(K) ≤ 1 1 α ∈ 0 ∈ and, there are λ ,C > 0 such that 2 2 f C e−M(λ2n), n N. n L∞(K) 2 k k ≤ ∈ Let β Nd, β = 0. Applying (4.2) with k = β and m = 2 β , we obtain ∈ 0 6 | | | | f(β) k n kL∞(K′) f (2 β )! 1/2 4(2e2)|β| f 1/2 max d2|β| max f(α) , k nkL∞(K) | | . ≤ k nkL∞(K) |α|=2|β|k n kL∞(K) δ2|β| (cid:18) (cid:26) (cid:27)(cid:19) Combining this with the above inequalities and taking λ := λ /H, one finds by (M.2) 1 2 and (NE) that there are h,C > 0 such that f(β) Ch|β|M e−M(λ2n/H2), β Nd,n N. k n kL∞(K′) ≤ β ∈ 0 ∈ Thus (f ) {Mp}(Ω). n n ∈ EN We now show (4.2). The one-dimensional Gorny inequality [17, p. 324] states that m k g(k) 4e2k g 1−k/m k kL∞([a,b]) ≤ k k kL∞([a,b])× (cid:16) (cid:17) g 2mm! k/m (4.3) max g(m) , k kL∞([a,b]) , k kL∞([a,b]) (b a)m (cid:18) (cid:26) − (cid:27)(cid:19) for all g Cm([a,b]) and 0 < k < m. Denote by ∂/∂ξ the directional derivative in the ∈ direction ξ, where ξ Rd is a unit vector. It is shown in [4, Thm. 2.2] that ∈ ∂kf (4.4) max f(α) sup |α|=kk kL∞(K′) ≤ |ξ|=1(cid:13)∂kξ(cid:13)L∞(K′) (cid:13) (cid:13) for all f Ck(K′), k N – it is for this inequalit(cid:13)y that(cid:13)we need the compact K′ to be (cid:13) (cid:13) ∈ ∈ regular. We write l(x,ξ) for the line in Rd with direction ξ passing through the point x K′. Define g (t) = f(x+tξ) for t t R : x+tξ K . The latter set always x,ξ ∈ ∈ { ∈ ∈ } 10 A.DEBROUWERE,H.VERNAEVE,ANDJ. VINDAS contains a compact interval I 0 of length at least 2δ. Inequality (4.3) therefore x,ξ ∋ implies that ∂kf (k) (k) = sup g (0) sup g ∂kξ | x,ξ | ≤ k x,ξkL∞(Ix,ξ) (cid:13) (cid:13)L∞(K′) x∈K′ x∈K′ (cid:13) (cid:13) (cid:13)(cid:13) (cid:13)(cid:13) ≤ xs∈uKp′4e2k mk kkgx,ξk1L−∞k(/Imx,ξ)× (cid:16) (cid:17) g m! k/m max g(m) , k x,ξkL∞(Ix,ξ) k x,ξ kL∞(Ix,ξ) δm (cid:18) (cid:26) (cid:27)(cid:19) k/m m k ∂mf f m! 4e2k f 1−k/m max , k kL∞(K) . ≤ k k kL∞(K) ∂mξ δm ((cid:13) (cid:13)L∞(K) )! (cid:16) (cid:17) (cid:13) (cid:13) (cid:13) (cid:13) The result now follows from (4.4) and the fact that (cid:13) (cid:13) ∂mf d d ∂mf(x) = sup ξ ξ dm sup f(α) . ∂mξ ··· ∂x ∂x j1··· jm ≤ k kL∞(K) (cid:13)(cid:13) (cid:13)(cid:13)L∞(K) x∈K(cid:12)(cid:12)jX1=1 jXm=1 j1 ··· jm (cid:12)(cid:12) |α|=m (cid:12) (cid:12) (cid:13) (cid:13) (cid:12) (cid:12) (cid:3) (cid:13) (cid:13) (cid:12) (cid:12) Next, we discuss the sheaf properties of {Mp}(Ω). Given an open subset Ω′ of Ω G and f = [(f ) ] {Mp}(Ω), the restriction of f to Ω′ is defined as n n ∈ G f = [(f ) ] {Mp}(Ω′). |Ω′ n|Ω′ n ∈ G Clearly, the assignment Ω {Mp}(Ω) is a presheaf on Rd. Our aim in the rest of this → G section is to show that it is in fact a sheaf. The idea of the proof comes from the theory of hyperfunctions in one dimension: the fact that the hyperfunctions are a sheaf on R is a direct consequence of the Mittag-Leffler Lemma (= solution of the Cousin problem in one dimension) [33]. Similarly, the solvability of the Cousin problem for the spaces {Mp}(Ω) would imply that {Mp} is a sheaf on Rd. To show the latter, we identify EN G the space {Mp}(Ω) with a space of vector-valued ultradifferentiable functions of class EN M with values in an appropriate sequence space and then use Proposition 2.3. We p { } need some preparation: For λ > 0 we define the following Banach space sMp,λ = (c ) CN : σ ((c ) ) := sup c eM(λn) < . n n λ n n n { ∈ | | ∞} n∈N Set s{Mp} = lim sMp,λ, λ−→→0+ a (DFS)-space. Given a sequence r we denote by M the associated function of j ∈ R rj the sequence M p r . In [10, Cor. 3.1] we have shown that a sequence (c ) CN p j=0 j n n ∈ belongs to s{Mp} if and only if Q σrj((cn)n) := sup|cn|eMrj(n) < ∞ n∈N