A Non-Linear Theory of Infrahyperfunctions

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Mp coincides with their pointwise product. Moreover, by establishing a Schwartz {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 Mp = p! corresponds to the hyperfunction case, thereby fully settling a question of M. Oberguggenberger [34, p. 286, Prob. 27.2]. We mention the pioneer article [37] by S. Pilipović, where a so-called algebra of megafunctions in dimension 1 was introduced. 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. Hörmander1 [25] but we give a short proof of their existence based on Hörmander’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 nontrivial 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 algebras and show { p} that the multiplication of ultradifferentiable functions of class Mp 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 mp := Mp/Mp−1, p N. We shall always assume that M0 = 1 and that mp as p . Furthermore,∈ we will make use of various of the following conditions:→ ∞ →∞2 (M.1) Mp Mp−1Mp+1, p N, ′ ≤ p+1 ∈ (M.2) Mp+1 AH Mp, p N0, for some A, H 1, (M.2) M ≤ AHp+qM M ,∈p, q N , for some A,≥ H 1, p+q ≤ p q ∈ 0 ≥ 1He uses the name quasianalytic distributions in [25]; the terminology infrahyperfunctions comes from [14, 35]. NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 3

∗ (M.2) 2mp CmpQ, p N, for some Q N, C > 0, ∞ ≤ ∈ ∈ 1 (QA) = , m ∞ p=1 p (NE) Xp! 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, p∈N0 Mp and M(0) = 0. We define M on Rd as the radial function M(x) = M( x ), x Rd. | | ∈ As usual, the relation Mp Np between two weight sequences means that there are 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 = Ch > 0. Condition (NE) can therefore be formulated as p! Mp. We refer to [23, 1] for the 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) + log A, t 0. ≤ ≥ Suppose K is a regular compact subset of Rd, that is, int K = 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 K,h := sup sup | |α| | < . k k d h M ∞ α∈N0 x∈K α For an open set Ω in Rd, we define {Mp}(Ω) = lim lim Mp,h(K). E E K←−⋐Ω h−→→∞

{Mp} The elements of (Ω) are called ultradifferentiable functions of class Mp (or E {Mp} { } Roumieu type) in Ω. When Mp = p! the space (Ω) coincides with the space (Ω) of real analytic functions in Ω. By the Denjoy-CarlemanE theorem, (QA) meansA that there are no nontrivial compactly supported ultradifferentiable functions of class M , { p} or equivalently, that a function ϕ {Mp}(Ω) which vanishes on an open subset that meets every connected component∈ of E Ω, is necessarily identically zero. It should be mentioned that, under (M.1), (M.2) and (NE), a weight sequence Mp satisfies (M.2)∗ [1, Thm. 14] if and only if ω = M is a Braun-Meise-Taylor type weight {Mp} function [3] (as defined in [1, p. 426]). In such a case, we have (Ω) = {ω}(Ω) as locally convex spaces. E E

We write for the family of all positive real sequences (rj)j∈N0 with r0 = 1 that increase (notR necessarily strictly) to inﬁnity. This set is partially ordered and directed by the relation rj sj, which means that there is a j0 N0 such that rj sj for all j j . Following Komatsu [24], we removed the non-quasianalyticity∈ assump≤ tion ≥ 0 in his projective description of {Mp}(Ω) and we found in [12] an explicit system of E seminorms 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 Mp be a weight function satisfying (M.1), (M.2) , and (NE). A function ϕ C∞(Ω) belongs to {Mp}(Ω) if and only if ∈ E ϕ(α)(x) ϕ K,rj := sup sup | |α| | < , k k d x∈K ∞ α∈N0 Mα j=0 rj

{Mp} for all K ⋐ Ω and rj . Moreover, the topologyQ of (Ω) is generated by the system of seminorms ∈ R : K ⋐ Ω,r . E {kkK,rj j ∈ R} ′ Suppose that the weight sequence Mp satisfies (M.1), (M.2) ,(QA), and (NE). The ′{Mp} elements of the dual space (Ω) are called quasianalytic functionals of class Mp (or Roumieu type) in Ω, whileE those of ′(Ω) are called analytic functionals in Ω.{ The} properties (M.1), (M.2)′, and (NE) implyA 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 E ⊆ p 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). An ultradifferentialE operator⊆A of class M is an infinite order differential operator { p} α P (D)= aαD , aα C, d ∈ αX∈N0 (Dα =( i∂)α) where the coefficients satisfy the estimate − CL|α| aα | | ≤ M|α| for every L > 0 and some C = CL > 0. If Mp satisfies (M.2), then P (D) acts continuously on {Mp}(Ω) and hence it can be defined by duality on ′{Mp}(Ω). Next, we discussE the notion of support for quasianalytic functionals.E 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 isomorphism of locally convexE spaces E ′{Mp}(Rd) = lim ′{Mp}[K]. E ∼ E K−→⋐Rd Let f ′{Mp}(Rd). A compact K ⋐ Rd is said to be an M -carrier of f if ∈ E { p} f ′{Mp}[K]. It is well known that for every f ′(Ω) there is a smallest compact∈ EK ⋐ Rd among the p! -carriers of f, called∈ the Asupport of f and denoted by { } suppA′ f. This essentially follows from the cohomology of the sheaf of germs of analytic 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]. Hörmander noticed that a similar result holds for NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 5 quasianalytic functionals of Roumieu type. More precisely, he showed the following important result:

′ Proposition 2.2. ([25, Cor. 3.5]) Let Mp be a weight sequence satisfying (M.1), (M.2) , (QA) and (NE). For every f ′{Mp}(Ω) there is a smallest compact set among the ∈ E Mp -carriers of f and that set coincides with suppA′ f. We simply denote this set by supp{ }f. 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 seminormE q on F , K ⋐ Ω, and r it holds∈ that j ∈ R q(ϕ(α)(x)) ϕ qK,rj ( ) := sup sup |α| < . d x∈K ∞ α∈N0 Mα j=0 rj We endow it with the locally convex topology generatedQ by the system of seminorms q : q continuous seminorm on F,K ⋐ Ω,r . { K,rj j ∈ R} A Fr´echet space E with a generating system of seminorms k : k N has the property (DN) [32, p. 368] if {kk ∈ } ( 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 Fβ for F endowed with the strong topology.

Proposition 2.3. ([12, Thm. 6.7]) Let Mp be a weight sequence satisfying (M.1), ′ d (M.2) , (QA), and (NE). Let Ω R be open, let = Ωi : i I be an open ⊆ ′ M { ∈ } covering of Ω, and let F be a (DFS)-space such that Fβ has the property (DN). Suppose ϕ {Mp}(Ω Ω ; F ), i, j I, are given F -valued functions such that i,j ∈E i ∩ j ∈ 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 ∈E i ∈ ϕ = ϕ ϕ on Ω Ω , i,j i − j i ∩ j for all i, j I. ∈ Naturally, rephrased in the language of cohomology groups [33] Proposition 2.3 says that the ﬁrst cohomology group of the open covering with coeﬃcients in the sheaf of F -valued quasianalytic functions {Mp}( ; F ) vanishes,M that is, H1( , {Mp}( ; F )) = 0. E · M E ·

3. Sheaves of infrahyperfunctions In [25] Hörmander constructed a flabby sheaf {Mp} with the property that its set of sections with support in K, K ⋐ Rd, coincidesB with the space of quasianalytic functionals of class M supported in K. In this section, we give a short proof of { p} Hörmander’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 A U we write Γ (U, ) for the set of sectionsF over U with support in⊆A. Define ⊆ A F Γc(U, )= ΓK(U, ). F ⋐ F K[U ′ Proposition 3.1. ([25, Sect. 6]) Let Mp be a weight sequence satisfying (M.1), (M.2) , (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). B { p} For Mp = p! this is exactly the sheaf of hyperfunctions . Our proof of Proposition 3.1 relies on the following general method for the constructionB 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 Fréchet space FK is given and that for each two compacts K1,K2 ⋐ X, K1 K2, there is an injective linear continuous 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 ιK1,K1 = id. We shall identify FK1 with its image under the mapping ιK1,K2 . Suppose 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 K2 meets K1, then FK1 is dense in FK2 . (FS2) For K1,K2 ⋐ X the mapping F F F :(f , f ) f + f K1 × K2 → K1∪K2 1 2 → 1 2 is surjective. (FS3) (i) For K1,K2 ⋐ X it holds that F = F F . K1∩K2 K1 ∩ K2 (ii) Let K1 K2 ... be a decreasing sequence of compacts in X and set K = K .⊇ Then, ⊇ ∩n n FK = FKn . N n\∈ (FS4) F = 0 . ∅ { } Then, there exists an (up to sheaf isomorphism) unique flabby sheaf over X such that F Γ (X, )= F , K ⋐ X. K F K Moreover, for every relatively compact open subset U of X, one has (U)= F /F . F U ∂U NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 7

d ′{Mp} Proof of Proposition 3.1. We use Lemma 3.2. Set X = R , FK = β [K], and t E ιK2,K1 = rK2,K1 with rK2,K1 the canonical restriction mapping {Mp}[K ] {Mp}[K ]. E 2 →E 1 Condition (FS1) is a consequence of (QA) while (FS3) and (FS4) are satisﬁed because of Proposition 2.2. By the well known criterion for surjectivity of continuous linear mappings between Fr´echet spaces [43, 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. 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 ρc : Γc(X, ) Γc(X, ) be a linear mappingF such thatG F → G supp ρ (T ) supp T, T Γ (X, ). c ⊆ ∈ c F Then, there is a unique sheaf morphism ρ : such that, for every open set U in X, we have ρ (T )= ρ (T ) for all T Γ (U,F →). If, G moreover, U c ∈ c F supp ρ (T ) = supp T, T Γ (X, ), c ∈ c F then ρ is injective.

′ Proposition 3.4. Let Mp be a weight sequence satisfying (M.1), (M.2) , (QA) and (NE). ′ (i) Let Np be another weight sequence satisfying (M.1), (M.2) , (QA) and (NE). {Mp} {Np} {Mp} If Np Mp, then is a subsheaf of . In particular, is always a subsheaf⊂ of the sheafB of hyperfunctions B. B (ii) The sheaf of distributions ′ is a subsheafB of {Mp} and the following diagram commutes D B ′ D B

{Mp} B (iii) Suppose that, in addition, Mp satisﬁes (M.2). Then, for every ultradiﬀerential {Mp} operator P (D) of class Mp there is a unique sheaf morphism P (D): {Mp} { } d {Mp} ′{Mp} d B → that coincides on Γc(R , ) = (R ) with the usual action of BP (D) on quasianalytic functionalsB of classE 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 supp P (D)f supp f, f ′{Mp}(Rd). ⊆ ∈E 8 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS

4. Algebras of generalized functions of class M { p} We now introduce differential algebras {Mp}(Ω) of generalized functions of class M as quotients of algebras consisting of sequencesG of ultradifferentiable functions of { p} class Mp satisfying an appropriate growth condition. Furthermore, we provide a null characterization{ } of the negligible sequences and discuss the sheaf-theoretic properties {Mp} of the functor Ω (Ω). Unless otherwise stated, throughout this section Mp stands for a weight→ sequence G satisfying (M.1), (M.2), (QA), and (NE). We define the space of M -moderate sequences as { p} N {Mp}(Ω) = (f ) {Mp}(Ω) :( K ⋐ Ω)( λ> 0)( h> 0) EM { n n ∈E ∀ ∀ ∃ −M(λn) sup fn K,he < , n∈N k k ∞} and the space of M -negligible sequences as { p} N {Mp}(Ω) = (f ) {Mp}(Ω) :( K ⋐ Ω)( λ> 0)( h> 0) EN { n n ∈E ∀ ∃ ∃ M(λn) sup fn K,he < . n∈N k k ∞}

{Mp} Notice that M (Ω) is an algebra under pointwise multiplication of sequences, as E {Mp} {Mp} follows from (M.1) and (M.2), and that N (Ω) is an ideal of M (Ω). Hence, we {Mp} E E can define the algebra (Ω) of generalized functions of class Mp as the factor algebra G { } {Mp}(Ω) = {Mp}(Ω)/ {Mp}(Ω). G EM EN {M } We denote by [(f ) ] the equivalence class of (f ) p (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)((fn)n) := (P (D)fn)n. Consequently, every ultradifferential operator P (D) of 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 {Mp} {Mp} Lemma 4.1. Let (fn)n M (Ω). Then, (fn)n N (Ω) if and only if for every K ⋐ Ω there is λ> 0 such∈ E that ∈ E M(λn) sup sup fn(x) e < . n∈N x∈K | | ∞ 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) = NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 9

δ > 0. Then,

k (α) 2k m 1−k/m max f L∞(K′) 4e f L∞(K) |α|=k k k ≤ k k k × k/m m (α) f L∞(K)m! (4.2) max d max f L∞(K), k k |α|=m k k δm for all f Cm(K) and 0 0 there are h1,C1 > 0 such that

(α) |α| M(λ1n) d f ∞ C h M e , α N , n N, k n kL (K) ≤ 1 1 α ∈ 0 ∈ and there are λ2,C2 > 0 such that

−M(λ2n) f ∞ C e , n N. k nkL (K) ≤ 2 ∈ Let β Nd, β = 0. Applying (4.2) with k = β and m =2 β , we obtain ∈ 0 6 | | | | (β) f ∞ ′ k n kL (K ) 1/2 2 |β| 1/2 2|β| (α) fn L∞(K)(2 β )! 4(2e ) fn L∞(K) max d max fn L∞(K), k k | | . ≤ k k |α|=2|β| k k δ2|β| Combining this with the above inequalities and taking λ1 := λ2/H, one ﬁnds by (M.1), (M.2), and (NE) that there are h,C > 0 such that

2 (β) |β| −M(λ2n/H ) d f ∞ ′ Ch M e , β N , n N. k n kL (K ) ≤ β ∈ 0 ∈ {Mp} Thus (fn)n N (Ω). We now show∈E (4.2). The one-dimensional Gorny inequality [17, p. 324] states that

k (k) 2k m 1−k/m g ∞ 4e g k kL ([a,b]) ≤ k k kL∞([a,b])× m k/m (m) g L∞([a,b])2 m! (4.3) max g ∞ , k k , k kL ([a,b]) (b a)m − for all g Cm([a, b]) and 0

k ′ ′ for all f C (K ), k N – it is for this inequality that we need the compact K to be 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 Ix,ξ 0 of length at least 2δ. Inequality (4.3) therefore implies that ∋ k ∂ f (k) (k) ∞ k = sup gx,ξ (0) sup gx,ξ L (Ix,ξ) ∂ ξ ∞ ′ ′ | | ≤ ′ k k L (K ) x∈K x∈K k 2k m 1−k/m sup 4e g ∞ x,ξ L (Ix,ξ) ≤ x∈K′ k k k × k/m ∞ (m) gx,ξ L (Ix,ξ)m! max g ∞ , k k k x,ξ kL (Ix,ξ) δm k/m k m 2k m 1−k/m ∂ f f L∞(K)m! 4e f L∞(K) max m , k k m . ≤ k k k ∂ ξ ∞ δ ( L (K) )!

The result now follows from (4.4) and the fact that m d d m ∂ f ∂ f(x) m (α) ∞ m = sup ξj1 ξjm d sup f L (K). ∂ ξ L∞(K) x∈K ··· ∂xj1 ∂xjm ··· ≤ |α|=m k k j1=1 jm=1 ··· X X

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 {Mp} ′ f ′ = [(f ′ ) ] (Ω ). |Ω n|Ω n ∈ G Clearly, the assignment Ω {Mp}(Ω) is a presheaf on Rd. Our aim in the rest of this section is to show that it is→ in G 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 {M } p (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 Mp with values in an appropriate sequence space and then use Proposition 2.3. We {need} some preparation: For λ> 0 we define the following Banach space

Mp,λ N M(λn) s = (cn)n C : σλ((cn)n) := sup cn e < . { ∈ n∈N | | ∞} Set s{Mp} = lim sMp,λ, λ−→→0+ a(DFS)-space. Given a sequence rj we denote by Mrj the associated function of p ∈ R the sequence M 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 Mrj (n) σrj ((cn)n) := sup cn e < n∈N | | ∞ NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 11 for all r . Moreover, the topology of s{Mp} is generated by the system of seminorms j ∈ R σ : r . The strong dual of s{Mp} is given by { rj j ∈ R} ′{Mp} Mp,−λ sβ = lim s , λ←−→0+ where

Mp,−λ N ′ −M(λn) s = (cn)n C : σλ((cn)n) := sup cn e < , λ> 0. { ∈ n∈N | | ∞} Proposition 4.2. We have

{Mp}(Ω) = {Mp}(Ω; s′{Mp}) and {Mp}(Ω) = {Mp}(Ω; s{Mp}). EM E EN E The proof of Proposition 4.2 is based on the ensuing lemma.

Lemma 4.3. Let (ap,n)p,n∈N be a double sequence of positive real numbers. Then,

M(λn) ap,ne sup p < p,n∈N h ∞ for some h,λ > 0 if and only if

Ms (n) ap,ne j (4.5) sup p < p,n∈N j=0 rj ∞ for all rj,sj . Q ∈ R Proof. The direct implication is clear. Conversely, suppose (4.5) holds for all rj,sj . Equivalently, ∈ R q ap,nn sup p q < , p,n,q∈N j=0 rj j=0 sjMq ∞ for all r ,s . Deﬁne j j ∈ R Q Q 1 q bp,q = sup ap,nn , p,q N. Mq n∈N ∈ Then, bp,q sup p q < , p,q∈N j=0 rj j=0 sj ∞ for all r ,s . Hence, [36, Lemma 1] implies that there is h> 0 such that j j ∈ R Q Q q M(n/h) bp,q ap,nn ap,ne > sup p+q = sup p+q = sup p . ∞ p,q∈N h p,n,q∈N h Mq p,n∈N h

Proof of Proposition 4.2. We only show the second equality. The ﬁrst one can be shown by a similar argument (but by using [24, Lemma 3.4] instead of Lemma 4.3). Clearly, 12 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS

N {Mp} {Mp} {Mp} the space (Ω; s ) consists of all (fn)n (Ω) such that for all K ⋐ Ω and r ,s E it holds that ∈ E j j ∈ R (α) (α) Msj (n) σsj ((fn (x))n) fn L∞(K)e sup sup |α| = sup sup k k |α| < . d x∈K d n∈N ∞ α∈N0 Mα j=0 rj α∈N0 Mα j=0 rj We find that (f ) {Mp}(Ω)Q by applying Lemma 4.3Q to n n ∈EM (α) fn L∞(K) ap,n = max k k , p,n N. |α|≤p Mα ∈ The converse inclusion can be shown similarly. Corollary 4.4. We have N {Mp}(Ω) = (f ) {Mp}(Ω) :( K ⋐ Ω)( r )( s ) EM { n n ∈E ∀ ∀ j ∈ R ∃ j ∈ R −Msj (n) (4.6) sup fn K,rj e < , n∈N k k ∞} and N {Mp}(Ω) = (f ) {Mp}(Ω) :( K ⋐ Ω)( r )( s ) EN { n n ∈E ∀ ∀ j ∈ R ∀ j ∈ R Msj (n) (4.7) sup fn K,rj e < . n∈N k k ∞} Remark 4.5. Since the structure (choice and order of quantifiers) of the spaces occurring in (4.6) and (4.7) coincides with the structure of the widely accepted definition for spaces of moderate and negligible sequences based on an arbitrary locally convex space [13], Corollary 4.4 may serve to clarify our definition of {Mp}(Ω). G In [29, Prop. 4.1] it is shown that for a weight sequence Mp satisfying (M.1) and ′ ′{Mp} ∗ (M.2) , the space sβ satisfies (DN) if and only if (M.2) holds for Mp. We have set the ground to show that {Mp} is a sheaf. G ∗ Theorem 4.6. Let Mp be a weight sequence satisfying (M.1), (M.2), (M.2) , (QA), and (NE). The functor Ω {Mp}(Ω) is a sheaf of differential algebras on Rd. Fur- → G {Mp} {Mp} thermore, every ultradifferential operator of class Mp P (D): is a sheaf morphism. { } G → G Proof. It is clear that the presheaf {Mp} satisfies the first sheaf axiom (S1). We now show that it also satisfies the patchingG condition (S2). Let Ω Rd be open ⊆ and let Ω : i I be an open covering of Ω. Suppose f = [(f ) ] {Mp}(Ω ) { i ∈ } i i,n n ∈ G i are given such that fi = fj on Ωi Ωj, for all i, j I. This means that there are {Mp} ∩ ∈ (gi,j,n)n N (Ωi Ωj) such that gi,j,n = fi,n fj,n on Ωi Ωj, for all i, j I, n N. {M∈Ep} ∩ ′{Mp} − ∩ ∈ ∈ Since s is a (DFS)-space and sβ has the property (DN), Propositions 2.3 and 4.2 imply that there are (g ) {Mp}(Ω ) such that g = g g on Ω Ω , for i,n n ∈EN i i,j,n i,n − j,n i ∩ j all i, j I, n N. Hence fn(x)= fi,n(x) gi,n(x) if x Ωi, is a well-defined function on ∈ ∈ {M } − ∈ Ω, for each n N, and (f ) p (Ω). Set f = [(f ) ] {Mp}(Ω). Then, f = f ∈ n n ∈EM n n ∈ G |Ωi i NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 13 for each i I. Finally, it is clear from its definition that P (D): {Mp} {Mp} is a sheaf morphism.∈ G → G In the next section we establish a very important property of {Mp}, namely, that it is a soft sheaf. G

5. Embedding of the sheaf of infrahyperfunctions In this section we shall embed {Mp}(Ω) into {Mp}(Ω) in such a way that the mul- B G tiplication of ultradifferentiable functions of class Mp is preserved. We proceed in { } d {Mp} various steps. We first construct a support preserving embedding of Γc(R , ) = ′{Mp} d d {Mp} B (R ) into Γc(R , ) by means of convolution with a special analytic mollifier sequenceE that we constructG below. Next, we use Lemma 3.3 together with the sheaf properties of {Mp} and {Mp} to extend this embedding to the whole space {Mp}. This requiresB proving thatG {Mp} is soft. We are very indebted to HörmanderB for his “hard analysis” type treatmentG of quasianalytic functionals. A lot of the techniques used in this section are modifications of the ideas from [25]. We start with a discussion about the mollifier sequences that will be used to embed ′{Mp} d d {Mp} (R ) into Γc(R , ). A sequence (χn)n∈N in (Ω) is called an analytic cut-off Esequence supported in ΩG [26] if D (a) 0 χ 1, n N, ≤ n ≤ ∈ (b) (χn)n is a bounded sequence in (Ω), (c) There is L 1 such that D ≥ (α) |α| χ ∞ d L(Ln) , n N, α n. k n kL (R ) ≤ ∈ | | ≤ We call (χn)n an analytic cut-off sequence for K ⋐ Ω if (d) there is an open neighborhood V of K such that χ 1 on V for all n N. n ≡ ∈ It is shown in [26, Thm. 1.4.2] that for every K ⋐ Rd and every open neighborhood Ω of K there is an analytic cut-off sequence for K supported in Ω. The following two simple lemmas are very useful. We fix the constants in the Fourier transform as

(ϕ)(ξ)= ϕ(ξ)= ϕ(x)e−ixξdx. F d ZR Lemma 5.1. (Proof of [25, Thm. 3.4])b Let Mp be a weight sequence satisfying (M.1) d and (NE). Let Ω be a relatively compact open subset in R and let (χn)n be an analytic cut-oﬀ sequence supported in Ω. Let h> 0. Then, ξ n χ ϕ(ξ) CL Ω ϕ (√d(h + Lk))nM , n N, ξ Rd, | | | n | ≤ | |k kΩ,h n ∈ ∈ for all ϕ Mp,h(Ω), where C,k > 0 are chosen in such a way that ∈E d pp CkpM , p N . ≤ p ∈ 0 Lemma 5.2. ([23, Lemma 3.3]) Let h> 0 and let ϕ L1(Rd). If ∈ M(ξ/h) µh(ϕ) := ϕ(ξ) e dξ < , d | | ∞ ZR b 14 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS then ϕ C∞(Rd) and ∈ µ (ϕ)h|α|M (α) h α Nd sup ϕ (x) d , α 0. x∈Rd | | ≤ (2π) ∈

Let (χn)n be an analytic cut-off sequence for B(0, 1) consisting of even functions. Define θ (x)= nd −1(χ )(nx)= −1(χ ( /n))(x), n N. n F n F n · ∈ {Mp} d By Lemma 5.2, (θn)n M (R ). The sequence (θn)n is called an Mp -mollifier sequence if, in addition,∈ the E following property holds: for every c> 0 there{ are} S,δ,γ > 0 such that (α) −M(δn) |α| d (5.1) sup θn (x) Se γ Mα, α N0, n N. |x|≥c | | ≤ ∈ ∈ The next lemma establishes the existence of such mollifier sequences; in fact, we provide an explicit construction of a p! -mollifier sequence in its proof. { } Lemma 5.3. For every weight sequence Mp satisfying (M.1) and (NE) there exists an M -mollifier sequence. { p} Proof. It suffices to consider the case Mp = p!. Moreover, if (θn)n is a one-dimensional p! -mollifier sequence, then (θ ) with { } n n θ = θ θ n n ⊗···⊗ n d times

{Mp} is a d-dimensional p! -mollifier sequence,| since{z (θn})n M (R). Therefore we may { } ∈ E also assume d = 1. Let (κn)n be an analytic cut-off sequence for [ 2, 2] consisting of even functions. We denote by H the characteristic function of [ 1,−1]. Define − n n H = H(n ) H(n ), n N. n 2 · ∗···∗ · ∈ n times Notice that supp H [ 1, 1] and H (x)dx = 1 for all n N. Set χ = κ H . n ⊆ − | R n {z } ∈ n n ∗ n Then, (χn)n is an analytic cut-off sequence for [ 1, 1]. We now show that additionally (5.1) is satisfied (for M = p! and, thus,R M(t) −t). Notice that p ≍ θ (z)= n −1(χ )(nz)= n −1(κ )(nz)(sinc(z))n, z C, n N, n F n F n ∈ ∈ where, as usual, sin z 1 sinc(z)= = H(z), z C. z 2 ∈ Let 0 0 such that { } sinc(x + iy) µ, x a, y r. | | ≤ | | ≥ | | ≤ NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 15

Since the sequence (κ ) is bounded in (R) there are C,D > 0 such that n n D z −1(κ )(z) CeD|y|, z = x + iy C, n N. | ||F n | ≤ ∈ ∈ Dr0 Choose 0 0 there are S,δ,γ > 0 such that (α) −δn α sup θn (x) Se γ α!, α N0, n N. |x|≥c | | ≤ ∈ ∈

We are ready to embed ′{Mp}(Rd) into Γ (Rd, {Mp}). Fix an analytic cut-off se- E c G quence (χn)n for B(0, 1) consisting of even functions and, say, supported in B(0,r) for some r > 1. In addition, suppose that the associated sequence θ (x)= nd −1(χ )(nx), n N, n F n ∈ is an M -mollifier sequence (we shall freely use the constant r, the constant L in { p} property (c), and the ones appearing in (5.1)). Since θn is an entire function, the convolution (f θ )(x)= f(t), θ (x t) , x Rd, ∗ n h n − i ∈ is well-defined for f ′{Mp}(Rd). ∈E Proposition 5.4. Let Mp be a weight sequence satisfying (M.1), (M.2), and (NE). Then, the mapping ι : ′{Mp}(Rd) Γ (Rd, {Mp}): f ι (f) = [(f θ ) ], c E → c G 7→ c ∗ n n is a linear embedding. Furthermore, supp ι (f) = supp f for all f ′{Mp}(Rd). c ∈E We need two lemmas in preparation for the proof of Proposition 5.4. As customary, we denote by m(t) := 1, t 0, ≥ m ≤t Xp the counting function of the sequence (mp)p∈N. Notice that, by (M.1), tm(t) tp = sup = eM(t), t 0. Mm(t) p∈N0 Mp ≥ Lemma 5.5. Let Ω Rd be open and let Ω′ be a relatively compact open subset of Ω. Let κ (Ω) be such⊆ that 0 κ 1 and κ 1 on a neighborhood of Ω′. Then, ∈D ≤ ≤ ≡ ((κϕ) θ ϕ) {Mp}(Ω′), ∗ n − n ∈EN for all ϕ {Mp}(Ω). In particular, ∈E (κϕ) θ ϕ, as n , in {Mp}(Ω′). ∗ n → →∞ E 16 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS

Proof. Fix ϕ {Mp}(Ω). We claim that ∈E (5.2) ((κϕ) θ (κ ϕ) θ ) {Mp}(Ω′) ∗ n − n ∗ n n ∈EN for any bounded sequence (κ ) in (Ω) such that 0 κ 1 and for which there n n D ≤ n ≤ is a neighborhood U of Ω′ such that κ 1 on U for all n N. Before proving the n ≡ ∈ claim, we show how it implies the result. Let (ρp)p be an analytic cut-oﬀ sequence for Ω′ supported in Ω. Lemma 5.1 implies that there are C,h > 0 such that ξ p ρ ϕ(ξ) ChpM , p N,ξ Rd. | | | p | ≤ p ∈ ∈ Set pn = m(n/(Hh)) + d + 1 and κn = ρpn , n N. By the claim it suﬃces to show that d ∈ ((κ ϕ) θ ϕ) {Mp}(Ω′). n ∗ n − n ∈EN For K ⋐ Ω′ we have

sup ((κnϕ) θn)(x) ϕ(x) sup ((κnϕ) θn)(x) (κnϕ)(x) x∈K | ∗ − | ≤ x∈K | ∗ − | 1 d κnϕ(ξ) (1 χn(ξ/n))dξ ≤ (2π) d | | − ZR AC(Hh)d+1M (Hh)m(n/(Hh))M dd+1 m(n/(Hh)) dξ ≤ (2π)d ξ m(n/(Hh))+d+1 Z|ξ|≥n | | De−M(n/(Hh)), ≤ where AC(Hh)d+1M 1 D = d+1 dξ < , (2π)d ξ d+1 ∞ Z|ξ|≥1 | | whence the result follows (cf. Lemma 4.1). We now show (5.2). We ﬁrst prove that ((κ ϕ) θ ) {Mp}(Rd). n ∗ n n ∈EM Since the sequence (κ ϕ) is bounded in (Rd) there is C > 0 such that n n D C κ ϕ(ξ) , ξ Rd, n N. | n | ≤ (1 + ξ )d+1 ∈ ∈ | | For λ> 0 we have d M(λξ) M(λrn) 1 µ1/λ((κnϕ) θn)= κnϕ(ξ) χn(ξ/n)e dξ Ce d+1 dξ, ∗ d | | ≤ d (1 + ξ ) ZR ZR | | and therefore the sequence is moderate by Lemma 5.2. Hence, by Lemma 4.1, it suﬃces d to show that for every K ⋐ Ω′ there is λ> 0 such that M(λn) sup sup ((κϕ) θn)(x) ((κnϕ) θn)(x) e < . n∈N x∈K | ∗ − ∗ | ∞

Choose c > 0 such that κ κn 0 on K + B(0,c) and let K0 ⋐ Ω be such that supp(κ κ ) K for all n− N.≡ Then, − n ⊆ 0 ∈ sup ((κϕ) θn)(x) ((κnϕ) θn)(x) sup (κ(x t) κn(x t))ϕ(x t)θn(t)dt | ∗ − ∗ | ≤ d − − − − x∈K x∈K ZR −M(δn) 2S ϕ L∞(K0) K0 e . ≤ k k | | NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 17

Mp,h d ∞ d For h > 0, we write ∞ (R ) for the Banach space consisting of all ϕ C (R ) such that E ∈ ϕ(α)(x) sup sup | |α| | < . d d h M ∞ α∈N0 x∈R α Lemma 5.6. (cf. ﬁrst part of [25, Thm. 3.4]) Let h> 0 and let Ω be a relatively compact d open subset of R . Suppose that (κp)p is an analytic cut-oﬀ sequence supported in Ω d and that (ψn)n is a uniformly bounded sequence of continuous functions on R such that supp(ψ ) B(0,r) and 1 ⊆ supp(ψ ) B(0, rn) B(0, n 1), n 2. n ⊆ \ − ≥ Then, there are a sequence (pn)n of natural numbers and k > 0 such that ∞ R(ϕ) := (κ ϕ) −1(ψ ) Mp,k(Rd), ϕ Mp,h(Ω). pn ∗F n ∈E∞ ∈E n=1 X Mp,k d Moreover, the convergence of the series R(ϕ) holds in the topology of ∞ (R ) and Mp,h Mp,k d E the mapping R : (Ω) ∞ (R ) is continuous. E →E Proof. By Lemma 5.1 there are C,k0 > 0 such that ξ p κ ϕ(ξ) C ϕ kpM , p N, ξ Rd, | | | p | ≤ k kΩ,h 0 p ∈ ∈ Mp,h for all ϕ (Ω). For p = m(t), t 0, and k0t ξ 2rk0t, we obtain ∈E d ≥ ≤| | ≤ κ\ϕ(ξ) C ϕ e−M(ξ/(2k0r)). | m(t) | ≤ k kΩ,h Set p = max 1, m((n 1)/k ) , n N. Hence, n { − 0 } ∈ κ[ϕ(ξ) C ϕ e−M(ξ/(2k0r)), (n 1) ξ rn, n k +1. | pn | ≤ k kΩ,h − ≤| | ≤ ≥ 0 ′ ′ 2 Choose C > 0 such that ψn L∞(Rd) C for all n N. Then, for k = 2H k0r, we have k k ≤ ∈ ∞ ∞ −1 M(ξ/k) µk (κpn ϕ) (ψn) ψn(ξ) κ[pn ϕ(ξ) e dξ ∗F ≤ Rd | || | n≥k0+1 ! n≥k0+1 Z X X ∞ ′ M(ξ/k)−M(ξ/(2k0r)) C C ϕ Ω,h e dξ ≤ k k (n−1)≤|ξ|≤rn n≥Xk0+1 Z D ϕ , ≤ k kΩ,h where ∞ D = A2C′C e−M(n/k) e−M(ξ/(2Hk0r))dξ < . Rd ∞ nX=k0 Z We also have

−1 ′ M(rk0/k) µ ((κ ϕ) (ψ )) C e B(0,rk ) Ω ϕ ∞ , k pn ∗F n ≤ | 0 || |k kL (Ω) for n k . The result now follows from Lemma 5.2. ≤ 0 18 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS

{Mp} d ′{Mp} d Proof of Proposition 5.4. STEP I: (f θn)n M (R ) for all f (R ): Let λ> 0 be arbitrary. Choose C > 0 such∗ that ∈ E ∈ E f(ξ) CeM(λξ/H), ξ Rd. | | ≤ ∈ We have b M(λξ) M(λξ/H)+M(λξ) M(Hλrn) µ1/λ(f θn)= f(ξ) χn(ξ/n)e dξ C e dξ De , ∗ d | | ≤ ≤ ZR Z|ξ|≤rn where b D = A2C e−M(λξ/H) dξ < . d ∞ ZR The result follows from Lemma 5.2. ′{Mp} d d STEP II: supp ιc(f) supp f for all f (R ). Let K ⋐ R supp f be arbitrary. Choose K′ ⋐ ⊆Rd such that a neighborhood∈ E of supp f is contained\ in K′ and K′ K = . Set c := d(K,K′) > 0. The continuity of f implies that for each h > 0 there∩ is C >∅ 0 such that

sup (f θn)(x) C θn K−K′,h. x∈K | ∗ | ≤ k k By Property (5.1) we have that

(α) θn (x) −M(δn) ′ θn K−K ,γ = sup sup | |α| | Se . k k d ′ γ M ≤ α∈N0 x∈K−K α We then obtain that ι (f) vanishes on Rd supp f from Lemma 4.1. c \ STEP III: supp f supp ι (f) for all f ′{Mp}(Rd). Set K = supp ι (f). We need ⊆ c ∈E c to show that f ′{Mp}(Ω) for every open set Ω ⋑ K. Fix such a set Ω. By Lemma ∈ E 2.1 and [38, Lemma 2.3] there is a weight sequence Np satisfying (M.1), (M.2), and ′{Np} d d Mp Np such that f (R ). Choose ρ (R ) such that 0 ρ 1 and ρ 1 on a≺ neighborhood of∈E supp f. Employing Lemma∈D 5.5, we deduce that≤ ≤ ≡

{Mp} d f,ϕ = lim f, (ρϕ) θn = lim (f θn)(x)ρ(x)ϕ(x)dx, ϕ (R ). h i n→∞h ∗ i n→∞ d ∗ ∈E ZR Next, choose κ (Ω) such that 0 κ 1 and κ 1 on a neighborhood of K. By ∈ D ≤ ≤ ≡ STEP II we have that κ ρ 0 on a neighborhood of K. Therefore, ιc(f)|Rd\K = 0 implies that f θ 0, as− n ≡ , uniformly on supp(κ ρ). Hence, ∗ n → →∞ −

{Mp} d (5.3) f,ϕ = lim (f θn)(x)κ(x)ϕ(x)dx, ϕ (R ). h i n→∞ d ∗ ∈E ZR We now invoke Lemma 5.6. Set ψ1 = χ1 and

ψ = χ · χ · , n 2. n n n − n−1 n 1 ≥ − Clearly, the sequence (ψn)n satisﬁes the requirements of Lemma 5.6. Choose a relatively ′ ′ compact open subset Ω such that K ⋐ Ω ⋐ Ω and an analytic cut-oﬀ sequence (κp)p NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 19

′ for K supported in Ω . According to Lemma 5.6 (applied to the weight sequence Np), there are a sequence (pn)n and k > 0 such that the mapping ∞ R : Np,1(Ω′) Np,k(Rd): ϕ R(ϕ)= (κ ϕ) −1(ψ ) E →E∞ → pn ∗F n n=1 X is continuous. Consider the following continuous inclusion mappings ι : {Mp}(Ω) Np,1(Ω′), ι : Np,k(Rd) {Np}(Rd), 1 E →E 2 E∞ →E and set T = ι R ι : {Mp}(Ω) {Np}(Rd). Equality (5.3) gives us 2 ◦ ◦ 1 E →E ∞ −1 f,ϕ = (f (ψn))(x)(κ(x) κpn (x))ϕ(x)dx + h i d ∗F − ZR n=1 ∞ X −1 (f (ψn))(x)κpn (x)ϕ(x)dx, d ∗F n=1 R X Z for all ϕ {Mp}(Rd). Since (κ κ ) is a bounded sequence in (Ω K), the assump- ∈E − pn n D \ tion ιc(f)|Rd\K = 0 yields ∞ g = (f −1(ψ ))(κ κ ) (Ω). ∗F n − pn ∈D n=1 X For the second term, we have ∞ ∞ −1 −1 (f (ψn))(x)κpn (x)ϕ(x)dx = f, (ψn) (κpn ϕ) = f, T (ϕ) , d ∗F h F ∗ i h i n=1 R n=1 X Z X for all ϕ {Mp}(Rd). Hence f = g + tT (f) ′{Mp}(Ω). ∈E ∈E ∗ Proposition 5.7. Let Mp be a weight sequence satisfying (M.1), (M.2), (M.2) , (QA), and (NE). Then, the sheaf {Mp} is soft. G Proof. Let K ⋐ Rd and K ⋐ Ω, Ω open, be arbitrary. Choose a relatively compact open ′ ′ {Mp} set Ω such that K ⋐ Ω ⋐ Ω. It suffices to show that for every f = [(fn)n] (Ω) {Mp} d ′ ∈ G there is g = [(gn)n] (R ) such that g = f on Ω . Let(κp)p be an analytic cut-off ′ ∈ G {Mp} sequence for Ω supported in Ω. Lemma 5.1 and (fn)n M (Ω) imply that there are C,h > 0 such that ∈E ξ p κ f (ξ) CeM(n)hpM , ξ Rd, p,n N. | | | p n | ≤ p ∈ ∈ Let ψ (B(0, 2)) be such that 0 ψ 1 and ψ 1 on B(0, 1). Set a = H2h and ∈D d ≤ ≤ ≡ ψ = ψ · , n N, n an ∈ and define g =(κ f ) −1(ψ ), n N, n pn n ∗F n ∈ 20 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS where p = m(Hn)+ d + 1. We first show that (g ) {Mp}(Rd). Since the sequence n n n ∈EM (κ ) is bounded in (Ω) and (f ) {Mp}(Ω), we have p p D n n ∈EM D eM(λn) κ f (ξ) λ , ξ Rd, p,n N, | p n | ≤ (1 + ξ )d+1 ∈ ∈ | | for every λ> 0 and suitabled Dλ > 0. Hence \ M(λξ) M(2Haλn) 1 µ1/λ(gn)= κpn fn(ξ) ψn(ξ)e dξ AD2aλe d+1 dξ d | | ≤ d (1 + ξ ) ZR ZR | | for all λ > 0. The sequence (gn)n is therefore moderate by Lemma 5.2. We still need to show that (g f ) {Mp}(Ω′). n − n n ∈EN We verify this via Lemma 4.1. For any K ⋐ Ω′, we have −1 sup gn(x) fn(x) sup ((κpn fn) (ψn))(x) (κpn fn)(x) x∈K | − | ≤ x∈K | ∗F − | 1 \ d κpn fn(ξ) (1 ψn(ξ))dξ ≤ (2π) d | | − ZR AC(Hh)d+1M eM(n) (Hh)m(Hn)M d+1 m(Hn) dξ ≤ (2π)d ξ m(Hn)+d+1 Z|ξ|≥an | | De−M(n), ≤ where A2C(Hh)d+1M 1 D = d+1 dξ < , (2π)d ξ d+1 ∞ Z|ξ|≥a | | whence the result follows. We have completed all necessary work to prove our main theorem. Recall that σ : {Mp}(Ω) {Mp}(Ω) stands for the constant embedding (4.1). Ω E → G ∗ Theorem 5.8. Let Mp be a weight sequence satisfying (M.1), (M.2), (M.2) , (QA), and (NE). Then, there is a unique injective sheaf morphism ι : {Mp} {Mp} such that for each open set Ω Rd the following properties hold: B → G ⊆ ′{Mp} (i) ιΩ(f)= ιc(f) for all f (Ω). (ii) For all ultradifferential∈E operators P (D) of class M we have { p} P (D)ι (f)= ι (P (D)f), f {Mp}(Ω). Ω Ω ∈ B (iii) ι (f)= σ (f) for all f {Mp}(Ω). Consequently, Ω Ω ∈E ι (fg)= ι (f)ι (g), f,g {Mp}(Ω). Ω Ω Ω ∈E Proof. Since {Mp} and {Mp} are soft sheaves (Propositions 3.1 and 5.7), the existence and uniquenessB of a sheafG embedding ι satisfying properties (i) and (ii) is clear from Lemma 3.3 and Proposition 5.4. We now show that property (iii) is also satisfied. It ′ ′ suffices to show that ιΩ(f)|Ω′ = σΩ(f)|Ω′ for all open sets Ω ⋐ Ω. Fix such a set Ω and choose κ (Ω) such that 0 κ 1 and κ 1 on a neighborhood of Ω′. Notice that ∈D ≤ ≤ ≡ NONLINEARTHEORYOFINFRAHYPERFUNCTIONS 21

ιΩ(f)|Ω′ = ιΩ′ (f|Ω′ ) = ιΩ′ ((κf)|Ω′ ) = ιΩ(κf)|Ω′ = ιc(κf)|Ω′ . Hence, it suffices to show that ((κf) θ f) {Mp}(Ω′), ∗ n − n ∈EN but this has already been proved in Lemma 5.5. We end this article with a remark concerning the optimality of the above embedding. It can be viewed as an analogue of Schwartz’s impossibility result [41] in the setting of infrahyperfunctions, which states that, under some natural assumptions, the property (iii) from Theorem 5.8 cannot be improved to also preserve the multiplication of all ultradifferentiable functions from a class with lower regularity than {Mp}(Ω). E ′ Remark 5.9. Let Np be another weight sequence satisfying (M.1) and (M.2) . When embedding {Mp}(Ω) into an associative and commutative algebra ( {Mp},{Np}(Ω), +, ), the followingB requirements seem to be natural: A ◦ (P.1) {Mp}(Ω) is linearly embedded into {Mp},{Np}(Ω) and f(x) 1 is the unity in B A ≡ {Mp},{Np}(Ω). (P.2)A For each ultradifferential operator P (D) of class M there is a linear opera- { p} tor P (D): {Mp},{Np}(Ω) {Mp},{Np}(Ω) satisfying the following generalized Leibniz’ ruleA → A 1 P (D)(q f)= Dβq (DβP )(D)f, ◦ β! ◦ β≤Xdeg q {Mp},{Np} for every f (Ω) and every polynomial q. Moreover, P (D) {Mp} ∈ A |B (Ω) coincides with the usual action of P (D) on infrahyperfunctions of class M . { p} (P.3) {Np} {Np} coincides with the pointwise product of functions. ◦|E (Ω)×E (Ω) The ensuing result imposes a limitation on the possibility of constructing such an {Mp} {Np} algebra if Mp Np (implying that (Ω) ( (Ω)). On the other hand, our ≺ {Mp} E E differential algebra (Ω) satisfies all the properties (P.1)-(P.3) for Mp = Np and therefore Theorem 5.8G is optimal in this sense.

Theorem 5.10. Let Mp be a weight sequence satisfying (M.1), (M.2), (QA), and (NE) and let N be a weight sequence satisfying (M.1) and (M.2)′. Suppose that M N . p p ≺ p Then, there is no associative and commutative algebra {Mp},{Np}(Ω) satisfying (P.1)- (P.3). A Proof. The proof is basically the same as that of the corresponding result for non- quasianalytic ultradistributions [11, Thm. 3.1], but we include it for the sake of com- pleteness. Suppose {Mp},{Np}(Ω) is such an algebra. We have that q P (D)g = qP (D)g A {N◦p} for every ultradiﬀerential operator P (D) of class Mp , g (Ω) and polynomial q; this follows by induction on the degree of{ q.} Assume∈ E for simplicity that 0 Ω. Write H(x) = H(x1,...,xd) := H(x1) H(xd), where H(xj) is the Heaviside∈ function (characteristic function of the⊗···⊗ positive half-axis), and p. v.(x−1) = −1 −1 −1 p. v.(x1 ) p. v.(xd ), where p. v.(xj ) is the principle value regularization of ⊗···⊗−1 −1 the function xj . Let f be either H(x) or p. v.(x ). By employing the global struc- tural theorem for infrahyperfunctions of class M [42], we ﬁnd g {Np}(Rd) and an { p} ∈E 22 A.DEBROUWERE,H.VERNAEVE,ANDJ.VINDAS

ultradiﬀerential operator P (D) of class M such that P (D)g = f. Set { p} ∂d ∂ = and q(x)= x x x . ∂x ∂x 1 2 ··· d 1 ··· d The observation made at the beginning of the proof now yields q ∂H = (q∂H) and ◦ q p. v.(x−1)= q p. v.(x−1). Since q∂H = 0 and q p. v.(x−1) = 1 in {Mp}(Ω), we obtain ◦ B ∂H = ∂H (q p. v.(x−1))=(∂H q) p. v.(x−1)=0, ◦ ◦ ◦ ◦ contradicting ∂H = δ = 0 in {Mp}(Ω) and the injectivity of {Mp}(Ω) {Mp},{Np}(Ω). 6 B B →A

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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent Uni- versity, Krijgslaan 281, 9000 Gent, Belgium E-mail address: [email protected] Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent Uni- versity, Krijgslaan 281, 9000 Gent, Belgium E-mail address: [email protected] Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent Uni- versity, Krijgslaan 281, 9000 Gent, Belgium E-mail address: [email protected]