Boundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator $P(D) = P(D_x, D_t)$ on $\mathbb{R}^{d+1}$. Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of several results of Langenbruch [23] about distributional boundary values of zero solutions of $P(D)$.


Introduction
The study of distributional and ultradistributional boundary values of holomorphic functions, which goes back to the seminal works of Köthe [16] and Tillmann [34] for distributions and of Komatsu [19] for ultradistributions, is an important subject in the theory of generalized functions. We refer to the survey article [28] and the books [4,5] and the references therein for an account of results on this topic. Similarly, the boundary value behavior of harmonic functions [1,8,22,35] and of zero solutions of the heat equation [6,18,26,27] in generalized function spaces have been thoroughly investigated. In the distributional case, Langenbruch [23] generalized the above results by developing a theory of distributional boundary values for the zero solutions of a hypoelliptic partial differential operator P (D) = P (D x , D t ) on R d+1 with constant coefficients. For ultradistributions much less is known in this general setting: In [24,25] Langenbruch studied spaces of formal boundary values of zero solutions of P (D), in the style of Bengel's approach to hyperfunctions [2]. In [25,Satz 2.4] he characterized the zero solutions f of P (D) that admit a boundary value in a given ultradistribution space in terms of the ultradistributional extendability properties of f (see also [20,Section 3]). Furthermore, he proved that, for certain semi-elliptic operators P (D), the space of formal boundary values of solutions of P (D) may be identified with the Cartesian product of some Gevrey ultradistribution spaces of Roumieu type [25,Satz 4.9 and Satz 4.10]. The aim of the present paper is to complement these results by extending Langenbruch's results from [23] for distributions to the framework of ultradistributions, both of Beurling and Roumieu type, defined via weight sequences [19]. We now describe the content of the paper and state a sample of our main results. For the sake of clarity, we consider here only Gevrey ultradistribution spaces. Let P (D) be a hypoelliptic partial differential operator on R d+1 . Let X ⊆ R d be open. For σ > 1 we write D (σ) (X) and D {σ} (X) for the spaces of compactly supported Gevrey ultradifferentiable functions of order σ of Beurling type and of Roumieu type, and endow these spaces with their natural locally convex topology. The spaces of Gevrey ultradistributions of order σ of Beurling type and of Roumieu type are defined as the strong dual spaces of D (σ) (X) and D {σ} (X). We denote them by D ′(σ) (X) and D ′{σ} (X). We use D ′[σ] (X) as a common notation for D ′(σ) (X) and D ′{σ} (X); a similar convention will be used for other spaces as well.
provided that bv(f ) ∈ D ′[σ] (X) exists. We have two main goals: Firstly, we wish to characterize the elements of C ∞ P (V \X) that admit a boundary value in D ′[σ] (X) in terms of their local growth properties near X. Secondly, we aim to study the uniqueness and the solvability of the Cauchy type problem P (D)f = 0 on V \X, bv(D j t f ) = T j on X for j = 0, . . . , m − 1, where f ∈ C ∞ (V \X), m = deg t P and the initial data T 0 , . . . , T m−1 belong to D ′[σ] (X). This will lead to the representation of the m-fold Cartesian product of D ′[σ] (X) by means of boundary values of elements of C ∞ P (V \X). Let us explicitly state the above two results for certain semi-elliptic partial differential operators. To this end, we introduce the following weighted spaces of zero solutions of P (D) where K ⋐ V denotes that K is a compact subset of V . We endow these spaces with their natural locally convex topology. We then have: Theorem 1.1. Let P (D) be a semi-elliptic partial differential operator on R d+1 with deg x1 P = · · · = deg x d P = n. Set m = deg t P and a 0 = n/m. Let X ⊆ R d be open and let V ⊆ R d+1 be open such that V ∩ R d = X.
(a) Let σ > max{1, 1/a 0 }. For f ∈ C ∞ P (V \X) the following statements are equivalent: (i) f ∈ C ∞ P,[a0σ] (V \X). In such a case, bv(D l t f ) ∈ D ′[σ] (X) exists for all l ∈ N 0 . Next, define Then, the sequence is exact and bv m is a topological homomorphism, provided that V is Pconvex for supports. In particular, this holds for all open sets of the form V = X × I, with I ⊆ R an open interval containing 0. (b) Let σ = 1/a 0 > 1. We have that bv(f ) ∈ D ′{σ} (X) exists for all f ∈ C ∞ P (V \X). Define the mapping bv m similarly as in (1). Then, the sequence is exact and bv m is a topological homomorphism, provided that V is Pconvex for supports. In particular, this holds for all open sets of the form V = X × I, with I ⊆ R an open interval containing 0.
Since a partial differential operator P (D) on R d+1 is elliptic precisely when it is semi-elliptic with deg x1 P = · · · = deg x d P = deg t P , Theorem 1.1(a) is applicable to every elliptic partial differential operator. Moreover, it is well-known that every open set V ⊆ R d is P -convex for supports in such a case. Therefore, Theorem 1.1(a) comprises two classical results of Komatsu about the boundary values of holomorphic functions [19,Section 11] and harmonic functions [22,Chapter 2] in ultradistribution spaces. Also the heat operator and, more generally, the k-parabolic operators in the sense of Petrowsky [ j=0 D ′{1/a0} (X) may be identified with the space C ∞ P (V \X)/C ∞ P (V ) of formal boundary values. This partially covers the above mentioned result of Langenbruch [25,Satz 4.9 and Satz 4.10] (the assumption deg x1 P = · · · = deg x d P is not needed there as Langenbruch considers anisotropic Gevrey ultradistribution spaces). Next, we comment on the main new technique used in this article. In [12, p. 64] Hörmander showed the existence of distributional boundary values of holomorphic functions by combining Stokes' theorem with almost analytic extensions. Petzsche and Vogt [30] (see also [29]) extended this method to ultradistributions by using descriptions of ultradifferentiable classes via almost analytic extensions [7,10,29,30]. We develop here a similar technique to establish the existence of ultradistributional boundary values of zero solutions of a hypoelliptic partial differential operator P (D). Namely, we combine a Stokes type theorem for P (D) (Lemma 5.4) with the description of tuples (ϕ 0 , . . . , ϕ m−1 ) of compactly supported ultradifferentiable functions via functions Φ ∈ D(R d+1 ) that are almost zero solutions of P (D) and satisfy D j t Φ( · , 0) = ϕ j for j = 0, . . . , m − 1 (Proposition 4.1). Petzsche [29] constructed almost analytic extensions of ultradifferentiable functions by means of modified Taylor series. We use here the same basic idea, starting from a power series Ansatz Φ that formally solves the Cauchy problem P (D)Φ = 0 on R d+1 \R d and D j t Φ( · , 0) = ϕ j on R d for j = 0, . . . , m − 1 [14,Section 4].
This paper is organized as follows. In the preliminary Section 2 we collect several results about partial differential operators and ultradistributions that will be used throughout this work. Next, in Section 3, we construct a fundamental solution of a hypoelliptic partial differential operator P (D) satisfying precise regularity and growth properties. This fundamental solution will play an essential role in our work. We obtain it by a careful analysis of the construction of a fundamental solution of P (D) by Langenbruch [23, p. 12-14]. Section 4 is devoted to the description of ultradifferentiable classes via almost zero solutions of P (D). By combining this description with a Stokes type theorem for P (D), we show in Section 5 that zero solutions of P (D) satisfying certain growth estimates near R d have ultradistributional boundary values. We also establish the continuity of the boundary value mapping here. Our main results are proven in Section 6. We adapt several classical techniques used in the study of ultradistributional boundary values of holomorphic functions and harmonic functions [19,22]. In Sections 4-6 we also indicate how our methods may be adapted to the distributional case, thereby providing new proofs of several results of Langenbruch [23]. Most notably, we obtain an elementary proof of the continuity of the boundary value mapping (compare with [23,Section 3]). Finally, in Section 7, we discuss our results for semi-elliptic operators and prove Theorem 1.1.

Preliminaries
always stands for a non-constant hypoelliptic polynomial. Moreover, we use the notation for suitable m ∈ N and polynomials Q k ∈ C[X 1 , . . . , X d ], 0 ≤ k ≤ m, with Q m = 0. Then, Q m = c ∈ C\{0} [13, Example 11.2.8] and we assume without loss of generality that c = 1. As customary, we associate to P the constant coefficient partial differential operator P (D) given by and endow this space with the relative topology induced by C ∞ (V ). We now introduce various indices of P that play an important role in this article. Definition 2.1. We define (γ 0 , µ 0 ) = (γ 0 (P ), µ 0 (P )) as the largest pair of positive numbers satisfying the following property: There are C, R > 0 such that for all The existence of the pair (γ 0 , µ 0 ) follows from the fact that P is hypoelliptic; see [13,Section 11.4] and [33,Section 7.4]. Furthermore, it holds that γ 0 , µ 0 ≤ 1. Let V ⊆ R d+1 be open. For σ, τ ≥ 1 we denote by Γ σ,τ (V ) the space consisting of all ϕ ∈ C ∞ (V ) such that for all K ⋐ V there is h > 0 such that By [33,Theorem 7.4] For a > 0 the following two statements are equivalent: (ii) There are C, R > 0 such that Remark 2.5. By evaluating the inequalities from Lemma 2.2(i) for t = 0, we obtain that there are C, R > 0 such that for all k = 1, . . . , m Hence, (with a different C) and Since a 0 ≥ 0, we also have that deg Q k ≤ deg Q 0 for all k = 1, . . . , m.
The main results of this article are valid under the assumption a 0 = b 0 . In Section 7 we calculate a 0 (P ), b 0 (P ), γ 0 (P ) and µ 0 (P ) for semi-elliptic polynomials P .
Remark 2.9. Let N = (N p ) p∈N0 be a sequence of positive numbers such that N ≍ 1, we define the associated function of N as Whenever we make use of this function we employ the convention 0 · ∞ = 0. This applies in particular to N = M a, * , where M is a weight sequence satisfying p! 1/a ≍ M .
Finally, we present a lemma that will be used later on.
Lemma 2.10. Let a > 0 and let M be a weight sequence. There are C, L > 0 such that Proof. Recall that M a satisfies (M.1) and (M.2) because M does so. Hence, there are C, L > 0 such that Note that ω M a (ρ) = aω M (ρ 1/a ) for ρ ≥ 0. By [19,Proposition 3.2] and the fact that M ≥ 1, we have that for all p ∈ N 0 We employ [M ] as a common notation for (M ) and {M }. In addition, we often first state assertions for the Beurling case followed in parenthesis by the corresponding ones for the Roumieu case. Let K ⋐ R d and let h > 0. We define D M,h K as the Banach space consisting of all K .
Following Komatsu [21,22], we will sometimes make use of an alternative description of the spaces of ultradifferentiable functions of Roumieu type. We denote by R the set of all non-decreasing sequences h = (h l ) l∈N0 of positive numbers such that h 0 = h 1 = 1 and lim l→∞ h l = ∞. This set is partially ordered and directed by the pointwise order relation ≤ on sequences. We remark that we will use h (and h ′ ) to denote both positive numbers and elements of R. In order to avoid confusion, we will always clearly indicate whether h > 0 or h ∈ R.
Let K ⋐ R d be such that int K = K and let h ∈ R. We write E M,h (K) for the Banach space consisting of all ϕ ∈ C ∞ (K) such that Lemma 2.11. Let M be a weight sequence.
as locally convex spaces. Later on, we will make use of the following result about the existence of parametrices, which is essentially shown in [11]. It improves a classical result of Komatsu [22], namely, the strong non-quasianalyticity condition (M. Proof. We first consider the Beurling case. By [3,Theorem 14], the function ω M is a weight function in the sense of [11, Definition 1.1]. Moreover, there are n ∈ N and C > 0 such that [3, Equation (5)] [11]. Hence, [11, Proposition 2.5] and conditions (M.1) and (M.2) imply that there are an ultradifferential operator G(D x ) of class (M ) and Finally, we introduce some notation from classical distribution theory.
For l ∈ N 0 we define the norm In this context, we sometimes also write E (X) = C ∞ (X). We denote by D ′ (X) and E ′ (X) the strong duals of D(X) and E (X), respectively. The space E ′ (X) may be identified with the subspace of D ′ (X) consisting of compactly supported elements.

A fundamental solution with good regularity and growth properties
The goal of this auxiliary section is to construct a fundamental solution of P (D) with precise regularity and growth properties. The next result and its proof are inspired by [23, p. 12-14].
satisfying the following property: There are A, L, S > 0 such that In particular, there is L > 0 such that for all l ∈ N 0 there is S > 0 such that for all r > 0 We need some preparation for the proof of Proposition 3.1. Definition 2.1 and Definition 2.3 imply that there are C, R ≥ 1 such that for all l ∈ N 0 Throughout this section the constants C and R will always refer to those occurring in the above inequality. We need the following lemma.
(ii) There are A > 0 and C 2 , L 2 > 0 such that for all l, p ∈ N 0 Proof. Note that |P (x, t)| ≥ 1/C for all |x| ≥ R and t ∈ R. We first show (i). Set from which (i) follows. Next, we prove (ii). Set S = max |x|≤R where ϕ denotes the Fourier transform of ϕ. It is clear that E is a fundamental solution of P (D). Since P is hypoelliptic, E is smooth in R d+1 \{0}. We set There are C 3 , L 3 > 0 such that and C 4 , L 4 > 0 such that Lemma 3.2(i) and (6) imply that which implies (3).

Characterization of ultradifferentiable functions via (almost) zero solutions of P (D)
In this section we characterize tuples (ϕ 0 , . . . , ϕ m−1 ) of compactly supported ultradifferentiable functions via functions Φ ∈ D(R d+1 ) that are (almost) zero solutions of P (D) and satisfy D j t Φ( · , 0) = ϕ j for j = 0, . . . , m − 1. We start with the following result, which is essential for this article.
The same convention will be tacitly used in the rest of this article.
The proof of Proposition 4.1 requires some preparation. In [29] Petzsche constructed almost analytic extensions of ultradifferentiable functions by means of modified Taylor series. If p! 1/b0 ≺ M , we use here a similar idea to prove Proposition 4.1, starting from a power series Ansatz Φ that formally solves the Cauchy problem Section 4] (see also the proof of [24, Satz 4.1]). If p! 1/b0 ≍ M , we even show that Φ converges. We now recall the definition and some basic properties of this formal power series solution; see [14,Section 4] for details. For l ∈ N 0 we recursively define the mapping C l : Moreover, as C l is a constant coefficient partial differential operator, supp solves the Cauchy problem P (D)Φ = 0 on R d+1 \R d and D j t Φ( · , 0) = ϕ j on R d for j = 0, . . . , m − 1 [14,Proposition 4.4]. We need the following lemma. Lemma 4.3. Let M be a weight sequence and let K ⋐ R d . There is L 1 > 0 such that for all h > 0 the following property holds: For all α ∈ N d there is C > 0 such that for all p ∈ N 0 By (M.2), it suffices to consider the case α = 0. For l ∈ N 0 we define the auxiliary norm There is R ≥ 1 (only depending on K) such that for all l ∈ N 0 By Lemma 2.10, there are C, L ≥ 1 such that for all l ∈ N 0 and every ϕ ∈ D M,h The recursively defined operators C m−1+l , l ∈ N 0 , have the following explicit representation [14,Proposition 4.5] Let h > 0 and ϕ ∈ D M,h K be arbitrary. For all β ∈ N m 0 with σ(β) = l it holds that Hence, Since C l = 0 for l < m − 1, this shows the result.
Proof of Proposition 4.1. Lemma 2.8(ii) yields that either p! 1/b0 ≺ M or p! 1/b0 ≍ M . Suppose first that p! 1/b0 ≺ M . By Lemma 2.8(i), we may assume without loss of generality that m b0, * p ր ∞. Throughout this proof C will denote a positive constant that is independent of ϕ 0 , . . . , ϕ m−1 but may vary from place to place. Let . Let now h > 0 be given, arbitrary but fixed. Define λ p = Ah b0 m b0, * p+1 for p ∈ N 0 . Note that λ p ր ∞. For ϕ ∈ D M,h K we define Since λ p ր ∞, the above series is finite on R d × {t ∈ R | |t| ≥ ε} for each ε > 0. Hence, S ∈ C ∞ (R d × (R\{0})). We claim that for all α ∈ N d 0 and l ∈ N 0 Before we prove these claims, let us show how they entail the result. Property (10) implies that S ∈ D(R d+1 ) with supp S ⊆ K × R such that Then, Φ ∈ D(R d+1 ) with supp Φ ⊆ K × R. Moreover, (9) implies that Φ satisfies (ii). Next, we show (i). In [14,Proposition 4.2] it is shown that Combining this with (11) and the fact that C l = 0 for l < m − 1, we obtain that for all n = 0, . . . , m − 1 We now show (9) and (10). Let α ∈ N d 0 and l ∈ N 0 be arbitrary. We have that We first prove (9). Note that and P (D) and We now show suitable estimates for the above expressions. To this end, following Dyn'kin [7] (see also [29,32]), we introduce the auxiliary function Since lim p→∞ m b0, * p = ∞, it holds that Γ(ε) < ∞ for each ε > 0. Fix ε > 0. By definition of Γ, we have that εm b0, * p < 1 for all p ∈ N with p ≤ Γ(ε). Since (m b0, * p ) p∈N is non-decreasing, we also have that εm b0, * p ≥ 1 for all p ∈ N with p > Γ(ε). As M b0, * 0 = 1, we find that the sequence p → ε p M b0, * p is decreasing for 0 ≤ p ≤ Γ(ε) and non-decreasing for p ≥ Γ(ε). Consequently, ε Γ(ε) M b0, * Γ(ε) = e −ω M b 0 , * ( 1 ε ) . We remark that these properties of Γ, which will be frequently used in the rest of the proof, depend crucially on the assumptions p! 1/b0 ≺ M and (M.4) b0 . Note that, for all t ∈ R\{0} and p ∈ N it holds that ψ(λ p t) = 1 if p < Γ(Ah b0 |t|) and ψ(λ p t) = 0 if p ≥ Γ(Ah b0 |t|/2). For all x ∈ R d and t ∈ R\{0} with |t| small enough it thus follows and, by (7), In order to estimate the inner sums of P (D)S 1 , T 1 and T 2 , let n ≤ l, k ≤ m and j ≤ k be arbitrary. Note that For all x ∈ R d and t ∈ R\{0} with |t| small enough it holds that where we recall that L = H b0 A. This implies (9). Finally, we show (10). For all x ∈ R d and t ∈ R\{0} with |t| small enough it holds that Let n ≤ l be arbitrary. By using similar arguments as above, we find that for all x ∈ R d and t ∈ R\{0} with |t| small enough which implies (10) and finishes the proof of the claim.
Next, we suppose that p! 1/b0 ≍ M . For ϕ ∈ D M,h K we define Lemma 4.3 implies that there is L > 0 (independent of h > 0) such that the series S(x, t) converges in C ∞ (R d × (−2/(Lh b0 ), 2/(Lh b0 ))) and that for all α ∈ N d 0 and l ∈ N 0 there is C > 0 such that Furthermore, S satisfies (11) and P (D)S = 0 on R d × (−2/(Lh b0 ), 2/(Lh b0 )) by (7). Choose χ ∈ D(R) with supp χ ⊂ (−2/(Lh b0 ), 2/(Lh b0 )) and such that χ = 1 Then, Φ ∈ D(R d+1 ) with supp Φ ⊂ K × R. Moreover, P (D)Φ = 0 on R d × (−1/(Lh b0 ), 1/(Lh b0 )) and the exact same argument as in the first part of the proof shows that Φ satisfies (i).  There is L > 0 such that for all h > 0 the following property holds: Let Φ ∈ D(R d+1 ) be such that Remark 4.6. We would like to point out that Proposition 4.5 is not needed to prove the main results of this article in Section 5 and Section 6 below. However, we believe this result is interesting in its own right as it provides a complete characterization of ultradifferentiable classes in terms of (almost) zero solutions of P (D) (provided that a 0 = b 0 ); see Theorem 4.8 and Remark 4.9 below.
We need the following lemma to prove Proposition 4.5.
Proof of Proposition 4.5. By Lemma 2.8(ii), we have that either p! 1/a0 ≺ M or p! 1/a0 ≍ M . We only consider the case p! 1/a0 ≺ M as the case p! 1/a0 ≍ M can be treated similarly. Let E be the fundamental solution of P (D) constructed in Proposition 3.1. Let l ∈ N 0 be arbitrary. Choose r > 0 such that supp Φ ⊂ R d × [−r, r]. There are C, S > 0 such that There are C 1 , L 1 > 0 such that The above two inequalities and (13) yield that there are C 3 , L 3 > 0 (with L 3 independent of l) such that (19) sup Lemma 4.7 implies that for all x ∈ R d and α ∈ N d The inequalities (18) and (19) imply that for all α ∈ N d Furthermore, there are l ∈ N 0 and C > 0 such that Proof. Let ψ ∈ D [−ε,ε] be such that ψ = 1 on a neighborhood of 0. For n ∈ N and ϕ ∈ D K we define The result can now be shown in a similar way as Proposition 4.1 but starting from S = S n with n large enough instead of the function defined in (8).

Boundary values of zero solutions of P (D)
In this section we show that zero solutions of P (D) satisfying suitable growth estimates near R d have boundary values in a given ultradistribution space. We start with the following fundamental definition.
provided that bv(f ) ∈ D ′[M] (X) exists. Since The second statement is clear. We This definition is independent of the chosen sequence (V l ) l∈N0 . Obviously, it holds Since Q m = 1, we have the recursion relations (21) t j = P (m) , j = 0, We are ready to discuss the existence of ultradistributional boundary values of zero solutions of P (D).
Proof. This follows by multiple partial integrations.
Finally, we discuss the existence of distributional boundary values of zero solutions of P (D).
We then have:

Main results
This section is devoted to the two main results of this article. Firstly, we give various characterizations of zero solutions of P (D) that admit a boundary value in a given ultradistribution space. Secondly, we represent the m-fold Cartesian product of an ultradistribution space as the quotient of certain spaces of zero solutions of P (D).

Proof. It suffices to show that for every relatively compact open subset
The claim yields that f * x u n ∈ C ∞ P (Y × (−r, r)) for n = 1, 2. Since p! 1/γ0 ≺ M (p! 1/γ0 ⊂ M ), we have that G(D x )(f * x u 1 ) ∈ C ∞ P (Y × (−r, r)) (cf. the proof of Lemma 5.2(ii)). Hence, −r, r)). We now show the claim. Let H be the constant occurring in (M.2). Our assumption yields that there are T j ∈ D ′[M] (X), j = 0, . . . , m − 1, such that K2s is an (FS)-space ((DFS)-space) and the set K2s ) ′ , we obtain that there is h ′ > 0 (h ′ ∈ R) such that (22) holds in the topology of uniform convergence on the unit ball of · E M,h ′ (K2s) (in the Roumieu case we used Lemma 2.11(ii)). Furthermore, by Lemma 2.12, we may assume without loss of generality that the weight sequence ( p l=0 h ′ l M p ) p∈N0 satisfies (M.2) (with 2H instead of H). We obtain that there is a net (c t ) 0<t<r of positive numbers tending to zero such that for all ψ ∈ D
), we infer from (23) that for j = 0, . . . , m − 1 r)). Proposition 6.2 implies the next result; it can also be shown directly and easier by using the Schwartz parametrix method instead of Lemma 2.13.
is contained in the range of bv m n for n = 1, 2 (provided that a 0 = b 0 ). It suffices to consider the case X = R d .

Lemma 4.7 implies that
which yields the result.
Proof of Proposition 6.4. Since γ 0 ≤ a 0 , we have that p! 1/a0 ≺ M in the Beurling case. The bound (4) implies that for all l ∈ N 0 it holds that fixed. Hence, the dual pairing in the definition of f makes sense. It is clear that f ∈ C ∞ P (R d+1 \R d ). We now show that f ∈ C ∞ P,[M],a0 (R d+1 \R d ). By Lemma 2.8(ii), we have that either p! 1/a0 ≺ M or p! 1/a0 ≍ M . If p! 1/a0 ≍ M , by the definition of C ∞ P,[M],a0 R d+1 \R d , there is nothing left to show. Therefore, we assume that p! 1/a0 ≺ M . There is K ⋐ R d such that for some h > 0 (for all h > 0) there is C > 0 such that for all j = 0, . . . , m − 1 The bound (4) yields that there are C 1 , L 1 , S > 0 such that for all α ∈ N d 0 and j = 0, . . . , m − 1 Furthermore, (M.2) implies that there are C 2 , L 2 > 0 such that For all x ∈ R d and t ∈ R\{0} with |t| small enough it holds that The corresponding statement for bv m 2 follows from it by (21).
We are ready to prove the two main results of this article.
The following statements are equivalent: ( Proof. is exact and bv m n is a topological homomorphism. Proof. The mappings bv m 1 and bv m 2 are continuous by Proposition 5.3 and Proposition 5.5, respectively. Proposition 6.2 yields that ker bv m n = C ∞ P (V ). Next, we show that bv m n is surjective. To this end, we shall use some basic facts about the derived projective limit functor; we refer to the book [36] for more information. Choose a sequence (V l ) l∈N0 of relatively compact open subsets of V such that V l ⋐ V l+1 and V = l∈N0 V l . Set X l = V l ∩ X and bv m n,l = bv m n : We define the projective spectra (with restriction as spectral mappings) and the morphism (bv m n,l ) l∈N0 : Y → Z . We need to show that the mapping We end this section by giving the analogues of Theorem 6.6 and Theorem 6.7 for distributions.
The following statements are equivalent: Proof. This can be shown in a similar way as Theorem 6.6.
The mappings bv m 1 and bv m 2 for distributions are defined similarly as in Definition 6.1. We then have: is exact and bv m n is a topological homomorphism. Proof. This can be shown in a similar way as Theorem 6.7.
Next, given a semi-elliptic polynomial P , we present a sufficient geometric condition on an open set V ⊆ R d+1 to be P -convex for supports [9,15]. A function f : V → R is said to satisfy the minimum principle in a closed subset F of R d+1 if for all K ⋐ F ∩ V it holds that min x∈K f (x) = min where ∂ F K denotes the boundary of K in F (in case of F ∩ V = ∅ the condition is vacuous).
Corollary 7.4 implies that Theorem 6.6 and Theorem 6.7 are applicable to every semi-elliptic polynomial with deg x1 P = · · · = deg x d P = n. Moreover, Proposition 7.5 gives a sufficient condition on the open sets V ⊂ R d+1 for which Theorem 6.7 is valid. We now discuss this class of polynomials in some more detail. We distinguish three cases: (i) m = n = deg P : Then, P is elliptic and, conversely, every elliptic polynomial is of this form. Corollary 7.4 implies that Finally, note that Theorem 1.1 stated in the introduction follows from Theorem 6.6, Theorem 6.7 and the above remarks.