The Overdetermined Cauchy Problem for $\omega$-ultradifferentiable Functions

In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $\omega$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for non-quasianalytic weight functions $\omega$. We show that existence of solutions of the Cauchy problem is equivalent to the validity of a Phragm\'en-Lindel\"of principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.


Introduction
In this paper we consider the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some classes of ω−ultradifferentiable functions, in the sense of Braun, Meise and Taylor [BMT].
We consider a continuous increasing weight function ω : [0, +∞) → [0, +∞) satisfying With respect to weight functions considered in [BMT], we weakened their condition (γ) lim t→∞ log(1 + t) ω(t) = 0, by condition (γ) ′ above, in the spirit of the original paper of Björck [Bj]. For this reason in Section 2 we briefly retrace the paper of Braun, Meise and Taylor [BMT], defining the spaces E {ω} and E (ω) of ω-ultradifferentiable functions of Roumieu and Beurling type, but enlightening the results that are still valid with the weaker condition (γ) ′ and those ones which need the stronger condition (γ) (cf. also [Fi, G]). It comes out that condition (γ) is needed in the Roumieu case E {ω} , while condition (γ) ′ is sufficient in the Beurling case E (ω) ; in particular, the space E(Ω) of C ∞ functions on an open set Ω ⊂ R N can be viewed as E (ω) (Ω) for ω(t) = log(1 + t). The utility of weakening condition (γ) by condition (γ) ′ is clear, for instance, in the forthcoming paper [BJO], for the description of the space S ω of ω-ultradifferentiable Schwartz functions. In Section 3 we investigate the overdetermined Cauchy problem in the Beurling case. To be more precise, we settle the Cauchy problem in the frame of Whitney ω−ultradifferentiable functions, in the spirit of [N2,BN1,BN3], in order to bypass the question of formal coherence of the data, which naturally arises in the overdetermined case.
Indeed, in the classical Cauchy problem for a linear partial differential equation with initial data on a hypersurface, smooth initial data together with the equation allow to compute the Taylor series of a smooth solution at any given point of the hypersurface.
This leads, in the case of systems of linear partial differential equations, to the notion of formally noncharacteristic hypersurface that was considered in [AH LM, AN,N2].
In the case of overdetermined systems, the question of the formal coherence of the data would be particularly intricate, so that the above remarks suggest further generalizations of the Cauchy problem, where the assumption that the initial data are given on a formally non-characteristic hypersurface is dropped, and we allow formal solutions (in the sense of Whitney) of the given system on any closed subset as initial data.
Using Whitney functions, we can thus consider a more general framework in which two quite arbitrary sets are involved. We take K 1 and K 2 closed convex subsets of R N with K 1 K 2 andK j = K j for j = 1, 2, thinking at K 1 as the set where the initial data are given, and at K 2 as the set where we want to find a solution of the following Cauchy problem: where A 0 (D) is an a 1 ×a 0 matrix of linear partial differential operators with constant coefficients, ϕ ∈ W are the given Cauchy data in the Whitney classes of ω−ultradifferentiable functions of Beurling type on K 1 and K 2 respectively, and u| K1 ≡ ϕ means that they are equal in the Whitney sense, i.e. with all their derivatives. It comes out (see Section 3) that, in order to find a solution u ∈ W (ω) K2 a0 of the Cauchy problem (1.1), the function f must satisfy some integrability conditions. These may be written as for a matrix A 1 (D) of linear partial differential operators with constant coefficients obtained by a Hilbert resolution of M = coker ( t A 0 (ζ) : P a1 → P a0 ) : where P = C[ζ 1 , . . . , ζ N ]. The rows of the matrix A 1 (D) give a system of generators for the module of all integrability conditions for f that can be expressed in terms of partial differential operators, and if A 1 (ζ) ≡ 0 we say that the Cauchy problem is overdetermined.
We prove in Theorem 3.18 that the Cauchy problem (1.1), for f satisfying (1.2), admits at least a solution if and only if the following Phragmén-Lindelöf principle holds for all ℘ ∈ Ass(M) and V = V (℘): then it also satisfies: where H K is the supporting function of the compact set K, K j α α is a sequence of compact subsets of K j with K j α ⊂K j α+1 and K j = ∪ α K j α for j = 1, 2, V is the complex characteristic variety of P/℘ defined by and psh(V ) is the set of all plurisubharmonic functions on V (in the sense of Definition 3.19).
Relating the existence of solutions of the Cauchy problem to the validity of a Phragmén-Lindelöf principle may be very useful. For instance, in the case of a Gevrey weight ω(t) = t 1/s , s > 1, it was found in [BM] a complete characterization of algebraic curves V that satisfy the Phragmén-Lindelöf principle, by means of Puiseux series expansions on the branches of V at infinity: it comes out that the exponents and coefficients of the Puiseux series expansions are strictly related to the Gevrey order s. This implies that, looking at the Puiseux series expansions at infinity of the complex characteristic variety associated to the system A 0 (D), we can establish in which (small) Gevrey classes the Cauchy problem admits at least a solution and in which classes it doesn't work. Since Puiseux series expansions can be computed by several computer programs, such as MAPLE for instance, this characterization may be very useful.

Ultradifferentiable functions
In the present section we follow [BMT], enlightening when condition (γ) below can be weakened by condition (γ) ′ of Definition 2.3.
We recall, from [BMT], the following: be a continuous increasing function. It will be called a nonquasianalytic weight function ω ∈ W if it has the following properties: For z ∈ C N we write ω(z) for ω(|z|), where |z| = N j=1 |z j |. Remark 2.2. Condition (β) is the condition of non-quasianalyticity and it will ensure, in the following, the existence of functions with compact support (cf. Remark 2.20).
(1) If ω is a quasianalytic weight, i.e. ω ∈ W is a weight function except that it doesn't satisfy (β), then the function σ can be constructed as above, except that it may not satisfy (β).
(2) We can construct σ so that it coincides with ω on a given arbitrarily large bounded interval.
Proposition 2.11. Let ω ∈ W and, for j ∈ N, let g j : [0, +∞) → [0, +∞) satisfying g j (t) = o(ω(t)) as t tends to ∞. Then there exists σ ∈ W with the following properties: Lemma 2.12. Let ω ∈ W. Then there exists a nonzero function g ∈ S(R) with support in (−∞, 0] for which the Fourier transform g satisfies However, the following two propositions for the existence of functions with compact support are valid also for ω ∈ W ′ : Proposition 2.14. Let ω ∈ W ′ . Then for each N ∈ N there exists δ N > 0 such that for every ε > 0 there exists H ∈ C ∞ (R N ), H = 0, with Proof. See [BMT], Corollary 2.5 and Remark after Corollary 2.6.
The difference, in the next two lemmas, between taking ω ∈ W or ω ∈ W ′ , will be crucial in the sequel for the choice of ω ∈ W when defining the space of ω-ultradifferentiable functions of Roumieu type and ω ∈ W ′ for defining the space of ω-ultradifferentiable functions of Beurling type.
Lemma 2.16. Let ω ∈ W and let f ∈ D(R N ). If there exists B > 0 such that If (2.3) holds for f ∈ D(R N ) and B > 0 then there is D > 0, depending only on ω, N and B, and there is L > 0 depending only on ω and N , such that for K = supp f and m N (K) its Lebesgue measure, we have that Proof. See [BMT], Lemma 3.3.
Lemma 2.17. Let ω ∈ W ′ and f ∈ D(R N ). If there is B > 0 such that If (2.4) holds for f ∈ D(R N ) and B > 0 then there is D > 0, depending only on ω, N and B, and there is L > 0 depending only on ω and N , such that for K = supp f and m N (K) its Lebesgue measure, we have that Proof. The proof of (2.4) is the same of that of (2.3) in Lemma 2.16 (see [BMT,Lemma 3.3]). So we prove (2.5). By condition (α) there is L > 0 such that Let now z ∈ C N be given, let l be the index with and assume |z l | > 1. Write then In view of (2.4), this implies that, for all j ∈ N 0 : Now, note that for every x > 0 there exists j ∈ N 0 such that j ≤ Bx < j + 1, and hence from (2.6) and (γ) By passing to the infimum over all j ∈ N 0 in (2.7) and by using (2.8) we obtain: Definition 2.18. Let ω ∈ W and let K ⊂ R N be a compact set. For λ > 0 we define the Banach space endowed with the topology of the inductive limit.
For an open set Ω ⊂ R N we define then where the inductive limit is taken over all compact subsets K of Ω. We endow D {ω} (Ω) with the inductive limit topology.
The elements of D {ω} (Ω) are called ω−ultradifferentiable functions of Roumieu type with compact support.
Definition 2.19. Let ω ∈ W ′ and let K ⊂ R N be a compact set. For D λ (K) defined as in (2.9), we set endowed with the topology of the projective limit.
For an open set Ω ⊂ R N we define where the inductive limit is taken over all compact subsets of Ω. We endow D (ω) (Ω) with the inductive limit topology.
The elements of D (ω) (Ω) are called ω−ultradifferentiable functions of Beurling type with compact support.
Remark 2.20. As in [BMT], we have the following: (3) We say that two functions ω and σ are equivalent if ω = O(σ) and σ = O(ω). Note that if ω ≤ σ ≤ Cω for some C > 0 and if ψ(x) = σ(e x ), then With this formula, it's easy to see that definitions and most theorems in the sequel don't change if ω is only equivalent to a weight function.
(3) ⇒ (1) : By (3) and (γ) ′ we have that for all λ > 0, taking k ∈ N with k > λ, there exist C λ , C ′ λ > 0 such that For k > N + 1 b + λ the above integral is finite and hence there exists C ′′ λ > 0 such that To prove that f ∈ D(K) note that (3) and (γ) ′ imply that for every k ∈ N there exists C k > 0 such that Therefore for every n ∈ N there exists C n > 0 such that By the classical Paley-Wiener Theorem we finally have that f ∈ D (K) and hence the theorem is proved.
Remark 2.23. The inequality (2.12) enlightens the sufficiency of condition (γ) ′ on the weight ω: by the arbitrariety of λ we can allow a fixed b > 0 to make the integral convergent. On the contrary, in the Roumieu case (Theorem 2.21) we need condition (γ), i.e. log( For a sequence P = (p n ) n∈N of continuous functions p n : C N → R, we define Let ω be a weight function and K ⊂ R N a convex compact set. Define For an open convex set Ω ⊂ R N and a convex compact exhaustionK 1 ⊂K 2 ⊂K 3 ⊂ . . . of Ω, define also From the Paley-Wiener Theorems 2.21-2.22 (cf. also [BMT,Prop. 3.5]) we get: Proposition 2.24. We have the following: The isomorphisms are given by the Fourier-Laplace transform.
As in [BMT], we can collect some more properties on these spaces of ω-ultradifferentiable functions with compact support, taking ω ∈ W ′ in the Beurling case and ω ∈ W in the Roumieu case.
Corollary 2.25. Let K ⊂ R N be compact and Ω ⊂ R N be open.
. Then we have: Proposition 2.28. Let ω, σ ∈ W with σ = o(ω). Then the inclusions are continuous and sequentially dense for each open set Ω ⊂ R N .
Let us now introduce the algebras of ω-ultradifferentiable functions of Beurling and of Roumieu type with arbitrary support. Here again we need ω ∈ W in the Roumieu case, while we can allow ω ∈ W ′ in the Beurling case.
Definition 2.29. For ω ∈ W and an open set Ω ⊂ R N , we define For ω ∈ W ′ and an open set Ω ⊂ R N we define The topology of E {ω} (Ω) is given by first taking the inductive limit over all m ∈ N for each compact K ⊂ Ω and then taking the projective limit for K ⊂ Ω, while E (ω) (Ω) carries the metric locally convex topology given by the seminorms p K,m where K is a compact subset of Ω and m ∈ N.
Notation. We shall write E * (resp. D * ) if a statements holds for both E {ω} (resp. D {ω} ) and E (ω) (resp. D (ω) ), taking ω ∈ W in the Roumieu case and ω ∈ W ′ in the Beurling case. Remark 2.31. In general the spaces of ω-ultradifferentiable functions defined as in Definition 2.29 are different from the Denjoy-Carleman classes of ultradifferentiable functions as defined in [K] (cf. [BMM]).
As in [BMT], we have the following properties of the spaces E * (Ω): Proposition 2.32. E * (Ω) is a locally convex algebra with continuous moltiplication.
Let us now introduce the ω−ultradistributions of Beurling and of Roumieu type with compact and arbitrary support, taking ω ∈ W in the Roumieu case and ω ∈ W ′ in the Beurling case. (1) The elements of D ′ {ω} (Ω) are called ω−ultradistributions of Roumieu type.
(2) For an ultradistribution T ∈ D ′ {ω} (Ω) its support supp T is the set of all points such that for every neighbourhood U there is ϕ ∈ D {ω} (U ) with < T, ϕ > = 0.
Definition 2.39. Let ω ∈ W ′ and Ω ⊂ R N an open set.
(2) For an ultradistribution T ∈ D ′ (ω) (Ω) its support supp T is the set of all points such that for every neighbourhood U there is ϕ ∈ D (ω) (U ) with < T, ϕ > = 0.
Remark 2.40. By Proposition 2.28, the definition of support of an ultradistribution T doesn't depend on the choice of the class D {ω} (Ω) for ω ∈ W (resp. D (ω) (Ω) for ω ∈ W ′ ) as long as it contains T . In particular, if T is a distribution T ∈ D ′ (Ω), then the support defined above is the usual one.
As in [BMT,Prop. 5.3], the elements of E ′ * (Ω) can be identified with distributions in D ′ * (Ω) with compact support: Proposition 2.41. An ultradistribution T ∈ D ′ * (Ω) can be extended continuously to E * (Ω) iff supp T is a compact subset of Ω.
Definition 2.43. For an ultradistribution µ ∈ E ′ * (R N ), and for f ∈ E * (R N ) we define the convolution As in [BMT,Prop. 6.3]: Proposition 2.44. The convolution map is continuous.
This holds, in particular, for K equal to the convex hull of supp µ. Moreover, Conversely, if g is an entire function on C N that satisfies (2.14), i.e.

Proof. Let us first prove that if
To this aim we assume, without loss of generality, that 0 ∈K and define Then µ, f t = 0. Let us to prove that We have that µ ∈ E ′ (ω) (R N ), so µ is a linear and continuous function on E (ω) (R N ) and to prove (2.16) it's sufficient to prove that f t → f in E (ω) (R N ). Therefore, fixK ⊂ R N compact, m ∈ N and prove that (2.17) sup Indeed, We observe that (1−t α ) → 0 for t → 1 − and sup To estimate also the first addend of (2.18) let us remark that it's not restrictive to assume 0 ∈K, since we can enlargeK. Therefore, denoting by ch(K) the convex hull ofK, by the Lagrange Theorem we have that there exists ξ ∈ ch(K) on the segment of extremes x and tx, such that for some C > 0. However, . Then, from (2.18), we have obtained Therefore (2.16) holds true.

The Cauchy problem for overdetermined systems.
In this section we consider the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in the classes of ω−ultradifferentiable functions of Beurling type defined in the previous section.
To bypass the question of formal coherence of the initial data, that could be especially intricate in the overdetermined case (cf. [AH LM], [AN], [N2]), we consider initial data in the Whitney sense, in the spirit of [N2], [BN1], [BN3].
Let K 1 and K 2 be closed and convex subsets of R N such that K 1 K 2 andK j = K j for j = 1, 2.
For ω ∈ W ′ we denote by I (ω) (K 2 , Ω) the subspace of functions in E (ω) (Ω) which vanish of infinite order on K 2 : Definition 3.1. Let ω ∈ W ′ . We define the space W (ω) K2 of Whitney ω-ultradifferentiable functions on K 2 by the exact sequence K2 ≃ E (ω) (Ω) I (ω) (K 2 , Ω). In the same way we define W by the formal substitution ζ j ↔ 1 i ∂ j . Given an a 1 × a 0 matrix A 0 (D) with polynomial entries, we can thus consider the corresponding operator A 0 (D). We want to solve, in the Whitney's sense, the Cauchy problem where u| K1 ≡ 0 means that u vanishes with all its derivatives on K 1 .
Let us remark that if t Q(ζ) : P a1 → P is such that then, in order to solve the Cauchy problem (3.1), f must satisfy the integrability condition Since P is a Noetherian ring, the collection of all vectors t Q(ζ) satisfying (3.2) form a finitely generated P−module. So we can insert the map t A 0 (ζ) : P a1 → P a0 into a Hilbert resolution: where M = coker t A 0 (ζ) = P a0 t A 0 (ζ)P a1 and the matrix t A 1 (ζ) is obtained from a basis of the integrability conditions (3.2). The sequence is exact, i.e. Im t A j = Ker t A j+1 . Therefore a necessary condition to solve (3.1), is that f satisfies the following integrability condition: Moreover, every necessary condition for the solvability of (3.1), which can be expressed in terms of linear partial differential equations, is a consequence of (3.4) (cf. [N2]).
Definition 3.2. If A 1 (ζ) ≡ 0 then the Cauchy problem (3.1) is called overdetermined and, to solve it, the condition (3.4) has to be satisfied.
Let us remark that if u solves (3.1), then also f must vanish with all its derivatives on K 1 , so that we look (3.5) Remark 3.3. By Whitney's extension theorem it's not restrictive to consider zero Cauchy data (see [B], [BBMT], [MT], [M]).
Let us denote by I (ω) (K 1, K 2 ) the space of Whitney ω-ultradifferentiable fuctions on K 2 which vanish of infinite order on K 1 : The Cauchy problem (3.1)-(3.5) is then equivalent to: Remark 3.4. By the isomorphisms we have: (1) uniqueness of solutions of the Cauchy problem (3.6) is equivalent to the condition Ext 0 P (M, I (ω) (K 1 , K 2 )) = 0; (2) existence of solutions of (3.6), is equivalent to the condition Ext 1 P (M, I (ω) (K 1 , K 2 )) = 0; (3) existence and uniqueness of a solution of (3.6), is equivalent to the condition Ext 0 P (M, I (ω) (K 1 , K 2 )) = Ext 1 P (M, I (ω) (K 1 , K 2 )) = 0. Remark 3.5. Remark 3.4 enlightens the algebraic invariance of the problem: uniqueness and/or existence of solutions of the Cauchy problem (3.6) depend only on the module M and not on it's presentation by a particular matrix t A 0 (D).
Note also that we have the short exact sequence As in [N1] (cf. also [B,BN1,BN3]) we have that W (ω) Ki , for i = 1, 2, are injective P−modules, i.e. the following holds: Lemma 3.6. Let ω ∈ W ′ , M a P−module of finite type and K a compact convex subset of R N . Then Ext j P (M, W Ki ) = 0 for i = 1, 2 and for all j ≥ 1. By Lemma 3.6, the complex (3.7) reduces to: Remark 3.7. From Remark 3.4 and the above considerations, it follows that uniqueness and/or existence of solutions of the Cauchy problem (3.6) is related to injectivity and/or surjectivity of the homomorphism The injectivity of (3.9) is equivalent to the fact that the dual homomorphism has a dense image. Moreover, surjectivity is equivalent to have a dense and closed image. But (3.9) has a closed image if and only if (3.10) has a closed image (cf. [Gr], Ch. IV, § 2, n. 4, Thm. 3), so that the surjectivity of (3.9) is equivalent to the fact that the dual homorphism (3.10) is injective and has a closed image.
By Remarks 3.4 and 3.7, and [N2, Prop. 1.1-1.2], we have that: Proposition 3.8. Let ω ∈ W ′ and K 1 , K 2 closed convex subsets of R N with K 1 K 2 ,K j = K j for j = 1, 2. Let M be a unitary P−module of finite type and denote by Ass(M) the set of all prime ideals associated to M.
The overdetermined Cauchy problem (3.6) is thus reduced to the study of the dual homomorphism Let us start with some preliminary results.
Lemma 3.12. Let ω ∈ W ′ , ℘ a prime ideal of P and K ⊂ R N a convex and closed set withK = K. Then we have the following isomorphism: (3.14) Ext 0 P P ℘, W where p 1 (ζ), . . . , p r (ζ) are generators of ℘.
Proof. For any closed subspace F of a Fréchet space E, the dual F ′ of F is isomorphic (cf. [MV,Prop. 6.14]) to: where F 0 is the annihilator of F , defined by Then, since Ext 0 and, by Lemma 3.11, Observe that, for V (℘) defined as in (1.3), we have Therefore the Fourier-Laplace transformT (ζ) of an element T ∈ Ext 0 P P ℘, W (ω) K 0 is an entire function which satisfies:T (ζ) = T, e −i<·,ζ> = 0 ∀ζ ∈ V (℘).
Moreover we have that for every k 0 > 0 there exists k 1 > 0 such that Indeed,

Now observe that
for some c > 0 and Moreover, by definition of supremum, for all ε > 0 there existsx ∈ K such that x, Im ζ > H K (Im ζ) − ε.
So, choosing suchx in (3.20) we have for some c ′ > 0, hence there exists k 1 > 0 such that Furthermore, by Lemma 2.5 we have that for some K > 0 and hence for every k 0 > 0 there exists k ′ 1 > 0 such that Therefore (3.19) is proved. We can therefore apply the Ehrenpreis Fundamental Theorem (see [H,Thm. 7.7.13], and [B, G] for more details) and obtain that we can choose the entire functions F h satisfying so there exist C ′′ , C ′′′ > 0 and α ′ ∈ N such that |F h (ζ)| ≤ C ′′ e HK α (Im ζ)+α ′ ω(ζ) ≤ C ′′′ e HK α ′′ (Im ζ)+α ′′ ω(ζ) with α ′′ = max{α, α ′ }. Hence, by the Paley-Wiener Theorem 2.46: We have thus proved that if T ∈ Ext 0 P P ℘, W (ω) K 0 , then This result implies that T ∈ ℘(D) ⊗ E ′ (ω) (K), and so, by (3.15), Let us define O ψα (C N ) as the space of holomorphic functions u on C N which satisfy for some C > 0 and for all ζ ∈ C N : We can then consider the inductive limit From the Paley-Wiener Theorem 2.46, by Fourier-Laplace transform we have the following isomorphism: . Therefore, from Lemma 3.12: Let V be a reduced affine algebraic variety. Denote by O ψα (V ) the space of holomorphic functions on V (i.e. complex valued continuous functions on V which are restrictions of entire functions on C N ) that satisfy (3.21) for some α ∈ N, C > 0 and for all ζ ∈ V . Consider then the inductive limit We have the following: Proposition 3.13. Let ω ∈ W ′ , ℘ a prime ideal of P with associated algebraic variety V = V (℘), and K a closed convex subset of R N withK = K. Then we have a natural isomorphism: Proof. By (3.22) we have to prove the following isomorphism: First of all we prove that the homomorphism Since f satisfies (3.21) by assumption, from the Ehrenpreis Fundamental Theorem [H,Thm. 7.7.13] (see also [B], [G] for more details), we can choose f h satisfying (3.21) too, hence f h ∈ O ψ (C N ) and this implies that f ∈ ℘ ⊗ O ψ (C N ). So we have obtained that f is the zero element of O ψ (C N ) ℘(D) ⊗ O ψ (C N ), proving the injectivity of the homomorphism (3.23).
On the other hand, the homomorphism (3.23) is surjective: if f ∈ O ψ (V ), then f ∈ O(C N ) and satisfies (3.21) for some α ∈ N, C > 0 and for all ζ ∈ V . By the Ehrenpreis Fundamental Theorem [H,Thm. 7.7.13], there exist g ∈ O(C N ), with f = g on V , and two constants C ′ > 0 and n ∈ N such that Since the right-hand side is finite because f satisfies (3.21) on V , we have that ∀ζ ∈ C N for some C ′′ , C ′′′ > 0 and α ′ ∈ N. So g ∈ O ψ (C N ).
Proposition 3.13 will be crucial in the study of the homomorphism (3.12) related to the study of existence and/or uniqueness of solutions of the Cauchy problem (3.6).
To this aim we take K 1 and K 2 closed and convex sets, withK j = K j for j = 1, 2, and K 1 K 2 . Then we define, for j = 1, 2 : ψ j α (ζ) := H K j α (Im ζ) + αω(ζ) for K j α compact convex set with K j α ⊂K j α+1 and ∪ α K j α = K j , for each j = 1, 2. We consider the inductive limits From the above considerations we have the following: Remark 3.14. The study of the homomorphism (3.12) is reduced to the study of the homomorphism By Proposition 3.9 the existence of solutions of the Cauchy problem (3.6) is equivalent to the surjectivity of the homomorphism (3.11). But (3.11) has always a dense image, by the following: Lemma 3.15. Let ω ∈ W ′ , ℘ a prime ideal and K 1 , K 2 closed convex subsets of R N with K 1 K 2 ,K j = K j for j = 1, 2. Then the homomorphism has always a dense image.
Remark 3.16. By Proposition 3.9 and Lemma 3.15, the Cauchy problem (3.6) admits at least a solution if and only if the homomorphism (3.11) has a closed image, i.e. if and only if the dual homomorphism (3.12) has a closed image, by [Gr, Ch. IV, § 2, n. 4, Thm. 3] (see also Remark 3.7).
By Proposition 3.13 we thus have that the Cauchy problem (3.6) admits at least a solution if and only if the homomorphism (3.25) has a closed image.
Summarizing, by Remark 3.16 and Theorem 3.17, we have the following: Theorem 3.18 (Phragmén-Lindelöf principle for the existence of solutions). Let ω ∈ W ′ . The Cauchy problem (3.6) admits at least a solution if and only if the following Phragmén-Lindelöf principle holds for all ℘ ∈ Ass(M) and V = V (℘) : then it also satisfies: Let us now recall the definition of plurisubharmonic functions on an affine algebraic variety V ⊂ C N : at the singular points of V . By psh(V ) we denote the set of all functions that are plurisubharmonic on V .
Also the problem of existence of a unique solution of the Cauchy problem (3.6) can be easily treated by the study of the dual homomorphism (3.12). In particular, by Propositions 3.10 and 3.13, we have: is an isomorphism.
By Theorem 5.2 of [BN3] we can finally state the following: Theorem 3.22. Let ω ∈ W ′ . The Cauchy problem (3.6) admits one and only one solution if and only if, for all ℘ ∈ Ass(M) and V = V (℘), one of the following equivalent conditions holds: Remark 3.23. Clearly condition (i) (resp. (ii), (iii)) of Theorem 3.22 implies condition (i) (resp. (ii), (iii)) of Theorem 3.17.