Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 261–284

CHARACTERIZATION OF THE ω-HYPOELLIPTIC CONVOLUTION OPERATORS ON ULTRADISTRIBUTIONS

J. Bonet, C. Fernández, and R. Meise

Universidad Politécnica, Dpto. Matemática Aplicada E-46071 Valencia, Spain; jbonet@pleiades.upv.es Universidad de Valencia, Dpto. Análisis Matemático E-46100 Burjasot (Valencia), Spain; Carmen.Fdez-rosell@uv.es Heinrich-Heine-Universität, Mathematisches Institut D-40225 Düsseldorf, Germany; meise@cs.uni-duesseldorf.de

Abstract. We achieve characterizations of those ultradistributions μ ∈ ℰ′_{(ω)}(R^{N}) (resp. ℰ′_{{ω}}(R^{N})) with compact support such that for each ultradifferentiable function f in ℰ_{(ω)}(R^{N}) (resp. in ℰ_{{ω}}(R^{N})) each solution ν ∈ 𝒟′_{(ω)}(R^{N}) (resp. in 𝒟′_{{ω}}(R^{N}) ) of the convolution equation μ ∗ ν = f belongs to the same class as f. These characterizations extend classical results of Ehrenpreis and Hörmander for distributions and Björck and Chou for ultradistributions.

Introduction

Let 𝒟′_{(ω)}(R^{N}) denote the (ω)-ultradistributions of Beurling type on R^{N} in the sense of Beurling–Björck [1] or Braun, Meise and Taylor [5]. Then each (ω)-ultradistribution with compact support μ ∈ ℰ′_{(ω)}(R^{N}) induces a convolution operator S_{μ} on 𝒟′_{(ω)}(R^{N}). S_{μ} or μ is called (ω)-hypoelliptic if each solution ν ∈ 𝒟′_{(ω)}(R^{N}) of the convolution equation

S_{μ}(ν) = μ ∗ ν = f

belongs in fact to ℰ_{(ω)}(R^{N}), when f ∈ ℰ_{(ω)}(R^{N}). In the classical case of distributions, i.e. ω(t) = log(1 + t), the hypoelliptic convolution operators were characterized by Ehrenpreis [8] in terms of a slowly decreasing condition and the location of the zeros in C^{N} of the Fourier–Laplace transform μ̂ of μ, while Hörmander [9] showed that the singular support of the solution ν of μ ∗ ν = f can be controlled by the singular supports of f and μ and the support of μ. For a complete characterization in this case we refer to Hörmander [10, II, 16.6].

The results of Ehrenpreis and Hörmander were extended in part to ultradistributions by Björck [1] and Chou [7]. Björck characterized the (ω)-hypoelliptic differential operators P (D), while Chou gave necessary as well as sufficient conditions for hypoelliptic convolution operators μ acting on 𝒟′_{{M_{p}}}(R^{N}) and 𝒟′_{(M_{p})}(R^{N}). For results concerning pseudo-differential operators on Gevrey classes we refer to Rodino [19, 3.2, 3.3] and the references given there.

1991 Mathematics Subject Classification: Primary 46F05, 46E10, 46F10; Secondary 35R50.

In the present paper we show that the classical characterizations of hypoellipticity extend completely to convolution operators acting on 𝒟′_{(ω)}(R^{N}) or on 𝒟′_{{ω}}(R^{N}), the space of {ω}-ultradistributions of Roumieu type. Using the abbreviation (∗) for (ω) or {ω}, and denoting the convex hull of a set K by ch(K), our main result can be stated as follows:

Theorem. For μ ∈ ℰ′_{∗}(R^{N}) the following assertions are equivalent:

(1) Whenever ν ∈ 𝒟′_{∗}(R^{N}) and μ ∗ ν ∈ ℰ_{∗}(R^{N}), then ν ∈ ℰ_{∗}(R^{N}).

(2) μ is slowly decreasing for (∗) and lim_{z∈V (μ̂), |z|→∞} (| Im z|/ω(z)) = ∞ if (∗) = (ω) (resp. lim inf_{z∈V (μ̂), |z|→∞} (| Im z|/ω(z)) > 0 if (∗) = {ω}).

(3) Ker S_{μ} ⊂ ℰ_{∗}(R^{N}) and μ is slowly decreasing for (∗).

(4) There exists F ∈ ℰ′_{∗}(R^{N}) and ψ ∈ 𝒟_{∗}(R^{N}) such that μ ∗ F = δ + ψ.

(5) There exists a fundamental solution E for S_{μ} satisfying

ch(sing supp_{∗}(E)) = − ch(sing supp_{∗}(μ)).

(6) For each open set Ω ⊂ R^{N} and u ∈ 𝒟′_{∗}(Ω − supp(μ) + ch(sing supp_{∗}(μ))) satisfying μ ∗ u ∈ ℰ_{∗}(Ω + ch(sing supp_{∗}(μ))), it follows that u ∈ ℰ_{∗}(Ω).

For μ ∈ ℰ′_{(ω)}(R^{N}) the proof of the theorem is given by appropriate modifications of the classical arguments using a recent characterization of those convolution operators S_{μ} which are surjective on 𝒟′_{(ω)}(R^{N}), given by Bonet, Galbis and Meise [2]. For μ ∈ ℰ′_{{ω}}(R^{N}) the proof can be reduced essentially to the Beurling case, due to results of Braun, Meise and Taylor [5] and Meise, Taylor and Vogt [15]. As we show, they imply that for each open set Ω in R^{N}, ℰ_{{ω}}(Ω) coincides topologically with the intersection of the spaces ℰ_{(σ)}(Ω), where σ runs through all weight functions satisfying σ(t) = o(ω(t)) as t tends to infinity. Consequently,

sing supp_{\{ω\}}(u) = \overline{∪\{sing supp_{(σ)}(u) : σ = o(ω)\}}

holds for each u ∈ 𝒟′_{{ω}}(R^{N}). From this it follows that μ ∈ ℰ′_{{ω}}(R^{N}) is {ω}- hypoelliptic if and only if there exists a weight function σ_{0} satisfying σ_{0} = o(ω) such that μ is (σ)-hypoelliptic for each weight function σ ≥ σ_{0} satisfying σ = o(ω).

Note that our results apply in particular to Gevrey ultradistributions and more generally to ultradistributions μ in ℰ′_{{M_{p}}}(R^{N}) or ℰ′_{(M_{p})}(R^{N}), provided that the sequence (M_{p})_{p∈N_{0}} satisfies the conditions (M1), (M2) and (M3) of Komatsu [11] (see Meise and Taylor [13, 3.11]) or the weaker conditions in Braun, Meise and Taylor [5, 8.9].

Acknowledgement. The research of the first two authors is supported by DGESIC project no. PB97-0333.

Characterization of the ω -hypoelliptic convolution operators 263

1. Preliminaries

In this preliminary section we introduce the nonquasianalytic classes, the spaces of ultradistributions and most of the notation that will be used in the sequel.

1.1. Definition. Let ω: [0, ∞[→ [0, ∞[ be a continuous function which is increasing and satisfies ω(0) = 0 and ω(1) > 0. ω is called a weight function if it satisfies the following conditions:

(α) ω(2t) ≤ K(1 + ω(t)) for some K ≥ 1 and for all t.

(β) ∫_{−∞}^{∞}(ω(t)/(1 + t^{2})) dt < ∞.
(γ) log(1 + t^{2}) = o(ω(t)) as t tends to ∞.
(δ) φ(t) = ω(e^{t}) is convex in R.

For a weight function ω we define ω̃: C^{N} → [0, ∞[ by ω̃(z) = ω(|z|) and again call this function ω, by abuse of notation. The Young conjugate of φ is defined by φ^{∗}(x) = sup_{y>0}{xy − φ(y)}.

1.2. Remark. (a) Each weight function ω satisfies lim_{t→∞}(ω(t)/t) = 0 by the remark following 1.3 of [14].

(b) For each weight function ω there exists a weight function σ satisfying σ(t) = ω(t) for all large t > 0 and σ | [0, 1[ = 0. This implies φ_{σ}(y) = φ_{ω}(y) for all large y, φ_{σ}^{∗∗} = φ_{σ}. From this it follows that all subsequent definitions do not change if ω is replaced by σ. On the other hand they also do not change if ω is replaced by ω + c, c some positive number.

(c) For each weight function ω there exists a weight function σ ∈ 𝒞^{∞} having bounded derivative on [0, ∞[ such that σ and ω are equivalent in the sense that ω ≤ σ ≤ Aω + C for some constants A, C > 0.

(d) Let ω be as in 1.1. Then, for x, y ∈ C^{N} we have, by [5, 1.2], ω(x + y) ≤ K(1 + ω(x) + ω(y)). Consequently, ω(x − y) ≥ ω(|x| − |y|) ≥ ω(x)/K − 1 − ω(y), for arbitrary x, y ∈ C^{N}.

1.3. Definition. Let ω be a weight function.

(a) For a set K ⊂ R^{N} and λ > 0 let

ℰ_{ω}(K, λ) := \{f ∈ C^{∞}(K) : ∥f∥_{K,λ} := sup_{x∈K} sup_{α} |f^{(α)}(x)| exp(−λφ^{∗}(\frac{|α|}{λ})) < ∞\}.

(b) For an open set Ω ⊂ R^{N} define

ℰ_{(ω)}(Ω) := proj_{←K⊂⊂Ω}proj_{m∈N}ℰ_{ω}(K, m)

= {f ∈ C^{∞}(Ω) : ∥f∥_{K,m} < ∞ for each K ⊂⊂ Ω and each m ∈ N},

and

ℰ_{\{ω\}}(Ω) := proj_{←K⊂⊂Ω}ind_{→m∈N}ℰ_{ω}(K, \frac{1}{m})
= {f ∈ C^{∞}(Ω) : for each K ⊂⊂ Ω there is m ∈ N with ∥f∥_{K,1/m} < ∞}.

The elements of ℰ_{(ω)}(Ω) (resp. ℰ_{{ω}}(Ω)) are called ω -ultradifferentiable functions of Beurling (resp. Roumieu) type on Ω. We write ℰ_{∗}(Ω), where ∗ can be either (ω) or {ω} at all occuring places.

(c) For a compact set K in R^{N} we let

𝒟_{∗}(K) := {f ∈ ℰ_{∗}(R^{N}) : supp(f) ⊂ K},

endowed with the induced topology. For an open set Ω ⊂ R^{N} and a fundamental sequence (K_{j})_{j∈N} of compact subsets of Ω we let

𝒟_{∗}(Ω) := ind_{j→}𝒟_{∗}(K_{j}).

The dual 𝒟′_{∗}(Ω) of 𝒟_{∗}(Ω) is endowed with its strong topology. The elements of 𝒟′_{(ω)}(Ω) (resp. 𝒟′_{{ω}}(Ω)) are called ω -ultradistributions of Beurling (resp. Roumieu) type on Ω.

1.4. Remark. (a) By Meise, Taylor and Vogt [15, 3.3], for each open set Ω in R^{N}, the seminorms

∥ ∥_{K,σ}: f → sup_{x∈K} sup_{α∈N_{0}^{n}} |f^{(α)}(x)| exp(−φ_{σ}^{∗}(|α|)),

where K is any compact set in Ω and σ is a weight function satisfying σ = o(ω), form a fundamental sequence of semi-norms for ℰ_{{ω}}(Ω).

(b) For each compact set K in R^{N}, 𝒟_{{ω}}(K) is a (DFN)-space by Braun, Meise and Taylor [5, 3.6]. A fundamental system of bounded sets is given by

B_{m} :={φ ∈ 𝒟_{{ω}}(K) : |φ|_{K,m} := ∫_{R^{N}} |φ̂(t)|e^{ω(t)/m} dt ≤ 1},

where the Fourier transform φ̂: R^{N} → C is defined as

φ̂(t) = ∫ φ(x)e^{−i⟨x,t⟩} dx.

(c) For each compact set K in R^{N}, 𝒟_{(ω)}(K) is a nuclear Fréchet space, by Braun, Meise and Taylor [5, 3.6]. A fundamental system of seminorms on 𝒟_{(ω)}(K) is given by (∥ · ∥_{K,m})_{m∈N} defined in 1.3 but also by

|||φ|||_{m} := ∫_{R^{N}} |φ̂(t)|e^{mω(t)} dt, φ ∈ 𝒟_{(ω)}(K).

1.5. Example. The following functions ω: [0, ∞[→ [0, ∞[ are examples of weight functions:

(1) ω(t) = t^{α}, 0 < α < 1.
(2) ω(t) = (log(1 + t))^{β}, β > 1.
(3) ω(t) = t(log(1 + t))^{−β}, β > 1.

Note that for ω(t) = t^{α}, the classes ℰ_{(ω)} (resp. ℰ_{{ω}}) coincide with the Gevrey classes Γ^{(d)} (resp. Γ^{{d}}) for d := 1/α.

(4) Let (M_{p})_{p∈N_{0}} be a sequence of positive numbers which has the following properties:

(M1) M_{j}^{2} ≤ M_{j−1}M_{j+1} for all j ∈ N;

(M2) there exists A, H > 1 with M_{n} ≤ AH^{n} min_{0≤j≤n} M_{j}M_{n−j} for all n ∈ N;

(M3) there exists A > 0 with ∑_{q=j+1}^{∞} M_{q−1}/M_{q} ≤ AjM_{j}/M_{j+1} ;

and define ω_{M}: R → [0, ∞[ by

ω_{M}(t) = \{ \begin{matrix}sup_{j∈N_{0}} log \frac{|t|^{j}M_{0}}{M_{j}} & for |t| > 0, \\ 0 & for t = 0. \end{matrix}

Then ω_{M} is a continuous even function and by Meise and Taylor [13, 3.11] there exists a concave weight function κ with ω_{M}(t) ≤ κ(t) ≤ Cω_{M}(t) + C for some C > 0 and all t > 0 and such that for each open set Ω in R^{N} we have

ℰ_{(M_{j})}(Ω) = \{f ∈ 𝒞^{∞}(Ω) : sup_{α∈N_{0}^{N}} sup_{x∈K} \frac{|f^{(α)}(x)|}{h^{|α|}M_{|α|}} < ∞ for each h > 0

and each K ⊂ Ω compact}= ℰ_{(κ)}(Ω).

We have an analogous identity for the Roumieu spaces. Note that by [5, 8.9], these identities hold even under weaker hypotheses on (M_{p})_{p∈N_{0}} .

1.6. Convolution operators. Let μ ∈ ℰ′_{∗}(R^{N}), μ ≠ 0, and open sets Ω_{1}, Ω_{2} in R^{n} be given. If Ω_{1} − supp(μ) ⊂ Ω_{2} then we define:

(1)  S_{μ}^{t}: 𝒟_{∗}(Ω_{1}) → 𝒟_{∗}(Ω_{2}),  S_{μ}^{t}(φ) := μ̌ ∗ φ|_{Ω_{2}},

where μ̌ ∗ φ: x → μ̌(φ(x − ·)), x ∈ Ω_{2} , and where μ̌(ψ) := μ(ψ̌) and ψ̌(x) := ψ(−x), x ∈ R^{N}. Since S_{μ}^{t} is continuous and linear, so is its adjoint operator

S_{μ} := (S_{μ}^{t})^{t}: 𝒟′_{∗}(Ω_{2}) → 𝒟′_{∗}(Ω_{1}).

(2)  T_{μ}^{t}: ℰ′_{∗}(Ω_{1}) → ℰ′_{∗}(Ω_{2}),  T_{μ}^{t}(ν) := μ̌ ∗ ν|_{Ω_{2}},

where μ̌ ∗ ν(φ) := (μ̌ ∗ (ν̌ ∗ φ))(0). Again, T_{μ}^{t} is continuous and linear, so that its adjoint

T_{μ} := (T_{μ}^{t})^{t}: ℰ_{∗}(Ω_{2}) → ℰ_{∗}(Ω_{1})

is continuous and linear.

Note that S_{μ}(ν) = μ ∗ ν and T_{μ}(f) = μ ∗ f, so that it is reasonable to call the operators S_{μ} and T_{μ} convolution operators. Note further that T_{μ}^{t}|_{𝒟_{∗}(Ω_{1})} = S_{μ}^{t} and S_{μ}|_{ℰ_{∗}(Ω_{2})} = T_{μ} and that T_{μ}^{t} and S_{μ}^{t} are injective.

1.7. Definition. For u ∈ 𝒟′_{∗}(R^{N}) the ∗-singular support, denoted by sing supp_{∗}(u), is the set of points in R^{N} having no open neighbourhood U to which the restriction u|_{U} is in ℰ_{∗}(U).

1.8. Remark. Similar as Hörmander [10, 4.2.5] one proves that

sing supp_{∗}(u ∗ v) ⊂ sing supp_{∗}(u) + sing supp_{∗}(v)

whenever u, v ∈ 𝒟′_{∗}(R^{N}) and one has compact support.

1.9. Spaces of entire functions. Let A(C^{N}) denote the space of all entire functions on C^{N}, endowed with the Fréchet space topology of uniform convergence on all compact subsets of C^{N}. For an upper semi-continuous function v: C^{N} →]0, ∞[ we define

A(v, C^{N}) := {f ∈ A(C^{N}) : ∥f∥_{v} := sup_{z∈C^{N}} |f(z)|v(z) < ∞}

and note that A(v, C^{N}) is a Banach space.

1.10. Definition. Let K be a convex compact subset of R^{N}. Then we define the support functional H_{K} of K by

H_{K}(y) := sup_{x∈K}⟨x, y⟩.

1.11. Fourier–Laplace transform. For μ ∈ ℰ′_{∗}(R^{N}) its Fourier–Laplace transform μ̂ ∈ A(C^{N}) is defined as

μ̂(z) := μ(exp(−i⟨ · , z⟩)), z ∈ C^{N}.

To characterize its growth behaviour, fix a weight function ω and define the functions w_{j}, w_{j,k}, v_{j} and v_{j,k} by

w_{j}(z) := exp(−j(| Im z| + ω(z))),
w_{j,k}(z) := exp(−j| Im z| − \frac{1}{k} ω(z)),
v_{j}(z) := exp(−j| Im z| + \frac{1}{j}ω(z)),
v_{j,k}(z) := exp(−j| Im z| + kω(z)).

Then the Fourier–Laplace transform ℱ: μ → μ̂ is an isomorphism between the following spaces (see Braun, Meise and Taylor [5, 3.5, 7.4]:

ℰ′_{(ω)}(R^{N}) ⟶ ind_{j→}A(w_{j}, C^{N}),
ℰ′_{{ω}}(R^{N}) ⟶ ind_{j→}proj_{←k}A(w_{j,k}, C^{N}),
𝒟_{(ω)}(R^{N}) ⟶ ind_{j→}proj_{←k}A(v_{j,k}, C^{N}),
𝒟_{{ω}}(R^{N}) ⟶ ind_{j→}A(v_{j}, C^{N}).

Moreover, for a given μ ∈ ℰ′_{∗}(R^{N}) and a convex compact set K in R^{N}, supp(μ) ⊂ K if and only if there exists j ∈ N such that for each ε > 0 we find C > 0 so that |μ̂(z)| ≤ C exp(H_{K}(Im z) + ε| Im z| + jω(z)) for every z ∈ C^{N}.

Furthermore, for μ, ν ∈ ℰ′_{∗}(R^{N}) and φ ∈ 𝒟_{∗}(R^{N}) we have

ℱ (S_{μ}^{t}(ν)) = ℱ (μ̌)ℱ (ν)  and  ℱ (T_{μ}^{t}(φ)) = ℱ (μ̌)ℱ (φ),

hence ℱ ∘ S_{μ}^{t} ∘ ℱ^{−1} (resp. ℱ ∘ T_{μ}^{t} ∘ ℱ^{−1} ) is the operator of multiplication by ℱ (μ̌).

1.12. Definition. Given μ ∈ ℰ′_{∗}(R^{N}) we let

V (μ̂) := {z ∈ C^{N} | μ̂(z) = 0}.

1.13. Definition. (a) An ultradistribution μ ∈ ℰ′_{(ω)}(R^{N}) is called slowly decreasing for (ω) if there exists C > 0 such that for each x ∈ R^{N} with |x| > C there is ξ ∈ R^{N} with |x − ξ| ≤ Cω(x) and |μ̂(ξ)| ≥ exp(−Cω(ξ)).

(b) An ultradistribution μ ∈ ℰ′_{{ω}}(R^{N}) is called slowly decreasing for {ω} if for each m ∈ N there exists R > 0 such that for each x ∈ R^{N} with |x| > R there exists ξ ∈ R^{N} satisfying |x − ξ| ≤ ω(x)/m such that |μ̂(ξ)| ≥ exp(−ω(ξ)/m).

1.14. Remark. If μ ∈ ℰ′_{∗}(R^{N}) is slowly decreasing for ∗ in the sense of Definition 1.13 then μ is slowly decreasing in the sense of Bonet, Galbis and Meise [2, 2.3, 3.1]. Hence it follows from [2, 2.9, 3.4] that the convolution operator Sμ̌ and consequently S_{μ} is surjective on 𝒟′_{∗}(R^{N}). Note that by Bonet, Galbis and Momm [3] the two slowly decreasing definitions are in fact equivalent. For N = 1 and ∗ = (ω) this follows easily from Meise, Taylor and Vogt [14, 2.7].

1.15. Notation. For a subset A of R^{N}, we denote the convex hull of A by ch(A).

268 J. Bonet, C. Fernández, and R. Meise

2. (ω)-hypoelliptic convolution operators

In this section we define and characterize when the convolution operator S_{μ} which is induced by μ ∈ ℰ′_{(ω)}(R^{N}) is (ω)-hypoelliptic. To do this we use essentially arguments which go back to Ehrenpreis [8], and Hörmander [9], [10, Section 16.6] in the classical case of convolution operators on distributions. Throughout this section, ω denotes a fixed weight function.

2.1. Theorem. For μ ∈ ℰ′_{(ω)}(R^{N}) the following assertions are equivalent:

(1) Whenever ν ∈ 𝒟′_{(ω)}(R^{N}) and μ ∗ ν ∈ ℰ_{(ω)}(R^{N}), then ν ∈ ℰ_{(ω)}(R^{N}),

(2) lim_{z∈V (μ̂), |z|→∞}(| Im z|/ω(z)) = ∞ and there exist A, R > 0 such that

(2.1)  |μ̂(x)| ≥ exp(−Aω(x))  for each x ∈ R^{N}, |x| ≥ R,

(3) lim_{z∈V (μ̂),|z|→∞}(| Im z|/ω(z)) = ∞ and μ is slowly decreasing for (ω),

(4) Ker S_{μ} ⊂ ℰ_{(ω)}(R^{N}) and μ is slowly decreasing for (ω),

(5) there exist F ∈ 𝒟′_{(ω)}(R^{N}) and ψ ∈ 𝒟_{(ω)}(R^{N}) such that μ ∗ F = δ + ψ,

(6) there exists F ∈ 𝒟′_{(ω)}(R^{N}) satisfying μ ∗ F = δ + φ for some φ ∈ ℰ_{(ω)}(R^{N}) and ch(sing supp_{(ω)}(F)) = − ch(sing supp_{(ω)}(μ)).

(7) there exists a fundamental solution E for S_{μ} satisfying

ch(sing supp_{(ω)}(E)) = − ch(sing supp_{(ω)}(μ)),

(8) for each open set Ω ⊂ R^{N} and u ∈ 𝒟′_{(ω)}(Ω − supp(μ) + ch(sing supp_{(ω)}(μ))) satisfying μ ∗ u ∈ ℰ_{(ω)}(Ω + ch(sing supp_{(ω)}(μ))), it follows that u ∈ ℰ_{(ω)}(Ω). 2.2. Definition. (a) If μ ∈ ℰ′_{(ω)}(R^{N}) satisfies condition 2.1(1), then μ and

also S_{μ} are called (ω)-hypoelliptic.

(b) F ∈ 𝒟′_{(ω)}(R^{N}) is called a parametrix for μ ∈ ℰ′_{(ω)}(R^{N}) if μ ∗ F = δ + φ for some φ ∈ ℰ_{(ω)}(R^{N}).

The proof of Theorem 2.1 is prepared by several intermediate results and is given at the end of the section.

2.3. Proposition. If μ ∈ ℰ′_{(ω)}(R^{N}) satisfies Ker S_{μ} ∈ ℰ_{(ω)}(R^{N}), then

lim_{z∈V (μ̂), |z|→∞} \frac{| Im z|}{ω(z)} = ∞.

Proof. Proceeding by contradiction we assume there is a sequence (ζ_{j})_{j∈N} ⊂ V (μ̂) such that lim_{j→∞} |ζ_{j}| = ∞ and | Im ζ_{j}| ≤ Mω(ζ_{j}) holds for every j ∈ N and some M > 0. If a = (a_{j})_{j∈N} ∈ l_{1} and φ ∈ 𝒟_{(ω)}(R^{N}), by the Paley–Wiener– Komatsu theorem [5, 3.5] there is s ∈ N such that for all n ∈ N there is C_{n} > 0 with

|φ̂(z)| ≤ C_{n} exp(s| Im z| − nω(z))

for all z ∈ C^{N}. If n > sM + 1 we have, for each k ∈ N,

∑_{j=1}^{k} |a_{j}| |⟨e^{i⟨x,ζ_{j}⟩}, φ⟩| = ∑_{j=1}^{k} |a_{j}| |φ̂(−ζ_{j})| ≤ C_{n} ∑_{j=1}^{k} |a_{j}|.

This implies that the map η: l_{1} ⟶ (𝒟′_{(ω)}(R^{N}), σ(𝒟′_{(ω)}(R^{N}), 𝒟_{(ω)}(R^{N}))) defined by η(a) := ∑_{j=1}^{∞} a_{j} exp(i⟨x, ζ_{j}⟩) is well-defined, linear and continuous. By the closed graph theorem [17, 24.31], η: l_{1} ⟶ 𝒟′_{(ω)}(R^{N}) is continuous and the range of η is contained in Ker S_{μ}, since ζ_{j} ∈ V (μ̂) for each j ∈ N. By assumption η(l_{1}) ⊂ ℰ_{(ω)}(R^{N}) and we can apply again the closed graph theorem to conclude that η: l_{1} ⟶ 𝒞^{∞}(R^{N}) is also continuous. This yields C > 0 such that, for all a = (a_{j})_{j∈N} in l_{1} and x ∈ R^{N} with |x| ≤ 1, we have

∑_{|α|=1}|(∑_{j=1}^{∞} a_{j} exp(i⟨x, ζ_{j}⟩))^{α}| ≤ C ∑_{j=1}^{∞} |a_{j}|.

If ζ_{j} = (ζ_{j}^{1}, …, ζ_{j}^{N}) for each j ∈ N, we get |ζ_{j}^{k}| ≤ C for each j ∈ N and K = 1, …, N. This implies that (ζ_{j})_{j∈N} is bounded, in contradiction with the choice of (ζ_{j})_{j∈N} .

We need the following lemma which can be seen in Sampson and Zielezny [20, p. 141].

2.4. Lemma. Let (ξ_{j})_{j∈N} be a sequence in R^{N} satisfying |ξ_{j}| > 2|ξ_{j−1}| > 2^{j} for every j ∈ N and let (a_{j})_{j∈N} be a bounded sequence in C. Then the series ∑_{j=1}^{∞} a_{j} exp(i⟨ · , ξ_{j}⟩) converges in 𝒟′(R^{N}) to some η ∈ 𝒟′(R^{N}). If η ∈ 𝒞^{3} , then lim_{j→∞} |a_{j}| = 0.

2.5. Proposition. If μ ∈ ℰ′_{(ω)}(R^{N}) is (ω)-hypoelliptic, then there exist A, R > 0 such that (2.1) holds, i.e.,

|μ̂(x)| ≥ exp(−Aω(x)) for each x ∈ R^{N}, |x| ≥ R.

Proof. Arguing by contradiction we construct a sequence (ξ_{j})_{j∈N} ⊂ R^{N} such that |ξ_{j}| > 2|ξ_{j−1}| > 2^{j} for every j ≥ 2 and |μ̂(ξ_{j})| < exp(−jω(ξ_{j})), for each j ∈ N. By Lemma 2.4, η := ∑_{j=1}^{∞} exp(i⟨ · , ξ_{j}⟩) belongs to 𝒟′(R^{N}) and hence to 𝒟′_{(ω)}(R^{N}) but it is not a 𝒞^{∞} function on R^{N}. We show that μ ∗ η ∈ ℰ_{(ω)}(R^{N}) to reach a contradiction. First observe that, for φ ∈ 𝒟_{(ω)}(R^{N}), we have

⟨μ ∗ η, φ⟩ = ⟨η, μ̌ ∗ φ⟩ = ∑_{j=1}^{∞} μ̂(ξ_{j})φ̂(−ξ_{j}) = ∑_{j=1}^{∞} μ̂(ξ_{j})⟨exp(i⟨ · , ξ_{j}⟩), φ⟩.

Next note that the definition of φ^{∗} implies for j, m ∈ N, j > m large enough, α ∈ N_{0} :

|(μ̂(ξ_{j}) exp(i⟨ · , ξ_{j}⟩))^{(α)}| exp(−mφ^{∗}(\frac{|α|}{m})) = |μ̂(ξ_{j})| |ξ_{j}|^{|α|} exp(−mφ^{∗}(\frac{|α|}{m}))

≤ exp(−jω(|ξ_{j}|) + |α| log |ξ_{j}| − mφ^{∗}(\frac{|α|}{m}))

≤ exp(−j + m)ω(ξ_{j}) ≤ exp(−j + m).

From this it follows easily that μ ∗ η ∈ ℰ_{(ω)}(R^{N}).

2.6. Proposition. If μ ∈ ℰ′_{(ω)}(R^{N}) satisfies the following two conditions:

(a) lim_{z∈V (μ̂), |z|→∞}(| Im z|/ω(z)) = ∞,

(b) μ̂ is slowly-decreasing for (ω). Then there exist A, R > 0 such that (2.1) holds.

Proof. Since ω satisfies condition (α) and (γ) we can find k > 0 such that ω(x) ≤ kω(\frac{1}{2}x) if |x| > k. Because of (b), there exists B > 0 such that for all x ∈ R^{N} there is y ∈ R^{N} with |x − y| < Bω(x) and |μ̂(y)| > B^{−1} exp(−Bω(x)). Since ω(t) = o(t) as t goes to ∞, we can enlarge k to get also ω(x) < |x|/2B if |x| > k.

By condition (a), for each m ∈ N there exists C_{m} > 0 such that

| Im ζ| ≥ (m + 1)ω(|ζ|) if μ̂(ζ) = 0 and |ζ| ≥ C_{m}.

We select m_{0} ∈ N, m_{0} > kB. For x ∈ R^{N} with |x| > 2max(B, C_{m_{0}}, k) we define g(λ) := μ̂(x + λ(y − x)), λ ∈ C, where y ∈ R^{N} is the element associated with x in condition (b) (see above). Clearly g ∈ ℋ (C) and, if |λ| ≤ \frac{1}{2} , we have

|x + λ(y − x)| ≥ |x| − (\frac{1}{2}B)ω(x) ≥ \frac{1}{2}|x| > C_{m_{0}}

and

\begin{aligned}|Im(x + λ(y − x))| & = | Im λ| |y − x| ≤ \frac{1}{2}Bω(x) ≤ \frac{1}{2}kBω(\frac{1}{2}x) \\ & ≤ m_{0}ω(x + λ(y − x)), \end{aligned}

which implies g(λ) ≠ 0 for each λ ∈ C with |λ| ≤ \frac{1}{2} .

Now we apply the minimum-modulus theorem of Chou [7, II.2.1] with R = 1, r = \frac{1}{3} , λ_{0} = 1, η < \frac{1}{32} , H = 2 + log(3e/2η), to get

|g(0)| ≥ \frac{|g(1)|^{3(H+1)}}{(sup_{|λ|≤3e} |g(λ)|^{3H})(sup_{|λ|≤3/2} |g(λ)|^{2})}.

If |λ| ≤ 3e, we have |x+λ(y−x)| ≤ |x|+3eBω(x) ≤ 4e|x| and |Im(x+λ(y−x))| ≤ 3eBω(x). By the theorem of Paley–Wiener–Schwartz–Komatsu [5, 7.2], there is D > 0 such that for all |λ| ≤ 3e we have |μ̂(x + λ(x − y))| ≤ D exp(Dω(x)) for all |λ| ≤ 3e. This yields

|μ̂(x)| ≥ \frac{exp(−A(3H + 3)ω(x))}{(D exp(Dω(x)))^{3H+2}},

and the proof is complete.

For the convenience of the reader we recall the following result from Hörmander [9, Lemma 1].

2.7. Lemma. Given positive constants S, T and ε > 0 we can find L ∈ N such that for all 0 < s < S and 0 < r < R/L if u is harmonic when x^{2} + y^{2} < R^{2} and satisfies the inequalities

(i) u(x, 0) ≤ 0,

(ii) u(x, y) ≥ −S|y| − Tr if x^{2} + y^{2} < R^{2} ,

it follows that (iii) u(x, y) ≤ s|y| + εr if x^{2} + y^{2} < r^{2} .

2.8. Proposition. Let μ ∈ ℰ′_{(ω)}(R^{N}) satisfy the hypothesis of 2.6. There exists D > 0 such that for each m ∈ N and each convex compact neighbourhood K of ch(supp(μ)) there exists R_{m} > 0 such that

(∗)  \frac{1}{|μ̂(z)|} ≤ exp(H_{K}(− Im z) + Dω(Re z))

for z ∈ C^{N}, |z| ≥ R_{m} and | Im z| ≤ mω(Re z).

Proof. We first observe that, if z ∈ C^{N} satisfies | Im z| < mω(z) for some m ∈ N, then | Re z|/|z| goes to 1 as |z| goes to ∞. This follows from the estimate

1 = \frac{|z|}{|z|} ≤ \frac{| Re z|}{|z|} + \frac{| Im z|}{|z|} < \frac{| Re z|}{|z|} + m\frac{ω(z)}{|z|},

since ω(t) = o(t) as t → ∞.

Next fix a compact convex neighbourhood K of supp(μ) and m ∈ N and consider z = x + iy ∈ C^{N} with 0 < |y| < mω(z). Define the entire function

g_{z}(λ) := μ̂(x + λy/|y|), λ ∈ C.

We study g_{z} for |λ| < Mω(x) for some M ∈ N, depending on m but not on z, to be selected later.

By condition (a) in Proposition 2.6, for each s ∈ N there exists D̃_{s} such that

μ̂(z) ≠ 0 if |z| ≥D̃_{s}, | Im z| ≤ sω(z).

We claim there is D_{M} > 0 such that if z = x + iy ∈ C^{N} satisfies |z| > D_{M}, 0 < |y| < mω(z) and |λ| < Mω(x), then g_{z}(λ) ≠ 0 and

|g_{z}(t)| ≥ exp(−Aω(x + ty/|y|))

for all t ∈ R, |t| < Mω(x), where A is the constant in (2.1) (which holds by Proposition 2.6).

To prove the claim, recall there is k ∈ N such that ω(x) ≤ kω(\frac{1}{2}x) if |x| ≥ k and, given M, there is K_{M} > 0 such that t − Mω(t) > \frac{1}{2}t if t > K_{M}. For |x| > max(K_{M}, 2D̃_{kM}, 2k), we have

|x + λy/|y|| ≥ |x| − Mω(x) ≥ \frac{1}{2}|x| (and ≥ k).

In particular |x| ≤ 2|x + λy/|y||, hence

\begin{aligned}|Im(x + λy/|y|)| & = | Im λ| < Mω(x) ≤ Mω(2|x + λy/|y||) \\ & ≤ Mkω(|x + λy/|y||). \end{aligned}

Thus g_{z}(λ) = μ̂(x + λy/|y|) ≠ 0. On the other hand, if |x| > max(2R, K_{M}) (R as in (2.1)) and t ∈ R, |t| < Mω(x), we get

|x + ty/|y|| ≥ |x| − |t| ≥ |x| − Mω(x) ≥ \frac{1}{2}|x| > R,

hence

|g_{z}(t)| = |μ̂(x + ty/|y|)| ≥ exp(−Aω(x + ty/|y|)).

By the very first observation of the proof, there is D_{M} > 0 such that

|z| ≥ D_{M}, z = x + iy ∈ C^{N}, 0 < |y| < mω(z)

implies

|x| ≥ max(K_{M}, 2D̃_{kM}, 2k, 2R).

This completes the proof of the claim.

We write H_{K}(y/|y|) = α + β, H_{K}(−y/|y|) = α − β. Then H_{K}(Im yλ/|y|) = α|λ| + βλ. By the theorem of Paley–Wiener–Schwartz–Komatsu [5, 7.2], there is b > 0 such that

|μ̂(z)| ≤ exp(H_{K}(Im z) + bω(z)), if |z| ≥ b.

Now, if z = x + iy ∈ C^{N} satisfies |z| > max(b, K_{M}, k) and |λ| < Mω(x), then

|g_{z}(λ)| ≤ exp(α| Im λ| + β Im λ + bω(x + λy/|y|)).

Since |x + λy/|y|| ≤ |x| + |λ| ≤ |x| + Mω(x) ≤ 2|x|, we get

|g_{z}(λ)| ≤ exp(α| Im λ| + β Im λ + bkω(x)).

We define

u_{z}(λ) := log(\frac{e^{β Im λ}e^{−Akω(x)}}{|g_{z}(λ)|} ) = β Im λ − Akω(x) − log |g_{z}(λ)|,

which is harmonic if |z| > R_{m} := max(D_{M}, b, K_{M}, k) and |λ| < Mω(x), since g_{z}(λ) does not vanish. For t ∈ R, |t| < Mω(x), we get

ω(x + ty/|y|) ≤ ω(2|x|) ≤ kω(x),

hence

exp(−A(ω(x + ty/|y|))) ≥ exp(−Akω(x)).

By our claim above, if t ∈ R, |t| < Mω(x) and λ ∈ C, |λ| < Mω(x) we have

u_{z}(t) ≤ −Akω(x) − (−Akω(x)) = 0,

and

u_{z}(λ) ≥ β Im λ−Akω(x)−α| Im λ|−β Im λ−bkω(x) = −α| Im λ|−(Ak +bk)ω(x).

Now let

ε := \frac{1}{2(m + 1)},  T := \frac{(Ak + bk)}{(m + 1)},  S := sup_{|η|=1}(\frac{1}{2}((H_{K}(η) + H_{K}(−η)))

and choose L ∈ N according to Lemma 2.7. Observe that the constants depend on m but not on z. Next let M := L(m + 1) + 1, R := Mω(x) and r = (m + 1)ω(x). Then we conclude from Lemma 2.7. that

|u_{z}(λ)| ≤ α| Im λ| + ε(m + 1)ω(x), if |λ| ≤ r.

Setting λ = i|y|, we get

u_{z}(i|y|) = β|y| − Akω(x) − log |μ̂(z)| ≤ α|y| + ω(x).

This yields

− log |μ̂(z)| ≤ (α − β)|y| + (Ak + 1)ω(x).

Thus 1/|μ̂(z)| ≤ exp(H_{K}(−y) + (Ak + 1)ω(x)) if z = x + iy ∈ C^{N}, |z| ≥ R_{m} and |y| ≤ mω(x).

2.9. Lemma. If μ ∈ ℰ′_{(ω)}(R^{N}) satisfies the hypotheses of Proposition 2.6 and μ̂(x) does not vanish if |x| ≥ R_{0} , then

⟨F, φ⟩ := (\frac{1}{2π} )^{N} ∫_{|x|≥R_{0}} \frac{φ̂(−x)}{μ̂(x)} dx, φ ∈ 𝒟_{(ω)}(R^{N})

defines an element F ∈ 𝒟′_{(ω)}(R^{N}) satisfying sing supp_{(ω)}(F) ⊂ − ch(supp(μ)) and μ ∗ F = δ + G for some G which is real analytic on R^{N}.

Proof. By the theorem of Paley–Wiener–Schwartz–Komatsu [5, 7.2] and (2.1) (which follows from Proposition 2.6), it is easy to see that F defines an element of 𝒟′_{(ω)}(R^{N}).

Moreover, for φ ∈ 𝒟_{(ω)}(R^{N}), we have

\begin{aligned}⟨μ ∗ F, φ⟩ & := (\frac{1}{2π} )^{N} ∫_{|x|≥R_{0}} \frac{φ̂(−x)μ̂(x)}{μ̂(x)} dx = ∫_{|x|≥R_{0}} φ̂(x) dx \\ & = φ(0) − (\frac{1}{2π} )^{N} ∫_{|x|≤R_{0}} φ̂(x) dx = φ(0) + ∫_{R^{N}} φ(t)G(t) dt, \end{aligned}

where G(t) = −(2π)^{−N} ∫_{|x|≤R_{0}} e^{i⟨,x,t⟩} dx defines a real analytic function on R^{N}. Thus μ ∗ F = δ + G as desired.

The main step in the proof is to show that sing supp_{(ω)}(F) ⊂ − ch(supp(μ)). To see this we fix x_{0} ∉ − ch(supp(μ)). We select a compact convex neighbourhood K of − ch(supp(μ)) such that x_{0} ∉ K. By Proposition 2.8, there exists D > 0 such that for each m ∈ N there exists R_{m} ≥ R_{0} such that

\frac{1}{|μ̂(z)|} ≤ exp(H_{K}(Im z)+Dω(Re z)), if z ∈ C^{N}, |z| ≥ R_{m}, | Im z| ≤ mω(Re z).

Since x_{0} ∉ K, there are η ∈ R^{N}, |η| = 1, σ > 0 and a compact convex ball V around x_{0} of radius δ > 0 such that H_{K}(η) + H_{V} (−η) ≤ −σ. Applying an orthogonal and real change of variables, we may assume η = (1, 0, …, 0). We choose φ ∈ 𝒟_{(ω)}(V′) such that φ is identically 1 near x_{0} , with V′ = B_{δ/2}(x_{0}). If we prove that for each k ∈ N there exists C_{k} > 0 such that

(∗)  |φ̂F (ξ)| ≤ C_{k} exp(−kω(ξ))  for all ξ ∈ R^{N},

it follows that φF ∈ ℰ_{(ω)}(R^{N}) and F is of class ℰ_{(ω)} near x_{0} as desired. To prove (∗), we first observe that, for ξ ∈ R^{N},

φ̂F (ξ) = (2π)^{−N} ∫_{|x|≥R_{0}} \frac{φ̂(ξ − x)}{μ̂(x)} dx.

By 1.2(d) there is β ∈ N (depending only on ω) such that ω(ξ − x) ≥ β^{−1}ω(ξ) − ω(x)−1 for all x, ξ ∈ R^{N}. We may assume that ω is 𝒞^{1} on R and has a bounded derivative (see 1.2(c)). Now we fix k ∈ N and select t ∈ N such that σt > D+βk. We denote by ∥x∥ := max(|x_{i}|, i = 1, …, N) the sup-norm on R^{N}. We have

\begin{aligned}(2π)^{N}|φ̂F (ξ)| & ≤ |∫_{\{x||x|≥R_{0}, ∥x∥≤R_{t}\}} \frac{φ̂(ξ − x)}{μ̂(x)} dx| + |∫_{∥x∥≥R_{t}} \frac{φ̂(ξ − x)}{μ̂(x)} dx| \\ & =: A(ξ) + B(ξ). \end{aligned}

Since φ ∈ 𝒟_{(ω)}(R^{N}) and V is a compact convex neighbourhood of supp(φ), for each m ∈ N there is L_{m} > 0 such that

|φ̂(z)| ≤ L_{m} exp(H_{V} (Im z) − mω(Re z))

for each z ∈ C^{N}.

Consequently, there is C_{k} > 0 with

A(ξ) ≤ C_{k} exp(−kω(ξ)) ∫_{\{x||x|≥R_{0}, ∥x∥≤R_{t}\}} \frac{exp(βkω(x))}{|μ̂(x)|} dx ≤ C′_{k} exp(−kω(ξ));

since the integral is finite.

On the other hand, writing x = (x_{1}, x′) ∈ R^{N},

B(ξ) ≤ |∫_{∥x′∥≥R_{t}} ∫_{R} \frac{φ̂(ξ − x)}{μ̂(x)} dx|+|∫_{∥x′∥≤R_{t}} ∫_{|x_{1}|≥R_{t}} \frac{φ̂(ξ − x)}{μ̂(x)} dx| =: I_{1}(ξ)+I_{2}(ξ).

We treat the two integrals separately. In I_{1}(ξ) we change the path of integration in the second integral by Cauchy’s theorem. We define z = γ(x_{1}) = x_{1} + itω(x_{1}, x′), x_{1} ∈ R, x′ ∈ R^{N−1} . Then, if g_{t}(x_{1}) is the bounded derivative of γ, we have

I_{1}(ξ) = |∫_{∥x′∥≥R_{t}} ∫_{R} \frac{φ̂((−γ(x_{1}), −x′) + ξ)}{μ̂(γ(x_{1}), x′)}g_{t}(x_{1}) dx|

≤ ∫_{∥x′∥≥R_{t}} ∫_{R} \frac{|φ̂((−γ(x_{1}), −x′) + ξ)|}{|μ̂(γ(x_{1}), x′)|}|g_{t}(x_{1})| dx

≤ CL_{βk} ∫_{∥x′∥≥R_{t}} ∫_{R} exp(H_{K}(tω(x_{1}, x′)η) + Dω(x_{1}, x′)

+ H_{V} (−tω(x_{1}, x′)η) − kβω(−x + ξ)) dx_{1} dx′

≤ L′_{k} ∫_{R^{N−1}} ∫_{R} exp(((H_{K}(η) + H_{V} (−η))t + D + kβ)ω(x) − kω(ξ)) dx_{1} dx′

≤ L′_{k} exp(−kω(ξ)) ∫_{R^{N}} exp(−εω(x)) dx

for some ε > 0, and the integral is finite.

Since the integral I_{2}(ξ) can be estimated in the same way, the proof is complete.

2.10. Remark. Note that arguments similar to the ones given in the proof of 2.9 permit to show the following: For μ ∈ ℰ′_{(ω)}(R^{N}) and a compact convex set K in R^{N}, the inclusion sing supp_{(ω)}(μ) ⊂ K is equivalent to the existence of b > 0 such that for each m ∈ N there exists C_{m} > 0 such that

|μ̂(z)| ≤ C_{m} exp(H_{K}(Im z) + bω(z))

for all z ∈ C^{N} with | Im z| ≤ mω(z) and |z| ≥ C_{m}. Compare with [10, Volume I, 7.3.8].

2.11. Remark. Let μ ∈ ℰ′_{(ω)}(R^{N}) be (ω)-hypoelliptic and let F and G be parametrices for μ. Then sing supp_{(ω)}(F) = sing supp_{(ω)}(G). Indeed, μ ∗ F = δ + f, μ ∗ G = δ + g and f, g ∈ ℰ_{(ω)}(R^{N}). Then μ ∗ (F − G) = f − g ∈ ℰ_{(ω)}(R^{N}). Since μ is (ω)-hypoelliptic, it follows that F − G ∈ ℰ_{(ω)}(R^{N}), which yields the conclusion.

As a consequence of Theorem 2.1, every parametrix F of μ ∈ ℰ′_{(ω)}(R^{N}) which is (ω)-hypoelliptic satisfies ch(sing supp_{(ω)}(F)) = − ch(sing supp_{(ω)}(μ)).

Proof of Theorem 2.1. (1) ⇒ (2) ⇒ (3): The implication (1)⇒(2) is a direct consequence of Propositions 2.3 and 2.5. By the very definition of slowly decreasing for (ω), (3) follows from (2).

(3) ⇒ (5): By Propositions 2.6 and 2.8 and Lemma 2.9, (3) implies the existence of F ∈ 𝒟′_{(ω)}(R^{N}) such that μ ∗ F = δ + G, G is real analytic on R^{N} and sing supp_{(ω)}(F) ⊂ − ch(supp(μ)). Select ψ ∈ 𝒟_{(ω)}(R^{N}) which is identically 1 on a neighbourhood of − ch(supp(μ)) and set F_{1} = ψF ∈ ℰ′_{(ω)}(R^{N}). We have

μ ∗ F_{1} = δ + G − μ ∗ ((1 − ψ)F ).

Clearly φ = G − μ ∗ ((1 − ψ)F ) ∈ 𝒟_{(ω)}(R^{N}), and (3) implies (5).

(5) ⇒ (1): Let h ∈ 𝒟′_{(ω)}(R^{N}) satisfy μ ∗ h ∈ ℰ_{(ω)}(R^{N}) and choose F ∈ ℰ′_{(ω)}(R^{N}) as in (5). Then F ∗ (μ ∗ h) = (δ + ψ) ∗ h = h + ψ ∗ h implies h = −ψ ∗ h + F ∗ (μ ∗ h) ∈ ℰ_{(ω)}(R^{N}). Hence (1) holds.

(1) ⇒ (6): By Lemma 2.9 and the implications already proved, there is a parametrix F of μ with sing supp_{(ω)}(F) ⊂ − ch(supp(μ)). Next fix x_{0} ∉ − ch(sing supp_{(ω)}(μ)) and choose χ ∈ 𝒟_{(ω)}(R^{N}) identically 1 on a neighbourhood of ch(sing supp_{(ω)}μ) and such that −x_{0} ∉ ch(supp(χ)). Then let ν := χμ ∈ ℰ′_{(ω)}(R^{N}). Clearly x_{0} ∉ − ch(supp ν) and μ = ν + φ for some φ ∈ 𝒟_{(ω)}(R^{N}). We note that ν is (ω)-hypoelliptic. Indeed, if ν ∗ h ∈ ℰ_{(ω)}(R^{N}) for some h ∈ 𝒟′_{(ω)}(R^{N}), we have μ ∗ h = ν ∗ h + φ ∗ h ∈ ℰ_{(ω)}(R^{N}), hence h ∈ ℰ_{(ω)}(R^{N}). By Lemma 2.9, there is a parametrix F_{0} for ν such that sing supp_{(ω)}(F_{0}) ⊂ − ch(supp(ν)). In particular F_{0} is of class ℰ_{(ω)} on a neighbourhood of x_{0} . It is easy to see that F_{0} is also a parametrix for μ. By Remark 2.11, sing supp_{(ω)}(F_{0}) = sing supp_{(ω)}(F), which implies x_{0} ∉ sing supp_{(ω)}(F).

(6) ⇒ (7): The same argument used above to show (5)⇒(1), permits to conclude (1) from (6). By what is already proved, we get (3). In particular μ is slowly decreasing for (ω) and by [2, Corollary 2.9] there is a fundamental solution E ∈ 𝒟′_{(ω)}(R^{N}) with μ ∗ E = δ. In particular E is a parametrix for μ, which is (ω)-hypoelliptic. By Remark 2.11, ch(sing supp_{(ω)}(E)) = ch(sing supp_{(ω)}(F)) = − ch( sing supp_{(ω)}μ).

(7) ⇒ (8): Let Ω be an open subset of R^{N} and let u ∈ 𝒟′_{(ω)}(Ω − supp(μ) + ch(sing supp_{(ω)}μ)) satisfy μ ∗ u ∈ ℰ_{(ω)}(Ω + ch(sing supp_{(ω)}(μ))). We fix x_{0} ∈ Ω and we show that the restriction of u to B_{R}(x_{0}) belongs to ℰ_{(ω)}(B_{R}(x_{0})) for some R > 0. To do so, take R > 0 such that B_{2R}(x_{0}) ⊂ Ω, choose φ ∈ 𝒟_{(ω)}(Ω−supp(μ)+ch(sing supp_{(ω)}(μ))) which is identically 1 on a neighbourhood of B_{R}(x_{0}) − supp(μ) + ch(sing supp_{(ω)}μ) and let v := φu ∈ ℰ′_{(ω)}(R^{n}). Since u and v coincide on a neighbourhood of B_{R}(x_{0}) − supp(μ) + ch(sing supp_{(ω)}μ), we get that μ ∗ u and μ ∗ v coincide on B_{R}(x_{0}) + ch(sing supp_{(ω)}(μ)), hence μ ∗ v ∈ ℰ_{(ω)}(B_{R}(x_{0}) + ch(sing supp_{(ω)}(μ))). Therefore,

sing supp_{(ω)}(v) = sing supp_{(ω)}(E ∗ μ ∗ v) ⊂ sing supp_{(ω)}(E) + sing supp_{(ω)}(μ ∗ v)

⊂ − ch(sing supp_{(ω)}(μ)) + (R^{n}\(B_{R}(x_{0}) + ch(sing supp _{(ω)}(μ)))

⊂ R^{n}\B_{R}(x_{0}).

(8) ⇒ (1): This is obvious.

(1) ⇒ (4): First observe that (1) clearly implies Ker S_{μ} ⊂ ℰ_{(ω)}(R^{N}) and, by what is already proved it also implies that μ is slowly decreasing for (ω).

(4) ⇒ (1): Conversely, assume that (4) holds and take ν ∈ 𝒟′_{(ω)}(R^{N}) such that ν ∗ μ = g ∈ ℰ_{(ω)}(R^{N}). Since μ is slowly decreasing for (ω), we can apply [2, Corollary 2.9] to find f ∈ ℰ_{(ω)}(R^{N}) such that μ ∗ f = g. Accordingly ν − f ∈ Ker S_{μ} ⊂ ℰ_{(ω)}(R^{N}). This yields ν ∈ ℰ_{(ω)}(R^{N}) and (1) is satisfied.

2.12. Remark. Theorem 2.1 should be compared with the characterization of the (ω)-hypoelliptic differential operators given by [1, 4.1.1] and the results stated in Chou [7, IV.11, Remarque 4]. In both articles classes of ultradifferentiable functions are used which are defined in a different way than those considered here. However, in many cases these different definitions decribed the same classes. Note that there are more equivalences and more precise statements in Theorem 2.1 than in Chou [7].

2.13. Proposition. Let σ and ω be two weights such that ω(t) = o(σ(t)) as t → ∞. Then the following assertions hold.

(a) There exists μ ∈ ℰ′_{(ω)}(R^{N}) which is (σ)-hypoelliptic but not (ω)-hypoelliptic.

(b) There exists μ ∈ ℰ′_{(ω)}(R) which is (ω)-hypoelliptic but not (σ)-hypoelliptic.

Proof. (a) By Braun [4, Theorem 7], there exist an elliptic ultradifferential operator G(D) of class (σ) and f ∈ ℰ_{{σ}}(R^{N}) which is real analytic on R^{N}\{0} such that G(D)f = δ. We select φ ∈ 𝒟_{{σ}}(R^{N}) identically 1 on a neighbourhood of {0} and define μ := φf ∈ 𝒟_{{σ}}(R^{N}) ⊂ 𝒟_{(ω)}(R^{N}). Clearly μ is not (ω)-slowly decreasing, hence it is not (ω)-hypoelliptic. On the other hand G(D)μ = δ + χ with χ ∈ 𝒟_{(σ)}(R^{N}), and μ has a parametrix in 𝒟′_{(σ)}(R^{N}), hence it is (σ)- hypoelliptic.

(b) By [5, 1.7], there is m: [0, ∞[→ [0, ∞[ such that ω(t) = o(m(t)) and m(t) = o(σ(t)) as t → ∞. Proceeding by recurrence, with r_{1} ≥ 2, select a sequence (r_{j})_{j∈N} with 4r_{j} ≤ r_{j+1} , j ∈ N, such that, for all j ∈ N,

(i) 1 + σ(r_{j}) ≥ m(r_{j}),

(ii) j(ω(r_{j}) + 1) ≤ m(r_{j}),
(iii) (j + 1)^{2} ≤ inf_{t≥r_{j}}(ω(t)/log(t)).

By (iii), if n(t) = card {j ∈ N | r_{j} ≤ t}, t ≥ 0, we have n(t) log t = o(ω(t)) as t → ∞. For each j ∈ N, select z_{j} ∈ C with |z_{j}| = r_{j}, Im z_{j} = m(r_{j}) and let f(z) := ∏_{j∈N}(1 − (z/z_{j})), z ∈ C. The function f is entire, its zeros are the z_{j} ’s and

log |f(z)| ≤ n(|z|) log |z| + log 2 + \frac{4}{9},

for each z ∈ C by Braun, Meise, Vogt [6, 3.11]. Hence there is μ ∈ ℰ′_{(ω)}(R) with μ̂ = f, supp(μ) = {o} and T_{μ} is surjective on ℰ_{(ω)}(R), therefore on ℰ_{(σ)}(R). By our construction and Theorem 2.1, μ is (ω)-hypoelliptic, but not (σ)-hypoelliptic, since the first part of condition (2) in Theorem 2.1 is not satisfied for (σ).

2.14. Remark. Observe that the example μ ∈ ℰ′_{(ω)}(R^{N}) constructed in Proposition 2.11(a) satisfies Ker S_{μ} ⊂ ℰ_{(σ)}(R^{N}) ⊂ ℰ_{(ω)}(R^{N}) but it is not (ω)- slowly decreasing. Accordingly the condition Ker S_{μ} ⊂ ℰ_{(ω)}(R^{N}) alone does not imply that μ is (ω)-hypoelliptic.

We refer the reader Ehrenpreis [8, pp. 574–579] for interesting examples on hypoelliptic convolution operators in the case of classical distributions.

3. {ω}-hypoelliptic convolution operators

In this section we define and characterize when the convolution operator S_{μ} which is induced by μ ∈ ℰ′_{{ω}}(R^{N}) is {ω}-hypoelliptic. The statements look very similar, but the proofs require new ingredients. Some of them, as for example Proposition 3.5 and Corollary 3.7, might be of independent interest. Again in this section ω denotes a fixed weight function.

3.1. Theorem. For μ ∈ ℰ′_{{ω}}(R^{N}) the following assertions are equivalent:

(1) Whenever ν ∈ 𝒟′_{{ω}}(R^{N}) and μ ∗ ν ∈ ℰ_{{ω}}(R^{N}), then ν ∈ ℰ_{{ω}}(R^{N}),

(2) lim inf_{z∈V (μ̂), |z|→∞}(| Im z|/ω(z)) > 0 and for each m ∈ N there is R_{m} > 0 such that |μ̂(x)| ≥ exp(−ω(x)/m) for each x ∈ R^{N}, |x| ≥ R_{m}, (3) lim inf_{z∈V (μ̂), |z|→∞}(| Im z|/ω(z)) > 0 and μ is slowly decreasing for {ω},

(4) Ker S_{μ} ⊂ ℰ_{{ω}}(R^{N}) and μ is slowly decreasing for {ω},

(5) Ker S_{μ} ⊂ ℰ_{{ω}}(R^{N}) and T_{μ}: ℰ_{{ω}}(R^{N}) → ℰ_{{ω}}(R^{N}) is surjective,

(6) there exist F ∈ ℰ′_{{ω}}(R^{N}) and ψ ∈ 𝒟_{{ω}}(R^{N}) such that μ ∗ F = δ + ψ,

(7) there exists F ∈ 𝒟′_{{ω}}(R^{N}) satisfying μ ∗ F = δ + φ for some φ ∈ ℰ_{{ω}}(R^{N}) and the convex hull of sing supp_{{ω}}(F) and −sing supp_{{ω}}(μ) coincide,

(8) there exists a fundamental solution E for S_{μ} satisfying

ch(sing supp_{{ω}}(E)) = − ch(sing supp_{{ω}}(μ)),

(9) for each open set Ω ⊂ R^{N} and u ∈ 𝒟′_{{ω}}(Ω−supp(μ)+ch(sing supp_{{ω}}(μ))) satisfying μ∗u ∈ ℰ_{{ω}}(Ω+ch(sing supp_{{ω}}(μ))), it follows that u ∈ ℰ_{{ω}}(Ω).

3.2. Definition. (a) If μ ∈ ℰ′_{{ω}}(R^{N}) satisfies condition 3.1(1), then μ and also S_{μ} are called {ω}-hypoelliptic.

(b) F ∈ 𝒟′_{{ω}}(R^{N}) is called a parametrix for μ ∈ ℰ′_{{ω}}(R^{N}) if μ ∗ F = δ + φ for some φ ∈ ℰ_{{ω}}(R^{N}).

3.3. Proposition. If μ ∈ ℰ′_{{ω}}(R^{N}) satisfies Ker S_{μ} ⊂ ℰ_{{ω}}(R^{N}), then

lim_{z∈V (μ̂)} inf_{|z|→∞} \frac{| Im z|}{ω(z)} > 0.

Proof. Arguing by contradiction we assume that there exists a sequence (z_{j})_{j∈N} ⊂ C^{N}, |z_{j}| → ∞, μ̂(z_{j}) = 0 and | Im z_{j}| < ω(z_{j})/j for each j ∈ N. If a = (a_{j})_{j∈N} ∈ l_{1} and φ ∈ 𝒟_{{ω}}(R^{N}), we can apply the Paley–Wiener–Komatsu theorem [5, 3.5] to conclude

∑_{j=1}^{k} |a_{j}| |⟨e^{i⟨x,z_{j}⟩}, φ⟩| ≤ C ∑_{j=1}^{k} |a_{j}| exp((\frac{D}{j} − ε)ω(z_{j})),

for some ε > 0, D, C > 0 depending only on φ. Now an application of the closed graph theorem (see Meise and Vogt [17, 24.31]) and an obvious modification of the proof of 2.3 leads to a contradiction.

3.4. Proposition. Let μ ∈ ℰ′_{{ω}}(R^{N}) is {ω}-hypoelliptic, then for each m ∈ N there is R_{m} > 0 such that |μ̂(x)| ≥ exp(−ω(x)/m) for each x ∈ R^{N}, |x| ≥ R_{m}.

Proof. Proceeding by contradiction, we find m ∈ N and a sequence (ξ_{j})_{j∈N} in R^{N} such that |ξ_{j}| > 2|ξ_{j−1}| > 2^{j} for each j ≥ 2 such that |μ̂(ξ_{j})| ≤ exp(−ω(ξ_{j})/m) for each j ∈ N. By Lemma 2.4, η := ∑_{j=1}^{∞} exp(i⟨ · , ξ_{j}⟩) belongs to 𝒟′_{{ω}}(R^{N}) but it is not a 𝒞^{∞} function on R^{N}. Proceeding similarly as in the proof of 2.5 we show μ ∗ η ∈ ℰ_{{ω}}(R^{N}). This is a contradiction.

3.5. Proposition. Let ω and σ_{0} be two weight functions with σ_{0} = o(ω). Let S := {σ : σ is a weight function, σ_{0} ≤ σ, σ = o(ω)}. Then, for each open set Ω in R^{N}, the following topological equality holds

ℰ_{{ω}}(Ω) = ⋂_{σ∈S} ℰ_{(σ)}(Ω),

when the space on the right hand side is endowed with the corresponding projective limit topology.

Proof. To prove the equality as vector spaces, it suffices to show

𝒟_{{ω}}(B_{δ}(0)) = ⋂_{σ∈S} 𝒟_{(σ)}(B_{δ}(0)),

for each δ > 0, since the equality holds if it holds locally. By Meise, Taylor and Vogt [5, 3.9] the inclusion “⊂” is valid. To prove the converse inclusion, fix f ∈ ⋂_{σ∈S} 𝒟_{(σ)}(B_{δ}(0)) and choose (ρ_{ε})_{ε>0} in 𝒟_{{ω}}(R^{N}) such that ρ_{ε} → δ in 𝒟′_{{ω}}(R^{N}) as ε ↓ 0. Fix σ ∈ S and note that f ∗ ρ_{ε} → f in ℰ_{(σ)}(B_{δ}(0)) (see the proof of [5, 3.8]). Since this holds for all σ ∈ S, it follows from [15, 3.3] that (f ∗ ρ_{ε})_{ε>0} is a Cauchy net in ℰ_{{ω}}(B_{δ}(0)), which is a complete space by [5, 4.9]. Accordingly, there exists g ∈ ℰ_{{ω}}(B_{δ}(0)) such that f ∗ ρ_{ε} → g in ℰ_{{ω}}(B_{δ}(0)). Since f ∗ ρ_{ε} → f pointwise, it follows that f = g ∈ 𝒟_{{ω}}(B_{δ}(0)).

Note that the topological equality in the proposition follows from the algebraic one and [15, 3.3].

3.6. Remark. Under certain conditions on the sequence (M_{p})_{p≥0} , see [5, 8.9] and 1.5(4), the spaces ℰ^{{M_{p}}}(Ω) coincide with ℰ_{{ω_{M} }}(Ω) for a suitably defined weight function ω_{M}. In this situation the proposition above shows that in Chou [7, I.2.6] even the topological equality holds.

3.7. Corollary. For each u ∈ 𝒟′_{{ω}}(R^{N}) we have
sing supp_{\{ω\}}(u) = \overline{⋃_{σ∈S} sing supp_{(σ)}(u)}.

Proof. Let A := sing supp_{{ω}}(u). Then u|_{R^{N} \A} ∈ ℰ_{{ω}}(R^{N}\A). By Proposition 3.5, u|_{R^{N} \A} ∈ ℰ_{(σ)}(R^{N}\A) for each σ ∈ S. Hence sing supp_{(σ)}(u) ⊂ A for each σ ∈ S. Since A is closed, this implies

B := \overline{⋃_{σ∈S} sing supp_{(σ)}(u)} ⊂ A.

To prove the converse implication, fix x_{0} ∈ R^{N}\B. If no such x_{0} exists, we have B = R^{N}, hence A = R^{N}. Choose a neighbourhood V of x_{o} such that V ∩B = ∅. Then u|_{V} ∈ ℰ_{(σ)}(V) for every σ ∈ S and hence, by Proposition 3.5, u ∈ ℰ_{{ω}}(V). This implies x_{0} ∉ sing supp_{{ω}}(u). Therefore R^{N}\B ⊂ R^{N}\A and consequently A ⊂ B.

3.8. Lemma. If μ ∈ ℰ′_{{ω}}(R^{N}) satisfies the hypothesis of 3.1(3), then there is a weight function σ_{0} with σ_{0} = o(ω) such that for each weight function σ satisfying σ_{0} ≤ σ and σ = o(ω), μ belongs to ℰ′_{(σ)}(R^{N}) and μ is (σ)-hypoelliptic.

Proof. By [5, 7.6] there is a weight function σ_{1} with σ_{1} = o(ω) such that μ ∈ ℰ′_{(σ_{1})}(R^{N}). By [2, 3.2], there is a weight σ_{0} ≥ σ_{1} with σ_{0} = o(ω) and such that μ is (σ_{0})-slowly decreasing. By the very definition μ ∈ ℰ′_{(σ)}(R^{N}) and μ is (σ)-slowly decreasing for every weight function σ such that σ_{0} ≤ σ and σ = o(ω). By Theorem 2.1 it remains to show that Im z/σ(z) tends to ∞ as |z| → ∞, z ∈ V (μ̂), for every weight function σ. To see this observe that

\frac{Im z}{σ(z)} = \frac{Im z}{ω(z)} \frac{ω(z)}{σ(z)}, z ∈ C^{N}

and apply σ = o(ω) and the first part of the assumption 3.1(3).

3.9. Remark. If F and G are two parametrices of the {ω}-hypoelliptic operator μ ∈ ℰ′_{{ω}}(R^{N}), then sing supp_{{ω}}(F) and sing supp_{{ω}}(G) coincide. The proof is the same as the one of 2.11.

Proof of Theorem 3.1. (1) ⇒ (2): This follows directly from Propositions 3.3 and 3.4.

(2) ⇒ (3): This holds obviously.

(3) ⇒ (6): By 3.8, there is R_{0} > 0 such that μ̂(x) ≠ 0 for every x ∈ R^{N} with |x| ≥ R_{0} , since μ ∈ ℰ′_{(σ_{0})}(R^{N}) and is (σ_{0})-hypoelliptic. We define, for φ ∈ 𝒟_{{ω}}(R^{N}),

⟨F_{0}, φ⟩ := (\frac{1}{2π} )^{N} ∫_{|x|≥R_{0}} \frac{φ̂(−x)}{μ̂(x)} dx.

Since μ ∈ ℰ′_{(σ)}(R^{N}) and is (σ)-hypoelliptic for all weight functions σ with σ_{0} ≤ σ and σ = o(ω), the same definition of F_{0} for each φ ∈ 𝒟_{(σ)}(R^{N}) yields, by Lemma 2.9, that F_{0} ∈ 𝒟′_{(σ)}(R^{N}) ⊂ 𝒟′_{{ω}}(R^{N}), and F_{0} is a parametrix for μ such that μ ∗ F_{0} = δ + G for a real analytic function G on R^{N} and

sing supp_{(σ)}(F_{0}) ⊂ − ch(supp(μ)).

As supp(μ) is compact, using Corollary 3.7 we obtain

sing supp_{{ω}}(F_{0}) ⊂ − ch(supp(μ)).

We fix φ ∈ 𝒟_{{ω}}(R^{N}) identically 1 on a neighbourhood of ch(sing supp_{{ω}}(μ)) and we set F = φF_{0} ∈ ℰ′_{{ω}}(R^{N}). Since (1 − φ)F_{0} ∈ ℰ_{{ω}}(R^{N}), we have

μ ∗ F = δ + G − μ ∗ ((1 − φ)F_{0});

since G − μ ∗ ((1 − φ)F_{0}) ∈ ℰ_{{ω}}(R^{N}), we reach the conclusion.

(6) ⇒ (1): This follows with an argument similar to the one we used to check (5) ⇒ (1) in the proof of Theorem 2.1.

(1) ⇒ (7): Since (1) implies (2), we can proceed as in the proof of (3) ⇒ (6) to get that, for some R_{0} > 0,

⟨F, φ⟩ := (\frac{1}{2π} )^{N} ∫_{|x|≥R_{0}} \frac{φ̂(−x)}{μ̂(x)} dx

defines F ∈ 𝒟′_{(σ)}(R^{N}) with sing supp_{(σ)}(F) ⊂ − ch(supp(μ)) for every weight function σ ≥ σ_{0} and σ = o(ω), and F ∗ μ = δ + g, g real analytic in R^{N}. By the proof of (1)⇒(6) in Theorem 2.1,

sing supp_{(σ)}(F) ⊂ − ch(sing supp_{(σ)}(μ))

for every σ and we can apply Corollary 3.7 to conclude

sing supp_{{ω}}(F) ⊂ − ch(sing supp_{{ω}}(μ)).

If we apply (6), which follows from (1), we obtain G ∈ ℰ′_{{ω}}(R^{N}) with μ∗G = δ+ψ and with ψ ∈ ℰ_{{ω}}(R^{N}), that is, G is {ω}-hypoelliptic and μ is a parametrix for G. Since all the parametrices have the same {ω}-singular support,

sing supp_{{ω}}(μ) ⊂ − ch(sing supp_{{ω}}(F)).

(7) ⇒ (8): Multiplying by a cut-off function on 𝒟_{{ω}}(R^{N}) which is identically 1 on a neighbourhood of − ch(sing supp_{{ω}}(μ)), we conclude (6), which is already equivalent to (1) and (3). Consequently μ is {ω}-slowly decreasing, hence there is a fundamental solution E ∈ 𝒟′_{{ω}}(R^{N}) of μ by [2, 3.4]. Since E is a parametrix for μ, we can apply 3.9 to conclude (8).

(8) ⇒ (9): The proof is the same as the proof of (7) ⇒ (8) in Theorem 2.1.

(9) ⇒ (1) ⇒ (4): The first implication is obvious, the second one follows from Proposition 3.4.

(4) ⇒ (5): First observe that Ker S_{μ} is a Fréchet space and Ker S_{μ} = Ker T_{μ}. Since μ is slowly decreasing for {ω}, the operator T_{μ} is locally surjective by [2, 3.4]. Since Ker T_{μ} is a Fréchet space, a Mittag-Leffler argument shows that T_{μ} is surjective on ℰ_{{ω}}(R^{N}). Alternatively one can use the theory of the vanishing of the derived functor Proj^{1} as it is developed in Palamodov [18] and Vogt [21]: since Ker S_{μ} is a Fréchet space and Ker S_{μ} = Ker T_{μ}, we have Proj^{1} Ker T_{μ} = 0. Since Proj^{1}ℰ_{{ω}}(R^{N}) = 0, this implies that T_{μ} is surjective on ℰ_{{ω)}(R^{N}).

(5) ⇒ (1): We fix ν ∈ 𝒟′_{{ω}}(R^{N}) such that g = ν ∗ μ ∈ ℰ_{{ω}}(R^{N}). Since T_{μ} is surjective, we find f ∈ ℰ_{{ω}}(R^{N}) with f ∗ μ = g. This implies f − ν ∈ Ker S_{μ} ⊂ ℰ_{{ω}}(R^{N}), from where ν ∈ ℰ_{{ω}}(R^{N}) follows.

Characterization of the ω-hypoelliptic convolution operators 283

3.10. Remark. Theorem 3.1 extends the results of Chou [7, IV.2-1] in case that the classes of ultradifferentiable functions considered in Chou coincide with those considered in the present paper. In this case our results are more precise and give more equivalences.

Note that the proof of (3) ⇒ (6) in Theorem 3.1 shows that the converse of Lemma 3.8 holds, too.

3.11. Corollary. Let μ ∈ ℰ′_{{ω}}(R^{N}) be given. Then μ is {ω}-hypoelliptic if and only if there is a weight function σ_{0} with σ_{0} = o(ω) such that for each weight function σ satisfying σ_{0} ≤ σ and σ = o(ω), μ belongs to ℰ′_{(σ)}(R^{N}) and μ is (σ)-hypoelliptic.

3.12. Remark. Note that by [5, 8.8] the intersection of all spaces ℰ_{(ω)}(R^{N}) is equal to the intersection of all spaces ℰ_{{ω}}(R^{N}) and hence equal to the space of all real analytic functions on R^{N}, by the theorem of Bang–Mandelbrojt (see Chou [7, I.2.2]). Thus, each μ ∈ ℰ′_{(ω)}(R^{N}) (resp. μ ∈ ℰ′_{{ω}}(R^{N})) which is (σ)- hypoelliptic (resp. {σ}-hypoelliptic) for every non-quasianalytic weight function σ with σ ≥ ω, is elliptic in the sense of [7, IV.3].

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Received 29 May 1998