Bulletin T.CXXIII de l’Académie Serbe des Sciences et des Arts - 2002 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 27

ON HYPOELLIPTICITY IN G

M. NEDELJKOV, S. PILIPOVIĆ

(Presented at the 4th Meeting, held on May 31, 2002)

A b s t r a c t. We give a condition of sufficiency for the hypoellipticity of a family of equations with constant coefficients satisfied prescribed power growth rate with respect to ε ∈ (0, 1). The framework is Colombeau algebra of generalized functions.

AMS Mathematics Subject Classification (2000): 35H10, 46F30 Key Words: hypoellipticity, Colombeau generalized functions

1. Introduction

We have considered in [5] the hypoellipticity of a differential equation with generalized constant coefficients and have given several necessary conditions. In order to explain our approach, consider a family of equations with constant coefficients

P_{ε}(D)G = ∑_{|α|≤m} a_{α,ε}D^{α}G = F_{ε}, F_{ε} ∈ C^{∞}(Ω), ε ∈ (0, 1)

If P_{ε}(D) is hypoelliptic for fixed ε ∈ (0, 1), then the corresponding solution to the above equation, G_{ε}, is in C^{∞}(Ω). If we suppose that sup_{x∈K⊂⊂Ω} |D^{α}F_{ε}(x)| satisfies the power growth condition O(ε^{−N_{K}}) for every α (N_{K} depends on a compact set K), then one can ask whether the derivatives of G_{ε} satisfy similar estimates on compact sets.

In this paper we will repeat necessary conditions for the hypoellipticity ([5]) and give appropriate sufficient conditions for it.

Another type of hypoellipticity was considered by Hörmann and Oberguggenberger (personal communication) who pointed to us that our sufficient conditions in [5] need some additional assumptions. In this paper we reconsider the following condition:

“There exist N > 0 and q ∈ ℕ such that for every ϕ ∈ A_{q} A > 0 there exist η > 0 and B ∈ ℝ such that

|τ| ≥ A(log |σ| + N log ε) − B, σ + iτ ∈ V (P_{ϕ,ε}), ε ∈ (0, η), ”

and show that above we need

|τ| ≥ A log |σ| + N log ε − B, σ + iτ ∈ V (P_{ϕ,ε}), ε ∈ (0, η).

The proof of this fact is the main goal of this paper.

2. Colombeau algebras

Let

A_{0}(ℝ) = {ϕ ∈ C_{0}^{∞}| ∫ ϕ(x)dx = 1, diam(supp ϕ) = 1},
A_{q}(ℝ) = {ϕ ∈ A_{0}| ∫ x^{α}ϕ(x)dx = 0, 1 ≤ α ≤ q, α ∈ ℕ}, q ∈ ℕ

and A_{q}(ℝ^{n}) = {ϕ(x_{1}, …, x_{n}) = ϕ_{1}(x_{1}) · . . . · ϕ_{1}(x_{n})| ϕ_{1} ∈ A_{q}(ℝ)}. Put ϕ_{ε} = (1/ε)ϕ(·/ε), where ϕ ∈ A_{0}.

Let Ω be an open subset of ℝ^{n} and E(Ω) be the space of functions G : A_{0} × (0, 1) × Ω → ℂ which are C^{∞} for every ϕ ∈ A_{0} and ε ∈ (0, 1). We will use the notation G_{ϕ,ε} for (ϕ, ε, x) ↦ G_{ϕ,ε}(x), x ∈ Ω.

A family of smooth complex valued functions on Ω, G_{ϕ,ε}, ϕ ∈ A_{0}, ε ∈ (0, 1), belongs to E_{M}(Ω) if for every compact set K ⊂⊂ Ω and α ∈ ℕ_{0}^{n} there exist N ∈ ℕ and r = r(K, α) ∈ ℝ such that

sup_{x∈K} |∂^{α}G_{ϕ,ε}(x)| = O(ε^{r}), ε → 0, for every ϕ ∈ A_{N}.  (1)

On Hypoellipticity in G 49

If G_{ϕ,ε} does not depend on x and (1) holds for α = 0, then the space of corresponding families of complex numbers is denoted by ℂ_{M}. If g ∈ D′, the corresponding element in E_{M} is given by G_{ϕ,ε} = g ∗ δ_{ϕ,ε}, where we use the notation δ_{ϕ,ε} = ϕ_{ε}, ϕ ∈ A_{0} since it is a delta net.

The space of all elements G_{ϕ,ε} in E_{M}(Ω) which satisfy (1) independently of α ∈ ℕ_{0} is denoted by E_{M}^{∞}.

The space E_{0}(Ω) (resp. ℂ_{0}) is the subspace of E_{M}(Ω) (resp. ℂ_{M}) consisting of elements G_{ϕ,ε} with the property that for every K ⊂⊂ Ω, α ∈ ℕ_{0}^{n} and r ∈ ℝ there exists N ∈ ℕ such that (1) holds (resp. (1) holds for α = 0 and G_{ϕ,ε} does not depend on x) for every ϕ ∈ A_{N}.

The space of Colombeau’s generalized functions on an open set Ω ⊂ ℝ^{n} is defined by G(Ω) = E_{M}(Ω)/E_{0}(Ω) and \overline{ℂ} = ℂ_{M}/ℂ_{0} is the ring of Colombeau’s generalized complex numbers. Note that Ω → G(Ω), Ω ⊂ ℝ^{n}, is a sheaf.

[G_{ϕ,ε}] denotes the class in G (or \overline{ℂ}) determinated by the representative G_{ϕ,ε}.

Let G ∈ G(Ω). The complement of the largest open set of Ω in which G is equal to the zero generalized function is called the support of G, supp_{g} G.

The space of generalized functions with compact supports in the interior of Ω is denoted by G_{c}(Ω).

G^{∞}(Ω) (cf. [6]) is the space of all generalized functions which have a representative in E_{M}^{∞}. It is a subalgebra of G(Ω) and

G^{∞}(Ω) ∩ D′(Ω) = C^{∞}(Ω) (see [6]).

The space of tempered Colombeau’s generalized functions G_{t}(ℝ^{n}) is defined to be E_{t}(ℝ^{n})/E_{0t}(ℝ^{n}), where E_{t}(ℝ^{n}) is the set of all G_{ϕ,ε} ∈ E such that for every α ∈ ℕ_{0}^{n} there exist γ > 0, N ∈ ℕ and r ∈ ℝ such that

sup_{x∈ℝ^{n}} |∂^{α}G_{ϕ,ε}(x)|/(1 + |x|)^{γ} = O(ε^{r}), ε → 0 for every ϕ ∈ A_{N}, (2)

and E_{0t}(ℝ^{n}) is the space of all G_{ϕ,ε} ∈ E_{t} with the property that for every α ∈ ℕ_{0}^{n} there exists γ ∈ ℝ such that for every r ∈ ℝ there exists N ∈ ℕ such that (2) holds for every ϕ ∈ A_{N}.

Note that G_{t}(ℝ^{n}) is not a subspace of G(ℝ^{n}), but there is a canonical map G_{t}(ℝ^{n}) → G(ℝ^{n}).

Let G ∈ G(Ω). The complement of the largest open set of Ω in which G is in G^{∞}(Ω) is called the singular support of G. It is denoted by singsupp_{g} G.

The equality in G is often too strong for applications, so we shall use a notion of equality in generalized distribution (resp. generalized tempered distribution) sense =^{g.d.} (resp. =^{g.t.d.}) which is defined by:

G_{1} =^{g.d.} G_{2} (resp. G_{1} =^{g.t.d.} G_{2} for tempered generalized functions) if for every ψ ∈ D (resp. ψ ∈ S), ⟨G_{1}, ψ⟩ = ⟨G_{2}, ψ⟩ in \overline{ℂ}, where ⟨G, ψ⟩ means ∫ G(x)ψ(x)dx.

3. Hypoellipticity

Following [8], we define polynomials in n real variables as elements of the ring \overline{ℂ}[x_{1}, …, x_{n}]. A generalized polynomial function is a tempered generalized function of the form

∑_{|α|≤m} a_{α}x^{α}, x ∈ ℝ, a_{α} = [a_{α,ϕ,ε}] ∈ \overline{ℂ}, α ∈ ℕ_{0}^{n}.

It is of degree m if a_{α} = 0 for |α| > m and there exists β, |β| = m such that a_{β} ≠ 0.

If [H_{ϕ,ε}(x)] = ∑_{|α|≤m}[a_{α,ϕ,ε}]x^{α} is such a generalized function, then it can be written only in one way as a polynomial. In fact, if ∑_{|β|≤m} b_{β,ϕ,ε}x^{β} = N_{ϕ,ε}(x) ∈ E_{0t}(ℝ^{n}), then by making successive derivations and by putting x = 0 it follows b_{β,ϕ,ε} ∈ ℂ_{0}, |β| ≤ m ([8]).

Let us remind that in the classical distribution theory a fundamental solution of a differential operator is a distribution E such that P (D)E = δ.

Let

P (D) = ∑_{|α|≤m} a_{α}D^{α} = [P_{ϕ,ε}(i\frac{∂}{∂x})] (D^{α} = i^{|α|}∂^{α}),  (3)

where ∑_{|α|≤m} a_{α}x^{α} is a polynomial in G. In Colombeau’s theory, the fundamental solution of P is a generalized function E ∈ G satisfying P (D)E = [δ_{ϕ,ε}]. This means that its representatives E_{ϕ,ε} ∈ E_{M} satisfy

∑ a_{α,ϕ,ε}D^{α}E_{ϕ,ε}(x) = δ_{ϕ,ε}(x) + N_{ϕ,ε}(x), x ∈ ℝ^{n},

for some N_{ϕ,ε} ∈ E_{0}.

This fundamental solution allows us to solve the equation P (D)U =^{g.d.} G

for G ∈ G, because G ∗ [δ_{ϕ,ε}] =^{g.d.} G.

Proposition 5 ([5]) Let P (D) be a generalized differential operator of the form (4) with coefficients in \overline{ℂ} of degree m such that for some (c_{1}, c_{2}, …, c_{n}) ∈ ℝ^{n} there exist r > 0 and N ∈ ℕ such that for every ϕ ∈ A_{N} there exist C > 0 and η > 0 such that

| ∑_{|α|=m} a_{α,ϕ,ε}c^{α}| ≥ Cε^{r}, ε ∈ (0, η).  (4)

Then, P (D) admits a generalized fundamental solution.

In non-standard models of Colombeau’s theory this hypothesis can be replaced by ∑_{|α|=m} a_{α,ϕ,ε}c^{α} ≠ 0, since \overline{ℂ} is a field in such models (cf. Li Bang He, [4] and Oberguggenberger [7]).

Solution to

∑_{|α|≤m} a_{α,ϕ,ε}D^{α}G_{ϕ,ε} = F_{ϕ,ε}, F_{ϕ,ε} ∈ E_{M}, a_{α,ϕ,ε} ∈ ℂ_{M},  (5)

in E_{M} are constructed in [8], in a simplified version of Colombeau’s theory, by adapting the classical distributional method of solving a constant coefficients partial differential equation. Problems which are specific for (5) are connected with the growth rate of solutions with respect to ε which implies that the main assertions in [8] and in this paper are non-trivial generalization of the corresponding ones in the space of distributions.

In this paper we investigate the hypoellipticity of the families of equations (5). This family is hypoelliptic if F_{ϕ,ε} ∈ E_{M}^{∞} implies G_{ϕ,ε} ∗ δ_{ϕ,ε} ∈ E_{M}^{∞}.

Now we give the definition of hypoellipticity in the framework of Colombeau generalized functions. Let [P_{ϕ,ε}(D)] be of the form (3) and suppose that (4) holds for some c = (c_{1}, …, c_{n}) ∈ ℝ^{n}, C_{1} > 0, r > 0, N ∈ ℕ and η > 0. This operator is called hypoelliptic if for every open Ω ⊂ ℝ^{n} and every solution G ∈ G(Ω) to

P (D)G = 0,  (6)

the generalized function G ∗ [δ_{ϕ,ε}] is in G^{∞}(Ω).

Proposition 6 ([5])

a) P (D) is hypoelliptic if it admits a fundamental solution E with

singsupp_{g} E = {0}.  (7)

b) Let [P_{ϕ,ε}(D)] be hypoelliptic. Then, for every open set Ω and G ∈ G(Ω), P (D)G ∈ G^{∞}(Ω) implies G ∗ [δ_{ϕ,ε}] ∈ G^{∞}(Ω).

Let P_{ϕ,ε} be a hypoelliptic differential operator on Ω, Ω_{0}, W, and O are the same as in the proof of assertion a) in Proposition 6, G a solution of (6) and W ± O ⊂ Ω_{0}. Then we have the following

Proposition 7 ([5]) There exists N ∈ ℕ such that for every q ∈ N_{0}^{n} there exist C > 0 and η > 0 such that

max_{x∈W} |D^{q}G_{ϕ,ε} ∗ δ_{ϕ,ε}(x)| ≤ C max_{x∈Ω_{0}} |G_{ϕ,ε}(x)|ε^{−N}, ε < η.

4. Main assertion

The main prerequisite, Lemma 1 of [5], has the same proof as in [5].

Lemma 1 Let N, A, η and B be the same as in (8) below. Assume that a), b) and c) hold for ε < ε_{0}, ϕ ∈ A_{q_{0}} where

a) (σ_{1}, σ_{2}) ∈ ℝ^{2} such that A log |(σ_{1}, σ_{2})| + N log ε ≥ B + 1.

b) τ_{1} ∈ ℝ, |τ_{1}| ≤ (A log |(σ_{1}, σ_{2})| + N log ε − B)/2.

c) (σ̄_{1} + iτ̄_{1}) ∈ V (P_{ϕ,ε}) and |τ̄_{1}| ≥ (A log |(σ̄_{1}, σ_{2})| + N log ε − B).

Then, for every fixed ε < η, |σ̄_{1} − σ_{1}| ≥ ε^{N+1} or |τ̄_{1} − τ_{1}| ≥ ε^{N+1}.

Theorem 1 The operator P (D) is hypoelliptic if and only if there exist N > 0 and q ∈ ℕ such that for every ϕ ∈ A_{q} and A > 0 there exist η > 0 and B > 0 such that

|τ| ≥ A log |σ| + N log ε − B, σ + iτ ∈ V (P_{ϕ,ε}), ε ∈ (0, η).  (8)

Proof of sufficiency Without a loss of generality one can assume that η < ε_{0} and q > q_{0}, where η and q are used in the estimates bellow, while ε_{0} and q_{0} are from Lemma 1. The change of constant A bellow will cause an appropriate decrease of η, but we shall use the same letter η in all cases.

One needs to prove that (8) implies (5). We will assume, without loss of generality, that P_{ϕ,ε} is of the form P_{ϕ,ε}(s) = a_{m,ϕ,ε}s_{1}^{m} + lower order terms. The fundamental solution for P_{ϕ,ε} is given by

E_{ϕ,ε}(x) = (2π)^{−n} ∫_{T_{ϕ,ε}} \frac{e^{−i⟨x,s⟩}ϕ̂(εs)}{P_{ϕ,ε}(s)}ds
= (2π)^{−n} ∑_{k=1}^{∞} ∫_{ℝ} dσ_{1} ∫_{Γ_{k,ϕ,ε}} \frac{e^{−i⟨x,(σ_{1}+iτ_{1},σ′)⟩}ϕ̂(ε(σ_{1} + iτ_{1}, σ′))}{P_{ϕ,ε}((σ_{1} + iτ_{1}, σ′))}dσ′,

where T_{ϕ,ε} = ⋃_{j=1}^{∞}{(σ_{1} + iτ_{1}, σ′); σ_{1} ∈ ℝ, τ_{1} = k_{j} ∈ {0, …, m + 1}, σ′ = σ_{2} ∈ Γ_{j,ϕ,ε}} and Γ_{j,ϕ,ε} are bounded closed domains of ℝ^{n−1} such that for ϕ ∈ A_{q}, P_{ϕ,ε}(ξ) > C_{P} ε^{r}, ξ ∈ T_{ϕ,ε}, ε < η.

Only the proof for n = 2 is given bellow. Higher dimensions can be handled in a similar way.

Let the ball B_{h} = B((x_{1}, x_{2}), h), h > 0, does not contain the point 0 and a = dist(0, B_{h}). We will prove that E_{ϕ,ε} represents an element in G^{∞}(B_{h}).

Divide (σ_{1}, σ_{2})-plane into nine regions Ω_{j}, j = 1, …, 9 by the lines σ_{1} = ±μ, σ_{2} = ±μ and denote them by Ω_{1} = {|σ_{1}| ≤ μ, |σ_{2}| ≤ μ}, Ω_{2} = {σ_{1} ≥ μ, |σ_{2}| ≤ μ}, Ω_{3} = {σ_{1} ≥ μ, σ_{2} ≥ μ}, Ω_{4} = {|σ_{1}| ≤ μ, σ_{2} ≥ μ}, . . .. Now, choose μ > 0 such that

A log μ+N log ε−B = max{ max_{s∈T_{ϕ,ε}} |τ|, 1} = C_{0}, i.e. μ = e^{B+C_{0}/A}ε^{−N/A}, ε < η.

Without a loss of generality one can assume that B((x_{1}, x_{2}), h) ⊂ [0, ∞)× [0, ∞). Denote by T_{j,ϕ,ε} the projection of Ω_{j} on T_{ϕ,ε},

T_{j,ϕ,ε} = {(σ_{1} + iτ_{1}, σ_{2})| σ_{1} ∈ ℝ, τ_{1} ∈ {0, …, m + 1}, σ_{2} ∈ Γ_{k,ϕ,ε} ∩ Ω_{j}, k ∈ ℕ}.

Then E_{ϕ,ε} = ∑_{j=0}^{9} E_{j,ϕ,ε}, where

E_{j,ϕ,ε}(x) = (2π)^{−n} ∫_{T_{j,ϕ,ε}} \frac{e^{−i⟨(x_{1},x_{2}),(s_{1},s_{2})⟩}ϕ̂(εs)}{P_{ϕ,ε}(s)}ds, j = 1, …, 9, x ∈ ℝ^{2}.

We will show that there exist Ñ and q̃ such that for every ϕ ∈ Aq̃, α = (α_{1}, α_{2}) ∈ ℕ_{0}^{2} there exist C_{α} > 0 and η_{α} such that

sup_{x∈B_{h}} |∂^{α}E_{j,ϕ,ε}(x)| ≤ C_{α}εÑ, ε < η_{α}, j = 1, …, 9,  (9)

We always take q̃ = q and ϕ ∈ A_{q}.

Let j = 1 and A > 3|α|. Put η_{α} = η and take any ε < η. Using that mes(T_{1,ϕ,ε}) = (2μ)^{2} = 4(e^{(B+C_{0})/A}ε^{−N/A})^{2}, P_{ϕ,ε}(s) ≥ C_{P} ε^{r}, s ∈ T_{1,ϕ,ε} and |s_{1}^{α_{1}}s_{2}^{α_{2}}| ≤ C̃_{α}ε^{−(N/A)|α|} (ε < η), we have

sup_{x∈B_{h}} |∂^{α}E_{1,ϕ,ε}(x)| = (2π)^{−n} |∫_{T_{1,ϕ,ε}} \frac{(−i)^{|α|}s_{1}^{α_{1}}s_{2}^{α_{2}} ϕ̂(εs)}{P_{ϕ,ε}(s)}ds| (10)
≤ Cε^{−3N|α|/A−r},  (11)

for some C > 0. This proves (9) for j = 1.

Consider E_{2,ϕ,ε}, ε < η. The integration over the contour σ_{1} + iτ_{1}, μ ≤ σ_{1} ≤ ν, τ_{1} ∈ {0, …, m + 1} is changed by integration over the contour

\overline{Q(μ)Q_{1}(μ)} ∪ \overline{Q_{1}(μ)Q_{1}(ν)} ∪ −\overline{Q(ν)Q_{1}(ν)},

where \overline{Q(μ)Q_{1}(μ)} = \{μ+it| 0 ≤ t ≤ \frac{1}{2}(A log μ+N log ε−B)\}, \overline{Q_{1}(μ)Q_{1}(ν)} = \{σ_{1} + iτ_{1}| τ_{1} = \frac{1}{2}(A log |σ_{1}| + N log ε − B), σ_{1} ∈ [μ, ν]\}, \overline{Q(ν)Q_{1}(ν)} = \{ν + it| 0 ≤ t ≤ \frac{1}{2}(A log ν + N log ε − B)\}. We have

∂^{α}E_{2,ϕ,ε}(x) = (2π)^{−n} ∫_{−μ}^{μ}(∫\overline{Q(μ)Q_{1}(μ)} − ∫\overline{Q(ν)Q_{1}(ν)} + ∫\overline{Q_{1}(μ)Q_{1}(ν)})

\frac{s_{1}^{α_{1}}s_{2}^{α_{2}}e^{−i⟨(x_{1},x_{2}),(s_{1},s_{2})⟩}ϕ̂(εs)}{P_{ϕ,ε}(s)}ds = I_{1ε} + I_{2ε} + I_{3ε}, x ∈ B_{h}.

Since

|P_{ϕ,ε}(s_{1}, σ_{2})| = |a_{m,ϕ,ε}| ∏_{j=1}^{m}(|σ_{1} − \overline{σ}_{1}|^{2} + |τ_{1} − \overline{τ}_{1}|^{2})^{1/2},

where σ_{2} is fixed, \overline{σ}_{1} + i\overline{τ}_{1} ∈ V (P_{ϕ,ε}) and s_{1} = σ_{1} + iτ_{1} belongs to any of the quoted contours, Lemma 1 implies that |P_{ϕ,ε}(s)| ≥ C_{P} ε^{r} on these contours (for ε < η). Now, one can prove that I_{1ε} ≤ Cε^{−r−2}ε^{−N/A}, ε < η for some C > 0.

For every k > 0 there exists C_{k} > 0 such that

|ν + iτ|^{α_{1}}|σ_{2}|^{α_{2}}|ϕ̂(ε(ν + iτ, σ_{2}))|
≤ |ν + iτ|^{α_{1}}|μ|^{α_{2}} \frac{C_{k}e^{ε|τ|}}{(1 + ε(ν^{2} + τ^{2} + |σ_{2}|^{2})^{1/2})^{k}} on \overline{Q(ν)Q_{1}(ν)}

(see (1.4) in [2],Ch.2, Sec.2). Choosing ν = ε^{−2} and k large enough one gets I_{2ε} ≤ Cε^{−r−1}ε^{−N/A}, for some C > 0.

Consider I_{3ε}. Again,

|ϕ̂(ε(σ_{1} + i(log |σ_{1}|^{A}ε^{N} − B)/2, σ_{2}))|
≤ \frac{C_{k}e^{ε(log |σ_{1}|^{A}ε^{N})/2}}{(1 + ε(σ_{1}^{2} + (log |σ_{1}|^{A}ε^{N} − B)^{2}/4 + |σ_{2}|^{2})^{1/2})^{k}} ≤ C|σ_{1}|^{Aε/2},

on \overline{Q_{1}(μ)Q_{1}(ν)} for some C > 0. Thus, with suitable constants, we have

|I_{3ε}| ≤ C_{1} ∫_{−μ}^{μ} ∫_{μ}^{∞} \frac{|σ_{1} + i(log(|σ_{1}|^{A}ε^{N}) − B)/2|^{α_{1}}|σ_{2}|^{α_{2}}e^{−x_{1}(log(|σ_{1}|^{A}ε^{N})−B)/2}}{P_{ϕ,ε}(s)}

ϕ̂(εs)(1 + \frac{A}{2|σ_{1}|})dσ_{1}dσ_{2}

≤ C_{2}ε^{−r}(2μ)|μ|^{α_{2}}ε^{−aN/2} ∫_{μ}^{∞} |σ_{1}|^{Aε/2−aA/2}(|σ_{1}|^{α_{1}} + log^{α_{1}} |σ_{1}|^{A}ε^{N})dσ_{1}.

Taking A so large that −aA/4 dominates all exponents of σ_{1} under the integral sign and σ_{1} > μ, we have (with suitable constants)

|I_{3ε}| ≤ C_{3}ε^{−r−aN/2}|μ|^{α_{2}+1−aA/4+Aε/2+α_{1}} log^{α_{1}} |μ|^{A} ∫_{1}^{∞} |σ̃|^{−aN/4}dσ̃
≤ C_{4}ε^{−r−aN/2}(ε^{−N/A})^{α_{2}+2−aA/4+Aε/2+α_{1}}.

Now, it is easy to see that (4) holds in the case j = 2 if one takes A > a(|α| + 2)/4.

One can give similar estimate for E_{3,ϕ,ε}, ε < η with a change of the integration over T_{3,ϕ,ε} (for μ ≤ σ_{1} and μ ≤ σ_{2}) by the path consisting of lines connecting the boundary points of T_{3,ϕ,ε} and the points

((σ_{1} + i(A log |σ_{1}| + N log ε − B)/2), σ_{2})and

((σ_{1} + i(A log |σ_{1}| + N log ε − B)/2), σ_{2} + i(A log |σ_{2}| + N log ε − B)/2)).

The proof is based on the estimate on ϕ̂ as above.

The proof for each of E_{k,ϕ,ε}, j = 4, …, 9 is the same as for j = 2 or j = 3. □

The change of condition (8) implies the change of Theorem 2 in [5]. Again its proof has the same idea as in [5].

Theorem 2 An operator [P_{ϕ,ε}(D)] is hypoelliptic if and only if there exist N > 0 and q ∈ ℕ such that for every ϕ ∈ A_{q} and every A > 0 there exist h > 0, η > 0 and b ∈ ℝ such that

σ + iτ ∈ V (P_{ϕ,ε}) ⇒ |τ| ≥ ε^{hN}|σ|^{hA} − b, ε ∈ (0, η).

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