c Allerton Press, Inc., 2011. ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2011, Vol. 46, No. 6, pp. 313–325.  c H.G.Ghazaryan, 2011, published in Izvestiya NAN Armenii. Matematika, 2011, No. 6, pp. 11-30. Original Russian Text 

DIFFERENTIAL EQUATIONS

On Almost Hypoelliptic Polynomials Increasing at Infinity H. G. Ghazaryan1, 2* 1

Russian-Armenian (Slavonic) University, Yerevan, Armenia 2 Yerevan State University Received March 8, 2011

Abstract—It is proved that a polynomial (the symbol of a differential operator), the Newton polygon of which is a rectangular parallelepiped with a vertex at the origin, is almost hypoelliptic if and only if it is regular. Also some algebraic conditions of almost hypoellipticity are obtained for nonregular polynomials increasing at infinity. The results are unimprovable for polynomials of two variables. MSC2010 numbers : 12E10 DOI: 10.3103/S1068362311060057 Keywords: Almost hypoelliptic polynomial; regular polynomial; hypoelliptic operator (polynomial); Newton polyhedron.

1. INTRODUCTION Let Rn be for the n-dimensional Euclidean space, ξ = (ξ1 , . . . , ξn ) ∈ Rn for its elements, N be the set of natural numbers, N0 = {0, 1, 2, ...}, N0n be the set of n-dimensional multiindices, i.e. the points integer components. For any ξ ∈ Rn and α ∈ N0n , it is assumed that α = (α 1 , . . . , αn ) with nonnegative α1 α 2 2 |ξ| = ξ1 + · · · + ξn , ξ = ξ1 · · · ξ αn and Dα = D1α1 · · · Dnαn , where Dj = ∂/∂ξj (j = 1, . . . , n).  Further, for a linear differential operator P (D) = γα Dα with constantcoefficients, where the  sum is taken over a finite collection of multiindices (P ) = α : α ∈ N0n , γα = 0 , the sum P (ξ) = γα ξ α over the same set defines the characteristic polynomial, i.e. the complete symbol of the operator. Definition 1.1. We say that a polynomial P (ξ) is more powerful than the polynomial Q(ξ) and write Q < P , if for some constant C > 0   |Q(ξ)| ≤ C |P (ξ)| + 1 , ξ ∈ Rn . Definition 1.2. An operator P (D) and the corresponding polynomial P (ξ) are said to be almost hypoelliptic, if D ν P < P for any ν ∈ N0n . Note, that the notions of almost hypoelliptic operator and almost hypoelliptic polynomial ¨ generalize those of hypoelliptic operator and hypoelliptic polynomial introduced by L. Hormander [1]. Recall, that hypoelliptic polynomials are described by the condition that D ν P (ξ)/P (ξ) → 0

as |ξ| → ∞

(see [5, Theorem 11.1.3]), while hypoelliptic operators are described by the condition for any 0 = ν ∈ that all solutions which are of the distribution classes of differential equations, which correspond to these operators, are infinitely differentiable functions (see [2]-[4]). By the above descriptions, a hypoelliptic polynomial is almost hypoelliptic, while one can be convinced that the converse statement is not true by the example of the polynomial P (ξ) = ξ12 + ξ22 + ξ12 ξ22 (see also the later Lemma 2.1), which corresponds to the operator N0n

P (D) = − *

∂2 ∂4 ∂2 − + . ∂x21 ∂x22 ∂x21 ∂x22

E-mail: [email protected]

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¨ A rapid development of the theory of hypoelliptic equations started after L. Hormander’s paper [1], and especially after his monograph [6] and paper [7]. In particular, different authors found different algebraic conditions providing the hypoellipticity of equations (see, eg. [8]-[11]). These and many other works were related to the nondegenerating, i.e. regular polynomials (Definition 1.3), the character of which is close to that of elliptic ones. Later, some algebraic conditions providing the hypoellipticity were found also for degenerating (nonregular) polynomials, which strongly differ from elliptic ones (see eg. [12]-[15]). Despite the wideness of the class of hypoelliptic operators, which contains a large set of elliptic operators and stretches far beyond, the problem of description of those non-hypoelliptic differential operators, which have a “sufficient quantity” of infinitely differentiable solutions, naturally arose. On the way of solving this problem, the notions of partially hypoelliptic operator (see, eg. [16]-[19]), globally hypoelliptic operator (see, eg. [20]) and almost hypoelliptic operator (see, eg. [21]-[23]) were introduced. Let us give some additional definitions and notation. Let λ ∈ Rn , a polynomial R(ξ) = R(ξ1 , · · · ξn ) is said to be λ-homogeneous (generalized homogeneous) of an order d, if   R tλ ξ = R tλ1 ξ1 , . . . , tλn ξn = td R(ξ), ξ ∈ Rn , t > 0, or, which is the same, the polynomial R is representable in the form

rα ξ α . R(ξ) = (λ, α)=d

For λ-homogeneous polynomial R, denote

    and (λ, R) = η ∈ Rn,0 : R(η) = 0 . Rn,0 = ξ ∈ Rn : ξ1 ξ2 · · · ξn = 0   For η ∈ (R) = (λ, R) denote   ℵ(η, R) = ν ∈ N0n : Dν R(η) = 0 , Δ(η, R) =

min (λ, ν) ν∈ℵ(η,R)

(1.1)

(1.2) (1.3)

 and assume that Δ(η, R) = 0 if η ∈ Rn,0 \ (R). For a collection of multiindices ℵ = {α1 , · · · , αN }, the smallest convex polyhedron (ℵ) in Rn,0, which contains all multiindices αj ∈ ℵ (j = 1, · · · , N ), is said to be Newton polyhedron of the collection ℵ (see, [8] or [27]). A polyhedron  with vertices in N0n is called complete, if  has a vertex at the origin and a different vertex on each of the coordinate axis. If  is a complete polyhedron, then the set Γ ⊂  is called face of , if there are a vector λ = (λ1 , . . . , λn ) and a number d ≥ 0, such that (λ, α) = d for any α ∈ Γ and (λ, α) < d for any β ∈  \ Γ. Then, the vector λ is said to be an outer normal or -normal of the face Γ. We denote the set of unit outer normals to the face Γ by Λ(Γ), and the k-dimensional faces of the polyhedron  by ki (i = 1, . . . , Mk , k = 0, 1, . . . , n − 1). A face ki (1 ≤ i ≤ Mk , 0 ≤ k ≤ n − 1) of the polyhedron  is said to be principal, if among its normals there is one having at least one positive coordinate. If among the -normals of a principal face ki there is a vector with nonnegative (positive) coordinates, then this face is called regular (completely regular). A polyhedron  is said to be regular (completely regular), if  is complete and its all (n − 1)dimensional non-coordinate faces are regular (completely regular). The polyhedron  = (P ) constructed by the collection of multiindices (P ) of an operator P (D) or of the corresponding polynomial P (ξ) is said to be the Newton Polyhedron or the characteristic polyhedron of the operator P (D) or of the polynomial P (ξ). It can be easily seen in the proof of the statement (i) of Lemma 2.1 in this paper, that the Newton polyhedron of a hypoelliptic polynomial is completely regular, while that of an almost hypoelliptic polynomial is regular (see also Lemma 1.1 in [28]). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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To each principal face ki (i = 1, . . . , Mk , k = 0, 1, . . . , n − 1) of a polyhedron (P ), we put into correspondence the sub-polynomial

P i,k (ξ) = γα ξ α α∈ki

of the polynomial P (ξ). In [8], it is proved that the polynomial P i,k (ξ) corresponding to the face ki of the Newton polyhedron  = (P ) is λ-homogeneous for any λ ∈ Λ(ki ), i.e. there is a number d = di,k ≥ 0 such that

γα ξ α , P i,k (ξ) = (λ,α)=d

and d > 0, if the face ki is principal. Definition 1.3 (see [8]). A face ki (1 ≤ i ≤ Mk , 0 ≤ k ≤ n − 1) of a polyhedron  = (P ) is said to be non-degenerating, if P i,k (ξ) = 0 for any ξ ∈ Rn,0 . If P i,k (η 0 ) = 0 at a point η 0 ∈ Rn,0 , then we call the face ki degenerating. Let ki be some principal face of a Newton polyhedron  = (P ). Obviously, for any nonzero vector λ ∈ Λ(ki ) there are a natural number M = M (λ, ki ) and nonnegative numbers dj = d(λ, ki ) (j = 0, 1, . . . , M ), such that the polynomial P is representable as a sum of nonzero λ-homogeneous polynomials, i.e. P (ξ) =

M

Pj (ξ) =

j=0

M

Pdj (ξ) =

j=0

M



γα ξ α ,

(1.4)

j=0 (λ,α)=dj

where d0 > d1 > . . . > dM ≥ 0 and Pd0 (ξ) ≡ P i,k (ξ) for any λ ∈ Λ(ki ). If a principal face ki is degenerating, then in accordance with the notation (1.1) – (1.3) we put in correspondence to the generalized-homogeneous polynomial P i,k (ξ), i.e. to the face ki , and to the  vector λ  ∈ Λ(ki ) the sets (λ, Pj ), ℵ(η, λ, Pj ) and thenumbers Δ(η, λ, Pj ) (j = 0, 1, . . . , M ) with any η ∈ (λ, Pj ). For a multiindex ν ∈ N0n , the sets (λ, D ν Pj ), ℵ(η, λ, Dν Pj ) and the numbers Δ(η, λ, Dν Pj ) are defined similarly, taking into account that (see representation (1.4)) the polynomial D ν P is representable in the form Dν P (ξ) =

α∈(D ν P )

γα,ν ξ α =

M

Dν Pj (ξ).

(1.5)

j=0

V. P. Mikhailov [8] has proved that for any non-degenerating polynomial P with a complete Newton polyhedron (P ) there is a constant C = C(P ) > 0 such that

  |ξ α | ≤ C |P (ξ)| + 1 , ξ ∈ Rn . (1.6) α∈(P )

Moreover, the converse statement is true, which is proved in the below lemma. Lemma 1.1. If a polynomial P with a complete Newton polyhedron (P ) satisfies the condition (1.6), then P is non-degenerating.

Proof: We prove that if a principal face ki of the polyhedron (P ) is degenerating, then P does not satisfy the condition (1.6), whatever be the constant C. To this end, we suppose that λ ∈ Λ(ki ), η ∈  i,k (P ), besides (λ, α) = d0 is the equation of an (n − 1)-dimensional to (P ) supporting hyperplane JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 46

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passing through the face ki , and P i,k is the sub-polynomial of P , which corresponds to face ki . Then P i,k (η) = 0 and (1.4) implies that for any s = 1, 2 . . . P (ξ s ) ≡ P (sλ η) = P i,k (sλ η) +

M

Pj (sλ η)

j=1

= sd0 P i,k (η) +

M

sdj Pj (η) =

j=1

M

sdj Pj (η)

j=1

and |(ξ s )α | = sd0 |η1α1 · · · ηnαn |.  Note that η1α1 . . . ηnαn = 0 for the points η ∈ (P i,k ) and dj < d0 for all j = 1, . . . , M . Therefore, by the last two equalities we obtain that |P (ξ s )| = o(sd0 ) as s → ∞, while |(ξ s )α |/sd0 = |η1α1 · · · ηnαn | > 0,

s = 1, 2, . . .

Hence, the inequality (1.6) can not be true, and the proof is complete. Based on V. P. Mikhailov’s theorem and Lemma 1.1, henceforth we say that a polynomial P is nondegenerating, if it satisfies (1.6). 2. NON-DEGENERATING ALMOST HYPOELLIPTIC POLYNOMIALS The following lemma permits to consider only the polynomials with non-degenerating Newton polyhedrons. Lemma 2.1. If  = (P ) is the complete, n-dimensional Newton polyhedron of a regular polynomial P (ξ) = P (ξ1 , . . . , ξn ), then the polynomial P (ξ) is (i) hypoelliptic if and only if  is completely regular, (ii) almost hypoelliptic if and only if  is regular.

Proof: (i) First, assuming that P is a non-degenerating polynomial and  = (P ) is a completely regular polyhedron, we prove that P is hypoelliptic. By Theorem 11.11 in [5], it suffices to prove that for any 0 = ν ∈ N0n |Dν P (ξ)|/|P (ξ)| → 0

as |ξ| → ∞.

(2.1)

To prove this relation, observe that some simple geometric argument based on the completely regularity of the polyhedron  imply that for any 0 = ν ∈ N0n the points α ∈ (Dν P ) are non-principal for . Besides, in [8] it is proved that for any non-principal point β ∈  

|ξ α | → 0 as |ξ| → ∞. (2.2) |ξ β | α∈

Therefore, (2.1) follows from (1.6), (2.2) and the fact that a non-degenerating polynomial infinitely increases as |ξ| → ∞. Let us prove that a polynomial P , the polyhedron of which is not completely regular, can not be hypoelliptic, no matter P is non-degenerating or not. Observe that if the polyhedron  is not completely regular, then only the following two cases are geometrically possible: (a) if  is not a regular polyhedron, then the projection of a principal vertex of  on some coordinate hyperplane is out of , (b) if  is a regular polyhedron, then the projection of a principal vertex of  coincides with some principal vertex of . JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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For the case (a), we suppose e = (e1 , . . . , en ) is a vertex of , such that its projection e = (0, ..., 0, ek+1 , ..., en ) on the coordinate hyperplane α1 = . . . = αk = 0 (k ≤ n − 1) is out of . Then, we draw an (n − 1)-dimensional hyperplane passing through the point e , but not through the origin, and not intersecting with . We suppose that (λ, α) = d is the equation of this hyperplane, and λ is an outer with respect to  normal to that hyperplane. Then d > 0, (λ, e ) = d and (λ, α) < d for any α ∈ . Denoting ν = (e1 , . . . , ek , 0, . . . , 0), representing the polynomials P and Dν P = D1e1 · · · Dkek P by the vector λ in the forms (1.4) and (1.5) and considering them on the sequence ξ s = sλ η (s = 1.2, . . .) with a point η, chosen so that D ν P (η) = 0, we obtain Dν P (ξ) = Dν P (η)sd

and

s → ∞,

P (ξ s ) = o(sd ) as

(2.3)

i.e. the polynomial P is not hypoelliptic. For the case (b), we suppose that e is a principal vertex of , i.e.  is a regular, but not completely regular polyhedron. Then, drawing an (n − 1)-dimensional support hyperplane to , which passes trough the point e , but does not contain the origin and any other point of , except e , and repeating the argument used for the case (a), we obtain that there are constants C1 > 0 and C2 > 0 such that for any s = 1, 2, . . . |D ν P (ξ s )| ≥ C1 sd

and

|P (ξ s )| ≤ C2 sd .

Thus, again P is not hypoelliptic. (ii) The statement is proved by a literal repetition of the beginning of the proof of (i), where the estimates of the corresponding polynomials lead to the relations (2.3) implying |Dν P (ξ s )|/|P (ξ s )| → ∞ as |ξ s | → ∞. Thus, the proof is complete. Theorem 2.1. Let  = (P ) be a regular Newton polyhedron of a polynomial P and let all completely regular faces of  be non-degenerating. Then the polynomial P is almost hypoelliptic if and only if it is regular, i.e. all principal faces of  are regular.

Proof: To prove that under our conditions an almost hypoelliptic polynomial P can not have degenerating principal faces, observe that all 0-dimensional faces of  are regular. So, we start by 1-dimensional faces of . Assuming that Γ is a 1-dimensional principal, not completely regular face of , we are to prove that P is not almost hypoelliptic. We suppose that λ ∈ Λ(Γ) is any point and (λ, α) = d0 is the equation of an (n − 1)-dimensional support hyperplane to , which contains Γ and no point of  \ Γ. Then λ has at least one non-positive coordinate, because Γ is not a completely regular face. For definiteness, let λ1 ≤ 0. Then, setting m1 = max{α1 , α ∈ Γ} and

Γ0 = {α ∈ Γ, α1 = m1 }

we show that Γ0 consists of a unique point and consequently is a 0-dimensional subface of Γ, i.e. a vertex of . Indeed, if Γ contains two different points α1 and α2 such that α11 = α21 = m1 , then α1 = m1 for all points α ∈ Γ, since a 1-dimensional face is defined by its two points. Consequently, Γ is perpendicular to the axis 0α1 , i.e. λ1 > 0, which contradicts our assumption. Thus, Γ0 coincides with some principal vertex e = (m1 , e2 , . . . en ) of . Further, representing the polynomials P and D1m1 P by λ in the forms (1.4) and (1.5) and considering the behavior of these polynomials on the sequence ξ s = sλ η = (sλ1 η1 , . . . , sλn ηn ) (s = 1, 2, . . .), where  η ∈ (Γ). We get P (ξ s ) = Pd0 (η)sd0 +

M

Pdj (η)sdj =

j=1

and

⎡ D1m1 P (ξ s ) = D1m1 ⎣γe (ξ s )e +

M

Pdj (η)sdj

(2.4)

j=1



⎤ γα (ξ s )α ⎦ +

α∈Γ,α1
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=

γe (m1 !)η2e2

· · · ηnen sd0 −λ1 m1

+

M

D1m1 Pdj (η)sdj −λ1 m1 .

j=1

Note that d0 − λ1 m1 > d1 − λ1 m1 ≥ 0, since λ1 ≤ 0. Therefore, the above equalities imply that |P (ξ s )| = o(sd0 )

and |D1m1 P (ξ s )| = γe (m1 !)|η2e2 · · · ηnen |sd0 −λ1 m1 (1 + o(1))

as s → ∞. Besides, η ∈ Rn,0 , and hence η2e2 · · · ηnen = 0, and therefore the above relations prove that the polynomial P is not almost hypoelliptic. Now, supposing that Γ is a 2-dimensional principal, not completely regular face of  and, for definiteness, λ ∈ Λ(Γ) is any point and λ1 ≤ 0, we introduce m1 and Γ0 as above and show that either Γ0 consists of a single point, i.e. is a vertex of , or Γ0 is a 1-dimensional subface of a face of . To this end, we show that if Γ0 contains more than one point, then these points lie on a single straight line. Indeed, if this is no so, then there are three points α1 , α2 and α3 of Γ, which do not lie on a single straight line and are such that αj1 = m1 (j = 1, 2, 3). Such three points uniquely define a 2-dimensional face of Γ, and hence the plane passing through Γ is perpendicular to the 0α1 axis, and consequently λ1 > 0. Further, it is obvious that all subfaces of a principal face are principal, and hence either Γ0 is a 0dimensional principal face of , i.e. a vertex of , or it is a 1-dimensional principal face of . In both cases, the sub-polynomial Pd0 corresponding to the face Γ and the vector λ is of the form



γα ξ α + γα ξ α (2.5) Pd0 (ξ) = α∈Γ,α1 =m1

=

ξ1m1 q(ξ2 , . . . , ξn )

α∈Γ, α1


+

γα ξ α ,

α∈Γ,α1
and by definition of vertices q(η2 , ..., ηn ) = 0 for η ∈ Rn,0 in the 0-dimensional case. In the 1-dimensional case, Γ0 is a completely regular face, then q(η2 , . . . , ηn ) = 0 by the assumptions of our theorem. Besides, if the face Γ0 is regular, then it is not completely regular by the above proof.  Considering the polynomials P and D1m1 P on the sequence ξ s = sλ η (s = 1, 2, . . .), where η ∈ (Γ) is a fixed point, we come to the representation (2.4) for the polynomial P and, according to (2.5), to the representation D1m1 P (ξ s ) = (m1 !)q(η2 , . . . , ηn )sd0 −λ1 m1 +

M

D1m1 Pdj (η)sdj −λ1 m1 .

(2.6)

j=1

Taking into account that q(η2 , . . . , ηn ) = 0 and arguing as above, from (2.4) and (2.6) we obtain that the polynomial P is not almost hypoelliptic. If n ≥ 4, then by a similar argument for 3, 4, ..., (n − 1)-dimensional faces we obtain that the polynomial P is regular. Note that, for instance, for 4-dimensional faces the set Γ0 can be a 0-, 1- or 2dimensional face of . Thus, it remains to observe that the almost hypoellipticity of a non-degenerating polynomial P with a regular Newton polyhedron follows from Lemma 2.1. Theorem 2.1 is proved. Corollary 2.1. If n = 2, then only the completely regular faces of the Newton polyhedron of an almost hypoelliptic polynomial can be degenerating. This is obvious, since only 0-dimensional faces can be subfaces of a 1-dimensional face of , while the below corollary of Theorem 2.1 is one of the main results of the present paper. Corollary 2.2. If the Newton polyhedron of a polynomial P (ξ) = P (ξ1 , . . . , ξn ) is an n-dimensional rectangular parallelepiped with vertex at the origin, then the polynomial P is almost hypoelliptic if and only if it is non-degenerating. This statement follows from the geometrically obvious fact that only the 0-dimensional faces of a rectangular parallelepiped are completely regular. Also, the following corollary of Theorem 2.1 is true. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Corollary 2.3. If Γ is an l-dimensional (0 < l ≤ n − 1), regular but not completely regular face of the regular polyhedron  of an almost hypoelliptic polynomial P , all k-dimensional principal faces of which are non-degenerating for any k < l, then the face Γ is non-degenerating. 3. NONREGULAR ALMOST HYPOELLIPTIC POLYNOMIALS INCREASING AT INFINITY This section gives conditions providing the almost hypoellipticity of a class of degenerating polynomials with real coefficients. For describing this class of polynomials, some additional notions and notation are to be introduced. By In , we denote the class of polynomials P (ξ) = P (ξ1 , . . . , ξn ) such that |P (ξ)| → ∞ as |ξ| → ∞. Note that the paper [25] gives necessary and sufficient conditions under which a polynomial of two variables is from I2 . It is obvious that any non-degenerating or hypoelliptic polynomial of n variables is of the class In , but there are almost hypoelliptic polynomials of n variables, which are not of In . Such polynomials are studied in [28, 29]. In this section, we deal with degenerating polynomials P of the classes In , especially in the case when Newton’s regular polyhedron  = (P ) of P possesses a unique, completely regular, (n − 1)dimensional, degenerating face, while its all remaining principal faces are non-degenerating. Note that consideration of only (n − 1)-dimensional, degenerating faces of  is motivated by the fact that this is the only possible case for 2-dimensional polynomials, while consideration of only completely regular faces is motivated by Theorem 2.1 and Corollary 2.1. If P ∈ In , then it is obvious that adding a constant to P and multiplying it by a constant we can get P (ξ) > 0 for any ξ ∈ Rn . Therefore, henceforth we assume that P (ξ) > 0, for any ξ ∈ Rn , for any P ∈ In . The following statement is proved in [28]. Lemma 3.1. Let  = (P ) be a complete Newton polyhedron of a polynomial P ∈ In and let ki (i = 1, . . . , Mk , k = 0, 1, . . . , n − 1) be the principal faces of . Then: (a) P i,k (ξ) ≥ 0 for any ξ ∈ Rn (i = 1, . . . , Mk , k = 0, 1, . . . , n − 1). (b) If a pair of numbers (i, k) (1 ≤ i ≤ Mk , 0 ≤ k ≤ n − 1), a vector λ ∈ Λ(ki ) and a point  η ∈ (P i,k ) are fixed, and (see representation (1.5)) Pj (η) = 0 (j = 0, 1, . . . , l − 1) and Pl (η) = 0 (1 ≤ l ≤ M, l = l(η)), then Pl (η) > 0.  Let ki be some degenerating face of (P ), let λ ∈ Λ(ki ), η ∈ (P i,k ) and let the number l = l(η) be defined as in the statement (b) of Lemma 3.1. Then, by (1.1) – (1.3) we introduce the sets ℵ(η, λ, Pj ) and the numbers Δ(η, λ, Pj ) (j = 0, 1, . . . , l − 1) and, in addition to Lemma 3.1, we prove the following statement. Lemma 3.2. Let ki be a principal, degenerating face of the regular Newton polyhedron of an  almost hypoelliptic polynomial P and let λ ∈ Λ(ki ) and η ∈ (P i,k ) be any points. Then dj (λ) − Δ(η, λ, Pj ) ≤ dl ,

j = 0, 1, . . . , l − 1.

(3.1)

Proof: Suppose the converse statement is true, i.e. for some j (0 ≤ j ≤ l − 1) the inequality (3.1) is not true. Then, denoting the least such j by j0 , we get dj (λ) − Δ(η, λ, Pj ) ≤ dl ,

for j = 0, 1, ..., j0 − 1,

and dj0 (λ) − Δ(η, λ, Pj0 ) > dl .

(3.2)

Further, assuming that a multiindex β ∈ N0n is chosen so that Dβ Pj0 (η) = 0 and (λ, β) = Δ(η, λ, Pj0 ), we consider the values of the polynomials P and D β P on the sequence ξ s = sλ η (s = 1, 2, . . .). Note that (3.2) implies Δ(η, λ, Pj0 ) < Δ(η, λ, Pj ) (j = 0, 1, . . . , j0 − 1), since dj (λ) > dj0 (λ). Consequently, Pj (η) = Dβ Pj (η) = 0 (j = 0, 1, . . . , j0 − 1) and Dβ Pj0 (η) = 0, and therefore by the representations (1.4), (1.5) and the inequalities (3.2) we get P (ξ s ) = Pl (η)sdl + o(sdl ) as

s → ∞.

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For polynomial Dβ P we have for any s ∈ N M

D β P (ξ s ) = sdj0 (λ)−Δ(η,λ,Pj0 ) Dβ Pj0 (η) +

sdj (λ)−Δ(η,λ,Pj0 ) Dβ Pj (η).

j=j0 +1

Therefore, |D β P (ξ s )| = |Dβ Pj0 (η)|sdj0 (λ)−Δ(η,λ,Pj0 ) (1 + o(1))

as s → ∞,

since dj (λ) < dj0 (λ) (j = j0 + 1, . . . , M ). Now, it remains to see that these relations and (3.2) contradict the almost hypoellipticity of the polynomial P , since D β Pj0 (η) = 0, and the proof is complete. We call a generalized-homogeneous polynomial R(ξ) polynomial with isolated characteristics, if  for any point η ∈ (R) there exist its neighborhood U (η), smooth generalized-homogeneous functions q(ξ) = q(ξ, η), r(ξ) = r(ξ, η) and a natural number m = m(η), such that q(η) = 0, r(η) = 0, grad q(η) = 0 and R(ξ) = [q(ξ)]m r(ξ),

ξ ∈ U (η).

Remark 3.1. It follows from Lemma 2.1 of [12], that for n = 2 any generalized-homogeneous polynomial R(ξ1 , ξ2 ) is a polynomial with isolated characteristics, and if R1 and R2 are two generalized-homogeneous polynomials of two variables, such that R1 (η) = R2 (η) = 0 for some η ∈ Rn,0 and Ri (ξ) = [qi (ξ)]mi ri (ξ),

ξ ∈ Ui (η),

i = 1, 2,

then q1 (ξ) ≡ q2 (ξ), U1 (η) = U2 (η) and Dj q1 (η) = 0 (j = 1, 2). Below (in Theorem 3.1) we assume that if Γ is a (n − 1)-dimensional, completely regular, degenerating face with an outer normal μ and η ∈ (Γ), then (see (1.4)) the μ-homogeneous polynomials Pj (j = 0, 1, . . . , l − 1, l = l(η)) have isolated characteristics, and (see Remark 3.1) Pj (ξ) = [q(ξ)]mj rj (ξ) ≥ 0,

ξ ∈ U (η),

j = 0, 1, . . . , l − 1,

(3.3)

where q(η) = 0, rj (η) = 0 (j = 0, 1, . . . , l − 1) and Dn q(η) = 0 under the assumption μ1 ≥ μ2 ≥ . . . ≥ μn . Theorem 3.1. Let all principal faces of a regular Newton polyhedron  = (P ) of a polynomial P ∈ In be non-degenerating, except a single (n − 1)-dimensional, completely regular, but de with an outer normal μ. Assume that for any point η ∈ (Γ) the μgenerating face Γ = n−1 i0 for the almost homogeneous polynomials Pj (j = 0, 1, . . . , l − 1, satisfy conditions (3.3). Then  hypoellipticity of the polynomial P , it is necessary and sufficient that for any η ∈ (Γ) dj − Δ(η, Pj ) ≤ dl (η),

j = 0, 1, . . . , l − 1.

(3.4)

Proof: The proof of the necessity of inequalities (3.4) follows from Lemma 3.2. To prove the sufficiency, we apply a V. P Mikhailov’s method used in [8] in the case of non-degenerating polynomials, which is modified and adapted to degenerating polynomials (see, eg. [12], Theorem 2 or [28], Theorem 2.1). Suppose, the converse statement is true, i.e. there are a multiindex ν 0 and a sequence {ξ s } such that → ∞, as s → ∞ and

|ξ s |

0

|Dν P (ξ s )|/|P (ξ s )| → ∞. Note that it suffices to consider only the case |ν 0 | = 1 (see [28], Theorem 1.1). For definiteness we suppose that ν 0 = (1, 0, . . . , 0) and |D1 P (ξ s )|/|P (ξ s )| → ∞

as s → ∞.

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Without loss of generality we assume that ξis > 0 (i = 1, . . . , n, s = 1, 2, . . .) and set  

 n ln ξis ρs = exp  (ln ξjs )2 , λsi = , s = 1, 2, . . . ln ρs

321

(3.6)

j=1

s

λsi

Then ξ s = ρλs (ξis = ρs , i = 1, . . . , n) and |λs | = 1 (s = 1, 2, . . .). Further, a convergent subsequence can be extracted from {λs }. Thus, without loss of generality we assume that λs → λ as s → ∞, where |λ| = 1, and note that λ is a normal to the only face ki11 of , since the polyhedron  is convex. Now, we denote λ = e1,1 and choose n-dimensional vectors e1,2 , . . . , e1,n so that the system (e1,1 , e1,2 , . . . , e1,n ) is an orthonormal basis of Rn . Then λs =

n

κs1,j e1,j

and

κs1,1 → 1,

κs1,j = o(1)

(j = 2, . . . , n),

j=1

n s 1,j = → as s → ∞. If the basis (e1,1 , e1,2 , . . . , e1,n ) satisfies the condition since j=2 κ1,j e 0for s sufficiently large, then we denote that basis by e1 , . . . , en . Otherwise, we can assume that n s 1,j = 0 for any s ∈ N and j=2 κ1,j e  ⎤  ⎡  n  n



  s 1,j ⎦ s 1,j   ⎣ κ1,j e κ1,j e  → e2,2 , |e2,2 | = 1 as s → ∞.   j=2  j=2 λs

e1,1

We choose n-dimensional vectors e2,3 , . . . , e2,n to provide that the system (e2,2 , e2,3 , . . . , e2,n ) is an orthonormal basis in the (n − 1)-dimensional space generated by the basis (e1,2 , . . . , e1,n ). Then, for any n ≥ 3 λs = κs1,1 e1,1 + κs2,2 e2,2 +

n

κs2,j e2,j ,

s = 1, 2, . . . ,

j=3

where κs1,1 → 1, κs2,2 = o(κs1,1 ) and κs2,j = o(κs2,2 ) as s → ∞ for any j = 3, . . . , n.

 Continuing this process, we get an orthonormal basis (e1 , . . . , en ) such that λs = nj=1 κsj ej (s = 1, 2, . . .), where κs1 → 1 and κsj = o(κsj−1 ) (j = 2, . . . , n) as s → ∞. Obviously, there are natural numbers m ≤ n and s0 , such that λsj = 0 (j = 1, . . . , m) and λsj = 0 (j = m + 1, . . . , n, s = s0 , s0 + 1, . . .). Due to possibility of subsequence choice, we can assume that s0 = 1 and λsi > 0 (j = 1, . . . , m) for any s ∈ N . Now, for connecting the constructed basis with the k polyhedron , we choose the faces ki11 , . . . , kimm as follows: let for j = 1, . . . , m the face ijj lie on the k

k

j−1 or be its subface. support hyperplane of  and have the normal ej , and let ijj either coincide with ij−1

Then the ej -homogeneity of the sub-polynomial P ij ,kj implies that for some multiindex α, which k belongs to all faces ijj (j = 1, . . . , m),    n+1 s j    (α,κs1 e1 ) −ε1 κs1 j=2 κj e s i1 ,k1 (3.7) P ρs + o ρs P (ξ ) = ρs    n+1 s j    κ e (α,κs e1 +κs2 e2 ) −ε κs P i2 ,k2 ρs j=3 j + o ρs 2 2 = ρs 1     n+1 s j    −εr κs  (α, rj=1 κsj ej ) j=r+1 κj e ir ,kr r P ρs + o ρs = . . . = ρs as s → ∞, where ε1 , . . . , εr are positive numbers and en+1 is an arbitrary unit vector. It follows from the definition of the numbers {κsj } that (see (??))  n+1

ρs

s j j=r+1 κj e

→ be

r+1

≡ η ∈ Rn,0

as s → ∞.

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If (α, e1 ) = 0, then the face ki11 passes through the origin of N0n and, consequently, it is not a principal face. This case is considered as the corresponding one in the proof of Theorem 2 in [12], and we do not repeat it.   If (α, e1 ) > 0, then α, rj=1 κsj ej > 0 for s sufficiently large. If P ir ,kr (η) = 0, then by (3.7) κs (e1 ,α)

|P (ξ s )| = ρs 1

|P ir ,kr (η)|(1 + o(1))

as s → ∞.

(3.8)

If e11 < 0, then by a simple geometrical argument we obtain that the face ki11 , and hence all subfaces of the regular polyhedron  lie on the hyperplane α1 = 0, i.e. the polynomial P ir ,kr (ξ) is independent of ξ1 . Hence, D1 P i1 ,kr (ξ) = 0 for any ξ ∈ Rn and, in particular, D1 P ir ,kr (η) = 0. Arguing as in the proof of formula (3.8), we obtain κs [(e1 ,α)−ε]

|D1 P (ξ s )| ≤ C1 ρs 1

,

s = 1, 2, . . . ,

,

s = 1, 2, . . . ,

where C1 > 0 and ε > 0 are constants. If e11 ≥ 0, then arguing in the same way we get κs (e1 ,α)−e11

|D1 P (ξ s )| ≤ C1 ρs 1

where C2 > 0 is a constant. The last three relations contradict (3.5). Let, now, P ir ,kr (η) = 0. Then, the face kirr coincides with the (n − 1)-dimensional, degenerating face  Γ, and the equalities r = m = 1, kr = k1 = n − 1, e1 = μ are true along with the inclusion η ∈ (Γ). Using formulas (1.4)  and (1.5), we represent the polynomials P and D1 P by means of the vector μ = e1 and the point η ∈ (Γ). This gives P (ξ) =

l−1

(3.9)

Pj (ξ) + Pl (ξ) + Q(ξ),

j=0

D1 P (ξ) =

l−1

(3.10)

D1 Pj (ξ) + D1 Pl (ξ) + D1 Q(ξ),

j=0

where the number l = l(η) is defined for the point η as in the statement (b) of Lemma 3.1, Pj is a e1 homogeneous polynomial of order dj (j = 0, 1, . . . , l) such that d0 > d1 > . . . > dl−1 > dl . First, we show that for any j = 0, 1, . . . , l there is a constant Cj > 0 such that   (3.11) |D1 Pj (ξ s )| ≤ Cj |Pj (ξ s )| + |Pl (ξ s )| , s = 1, 2, . . . To this end, we assume that s

 n+1

η = ρs

j=2

κsj ej

and

κ s e1

ξ s = ρs 1 η s

for any  s = 1, 2, . . . and represent the polynomials Pj (j = 0, 1, . . . , l − 1) in the form (3.3) at the point η ∈ (Γ). This gives κs d

κs d

Pj (ξ s ) = ρs 1 j Pj (η s ) = ρs 1 j [q(η s )]mj rj (η s ), s

Pl (ξ ) =

j = 0, 1, · · · , l − 1,

κs d ρs 1 l Pl (η s )

for s sufficiently great and such that η s ∈ U (η). Correspondingly, for the polynomials D1 Pj (j = 0, 1, . . . , l) we get  κs (d −μ )  D1 Pj (ξ s ) = ρs 1 j 1 mj [q(η s )]mj −1 D1 q(η s ) + [q(η s )]mj D1 rj (η s ) , κs (dl −μ1 )

D1 Pl (ξ s ) = ρs 1

D1 Pl (η s ),

where κs1 → 1 as s → ∞. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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ON ALMOST HYPOELLIPTIC POLYNOMIALS INCREASING AT INFINITY

323

Observe that η s → η and q(η s ) → q(η) = 0 as s → ∞ and also that rj (η) = 0 and Pl (η) = 0. By the last four representations we conclude that for some positive constants a1 , a2 and the same values of s κs d

|Pj (ξ s )| ≥ a1 ρs 1 j |q(η s )|mj , |Pl (ξ s )| ≥

j = 0, 1, . . . , l − 1,

(3.12)

κs d a1 ρs 1 l ,

(3.13)

|D1 Pj (ξ s )| ≤

κs (d −μ ) a2 ρs 1 j 1 |q(η s )|mj −1 ,

|D1 Pl (ξ s )| ≤

κs (d −μ ) a2 ρs 1 l 1 .

j = 0, 1, . . . , l − 1,

(3.14) (3.15)

If mj0 = 1 for some j0 (0 ≤ j0 ≤ l − 1), then we get Δ(η, Pj0 ) = μn , since Dn q(η) = 0 by the conditions of our theorem. Then, for the polynomial Pj0 the condition (3.4) of our theorem takes the form dj0 − μn ≤ dl , and for Pj0 the inequality (3.11) follows by (3.14) and (3.13), since dj0 − μ1 ≤ dj0 − μn . Therefore, in what follows we shall assume that mj > 1 (j = 0, 1, . . . , l − 1) and arguing as above we get Δ(η, Pj ) = mj μn ≤ mj μj , and the conditions (3.4) imply dj − mj μ1 ≤ dj − mj μn ≤ dl (j = 0, 1, . . . , l − 1). κs d

Applying Lemma 1.3 of [28] to the case when xs = ρs 1 j , ys = |q(η s )|, a = κs1 (dj − μ1 ), b = mj − 1, c = κs1 dj , d = mj and e = κs1 dl (j = 0, 1, . . . , l − 1; s = 1, 2, . . .), by (3.12)-(3.15) we come to the inequality (3.11). To this end it only is to be noted that xs ≥ 1 and ys ∈ [0, 1] for s sufficiently great, which permits to apply the mentioned lemma for the considered {xs , ys }. By definition of Q,

 s κ d |Q(ξ s )| = o ρs 1 l

and

 s κ (d −μ ) |D1 Q(ξ s )| = o ρs 1 l 1

as s → ∞,

(3.16)

and it remains to observe that relations (3.11) and (3.16) contradict (3.5), since Pj (ξ s ) ≥ 0 (j = 0, 1, . . . , l) for s great enough. This completes the proof of Theorem 3.1. Below, we obtain an almost hypoellipticity criterion for polynomials P ∈ In , without the restriction (3.3) on the isolated characteristics of the corresponding generalized-homogeneous polynomials and on their nonnegativity. Note that this nonnegativity is essential for the almost hypoellipticity of nonregular polynomials, which can be seen by the following example. Example 3.1. Let n = 2 and let P ± (ξ) = (ξ1 − ξ2 )6 ± (ξ1 − ξ2 )2 ξ22 + ξ22 + 1. In this example, (P ± ) is a completely regular triangle with a unique completely regular, 1dimensional side 11 = {(6, 0) − (0, 6)} which is degenerating, and, due to the representation formula √ √ (1.4), Λ(11 ) = (1/ 2, 1/ 2) and P0± (ξ) = (ξ1 − ξ2 )6 , P1± (ξ) = ±(ξ1 − ξ2 )2 ξ22 , P2± (ξ) = ξ22 ,  √



√ √ √  (P1± ) = (1/ 2, 1/ 2), (−1/ 2, −1/ 2) , (P0± ) = d± 0 = 6,

d± 1 = 4,

d± 2 = 2,

l = 2.

Obviously, the polynomial P + satisfies conditions (3.3) and (3.4) of Theorem 3.1 and P + is almost hypoelliptic. Besides, the polynomial P − satisfies the condition (3.4), but not the condition (3.3) on / I2 . nonnegativity. Let us show that the polynomial P − is not almost hypoelliptic and, moreover, P − ∈ / I2 . To this end, we suppose that ξ1s = s + 1 and ξ2s = s. Then P − (ξ s ) = 2 (s = 1, 2, . . .), i.e. P − ∈ The polynomial P − is not almost hypoelliptic, since D12 P − (ξ s ) = 24 + 2s2

and

P − (ξ s ) = 2,

s = 1, 2, . . . ,

and |ξ s | → ∞ as s → ∞. The next theorem gives the mentioned almost hypoellipticity criterion. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 46

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GHAZARYAN

Theorem 3.2. Let all principal faces of the regular Newton polyhedron  = (P ) of a polynomial P ∈ In be non-degenerating, except a single (n − 1)-dimensional face Γ, let μ be an outer normal  to Γ, and let the polynomial P be represented in the form (1.4). Let for the point η ∈ (Γ) α the number l = l(η) be that in  the statement (b)n of Lemma 3.1. Further, let D Pj < P0 (j = 1, . . . , l − 1) for any point η ∈ (Γ) and any α ∈ N0 . Then, the polynomial P is almost hypoelliptic if and only if conditions (3.4) are satisfied for all η ∈ (Γ).

Proof: Assuming that the converse statement is true and repeating the argument of the proof of Theorem 3.1 we come to a contradiction in the case when P ir ,kr (η) = 0, while in the case P ir ,kr (η) = 0 the face kirr coincides with the face Γ. Representing the polynomials P and D1 P in the forms (3.9) and (3.10), we separately consider the cases when the sequence {P0 (ξ s )} is bounded or unbounded. If {P0 (ξ s )} is a bounded sequence, then by our conditions also {Pj (ξ s )} is bounded for any j = 1, . . . , l − 1. Besides, κs d

κs d

|Pl (ξ s )| = ρs 1 l |Pl (η s )| ≥ a3 ρs 1 l ,

s = 1, 2, . . . ,

where a3 > 0 is some constant, since Pl (η s ) → Pl (η) = 0 as s → ∞. Further, in the considered case the relations (3.16) are true for the polynomials Q and D1 Q. Besides, P ∈ In , i.e. P0 (ξ s ) ≥ 0 and Pl (ξ s ) ≥ 0 by Lemma 3.1. Therefore, κs d

|P (ξ s )| ≥ a4 ρs 1 l ,

s = 1, 2, . . . ,

where a4 > 0 is a constant. On the other hand, D1 Pj < P0 (j = 1, · · · , l − 1) by our conditions, and therefore by a similar argument we get κs d

|D1 P (ξ s )| ≤ a5 ρs 1 l ,

s = 1, 2, . . . ,

where a5 > 0 is a constant. The last two inequalities contradict (3.5). If {P0 (ξ s )} is an unbounded sequence, then a suitable choice of a subsequence permits to assume that P0 (ξ s ) → ∞ as s → ∞. Besides, Pj < P0 and D1 Pj < P0 (j = 1, . . . , l − 1), and therefore by Lemma 1 of [26]   |Pj (ξ s )| + |D1 Pj (ξ s )| /P0 (ξ s ) → ∞ as s → ∞. By a similar argument we obtain that there are constants a6 , a7 > 0 such that for any s = 1, 2, . . .     κs d κs d and |D1 P (ξ s )| ≤ a7 P0 (ξ s ) + ρs 1 l . |P (ξ s )| ≥ a6 P0 (ξ s ) + ρs 1 l These inequalities contradict (3.5), and the proof is complete. REFERENCES ¨ L. Hormander, ”On the theory of general partial differential operators”, Acta Math., 94, 161-248 (1955). L. Schwartz, Theorie des Distributions (Hermann, Paris, vol. 1, 1957, vol. 2, 1959). I. M. Gelfand, G. G. Shilov, Generalized Functions, vol. 1 (Fizmatgiz, Moscow, 1958). H. Bremerman, Distributions, Complex Variables and Fourier Transforms (Berkeley, 1965). ¨ L. Hormander, ’The Analysis of Linear Partial Differential Operators, vol. 2 (Springer-Verlag,1983). ¨ L. Hormander, Linear Partial Differential Operators (Springer-Verlag, Berlin-Heidelberg, 1963). ¨ L. Hormander, “Hypoelliptic differential operators”, Ann. Inst. Fourier (Grenoble), 11, 477-492 (1961). V. P. Mikhailov, “On the behavior at infinity of a class of polynomials”, Trudy MIAN SSSR, 91, 59-81 (1961). B. Malgrange, “’Sur une classe d’operateurs differentiels hypoelliptiques”. Bull. Soc. Math. France, 85 (3), 283-306 (1957). 10. L. Cattabriga, “Su une classe di polinomi ipoellittici”, Rend. Sem. Mat. Univ. Padova, 36, 285-309 (1966). 11. L. R. Volevich, S. G. Gindikin, “On a class of hypoelliptic operators”, Mat. Sbornik, 75 (3), 400-416 (1968). 12. H. G. Ghazaryan, “On a family of hypoelliptic polynomials”, Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 9 (3), 189-211 (1974). 1. 2. 3. 4. 5. 6. 7. 8. 9.

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13. H. G. Ghazaryan, “Comparison of the power of polynomials and their hypoellipticity”, Trudy MIAN SSSR, 150, 143-159 (1979). 14. V. N. Markaryan, “Addition of minor terms maintaining the hypoellipticity of an operator”, Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 15 (6), 443-460 (1980). 15. O. R. Gabrielyan, “On the prevailing order of degenerating polynomials and on their hypoellipticity”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 38 (6), 63-82 (2003). 16. Ya. S. Bugrov, “Embedding theorems for some functional classes”, Trudy MIAN SSSR, 77, 45-64 (1965). ¨ 17. L. Garding, B. Malgrange, “Operateurs differentiels partillement hypoelliptiques”. Rend. Acad. Sci. Paris, 247 (23), 2083-2085 (1958). 18. L. Ehrenpreis, “’Solutions of some problems of division 4”, Amer. J. Math., 82, 522-588 (1960). 19. E. A. Gorin, “On asymptotic properties of polynomials of several variables”, UMN, 16 (1), 91-118 (1961). 20. V.I. Burenkov, ”Conditional hypoellipticity and Fourier multipliers in weighted Lp -spaces with an exponential weight”, in: Proceedings of the Summer School “Function Spaces, Differential Operators, Nonlinear Analysis” Held in Fridrichroda in 1993, 133, 256-265 (B.G.Teubner, Stutgart-Leipzig, 1993). 21. G. G. Kazaryan, ”On almost hypoelliptic polynomials”. Doklady Ross. Acad. Nauk., 398 (6), 701-703 (2004). 22. H. G. Ghazaryan, V. N. Markaryan, “On a class of almost hypoelliptic operators”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 41 (6), 39-56 (2006). 23. V. N. Margaryan, H. G. Ghazaryan, “On the smoothness of solutions of a class of almost hypoelliptic equations”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 43 (3), 3964 (2008). 24. H. G. Ghazaryan , V. N. Margaryan, “On infinite differentiability of solutions of nonhomogeneous almost hypoelliptic equations”, Eurasian Mathematical Journal, 1 (1), 54-72 (2010). 25. H. G. Ghazaryan, V. N. Markaryan, “Behavior at infinity of non-elliptic polynomials”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 39 (3), 21-38 (2004). 26. H. G. Ghazaryan, V. N. Margaryan, “hypoellipticity criterion in terms of power and force of operators”, Trudy MIAN SSSR, 150, 128-142 (1979). 27. S. Gindikin , L. Volevich, “The Method of Newton’s Polyhedron in the Theory of Partial Differential Equations (Kluwer, 1992). 28. H. G. Ghazaryan, “On estimates of derivatives of polynomials of several variables”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 34 (3), 46-65 (1999). 29. H. G. Ghazaryan, “Almost hypoelliptic polynomials, non-increasing at infinity”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 36 (2), 15-26 (2001).

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