p(v) (E)

Volume1, Number 1(2010), 54 72

ON INFINITE DIFFERENTIABILITY OF SOLUTIONS

OF NONHOMOGENEOUS ALMOST

HYPOELLIPTIC EQUATIONS

H.G. Ghazaryan, V.N. Margaryan

Communiated by T.V. Tararykova

Keywords and phrases: hypoellipti operator (polynomial),almost hypoellipti

operator (polynomial),weighted Sobolevspaes.

Mathematis Subjet Classiation: 12E10.

Abstrat. A linear dierential operator

P (D)

^{with onstant oeients is alled}

almost hypoellipti if all derivatives

P ^(ν) (ξ)

^{of the} harateristi polynomial

P (ξ)

an be estimated abovevia

P (ξ)

^{. In this paperitis proved that allsolutionsof the}

equation

P (D)u = f

^where

f

^{and all its derivatives are square integrable with a}

ertain exponential weight, whih are square integrable with the same weight, are

alsosuhthatalltheirderivativesare squareintegrablewith thisweight,ifandonly

if the operator

P (D)

^{is almost} hypoellipti.

1 Introdution

After in 1950's L. Hormander introdued the onept of a hypoellipti dierential

equation

P (D)u = f

^all distributional solutions

u

^{of whih with an innitely}

dierentiableright-handside

f

^areinnitelydierentiable(see[15℄,[16℄),aproblem arose of nding additional assumptions on solutions

u

^{of more general, non-}

hypoellipti equationsensuring that these solutions are innitely dierentiable.

In [14℄, [8℄ L. Garding, B. Malgrange and L. Ehrenprie studied the lass of

partiallyhypoelliptiequationsallsolutionsofwhihwithaninnitelydierentiable

right-handside are innitely dierentiable under the a priori assumption that they

are innitely dierentiable with respet to aertain group ofthe variables.

In [3℄ Ya.S. Bugrov onstruted an example of a non-hypoellipti equation, all

solutions of whih in the half-spae are innitely dierentiable provided they are

square integrable inthe half-spae together with some of their derivatives.

In [4℄, [6℄ V.I. Burenkov onsidered the equation

P (D)u = f

^{in the ylinder}

Ω = Ω m × E ^{n − m}

^with

0 ≤ m < n

^where

Ω m

^{is an open set in}

E ^m

^(if

m = 0

then

Ω = E ⁿ

^{) and}

f

^{and all its} derivatives are

m

^{-loally square integrable on}

Ω

^,

i.e. square integrable on

Ω m × E ^{n − m}

^{for all ompats}

Q m ⊂ Ω m

^(if

m = 0

^square

integrable on

E ⁿ

^{). Neessary and suient onditions on}

P

^{were found ensuring}

that all solutions

u

^{of this equation with any suh}

f

^{, whih are}

m

^{-loally square}

integrableon

Ω

^{together withsome of theirderivativeare of the samelass as}

f

⁽ⁱⁿ

partiular are innitely dierentiable).

The lass of suh operators is essentially wider than the lass of hypoellipti

operators.

In [10℄, [17℄ the notionof an almost hypoellipti polynomial wasintroduedand

some suient onditions for almost hypoelliptiity were found in terms of the

homogeneity orders and multipliityof the roots of ertainsubpolynomials.

In [9℄ O.R. Gabrielyan obtained neessary and suient onditions for almost

hypoelliptiptiity of two-dimensional polynomials in terms of multipliity of the

rootsof the ertain homogeneoussubpolynomials.

We use the following standard notations:

N

^{the set of all natural numbers,}

N 0 = N ∪ { 0 }

^,

N ₀ ⁿ = N 0 × . . . × N 0

^{the set of all}

n

-dimensionalmulti-indies,

E ⁿ

and

R ⁿ

^the

n

-dimensional eulidean spaes of points (vetors)

x = (x 1 , . . . , x n )

and

ξ = (ξ 1 , . . . , ξ n )

respetively. For

ξ ∈ R ⁿ

^,

x ∈ E ⁿ

^and

α ∈ N ₀ ⁿ

^{we put}

| ξ | = p ξ ₁ ² + . . . + ξ _n ² , | α | = α 1 + . . . + α n

^,

ξ ^α = ξ ₁ ^{α 1} ...ξ _n ^{α n} , D ^α = D ^α ₁ ¹ ...D ^α _n ⁿ

^,where

D j = _∂ξ ^∂

j

or

D j = ¹ _{i ∂x} ^∂ _j (j = 1, ...n)

^{. Finallywe put}

C ⁿ = R ⁿ × iR ⁿ

^.

For alinear dierentialoperator with onstant oeients

P (D) = P

α

γ α D ^α

^{, let}

P (ξ) = P

α

γ _α ξ ^α

^{be its} harateristi polynomial (omplete symbol),where the sum extends over a nite olletion of multi-indies

(P ) = { α ∈ N ₀ ⁿ , γ _α 6 = 0 }

^{and let}

m = ord P = max {| α | , α ∈ (P ) }

^.

An operator

P (D)

^{(a polynomial}

P (ξ))

^{is alled} hypoellipti (see [15℄), if all solutions

u ∈ D ^′

⁽

D ^′ = D ^′ (E ⁿ )

^{isthe set of}distributions)of theequation

P (D)u = f

^{are innitely} dierentiable(belong to

C ^∞ = C ^∞ (E ⁿ ))

^{for all}

f ∈ C ^∞ .

This holds if and only if (see [16℄, Theorem 11.1.1) all solutions

u ∈ D ^′

^{of the}

equation

P (D)u = 0

^{are innitely} dierentiable.

L.Hormanderhasproved(see[16℄,Theorem11.1.1andTheorem11.1.3),thatan

operator

P (D)

^ishypoelliptiifandonlyifanyofthefollowingequivalentonditions issatised.

1)

Sing Supp u = Sing Supp P (D)u

^{for any open set}

Ω ⊂ E ⁿ

^and

u ∈ D ^′ ,

2)if

0 6 = ν ∈ N ₀ ⁿ ,

^then

P ^{(ν )} (ξ)/P (ξ) ≡ D ^ν P (ξ)/P (ξ) → 0

^as

| ξ | → ∞

^,

3)

d P (ξ) → ∞

^as

| ξ | → ∞ ,

^where

d P (ξ)

^{is the distane from the point}

ξ ∈ R ⁿ

tothe surfae

{ ζ ; ζ ∈ C ⁿ , P (ζ) = 0 }

^.

The following question naturally arises. Let the symbol

P (ξ)

^{of the operator}

P (D)

^{satisfy aweaker ondition thanondition 2):}

| P ^ν (ξ) | / [1 + | P (ξ) | ] ≤ C < ∞ ∀ ξ ∈ R ⁿ , ∀ ν ∈ N ₀ ⁿ , (1.1)

orlet 3)be replaed by the requirementthat

d P (ξ) ≥ ε > 0 (1.2)

forallsuiently large

ξ ∈ R ⁿ

^{.Whih onditionsshould beimposed ona funtion}

f ∈ C ^∞

^{and on solutions}

u ∈ D ^′ (E ⁿ )

^{of the equation}

P (D)u = f

^{toensure innite}

dierentiability of

u

ⁱⁿ

E ⁿ

^?

Weshall use the followinglasses of operatorsand funtions.

Denition 1.1. An operator

P (D)

^{(and a polynomial}

P (ξ))

^{is alled almost}

hypoelliptiifthepolynomial

P

^satisesoneof(equivalent)onditions(1.1),(1.2).

By Lemma 11.1.4 of [16℄ for any polynomial

P (ξ)

^{there exists a onstant}

σ 1 = σ 1 (P ) > 0

^{suh that the following inequality holds:}

X

| ν | >0

d ^| _P ^{ν |} (ξ) | P ^(ν) (ξ) | ≤ σ 1 | P (ξ) |∀ ξ ∈ R ⁿ . (1.3)

On the other hand if a polynomial

P ∈ I _n ,

^i.e

| P (ξ) | → ∞

^as

| ξ | → ∞

⁽ⁱⁿ

this onnetion we note that in [11℄ - [12℄ neessary and suient onditions in

terms of the oeients of

P

^{ensuring that}

P ∈ I 2

^{were found), then it is almost}

hypoelliptiif and only if

ρ P = lim

t →∞ inf

| ξ | = t d P (ξ) > 0 · (1.4)

For any

δ > 0

^and

m ∈ N,

^let

L 2,δ ≡ L 2,δ (E ⁿ )

^{be the set of all funtions}

u

loallyintegrableon

E ⁿ

^{with nitenorms}

|| u || L 2,δ =



 Z

E ⁿ

| u(x) | ² e ^{− 2δ | x |} dx





1/2

(1.5)

and

H _δ ^m = H _δ ^m (E ⁿ )

^{bethe set of allfuntions}

u ∈ L 2,δ

^{withnite norms}

|| u || H _δ ^m = X

| α |≤ m

|| D ^α u || L 2,δ . (1.6)

Finally weput

H _δ ^∞ =

\ ∞ m=1

H _δ ^m ·

It is obvious that

L 2, δ

^and

H _δ ^m

^{are Banah spaes and}

H _δ ^∞

^{is aFrehet spae suh}

that

H _δ ^∞ ⊂ C ^∞

^{for any}

δ > 0

^.

Let

N (P, δ) = { u ∈ L 2,δ : P (D)u ∈ H _δ ^∞ } ,

where

P (D)u

^{is understoodin the} distributionalsense, i.e.

u ∈ N (P, δ)

^{means that}

u ∈ L 2,δ

^{and there exists}

f ∈ H _δ ^∞

^{suh that (here and the sequel}

R

means

R

E ⁿ

) Z

uP ( − D)ϕdx = Z

f ϕdx ¯ ∀ ϕ ∈ C ₀ ^∞ ,

where

C ₀ ^∞

^{is the set of allfuntions in}

C ^∞

^{with ompat support}

The followingstatement is the main result of the artile.

Theorem. An operator

P ∈ I n

^{is almost} hypoellipti if and only if there exists a number

δ > 0

^suhthat

N (P, δ) ⊂ H _δ ^∞

^.

Remark.The ase

δ = 0

^{is quitedierent from the ase}

δ > 0.

^{In [4,6℄ it isproved}

that

N (P, 0) ⊂ H ₀ ^∞

^{if and only if there exist}

c, M > 0

^{suh that}

| P (ξ) | ≥ c

^{for all}

ξ ∈ R ⁿ

^satisfying

| ξ | ≥ M ·

Let

ε > 0

^and

L 2,δ,ε

^{be the set of all funtion}

u

^{loally integrable on}

E ⁿ

^with

nite norms

|| u || ^L 2,δ,ε =



 Z

E ⁿ

| u(x) | ² e ^{− 2δ | x | ε} dx





1/2

·

Considerthespae

H _δ,ε ^∞

^andthesets

N(P, δ, ε)

^{obtainedbyreplaing}

L _2,δ

^by

L _2,δ,ε

ⁱⁿ

theabovedenitions.In[7℄V.I.Burenkovproved thatif

ε > 1

^then

N (P, δ, ε) ⊂ H δ,ε

if and only if the operator

P

^is hypoellipti. An interesting question arises: what happens in the ase

0 < ε < 1?

2 Estimates for funtions in

H _δ ^∞

Lemma 2.1. Let

P ∈ I n

^{be an almost} hypoellipti polynomial, the number

ρ P

^be

dened by formula (1.4) and

σ ₁

^{be the minimal number for whih inequality (1.3)}

is satised. Then for any

ρ ∈ (0, ρ P )

^{there exists a number}

σ 2 = σ 2 (ρ, P ) > 0

^suh

that

X

| α | >0

ρ ^{| α |} | P ^(α) (ξ) | ≤ σ 1 | P (ξ) | + σ 2 ∀ ξ ∈ R ⁿ . (2.1)

Proof. Let

ρ ∈ (0, ρ P )

^{. Then by the denition of the number}

ρ P

^{there exists a}

number

M = M (ρ) > 0

^suhthat

d _P (ξ) ≥ ρ ∀ ξ ∈ R ⁿ ; | ξ | ≥ M. (2.2)

Let usput

σ ₂ = σ ₂ (ρ, P ) = max

| ξ |≤ M

X

| ν | >0

ρ ^{| ν |} | P ^(ν) (ξ) | ,

then estimate(2.1) follows by estimates (1.3) and (2.2).

For onveniene instead of the weight funtion

e ^{− δ | x |}

^{we onsider the equivalent}

smooth weightfuntion

g δ ∈ C ^∞

^.

Weassume that

g ∈ C ^∞

^{is axed positive funtion suh that for some}

κ > 0 κ ^{− 1} e ^{−| x |} ≤ g(x) ≤ κe ^{−| x |} ∀ x ∈ E ⁿ ·

Moreover, we suppose that for any

α ∈ N ₀ ⁿ

^{there exists a number}

κ α > 0 (κ 0 = κ)

for whih

| D ^α g(x) | ≤ κ α 2ptg(x) ∀ x ∈ E ⁿ ·

Note, that for example

g

^{may be onstruted as the} regularization (i.e. the averaging) of the funtion

H

^{, dened by}

H(x) = e ^{−| x |}

^for

| x | > 1

^and

H(x) = e ^{− 1}

for

| x | ≤ 1

^{by means of a xed} nonnegative weight funtion

ϕ ∈ C ₀ ^∞

^{suh that}

R ϕ(x)dx = 1

^.

For

δ > 0

^{,we set}

g δ (x) = g(δx)

^{. Then by the denition of}

g

κ ^{− 1} e ^{− δ | x |} ≤ g δ (x) ≤ κe ^{− δ | x |} ∀ x ∈ E ⁿ , (2.3)

| D ^α g δ (x) | ≤ κ α δ ^{| α |} g δ (x) ∀ α ∈ N ₀ ⁿ , ∀ x ∈ E ⁿ . (2.4)

We start with the statements proved in [13℄ (see Lemma 1.1and Lemma 1.2in

[13℄)

Lemma 2.2. Let

m ∈ N, a i , b i ≥ 0 (i = 0, 1, ..., m)

^,

d > 0

^and

t > 1

^{. Then}

a) if

a ₀ = b ₀

^and

a k ≤ b k + d

k − 1

X

j=0

t ^j a j (k = 1, 2, ..., m), (2.5)

then for

σ 3 = 2[2dt ^{m − 1} + 1] ^m

X m k=0

a k ≤ σ 3

X m k=0

b k , (2.6)

b) if

a m = b m

^and

t ^k a k ≤ t ^k b k + d X m j=k+1

t ^j a j (k = 0, 1, ..., m − 1), (2.7)

then

X m k=0

t ^k a k ≤ X m k=0

(1 + d) ^k t ^k b k . (2.8)

Lemma 2.3. If the set

G ⊂ E ⁿ

^{is ontained in the ball}

S T = { x ∈ E ⁿ , | x | ≤ T } ,

then for any

δ > 0

sup

y ∈ G

g δ (x + y) ≤ σ 4 g δ (x) ∀ x ∈ E ⁿ , (2.9) sup

y ∈ G | g δ (x + y) − g δ (x) | ≤ σ 5 g δ (x) ∀ x ∈ E ⁿ , (2.10)

where the funtion

g δ

^{is as dened above,}

σ 4 = κ ² e ^δT

^and

σ 5 = κ ² (δT )e ^δT (max

| α | =1 κ _α ) √ n

^.

In the sequel for a xed almost hypoellipti polynomial

P

^{of order}

m

^{, xed}

funtion

g

^{, and for any}

ρ ∈ (0, ρ P )

^{, let positive numbers}

κ α

^,

| α | ≤ m

^{, satisfy}

inequalities(2.3), (2.4), numbers

σ 1 (ρ), σ 2 (ρ)

^{, satisfy inequality (2.1) and}

h = h(g, m) = X

0< | α | <m

κ α

α! , (2.11)

∆ 0 = ∆ 0 (g, m, ρ) = ρ min

0< | α |≤ m

( α!

2κ α

1/ | α |

[(1 + h) ^m σ 1 ] ^{1/ | α |} )

· (2.12)

Lemma 2.4. Let

P (D)

^{be an almost} hypoellipti operator of degree

m

^{and the}

number

ρ P

^{be dened by formula (1.4), then it follows that for any}

ρ ∈ (0, ρ P )

^,

δ ∈ (0, ∆ 0 (ρ)]

^and

u ∈ H _δ ^∞

X

| α | >0

ρ ^{| α |} k [P ^(α) (D)u]g δ k ^L 2 ≤ 2(1 + h) ^m [σ 1 (ρ) k [P (D)u]g δ k ^L 2 +

+σ 2 (ρ) k ug δ k ^L 2 ]. (2.13)

Proof. First we prove that for any

t > 0

^,

δ ∈ (0, t]

^and

u ∈ H _δ ^∞ X

| α | >0

t ^{| α |} k [P ^(α) (D)u]g _δ k L 2 ≤ (1 + h) ^m X

| α | >0

t ^{| α |} k{ P ^(α) (D)[ug _δ ] k L 2 . (2.14)

Put for

j = 1, 2, ..., m a _j = X

| α | =j

t ^{| α |} k [P ^(α) (D)u]g _δ k L 2 , b _j = X

| α | =j

t ^{| α |} k P ^(α) (D)[ug _δ ] k L 2 .

It isobvious that

a m = b m

^{.Then by the Leibnitz formulaitfollows thatfor any}

t > 0

^and

j = 1, 2, ..., m − 1 a j = X

| α | =j

t ^{| α |} k [P ^(α) (D)u k ^L 2 = X

| α | =j

t ^{| α |} {k P ^(α) (D)[ug δ ] −

− X

| β | >0

1

β! [P ^(α+β) (D)u]D ^β g δ k L 2 } ≤ b j +

+ X

| α | =j

t ^{| α |} X

| β | >0

1

β! [P ^(α+β) (D)u]D ^β g δ k ^L 2 .

Applying property (2.4) of the funtion

g δ

^{, we get that for any}

δ ∈ (0, t]

a j ≤ b j + X

| α | =j

t ^{| α |} X

| β | >0

κ β

β! δ ^{| β |} k [P ^(α+β) (D)u]g δ k ^L 2 ≤

≤ b _j + X m l=j+1

t ^l X

| γ | =l

k [P ^(γ) (D)u]g _δ k L 2

X

0<ν<γ

κ ν

ν! =

= b j + h X m l=j+1

a j j = 1, ..., m − 1 ·

This means that the numbers

α j , b j j = 1, ..., m

^{satisfy ondition b) of Lemma}

2.2, whih proves estimate (2.14). Hene, by Lemma 2.1, inequality (2.14) and the

Parsevalequality we onlude that

X

| α | >0

ρ ^{| α |} k [P ^(α) (D)u]g δ k L 2 ≤ (1 + h) ^m X

| α | >0

ρ ^{| α |} k P ^(α) (D)[ug δ ] k L 2 =

= (1 + h) ^m X

| α | >0

ρ ^{| α |} k P ^(α) (ξ)F (ug δ )(ξ) k ^L 2 ≤

≤ (1 + h) ^m [σ 1 k P (ξ)F (ug δ ) k ^L 2 + σ 2 (ρ) k F (ug δ ) k ^L 2 ] =

= (1 + h) ^m [σ 1 k P (D)(ug δ ) k L 2 + σ 2 (ρ) k ug δ k L 2 ] , (2.15)

where

F

^{is the Fourier transform.}

Using the Leibnitz formula and estimates (2.4), for the rst summand in the

right-handside weobtain

k P (D)(ug δ ) k L 2 ≤ k [P (D)u]g δ k L 2 + X

| α | >0

1

α! k [P ^(α) (D)u]D ^α g δ k L 2 ≤

≤ k [P (D)u] g δ k L 2 + X

| α | >0

κ α

α! δ ^{| α |} k [P ^(α) (D)u]g δ k L 2 . (2.16)

It follows from(2.15) - (2.16) that

X

| α | >0

ρ ^{| α |} k [P ^(α) (D)u]g δ k L 2 ≤ (1 + h) ^m σ 1 [ k [P (D)u]g δ k L 2 +

+ X

| α | >0

κ α

α! δ ^{| α |} k [P ^(α) (D)u]g δ k L 2 ] + (1 + h) ^m σ 2 (ρ) k ug δ k L 2 .

Then it islear that

X

| α | >0

ρ ^{| α |} k [P ^(α) (D)u]g δ k L 2 [1 − (1 + h) ^m σ 1

κ α

α!

δ ρ

_| α |

] ≤

≤ (1 + h) ^m σ 1 k [P (D)u]g δ k ^L 2 + (1 + h) ^m σ 2 k ug δ k ^L 2 .

Sine

1 − (1 + h) ^m σ 1

κ α

α!

δ ρ

| α |

≥ 1

2 ∀ α ∈ N ₀ ⁿ , 0 < | α | ≤ m

for any

δ ∈ (0, ∆ 0 (ρ))

^{, this leads toinequality (2.13).}

For any

ρ ∈ (0, ρ P )

^{we denote by}

∆ 1 = ∆ 1 (ρ, g)

^{the greatest of numbers}

δ ∈ (0, ∆ 0 )

^{for whih}

1 − σ 1

"

κ α

α!

δ ρ

_| α |

+ X

0<γ<α

κ γ

γ!

δ ρ

_| γ | #

≥ 1

2 (2.17)

for any

α ∈ N ₀ ⁿ

^;

0 6 = | α | < m

^.

Lemma 2.5. Let

P ∈ I _n

^{be an almost} hypoellipti operator and

ρ ∈ (0, ρ _P )

^{, then}

for any

k ∈ N 0

^{there exist positive numbers}

A k,j

^,

j = 0, 1, ..., k

^{, and}

B k

^{suh that}

for all

δ ∈ (0, ∆ 1 ]

^and

u ∈ H _δ ^∞ X

| β |≤ k

X

| α | >0

ρ ^{| α |} k [D ^β P ^(α) (D)u]g δ k ^L 2 ≤

≤ X k

j=0

A k,j

X

| β | =j

k [D ^β P (D)u]g δ k ^L 2 + B k k ug δ k ^L 2 . (2.18)

Proof. We prove the result by indution in

k

^{. For}

k = 0

^{inequality (2.18) follows}

by (2.13) with the onstants

A 0,0 = 2(1 + h) ^m σ 1

^and

B 0 = 2(1 + h) ^m σ 2

^.

Assuming thatinequalities (2.18)hold for

k ≤ r

^{,let usprovethat they hold for}

k = r + 1

^.

Bythe indutiveassumption itfollows that forany

δ ∈ (0, ∆ 1 )

^and

u ∈ H _δ ^∞

^the

following inequality holds:

X

| β |≤ r+1

X

| α | >0

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ

L 2 ≤

≤ X

| β | =r+1

X

| α | >0

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ

L 2 +

+ X r

j=0

A _r,j X

| β | =j

[D ^β P (D)u]g _δ

L 2 + B _r k ug _δ k L 2 . (2.19)

If

β, ν ∈ N ₀ ⁿ

^and

0 6 = ν ≤ β

^{, then}

| β − ν | ≤ r

^{. Hene, by the Leibnitz formula}

and estimates (2.4) we onlude that for some positive onstants

C 1 = C 1 (g, ρ)

^,

C 2 = C 2 (g, ρ)

^{and for any}

δ ∈ (0, ρ]

^{the following}inequalitieshold

X

| β | =r+1

X

| α | >0

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ

L 2 = X

| β | =r+1

X

| α | >0

ρ ^{| α |}





D ^β P ^(α) (D)[ug δ ] −

− X

γ+ν 6 =0,ν ≤ β

C _β ^ν

γ! [D ^{β − ν} P ^(α+γ) (D)u]D ^γ+ν g _δ L 2



 ≤

≤ X

| β | =r+1

X

| α | >0

ρ ^{| α |} D ^β P ^(α) (D)[ug δ ]

L 2 +

+ X

| β | =r+1

X

| α | >0

ρ ^{| α |} X

γ 6 =0

1 γ!

[D ^β P ^(α+γ) (D)u]D ^γ g δ ]

L 2 +

+ X

| β | =r+1

X

| α | >0

ρ ^{| α |} X

0 6 =ν ≤ β

X

γ

C _β ^ν γ!

[D ^{β − ν} P ^(α+γ) (D)u]D ^γ+ν g δ ]

L 2 ≤

≤ X

| β | =r+1

X

| α | >0

ρ ^{| α |} D ^β P ^(α) (D)[ug δ ]

L 2 +

+ X

| β | =r+1

X

| α |≥ 2

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ ]

L 2

"

X

0 6 =ν<α

δ ρ

| γ | κ γ

γ!

# +

+C 1

X

| β | =r+1

X

| α | >0

ρ ^{| α |} X

0 6 =ν ≤ β

X

γ 6 =0

C _β ^ν

γ! κ ν+γ δ ^{| ν+γ |} [D ^{β − ν} P ^(α+γ) (D)u]D ^γ+ν g δ ]

L 2 ≤

≤ X

| β | =r+1

X

| α | >0

ρ ^{| α |} D ^β P ^(α) (D)[ug δ ]

L 2 +

+ X

| β | =r+1

X

| α |≥ 2

ρ ^{| α |} [D ^β P ^(α) (D)u]g _δ ]

L 2

"

X

0 6 =ν<α

δ ρ

_| γ | κ γ

γ!

# +

+C 2

X

| β |≤ r

X

| α |≥ 1

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ ]

L 2 ·

From this and by the indutive assumption we get that for any

δ ∈ (0, ρ]

^and

u ∈ H _δ ^∞

X

| β | =r+1

X

| α | >0

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ

_L

2 ≤

≤ X

| β | =r+1

X

| α | >0

ρ ^{| α |} D ^β P ^(α) (D)[ug δ ]

L 2 +

+ X

| β | =r+1

X

| α |≥ 2

ρ ^{| α |} [D ^β P ^(α) (D)u]g δ ]

L 2

"

X

0 6 =ν<α

δ ρ

_| γ | κ γ

γ!

# +

+ X r

j=0

C 1 A r,j

X

| β | =j

[D ^β P ^(α) (D)u]g δ ]

L 2 + C 1 B r k ug δ k L 2 . (2.20)

Let

ρ ∈ (0, ρ P )

^{and let}

M = M(ρ)

^{be the minimal of numbers satisfying the}

inequality(2.2) and

σ ₆ = max

| ξ |≤ M

X

| β | =r+1

X

| α | >0

ρ ^{| α |} | ξ ^β P ^(α) (ξ) | .

Toevaluatetherstsummandintheright-handsideof(2.20)weusetheParseval

equality,estimates (2.4) and Lemma 2.1. We onlude that for any

ρ ∈ (0, ρ _P ) X

| β | =r+1

X

| α | >0

ρ ^{| α |} D ^β P ^(α) (D)[ug δ ]

L 2 =

= X

| β | =r+1

X

| α | >0

ρ ^{| α |} ξ ^β P ^(α) (ξ)F (ug δ )]

L 2 ≤

≤ σ 1

X

| β | =r+1

ξ ^β P (ξ)F (ug δ )

L 2 + σ 6 k F (ug δ ) k L 2 =

= σ 1

X

| β | =r+1

D ^β P (D)[ug δ ]

L 2 + σ 6 k u g δ k L 2 . (2.21)

Toevaluatetherstsummandintheright-handsideof(2.21)weusetheLeibnitz

formula and estimates (2.3). Then by indutive assumption we onlude that for

somepositiveonstant

C 3 = C 3 (g, ρ)

^,andforany

δ ∈ (0, ρ]

^{the folowing}inequalities hold

X

| β | =r+1

|| D ^β P (D)[ u g _δ ] || L 2 ≤ X

| β | =r+1

|| [ D ^β P (D) u ] g _δ || L 2 +

+ X

| β | =r+1

X

γ+ν 6 =0; ν ≤ β

C _β ^ν

γ! || [D ^{β − ν} P ^(γ) (D)u ] D ^γ+ν g _δ || L 2 ≤

≤ X

| β | =r+1

|| [ D ^β P (D) u ] g δ || L 2 +

+ X

| β | =r+1

X

| γ | >0

κ γ

γ ! δ ^{| γ |} || [D ^β P ^(γ) (D)u ] g _δ || L 2 +

+ X

| β | =r+1

X

0<ν<β

(δ ^{| ν |} κ _ν ) · C _β ^ν || [D ^{β − ν} P (D)u] g _δ || L 2 +

+ X

| β | =r+1

X

| γ | >0

X

| ν | >0

C _β ^ν γ!

κ _γ+ν

γ! δ ^{| γ+ν |} || [D ^{β − ν} P ^(γ) (D)u] g _δ || L 2 ≤

≤ X

| β | =r+1

|| [ D ^β P (D) u ] g δ || L 2 +

+ X

| β | =r+1

X

| γ | >0

κ _γ γ!

δ ρ

_| γ |

ρ ^{| γ |} || [D ^β P ^(γ) (D)u ] g δ || L 2 +

+C 3



 X r

j=0

A j,r

X

| β | =j

|| [D ^β P (D)u ] g δ || ^L 2 + B r || ug δ || ^L 2



 . (2.22)

Applyingestimates(2.20)(2.22)weobtainthatforany

δ ∈ (0, ρ]

^and

u ∈ H _δ ^∞ X

| β | =r+1

X

| α | >0

ρ ^{| α |} || D ^β P ^(α) (D)[ u g δ ] || L 2 ≤

≤ σ 1

X

| β | =r+1

|| [ D ^β P (D) u ] g δ || L 2 +

+σ 1

X

| β | =r+1

X

| γ | >0

κ γ

γ!

δ ρ

| γ |

ρ ^{| γ |} || [D ^β P ^(γ) (D)u ] g δ || ^L 2 +

+ X

| β | =r+1

X

| α |≥ 2

ρ ^{| α |} || [ D ^β P ^(α) (D) u ] g δ || L 2

X

0<γ<α

κ γ

γ!

δ ρ

_| γ |

+

+[σ 1 (C 2 + C 3 ) + C 1 ] X r

j=0

A j,r

X

| β | =j

|| [D ^β P (D)u ] g δ || ^L 2 +

+ { [σ 1 (C 2 + C 3 ) + C 1 ] B r + σ 6 } || ug δ || ^L 2 . (2.23)

Applying estimates (2.19) and (2.23) we obtain that for any

δ ∈ (0, ρ]

^and

u ∈ H _δ ^∞

X

| β | =r+1

X

| α | >0

ρ ^{| α |} || D ^β P ^(α) (D)[ u g δ ] || L 2 ≤ σ 1

X

| β | =r+1

|| [ D ^β P (D) u ] g δ || L 2 +

+[σ 1 (C 2 + C 3 ) + C 1 + 1]

X r j=0

A j,r

X

| β | =j

|| [D ^β P (D)u ] g δ || L 2 +

+[σ 1 (C 2 + C 3 ) + C 1 + 1] B r + σ 6 } || u g δ || ^L 2 ]+

+σ 1

X

| β | =r+1

X

| γ | >0

κ γ

γ!

δ ρ

| γ |

ρ ^{| γ |} || [D ^β P ^(γ) (D)u ] g δ || ^L 2 +

+σ 1

X

| β | =r+1

X

| α |≥ 2

ρ ^{| α |} || [ D ^β P ^(α) (D) u ] g δ || L 2

X

0<γ<α

κ γ

γ!

δ ρ

_| γ |

·

Estimate (2.18) immediately follows by this inequality and by inequality (2.17)

inthe denition of

∆ 1 (ρ, g)

^{with the onstants}

A j,r+1 = 2[σ 1 (C 2 + C 3 ) + C 1 + 1] A j,r j = 0, 1, ..., r; A r+1,r+1 = 2σ 1 ; B r+1 = 2 { [σ 1 (C 2 + C 3 ) C 1 + 1] B r + σ 6 } ,

whihin turn ompletes the proof of the lemma.

3 Density of smooth funtions in weighted Sobolev spaes

In this setion we onsider almost hypoellipti operators in weighted Sobolev

funtion spaes

H _δ ^m = H _δ ^m (E ⁿ )

^and

H _δ ^∞ = H _δ ^∞ (E ⁿ ) .

^{We begin with a general}

resulton linear dierentialoperators with onstant oeients.

Lemma 3.1. For any linear dierential operator

P (D)

^{with onstant oeients}

and for any

δ > 0

^{the set}

H _δ ^∞

^{is dense in}

N (P, δ) = { u ∈ L 2,δ ; P (D)u ∈ H _δ ^∞ }

^with

respet to the topology, indued by the seminorms

k u k P,k,δ = k ug δ k L 2 + X

| α |≤ k

k [D ^α (P (D)u)]g δ k L 2 , k = 0, 1, . . . .

Proof. Assuming that

S 1 = { x ∈ E ⁿ : | x | < 1 } , ϕ ∈ C ₀ ^∞ (S 1 ), R

ϕ(x)dx = 1, u ∈ L 2, δ

^and

ε > 0,

^wedenote

ϕ ε (x) = ε ^{− n} ϕ(x/ε),

^{and set}

u ε (x) = u ∗ ϕ ε = Z

u(x − y)ϕ ε (y)dy = ε ^{− n} Z

u(x − y)ϕ(y/ε)dy ·

The funtion

u ε

^{is alled a} regularization (or molliation) of

u

^{(for the}

properties of

u ε

^{see for example [5℄, Chapter 1, or[1℄, Chapter 2,Setion 17).}

Firstweprovethat

u ε ∈ H _δ ^∞ ·

^{Weobserve thatthe followingtakesplae forany}

k ∈ N 0 ,

^{using property (2.10) of the funtion}

g δ

^{and Young's inequality}

X

| α |≤ k

|| (D ^α u ε ) g δ || ^L 2 = X

| α |≤ k

Z

u(x − y) D ^α ϕ ε (y) g δ (x)dy

L 2

≤

≤ X

| α |≤ k

"

|| (ug _δ ) ∗ D ^α ϕ _ε || L 2 +

Z

u(x − y) [ g _δ (x − y) − g _δ (x) ]D ^α ϕ _ε (y)dy

L 2

#

≤

≤ X

| α |≤ k

[ || (u g δ ) ∗ D ^α ϕ ε || L 2 + σ 5 (ε) || | u g δ | ∗ | D ^α ϕ ε | || L 2 ] ≤

≤ (1 + σ 5 (ε)) X

| α |≤ k

|| ug δ || ^L 2 || D ^α ϕ ε || ^L 1 =

= (1 + σ 5 (ε)) || ug δ || L 2

X

| α |≤ k

ε ^{−| α |} || D ^α ϕ || L 1 < ∞·

Sine

k ∈ N 0

^{is arbitrary, itfollows that}

u ε ∈ H _δ ^∞ ·

To omplete the proof it remainsto show that as

ε → 0

|| u ε − u || ^{P, k, δ} → 0 · (3.1)

Let a funtion

v ∈ L 2, loc (E ⁿ )

^{and a linear dierential operator}

Q(D)

^satisfy

the followingondition :

Q(D)v ∈ L _{2, loc} (E ⁿ ) ·

^{Then (see [2℄, 6.3(2))}

Q(D)v _ε (x) = [Q(D)v] ε (x)

^forall

x ∈ E ⁿ

^{and by the ontinuityinthe mean offuntions}

u ∈ L 2

|| v ε − v || ^L 2 → 0

as

ε → 0 .

Therefore,usingproperty(2.10)oftheweightfuntion

g

^{andYoung'sinequality}

the followingholds for any

k ∈ N 0

|| u ε − u || ^{P, k, δ} = || [ u ε − u ]g δ || ^L 2 + X

| α |≤ k

|| [ D ^α (P (D)u ε ) − D ^α (P (D)u) ]g δ || ^L 2 =

= || [ u ε − u ]g δ || L 2 + X

| α |≤ k

|| { [D ^α (P (D)u) ] ε − D ^α (P (D)u) } g δ || L 2 ≤

≤ || (u g δ ) ε − (u g δ ) || L 2 + || (u g δ ) ε − (u ε g δ ) || L 2 +

+ X

| α |≤ k

|| [ (D ^α P (D)u) g δ ] ε − (D ^α P (D)u) g δ || L 2 +

+ X

| α |≤ k

|| [ (D ^α P (D)u) g δ ] ε − (D ^α P (D)u) ε g δ || L 2 ≤

≤ || (u g δ ) ε − (u g δ ) || ^L 2 + X

| α |≤ k

|| [ (D ^α (P (D)u) g δ ] ε − [ D ^α (P (D)u) ] g δ || ^L 2 +

+

Z

u(x − y) [g δ (x − y) − g δ (x)]ϕ ε (y)dy

L 2

+

+ X

| α |≤ k

Z

[D ^α P (D)u](x − y) [ g δ (x − y) − g δ (x) ]ϕ ε (y)dy

L 2

≤

≤ || (u g δ ) ε − (u g δ ) || ^L 2 + X

| α |≤ k

|| [(D ^α (P (D)u) g δ ] ε − [ D ^α (P (D)u) ] g δ || ^L 2 +

+σ ₅ (ε) || | ug _δ | ∗ | ϕ _ε | || L 2 + σ ₅ (ε) X

| α |≤ k

|| [ | D ^α (P (D)u | ∗ | ϕ _ε | || L 2 ≤

≤ || (u g δ ) ε − (u g δ ) || L 2 + X

| α |≤ k

|| [(D ^α (P (D)u) g δ ] ε − [ D ^α (P (D)u) ] g δ || L 2 +

+σ 5 (ε) {|| ug δ || L 2 + X

| α |≤ k

|| [ D ^α (P (D)u) ] g δ || L 2 }·

Sine

σ 5 (ε) → 0

^as

ε → 0,

^{the right-hand side tends to zero as}

ε → 0.

^Thus,

(3.1) is true and the proof isomplete.

4 Proof of the main result

Theorem 4.1. Let

P ∈ I _n

^{be an almost} hypoellipti operator,

ρ ∈ (0, ρ _P )

^and

δ ∈ (0, ∆ 1 (ρ))

^{, then}

N (P, δ) ⊂ H _δ ^∞

^.

Proof.Let

u ∈ N (P, δ)

^,

ϕ ∈ C ₀ ^∞ (S ₁ )

^,

ϕ ≥ 0

^,

R

ϕ(x)dx = 1

^,

ε > 0

^,

ϕ _ε (x) = ε ^{− n} ϕ( ^x _ε )

and let

u ε = u ∗ ϕ ε

^{be a} regularization of

u

^{. Then (see the proof of Lemma 3.1)}

u ε ∈ H _δ ^∞

^{and by the Lemma 2.5for any}

δ ∈ (0, ∆ 1 (ρ))

^and

k ∈ N 0

X

| β |≤ k

X

| α | >0

ρ ^{| α |} [D ^α P ^(α) (D)u _ε ]g _δ

L 2 ≤

≤ X k

j=0

A k,j

X

| β | =j

[D ^β P (D)u ε ]g δ

L 2 + B k k u ε g δ k L 2 .

Notethat by Lemma 3.1

X k j=0

A k,j

X

| β | =j

|| [ D ^β P (D)u ε ] g δ || L 2 + B k || u ε g δ || L 2 →

→ X k

j=0

A k,j

X

| β | =j

|| [ D ^β P (D)u ] g δ || ^L 2 + B k || u g δ || ^L 2

as

ε → 0

^{. Hene, there existnumbers}

ε ₀ > 0

^and

C = C(ε ₀ ) > 0

^{suh that}

X

| β |≤ k

X

| α | >0

ρ ^{| α |} || [ D ^α P ^(α) (D)u ε ] g δ || ^L 2 ≤ C ∀ ε ∈ (0, ε 0 ).

On the other hand, sine

P ^{(α 0 )} (ξ) = const 6 = 0

^{for some} multi-index

α ⁰ 6 = 0,

by this inequality itfollows that for some onstant

C 1 > 0 X

| β |≤ k

|| (D ^β u _ε ) g _δ || L 2 ≤ C ₁ ∀ ε ∈ (0, ε ₀ ). (4.1)

This means that the set

{ u ε ; ε ∈ (0, ε 0 ) }

^{is uniformly bounded in}

H _δ ^∞

^.

Therefore, using Lemmas 2.5 and 2.6 we get that for some onstant

C 2 > 0

^and

any

ε 1 , ε 2 ∈ (0, ε 0 )

C 2

X

| β |≤ k

|| D ^β (u ε 1 − u ε 2 ) g δ || ^L 2 ≤

≤ X

| β |≤ k

X

| α | >0

ρ ^{| α |} || D ^β P ^(α) (D)(u ε 1 − u ε 2 ) g δ || ^L 2 ≤

≤ X k

j=0

A k,j

X

| β | =j

|| [ D ^β P (D)u ε 1 − D ^β P (D)u ε 2 ].g δ || L 2 +

+B k || (u ε 1 − u ε 2 ) g δ || ^L 2 ≤

≤ X

| β |≤ k

X

| α | >0

ρ ^{| α |} || [ D ^β P ^(α) (D)u ε 1 − D ^β P ^(α) (D)u) ] g δ || ^L 2 +

+ X

| β |≤ k

X

| α | >0

ρ ^{| α |} || [ D ^β P ^(α) (D)u ε 2 − D ^β P ^(α) (D)u) ] g δ || ^L 2 +

+B k || (u ε 1 − u) g δ || ^L 2 + B k || (u ε 2 − u) g δ || ^L 2 ·

Hene

X

| β |≤ k

|| D ^β (u _{ε 1} − u _{ε 2} ) g _δ || L 2 → 0 as ε ₁ , ε ₂ → 0 ⁺ ·

From thisand (3.2) weget thatforany bounded set

G ⊂ E ⁿ

^andany

k ∈ N 0

there exists a number

C 3 = C 3 (G, k) > 0

^{suh that}

X

| β |≤ k

|| (D ^β u _ε ) || L 2 (G) ≤ C ₃ ∀ ε ∈ (0, ε ₀ ),

and

X

| β |≤ k

|| D ^β (u ε 1 − u ε 2 ) || L 2 (G) → 0 as ε 1 + ε 2 → 0 ·

Sine the Sobolev spae

H ^k ≡ W ₂ ^k

^{is omplete and}

|| u _ε − u || L 2 (G) → 0

^as

ε → 0

^{(see the proof of Lemma 3.1) it follows that}

u ∈ H ^k ·

^{Moreover, sine}

G ⊂ E ⁿ

^and

k ∈ N 0

^{are arbitrary,we have}

u ∈ H _loc ^∞ (E ⁿ ) ·

Passing in (4.1) to the limit as

ε → 0

^{we onlude that}

u ∈ H _δ ^k

^{for any}

k ∈ N 0 .

^Hene

u ∈ H _δ ^∞ (E ⁿ ) ·

Remark 4.1Itfollowsby estimate(4.1)thatfor any

δ ∈ (0, ∆ ₁ (ρ))

^theset

N (P, δ)

is ontinuously embeddedin

H _δ ^∞

^{in the topologyof}

H _δ ^∞

^.

Sine

[D ^β (P (D)u)]

^.

g δ ∈ L 2

^{for any}

β ∈ N ₀ ⁿ

^and

u ∈ N (P, δ)

^{, the following}

orollary isan immediate onsequene of the aboveTheorem 4.1:

Corollary 4.1. Let

P (D)

^{be an almost} hypoellipti operator,

ρ ∈ (0, ρ P )

^,

δ ∈ (0, ∆ 1 (ρ))

^,

f ∈ H _δ ^∞

^{and let}

u ∈ L 2,δ

^{be a solution of the equation}

P (D)u = f

^.

Then

u ∈ H _δ ^∞

^.

Using Theorem 3.1and noting that

H _δ ^∞ (E ⁿ ) ⊂ H _loc ^∞ (E ⁿ ) ⊂ C ^∞ (E ⁿ ),

we an prove

Corollary 4.2. Let the assumptions of Corollary 4.1 hold. Then

u ∈ C ^∞ (E ⁿ )

^.

Theorem4.2. Let

P (D)

^{be a linear dierential operator with onstantoeients,}

suh that

N (P, δ) ⊂ H _δ ^∞

^{for a}

δ > 0

^{. Then}

ρ P ≥ δ

^{and operator}

P (D)

^{is almost}

hypoellipti.

Proof.Firstnotethatbythelosedgraphtheoremandtheondition

N(P, δ) ⊂ H _δ ^∞

there existnumbers

k ∈ N

^and

C 1 > 0

^{suh that}

X n

j=1

k (D _j u)g _δ k _{L 2} ≤ C ₁ k u k P,k,δ ∀ u ∈ N (P, δ). (4.2)

To prove the theorem itis sues toshow that

ρ P ≥ δ

^.

Suppose to the ontrary that

ρ P < δ

^{. Then by the denition of the number}

ρ P

it follows that there exists a sequene

{ ξ ^s }

^{of points in}

R ⁿ

^{suh that}

| ξ ^s | → ∞

^as

s → ∞

^and

d P (ξ ^s ) ≤ ρ P + δ

2 s = 1, 2, . . . . (4.3)

Let

ζ ^s ∈ D(P )

^{besuh that}

d P (ξ ^s ) = | ξ ^s − ζ ^s | s = 1, 2, . . . .

Then by (4.3)

| Im ζ ^s | ≤ d _P (ξ ^s ) ≤ ρ P + δ

2 < δ s = 1, 2, . . . . (4.4)

Weset

u s (x) = e ^{i(x, ζ s )} (s ∈ N )

^{. Then}

u s ∈ N (P, δ)

^{and by (4.2) we obtain that}

X n

j=1

k (D j u s )g δ k _{L 2} ≤ C 1 k u s k _P,k,δ s = 1, 2, . . . . (4.5)

By estimates (2.3), (4.5) and by the denition of the points

{ ζ ^s }

^{we obtain}

k u s k P,k,δ = k u s · g δ k L 2 + X

| β |≤ k

[D ^β P (D)u s ]g δ

L 2 =

= k u s · g δ k L 2 ≤ κ e ^{ρP 2 +δ | x |} e ^{− δ | x |}

L 2 ≡ C 2 , s = 1, 2, . . . . (4.6)

On the otherhand, by estimates (2.3) and (4.4)

X n j=1

k (D j u s )g δ k _{L 2} = X n

j=1

| ζ _j ^s | e ^{i(x,ζ s )} g δ

L 2 ≥

≥ X n

j=1

| ζ _j ^s | e ^{−| Imζ s |·| x |} g _δ

L 2 ≥ κ ^{− 1} X n

j=1

| ζ _j ^s | e ^{− ( | Imζ s | +δ) ·| x |}

L 2 ≥

≥ κ ^{− 1} X n j=1

| ζ _j ^s | e ^{− ρP 2 +δ | x |} e ^{− δ | x |}

L 2 ≡ C ₃ X n

j=1

| ζ _j ^s | . (4.7)

From (4.5) (4.7) itfollows that

X n j=1

| ζ _j ^s | ≤ C 4 , s = 1, 2, . . . ,

where

C ₄ = C ₁ C ₂ /C ₃

^.

From this and (4.3) weget

| ξ ^s | ≤ | ζ ^s − ξ ^s | + | ζ ^s | = d P (ξ ^s ) + | ζ ^s | ≤ ρ P + δ 2 + C 4 .

Therefore,thesequene

{ ξ ^s }

^{isbounded, whihontraditsthe}assumption.This

ontradition ompletes the proof.

UsingthestatementsofTheorems4.1and4.2,wearriveatthemainresultstated

inSetion 1.

Referenes

[1℄ R.A.Adams,Sobolevspaes.Aademipress,NewYorkSanFranisoLondon,1975.

[2℄ O.V. Besov, V.P. Il'in, S.M. Nikolskii, Integral representations of funtions and embedding

theorems.Nauka,Mosow,1975(inRussian).Englishtransl.JohnWileyandsons,NewYork,

v.1,1978,v.2,1979.

[3℄ Ya.S. Bugrov, Embedding theorems for some funtion spaes. Pro. SteklovInst.Math., 77

(1965),4564(inRussian).

[4℄ V.I. Burenkov, An analogue of Hormander's theorem on hypoelliptiity for funtions

onverging to 0 at innity. Pro. 7th Soviet Czehoslovak Seminar. Yerevan, 1982, 63 -

67(inRussian).

[5℄ V.I.Burenkov, Sobolev spaes on domains. B.G.Teubner, TeubnerTextezur Mathemati,

137,StuttgartLeipzig,1998.

[6℄ V.I.Burenkov,Investigationofspaesofdierentiablefuntionsdenedonirregulardomains.

Dotor'sdegreethesis.SteklovInst.Math.,Mosow,1982(in Russian).

[7℄ V.I.Burenkov,Conditionalhypoelliptiity andFouriermultipliersinweighted

L p

^-spaeswith

anexponential weight. Pro.oftheSummerShool"Funtionspaes,dierentialoperators,

nonlinear analysisheld in Fridrihrodain 1993. B.G. Teubner, Stuttgart-Leipzig. Teubner-

TextezurMathematik,133(1993),256265.

[8℄ L.Ehrenpreis, Solutionsof someproblems of division. 4. Amer. J. Math.,82 (1960),522

588.

[9℄ O.R.Gabrielyan,Comparisonofpowerandstrengthofpolynomialsin

R ²

^{.ComplexAnalysis,}

Dierential Equationsand Related Topis. Pro. ISAAC Conferene on Analysis,Yerevan,

2002,4151.

[10℄ H.G.Ghazaryan,Someestimatesofderivativesofpolynomialswithonstantoeients.Izv.

AN Armenii,Matematika,34, no.3(1999),4463(in Russian).English transl.Journalof

ContemporaryMathematialAnalysis(ArmenianAademyofSienes),34,no.3(1999).

[11℄ H.G. Ghazaryan, V.N. Margaryan, On the behaviour of nonellipti polynomials at innity.

Izv.ANArmenii,Matematika,39, no.3(2004),118(inRussian).English transl.Journal

ofContemporaryMathematialAnalysis(ArmenianAademyofSienes),39,no.3(2004).

[12℄ H.G. Ghazaryan, V.N. Margaryan, Behaviour at innity of polynomials in two variables.

Topisin Analysisandits Appliations,NATOSi.Series.KluwerAad.Publ.Dortreht

BostonLondon,147(2004),163190.

[13℄ H.G. Ghazaryan, V.N. Margaryan, On a lass of almost hypoellipti operators. Izv. AN

Armenii, Matematika, 41, no. 6 (2006), 39 56 (in Russian). English transl. Journal of

ContemporaryMathematialAnalysis(ArmenianAademyofSienes),41,no.6(2006),30

46.

[14℄ L. Garding, B. Malgrange, Operateurs dierentiels partiellement hypoelliptiques. Math.

Sand.,9(1961),521.

[15℄ L.Hormander,Onthetheoryofgeneralpartialdierentialoperators.AtaMath.,94(1955),

161248.

[16℄ L.Hormander,Theanalysis of linearpartialdierential operators2. SpringerVerlag,1983.

[17℄ G.G.Kazaryan,Onalmosthypoelliptipolynomials.DokladyRoss.Aad.Nauk.Matematika,

398,no.6(2004),701703(inRussian).

[18℄ A.N. Kolmogorov, S.V. Fomin, Elements of theory of funtions and funtional analysis.

Nauka,Mosow,1972(inRussian).

HaikGhazaryanandVahaganMargaryan

Departmentofmathematisandmathematialmodelling

RussianArmenian(Slavoni)StateUniversity

123OvsepEminSt

0051Yerevan,Armenia

E-mail:haikghazaryanmail.ru,mar tikoyahoo.om

Reeived:25.12.2009