c Allerton Press, Inc., 2008. ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2008, Vol. 43, No. 5, pp. 285–296.  c V.G.Karapetyan, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 5, pp. 43–59. Original Russian Text 

FUNCTIONAL ANALYSIS

On a Class of Differential Operators with Constant Coefficients V. G. Karapetyan1* 1

Yerevan State University, Armenia Received June 25, 2008

Abstract—A linear differential operator P (D) = P (D1 , . . . , Dn ) with constant coefficients is called almost hypoelliptic if all the derivatives Dα P of the characteristic polynomial P (ξ1 , . . . , ξn ) can be estimated by P . The paper proves that if P is an almost hypoelliptic operator and f is an infinitely differentiable function, square-summable with a definite exponential weight, then any square summable with the same weight solution u of the equation P (D)u = f is again an infinitely differentiable function and P (ξ) → ∞ as ξ → ∞. MSC2000 numbers : 35H10 DOI: 10.3103/S106836230805004X Key words: Hypoelliptic operator; Almost hypoelliptic operator; Embedding theorems; Regularity of solution.

1. NOTATION AND PROBLEM STATEMENT

 Everywhere below, we assume that N is the set of natural numbers, N0 = N {0}, N0n = N0 × · · · × N0 is the set of n-dimensional multiindices, E n and Rn are n-dimensional, real Euclidean spaces with points x = (x1 , . . . , xn ) and ξ = (ξ1 , . . . , ξn ) respectively and Cn0 = Rn × iRn (i2 = −1). Further, for any ξ ∈ Rn , ζ ∈ Cn , x ∈ E n and α ∈ N0n we denote |ζ| = (|ζ1 |2 + · · · + |ζn |2 )1/2 , ξ α = ξ1α1 . . . ξnαn ,

|α| = α1 + · · · + αn ,

Dα = D1α1 . . . Dnαn ,

where either Dj =

∂ ∂ξj

or

Dj =

We consider a linear differential operator P (D) =

1 ∂ , i ∂ξj



j = 1, . . . , n.

γα Dα ,

(1.1)

α

where the sum is taken over a finite collection (P ) = {α ∈ N0n : γα = 0} , and its complete symbol P (ξ) = The convex hull of the set (P ) P. *





γα ξ α .

α∈(P )

{0} is called characteristic polyhedron of the operator (or polynomial)

E-mail: [email protected]

285

286

KARAPETYAN

For a linear differential operator (1.1), an integer m ≥ 0 and any δ ≥ 0, we denote    2 1/2  loc −δ|x|  L2,δ ≡ u : u ∈ L2 , uL2,δ ≡ < +∞ ,  dx u(x)e

Wδm

⎧ ⎫ ⎨ ⎬  ≡ u : u ∈ L2,δ , uWδm ≡ Dα uL2,δ < +∞ , ⎩ ⎭ |α|≤m

  N (P, δ, m) ≡ u : u ∈ L2,δ , uN(P,δ,m) ≡ P (D)uWδm + uL2,δ < +∞ , D(P ) ≡ {ζ : ζ ∈ Cn ; P (ζ) = 0} ,

D(P, δ) = {ζ : ζ ∈ D(P ), |Im ζ| < δ} .

Besides, we set ∞ 

Wδ∞ ≡

Wδm

and N (P, δ) ≡

m=0

∞ 

N (P, δ, m).

m=0

In fact Wδm (N (P, δ, m)) becomes a Banach space and Wδ∞ (N (P, δ)) becomes a Freshet space, given the seminorms  · Wδm and P (D) · Wδm +  · L2,δ , m = 0, 1, 2, . . . . Clearly Wδ∞ ⊂ C ∞ , Wδ∞ ⊂ N (P, δ) for any δ > 0. V. I. Burenkov in [1] proved that N (P, 0) ⊂ W0∞ if P (ξ) = 0 for ξ ∈ Rn great enough. The present paper is mainly aimed at finding a condition on the symbol of the operator P (D), which ensures the existence of some δ > 0 for which N (P, δ) ⊂ Wδ∞ . Definition 1. (see Theorems 11.1.1 and 11.1.3 in [2]). A differential operator P (D) is called hypoelliptic if the following equivalent conditions is fulfilled: (i) for any α ∈ N0n , we have P (ν) (ξ)/P (ξ) ≡ Dν P (ξ)/P (ξ) → 0 as |ξ| → ∞, (ii) dP (ξ) → ∞ as |ξ| → ∞

(ξ ∈ Rn ),

where dP (ξ) is the distance from the point ξ to the surface D(P ). Theorems 11.1.1 and 11.1.3 of [2] prove that a linear differential operator P (D) = P (D1 , . . . , Dn ) is hypoelliptic if and only if any generalized solution u of the equation P (D)u = 0 is an infinitely differentiable function. Definition 2. (see [3]) The operator P (D) is called almost hypoelliptic if for any ν ∈ N0n there exists a constant Cν > 0 such that |P (ν) (ξ)| ≤ Cν (|P (ξ)| + 1),

ξ ∈ Rn .

(1.2)

Lemma 1. (Lemma 11.1.4 in [2]) For any polynomial Q(ξ) of n variables, there exists a constant H = H(n, ord Q) > 0 such that   Qα (ξ) 1/|α| −1   ≤H (1.3) H ≤ dQ (ξ)  Q(ξ)  α=0

whenever ξ ∈

Rn

and Q(ξ) = 0.

It immediately follows from (1.3) that if |P (ξ)| ≥ ,

|ξ| ≥ C,

(1.4)

with some constants > 0 and C > 0, then the polynomial P is almost hypoelliptic if and only if ρp ≡ lim inf min dP (ξ) > 0. t→∞ |ξ|=t

(1.5)

Note also that the polynomial P is hypoelliptic if and only if ρp = ∞. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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2. AUXILIARY STATEMENTS Proposition 1. (I) The polynomial P is hypoelliptic if and only if D(P, δ) is a bounded set for any δ > 0. (II) If the estimate (1.4) holds, then the polynomial P is almost hypoelliptic if and only if the set D(P, δ) is bounded for some δ > 0, and in this case ρp ≥ δ.

Proof: The statement (I) follows form (ii) in Definition 1 of hypoellipticity. To prove (II), we suppose that the set D(P, δ0 ) is bounded for some δ0 > 0 and show that under this condition ρp ≥ δ0 . If the converse statement is true, then ρp would be finite and ρp < δ0 . Hence, the definition of ρp s ∞ s implies that there exist some numbers |ts |∞ 1 and points {ξ }1 , |ξ | = ts s = 1, 2, . . . , such that ts → ∞ as s → ∞ and ρp + δ0 dP (ξ s ) ≤ s = 1, 2, . . . 2 Let the points ζ s ∈ D(P ), s = 1, 2, . . . be chosen so that dP (ξ s ) = |ξ s − ζ s |, s = 1, 2, . . . Then ρp + δ0 < δ0 s = 1, 2, . . . , (2.1) 2 s i.e. |ξ s |∞ 1 ⊂ D(P, δ0 ). By the requirement of (II) there exists a constant M > 0 for which |ζ | ≤ M , s = 1, 2, . . . Hence, by (2.1) |Im ζ s | ≤ |ξ s − ζ s | = dP (ξ s ) ≤

ts = |ξ s | ≤ |ξ s − ζ s | + |ζ s | ≤ δ0 + M,

s = 1, 2, . . . ,

which contradicts the condition that ts → ∞ as s → ∞. This contradiction proves that ρP ≥ δ0 , and hence the polynomial P is almost hypoelliptic. Now we prove the converse statement, i.e. that if the polynomial P satisfies (1.4) and is almost hypoelliptic, then there exists a number δ0 > 0 for which D(P, δ0 ) is a bounded set. To this end, we suppose that conversely, the set D(P, δ) is unbounded for any δ > 0, i.e. for any s = 1, 2, . . . there is a point ζ s ∈ D(P, 1/s) such that lims→∞ |ζ s | = ∞. Then, by Newton–Leibnitz formula and Definition 1 of hypoellipticity, for any s = 1, 2, . . .     (α) s s α   P (ζ )(−Im ζ )  s s  |P (Reζ )| = P (ζ ) +  α!   α=0     (α) s  (α) s s )α   P (ζ )(−Im ζ |P (ζ )| ≤ |−Im ζ s ||α| =   α! α!  α=0 α=0     (α+β) s     1 1 P (Re ζ ) s |β|  s  |Im ζ ≤ |   ≤ C1 (|P (Re ζ )| + 1) s , s α!β!  α=0  β where C1 = C1 (P ) > 0 is a constant. Hence, it follows that |P (Re ζ s )| ≤

C1 s(1 − c1 /s)

for s great enough. This contradiction to the estimate (1.4) proves that if the polynomial P is hypoelliptic and the estimate (1.4) is true, then the set D(P, δ0 ) is bounded for some δ0 > 0. The proof is complete. As proved in [4], there exists a nonnegative function g ∈ C ∞ such that k−1 e−|x| ≤ g(x) ≤ ke−|x| , x ∈ E n , |D α g(x)| ≤ kα g(x), x ∈ E n , JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 43

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KARAPETYAN

for some k and kα (α ∈ N0n ). For any δ > 0 by gδ we denote the function g(δ·). By (2.2) and (2.3) we conclude that k−1 e−δ|x| ≤ gδ (x) ≤ ke−δ|x| , x ∈ E n , (2.4) |D α gδ (x)| ≤ kα δ|α| gδ (x), In [4] it is proved that both norms uW m ≡ δ



α ∈ N0n , x ∈ E n .

(2.5)

(Dα u)gδ L2

|α|≤m

and uW m ≡ δ



Dα (ugδ )L2

|α|≤m

are equivalent to the initial norm of the space Wδm . Similarly, one can prove that the initial norm in N (P, δ, m) is equivalent to each of the following:  (Dα P (D)u)gδ L2 + ugδ L2 , uN (P,δ,m) ≡ |α|≤m

uN (P,δ,m) ≡



Dα (P (D)u)gδ )L2 + ugδ L2 .

|α|≤m

Theorem 1. If for an operator P (D) and some δ > 0, N (P, δ) ⊂ Wδ∞ , then ρP ≥ δ, and hence P is almost hypoelliptic.

Proof: We consider the mappings Dj : N (P, δ) → Wδ∞ ,

j = 1, . . . , n.

The generalized differential operator is closed and hence N (P, δ) ⊂ Wδ∞ . Besides, N (P, δ) and Wδ∞ are Freshet spaces. Therefore, by the closed graph theorem there exist some natural number m and a constant C > 0, such that n 

Dj uL2,δ ≤ cuN (P,δ,m) ,

u ∈ N (P, δ).

(2.6)

j=1

Let s1 ≡ {x : x ∈ E n , |x| < 1} and 1 ≤ φ ∈ C0∞ (s1 ) be some nonnegative function such that φ(x) = 1 for |x| < 1/2. Then uφ(x/M ) ∈ N (P, δ) for any M ≥ 1 and u ∈ N (P, δ), since N (P, δ) ⊂ Wδ∞ by the condition. By the estimate (2.5) for any m ≥ 0  x    x    x         α    gδ  + uφ gδ  =   D P (D) uφ uφ M N (P,δ,m) M M L2 L2 |α|≤m



≤ C1

|α|≤m+ordp

≤ C2



 x     x     α   gδ  + uφ gδ   D uφ M M L2 L2

 x    x      β   α−β gδ  + uφ gδ  φ (D u) D M M L2 L2

|α|≤m+ordp β≤α

≤ C3



(Dα u) gδ  + ugδ L2 < ∞,

|α|≤m+ordp

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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289

where C1 = C1 (P ), C2 = C2 (P, m) and C3 = C3 (P, m, φ) > 0 are some constants. Let 0 < δ0 < δ, ζ ∈ D(P, δ0 ) and uζ = ei . Then it is obvious that uζ ∈ N (P, δ) for any ζ ∈ D(P, δ0 ). Therefore, by (2.6) n    x   x        ≤ C uζ φ (2.7)   Dj uζ φ M M N (P,δ,m) L2,δ j=1

for all M ≥ 1 and ζ ∈ D(P, δ0 ). One can see that for the left-hand side of this inequality n  n    x   x          i g φ = φ (x) u D e   Dj ζ  j δ M M L2 L2,δ j=1

≥

j=1

n      x  x    i   i e (x) − e D φ (x) φ g g D     j j δ δ M M L2 L2 j=1

n      x x  1   i   i  |ζj | e . gδ (x) − gδ (x) φ (Dj φ) = e M M M L2 L2 j=1

Changing the variables as x/M = y and using the properties of the functions φ and g, by (2.4) we get    n  n 2 1/2  x       iM   −n/2 ≥ φ(y)gδM (y) dy M+ |ζj |  e Dj uζ φ M L2,δ j=1

j=1

−n/2

M − + M

≥

n 

  2 1/2  iM  Dj φ(y)gδM (y) dy e 

 −n/2 M+

|ζj |

j=1

k − M



|y|<1/2

e−2| Im ζ|M |y|−2δM |y| dy k2

1/2

1/2 e2| Im ζ|M |y|−2δM |y| dy

,

|y|<1

where k is the constant of the estimate (2.4). In the polar coordinates   n  n  x      |ζj | 1   −n/2  ≥ M+ 1 − e−M (δ+| Im ζ|) S1  Dj uζ φ M k L2,δ 2M (δ + | Im ζ|) j=1 j=1   k 1  − 1 − e−2M (δ−| Im ζ|) S1 , (2.8) M 2M (δ − |Im ζ|) where |S1 | is the surface of the unit sphere in Rn . For the right-hand side of (2.7), where we assume that ζ ∈ D(P, δ), P (ζ) = 0 and |Im ζ| < δ, one can demonstrate:      x    x  x       i   α  i g g φ = P (D) e φ (x) + φ (x) D   e   uζ δ δ M N (P,δ,m) M M L2 L2 |α|≤m    x x       γ (β) gδ (x) + ei φ gδ (x) ≤ C1 ζ P (ζ)ei Dα−γ+β φ M M L2 L2 |α|≤m β=0 γ≤α

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KARAPETYAN

   (β) i (D α−γ+β φ)( x )     γ P (ζ)e M = C1 g (x) ζ   δ   M (α−γ+β) |α|≤m β=0 γ≤α

L2

  x   gδ (x) , + ei φ M L2

where C1 = C1 (m, ord p) is a constant and M ≥ 1. Hence, changing the variable x/M = y and using the properties of the functions φ and g we conclude that 1/2  −n/2     x   M+   iM −δM |y| 2  |α| (1 + |ζ|) ≤ C2 k e  e  dy uζ φ M N (P,δ,m) M |y|≤1 |α|≤m+ordp

−n/2

+ M+ ≤ C2



−n −1 2

kM+

 k |y|≤1

(1 + |ζ|)|α| 

|α|≤m+ordp



   iM −δM |y| 2 e e  dy

1/2

1 2M (δ − |Im ζ|)

 1 − e−2M (δ−| Im ζ|) |S1 |   1 −n/2 1 − e−2M (δ−| Im ζ|) |S1 |, + M+ k  2M (δ − |Im ζ|) ×

where C2 = C2 (P, m, φ) > 0 is some constant. Thus, by (2.8) and (2.7)   n   1 |ζj | −n/2  1 − e−M (δ+| Im ζ|) |S1 | M+ k 2M (δ + |Im ζ|) j=1 1 k −  M 2M (δ − |Im ζ|)  ≤ C C2





−n/2−1

kM+

1−

e−M (δ−|

(1 + |ζ|)α 

|α|≤m+ord p



Im ζ|)



|S1 |

1 2M (δ − |Im ζ|)

 1 − e−2M (δ−| Im ζ|) |S1 |   k −n/2  1 − e−2M (δ−| Im ζ|) |S1 | . + M+ 2M (δ − |Im ζ|) ×

M− 2 and letting M → +∞, we obtain Multiplying this inequality by  |S1 | n

n  |ζj | j=1

k



1

Ck ≤ , 2(δ + |Im ζ|) 2(δ − |Im ζ|)

since |Im ζ| < δ. Hence n  j=1

 |ζj | ≤ Ck2

δ + δ0 , δ − δ0

since ζ ∈ D(P, δ0 ) (i.e. |Im ζ| < δ0 < δ). Thus, the set D(P, δ0 ) is bounded. Therefore, by Proposition 1 the operator P is almost hypoelliptic and ρP ≥ δ0 . So, ρP ≥ δ since δ0 ∈ (0, δ) was arbitrary, and the proof is complete. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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ON A CLASS OF DIFFERENTIAL OPERATORS

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Corollary 1. If P is a linear differential operator P and N (P, δ) ⊂ Wδ∞

or

(N (P, δn ) ⊂ Wδ∞

for all δ > 0 or for a sequence δn → +∞ (as n → ∞), then P is a hypoelliptic operator.

Proof: immediately follows from Theorem 1 since by that theorem ρp ≥ δ for any δ > 0, and ρp ≥ δn , n = 1, 2, . . . For an operator P (D) and any number δ > 0, we denote W (P, δ) = {u : u ∈ L2,δ , P (D)u ∈ L2,δ } . It is proved by Lemma 2.3 in [4] that if P ∈ In ≡ {Q : Q(ξ) = Q(ξ1 , . . . , ξn ) → ∞

as |ξ| → ∞}

and P is almost hypoelliptic, then some number Δ(P ) > 0 and a constant C = C(Δ, P ) > 0 exist, such that for all δ ∈ (0, Δ(p)) and u ∈ W (P, δ)  P (α) (D)(ugδ )L2 ≤ CuW (P,δ) ≡ C [(P (D)u)gδ L2 + ugδ L2 ] . (2.9) α

Lemma 2. If P (D) is a hypoelliptic operator, then there exists a constant C = C(P ) such that for any δ > 0 and u ∈ W (P, δ)     (α) (2.10) P (ξ)F (ugδ ) ≤ C(P (D)u)gδ L2 + C1 ugδ L2 , L2

α

where C1 = C1 (δ, P ) > 0 is another constant and F (·) is the Fourier transform.

Proof: By Corollary 2.2 in [4], Wδ∞ is dense in W (P, δ). Hence it suffices to prove (2.10) only for functions of Wδ∞ . To this end, assume that δ > 0 is an arbitrarily fixed number and ρ ≥ 1. We observe that the polynomial P is hypoelliptic, so there exists a number M = M (ρ) for which dP (ξ) ≥ ρ,

|ξ| ≥ M (ρ),

ξ ∈ Rn .

(2.11)

Besides, by the estimates (2.11) and (1.3) and the Parseval equality 1/2       (α)  (α) (α) (α) 2 ρ P (ξ)F (ugδ ) = ρ |P (ξ)F (ugδ (ξ)| dξ L2

α

=



+

ρ(α) |ξ|≥M (ρ)

1/2

ρ

|ξ|
+

 α

|P

(α)

2

(ξ)F (ugδ )(ξ)| dξ 1/2

   α

|P (α) (ξ)F (ugδ )(ξ)|2 dξ

 (α)

α

≤

1/2



α



Rn

α

|ξ|≥M (ρ)

|α| |dP (ξ)P (α) (ξ)F (ugδ )(ξ)|2 dξ

1/2

 ρ(α) |ξ|≤M (ρ)

|P (α) (ξ)F (ugδ )(ξ)|2 dξ

≤ HC2 P (D)(ugδ )L2 + ρordp C2 ugδ L2 , JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 43

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KARAPETYAN

where C2 = C2 (ord p) = card N (P ),  |P (α) (ξ)|, C2 = C2 (P, ρ) = max |ξ|≤M (ρ)

α

and N (P ) is the characteristic polynomial of the operator (or polynomial) P . By Leibnitz formula and the estimates (2.4) and (2.5)),      (P (α) (D)u)Dα gδ  P (D)(ugδ )L2 ≤     α! α L2     (α) ≤ (P (D)u)gδ L2 + (P (D)u)gδ  (k δ)|α| , L2

α=0

(2.13)

where k = max (kν /ν!)1/|ν| . |ν|≤ordp

Using Leibnitz formula and the properties of g we get  (P (α) (D)u)gδ L2 (k δ)|α| |α|=j



≤

(P (α) (D)(ugδ )L2 (k δ)|α| +

|α|=j

|α|=j β=0



≤





P (α) (D)(ugδ )L2 (k δ)(α) +

|α|=j

≤

  (P (α+β) (D)u)Dβ gδ L 2 (k δ)(α)  β! (P (α+β) (D)u)gδ L2 (k δ)|α+β|

|α|=j β=0

P

(α)



|α|

(D)(ugδ )L2 (k δ)

+ C3

|α|=j

ord p



(P (γ) (D)u)gδ L2 (k δ)Γ ,

(2.14)

Γ=j+1 |γ|=Γ

where C3 = C3 (P ) = card Ω(γ) ≡ card{(α, β) : α + β = γ, |α| = j}. Therefore, by Lemma 1.2 of [4]

   (α)  (P (D)u)gδ 

L2

α=0

≤

 

(k δ)|α|

   (α) (D)(ug ) P  δ 

|α| 

(1 + C3 )k δ

|α|=0

By (2.12), the estimates (2.13)–(2.15) imply that      ρ(α) P (α) (ξ)F (ugδ )

L2

α

+



(1 + C3 )k δ

for all u ∈

(2.15)

.

 ≤ HC2 (P (D)u)gδ L2

   P (α) (D)(ugδ )

|α| 

α=0

Wδ∞ .

L2

L2

+ ρordp C2 ugδ 

Taking here

   2(1 + C3 )k δ max{1, (HC2 )} , ρ = max 2,

we come to the estimate (2.10), and the proof is complete. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Corollary 2. (i) If P is a hypoelliptic operator and f ∈ L2,δ for some δ > 0, then any solution u of the equation P (D)u = f that belongs to L2,δ , necessarily belongs to       (α)  W (P!, δ) ≡ u : u ∈ L2,δ ,  P (D)u gδ  < ∞ . L2

α

(ii) If P is an almost hypoelliptic operator, δ ∈ (0, Δ(P )) and f ∈ L2,δ , then any solution u of the equation P (D)u = f belongs to W (P! , δ).

Proof: (i) and (ii) immediately follow from the estimates (2.10) and (2.9) respectively. 3. MAIN RESULTS Theorem 2. If P is a hypoelliptic operator, then for any δ > 0, N (P, δ) = Wδ∞ .

Proof: Obviously Wδ∞ ⊂ N (P, δ), hence it suffices to show that N (P, δ) ⊂ Wδ∞ , δ > 0. We observe that by a theorem proved in [2] N (P, δ) ⊂ C ∞ for any δ > 0. We shall use induction to prove that N (P, δ) ⊂ Wδm for any natural m. By the hypoellipticity condition and the Seidenberg–Tarskii theorem (see, eg. [2]), there exist some numbers a > 0 and A > 0 such that (3.1) 1 + |P (ξ)| ≥ A(1 + |ξ|a ), ξ ∈ Rn . For simplicity, we assume that a ≥ 1. Then by (2.10) n 

Di (ugδ )L2 ≤

i=1

C C1 (P (D)u)gδ L2 + ugδ L2 A A

(3.2)

for any δ > 0 and u ∈ N (P, δ) (⊂ W (P, δ)), i.e. N (P, δ) ⊂ Wδ1 for all δ > 0. To show that N (P, δ) ⊂ Wδm for all δ > 0 when N (P, δ) ⊂ Wδm−1 , m ≥ 2, we choose a multiindex α ∈ N0n of length m − 1 and observe that since P is hypoelliptic, V = D α u ∈ N (P, δ) when u ∈ N (P, δ). Hence, by the estimate (3.2) n 

Di ((Dα u) gδ )L2 ≤

i=1

where (P (D)D α u)gδ L2 assumption (D α u)gδ L2

C C1 (P (D)Dα u) gδ L2 + (Dα u)gδ L2 , A A

< ∞ since u ∈ N (P, δ). On the other hand, |α| = m − 1 and by the induction < ∞. Consequently, n 

Di ((Dα u)gδ )L2 < ∞

i=1

for any α ∈

N0n ,

|α| = m − 1. Hence, by Leibnitz formula

+∞ >

n 

Di ((Dα u)g)δ)L2 ≥

i=1

n 

(Di Dα u)gδ L2 − k δ

i=1

n 

(Dα u)gδ L2 .

i=1

Consequently, n 

(Di Dα u)gδ L2 < ∞,

i=1

The multiindex α ∈ N0n of length m − 1 being since (D α u)gδ L2 < ∞ by the induction " assumption. m m ∞ arbitrary, we obtain u ∈ Wδ , and u ∈ m Wδ = Wδ . Corollary 3. An operator P (D) is hypoelliptic, if and only if N (P, δ) = Wδ∞ for any δ > 0. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 43

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KARAPETYAN

Proof: follows from Theorems 1 and 2. Theorem 3. If an operator P ∈ In is almost hypoelliptic, then there exists a number Δ(P ) > 0 such that N (P, δ) = Wδ∞ for all δ ∈ (0, Δ(P )).

Proof: Let m be any natural number. Since Wδ∞ ⊂ N (P, δ), it suffices to show that N (P, δ) ⊂ Wδm when δ ∈ (0, Δ(P )). As P ∈ In , the Seidenberg–Tarskii theorem implies that the estimate (4) is true for some constants a, A > 0. For simplicity we assume that a ≥ 1. By the estimate (2.9) n 

# $ Dj (ugδ )L2 ≤ C (P (D)u)gδ L2 + ugδ L2 .

(3.3)

j=1

for any u ∈ N (P, δ) and δ ∈ (0, Δ(P )). % Note that if φ ∈ C0∞ (s1 ) is such that φdx = 1 and φ (x) = −n φ( x ) for any > 0, then u ≡ u ∗ φ ∈ Wδ∞ and lim u − uL2,δ = 0

→0+

by Lemma 2.2 of [4]. We now show that for any ∈ (0, 1) the function {u } is uniformly bounded in Wδr for any natural r. The inclusion u ∈ N (P, δ) implies that for any natural m there exists a constant Cm > 0 such that (P (D)u )Wδm = (P (D)u) Wδm ≤ Cm (P (D)u)Wδm .

(3.4)

Indeed, the properties of the function g, the Young inequality and the condition gδ ≤ k2 e δ gδ (x − y), imply that for any v ∈

Wδm

v Wδm =

 |α|≤m



=

|α|≤m

≤k e

Dα (v ∗ φ ) · gδ L2 =

|y| ≤ (y ∈ supp φ ) 

((Dα v) ∗ φ ) · gδ L2

|α|≤m

   2 1/2   α  D v(x − y)φ (y)dy  gδ (x) dx  



2 δ

∀x ∈ E n ,

|α|≤m

  2 1/2   α  (D v(x − y))gδ (x − y)φ (y)dy  dx  

≤ k2 e δvWδm φ L2 = k2 eδ vWδm . Now, we use induction in r to prove the uniform boundedness of {u }, ∈ (0, 1), in Wδr . For m = 0 the uniform boundedness of {u }, ∈ (0, 1), in Wδ1 follows from the inequalities (3.3) and (3.4), since by these inequalities n 

# $ Dj (u gδ )L2 ≤ C (P (D)u )gδ L2 + u gδ L2

(3.5)

j=1

$  # = C (P (D)u) gδ L2 + u gδ L2 ≤ C C1 P (D)uL2,δ + uL2 ,δ ,

j = 1, . . . , n.

Now we suppose that u ∈ N (P, δ), δ > 0 and that the set {u }, ∈ (0, 1), is uniformly bounded in Wδl for l ≤ r and prove that this set is uniformly bounded in Wδr+1 . So, let α ∈ N0n be any multiindex of the length r. Then by the inequalities (3.3) and (3.4) with m = r Dj [(Dα u )gδ ]L2 ≤ C[(P (D)(Dα u) gδ ]L2 + (Dα u) gδ L2 ] JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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ON A CLASS OF DIFFERENTIAL OPERATORS

295

# $ = C (Dα (P (D)u) gδ L2 + (Dα u) gδ L2 # $ ≤ C P (D)uWδr + uWδr < +∞,

since D β u = (Dβ u) for Dβ u ∈ L2,loc . The multiindex α ∈ N0n of the length r being arbitrary, by the equivalence of the norms  ·  and  ·  , Dj Dα u L2,δ ≤ const < ∞,

∈ (0, 1),

i.e. the set u , ∈ (0, 1), is uniformly bounded in Wδm . Using induction in m we show that lim

→0,ρ→0+

u − uρ Wδm = 0

(3.6)

for any u ∈ N (P, δ) and any natural number m. By (3.5) for m = 1 m 

# $ Dj ((u − uρ )gδ )L2 ≤ C (P (D)(u − uρ )gδ L2 + (u − uρ )gδ L2

j=1

# ≤ C |[(P (D)u) − (P (D)u)ρ ]gδ L2 + ((P (D)u)ρ − P (D)u)gδ L2 $ +(u − u)gδ  + (uρ − u)gδ L2 .

(3.7)

Hence n 

Dj ((u − uρ )gδ )L2 → 0

as

→0+

and ρ → 0+,

j=1

since (v − v)gδ  → 0

as

→ 0+

(3.8)

for any v ∈ L2,δ (see the relation (2.5) in [4]). Consequently, (Du )gδ − (Duρ )gδ L2 → 0,

as

, ρ → 0+

since the norms  · W m and  · W m are equivalent. Now, assuming that the relation (3.6) is true for δ δ m ≤ k − 1 (k ≥ 2) we prove its validity for m = k. By the estimate (3.5) n     Dj [(Dα u − Dα uρ )gδ ]  L2 j=1 |α|=k−1

 ≤C



(P (D)(Dα u − Dα uρ ))gδ L2

|α|=k−1

+



 (Dα u − Dα uρ )gδ L2 .

(3.9)

|α|=k−1

By the induction assumption the second summand in the right-hand side of (3.9) tends to zero as , ρ → 0+. Using (3.8) for the first summand in the right-hand side of (3.9) we get  (P (D)((Dα u ) − (Dα uρ )))gδ L2 |α|=k−1

=



((Dα P (D)u) − (Dα P (D)u)ρ )gδ L2 → 0

|α|=k−1

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No. 5

2008

296

KARAPETYAN

as , ρ → 0+, since Dα P (D)u ≡ v ∈ L2,δ by the condition u ∈ N (p, δ). By (2.8) and the equivalency of the norms  · W k and  · W k , δ

δ

n  

(Dj Dα u − Dj Dα uρ )gδ L2 → 0

as

, ρ → 0+

j=1 |α|=k−1

Thus, the relation (3.6) is true for any m. Now, observe that Wδm is a Banach space and the generalized differential operator is closed. Consequently, it immediately follows from the relations (2.6) and u − uL2,δ → 0 ( → 0+) that the generalized derivative D α u exists when |α| ≤ m and u ∈ Wδm . Therefore, u ∈ Wδ∞ by the arbitrariness of the number m ∈ N , hence N (P, δ) ⊂ Wδ∞ for all δ ∈ (0, Δ(P )). Corollary 4. The equality N (P, δ) = Wδ∞ is true for any δ > 0 if and only if the operator P is almost hypoelliptic.

Proof: follows from Theorems 2 and 1. REFERENCES 1. V. I. Burenkov, “Analog of Hoermander’s Theorem on Hypoellipticity for Functions Tending to Zero at the Infinity” (Russian), in: Collection of Reports of 7-th Soviet–Czechoslovakian Seminar, 63-67 (1982). 2. L. Hoermander, The Analysis of Linear Partial Differential Operators, vol. 2 (Springer-Verlag, 1983). 3. G. G. Kazaryan, “On almost Hypoelliptic Polynomials” (Russian) Reports of Russian Academy of Science 398 (6), 701-703 (2004). 4. G. G. Kazaryan and V. N. Markaryan, “On a Class of Almost Hypoelliptic Operators”, Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 41 (6), 3956 (2007). 5. O. V. Besov, V. P. Ilyin and S. M. Nikolsky, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996).

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

Vol. 43 No. 5 2008