On Borel's extension theorem for general Beurling classes of ultradifferentiable functions Текст научной статьи по специальности «Математика»

Владикавказский математический журнал январь-март, 2007, Том 9, Выпуск 1

UDC 517.51+517.98.

ON BOREL'S EXTENSION THEOREM FOR GENERAL BEURLING CLASSES OF ULTRADIFFERENTIABLE FUNCTIONS

Abanina D. A.

We obtain necessary and sufficient conditions under which general Beurling class of ultradifferentiable functions admits a version of Borel's extension theorem.

Mathematics Subject Classification (2000): 46E10,26E10,30D15. Key words: Ultradifferentiable functions, Borel's extension theorem.

1. Introduction

Definition 1.1. An increasing continuous function u : [0, to) ^ [0, to) is called a weight function if

logt = o(u(t)), t ^ to; u(t) = O(t); t ^ to;

(x) := u(ex) is convex on [x0, to).

00

A weight function u with J t-2u(t) dt < to is called nonquasianalytic.

1

Denote by Wf the set of all sequences Q = {un}00=i of weight functions with the folllowing property: for each n G N there exists a Cn > 0 such that

Un(t) + log(t + 1) ^ Un+i(t) + Cn for t ^ 0. (1)

By Wn denote the set of all sequences Q = {un}00=i of nonquasianalytic weight functions un. Without loss of generality we can assume that

un(t) ^ un+1 (t) for t ^ 0 and n G N.

The Young conjugate vW : [0, to) ^ [0, to) of is defined by

VW (y) := sup{xy - Vw(x) : x ^ 0}.

For A G (0, to) we define the space

{If (a)(x)l 1

f G Co(nN) : |f := sup sup lJ .j)1 < to ,

aeNN ||s||^A (|a|) J

where n^ := {x G RN : ||x|| ^ A}, ||x|| := max{|xj| : 1 ^ j ^ N} for x = (xi,..., xn) G

o|a| .p

Rn , |a| := ai + ... + «n for a = («i,..., «n ) G N N, f(a) := f

джа ... '

© 2007 D. A. Abanina

Next, for a weight sequence Q = {wn}ro=i G Wf we put

E(n) (nN) := f| (nN);

n=1

E(Q)(Rn) := {f G Cro(RN) : f |n„ G E^N) for each A > o} .

The elements of E(Q)(RN) are called Q-ultradifferentiable functions of Beurling type. Let us introduce now the corresponding spaces of sequences of complex numbers:

E°N := | d = (dQ)aeNnN G CnN : |d|w>N := sup < to \

and

ro

EN) := fl EN.

n=1

It is clear that the restriction operator p : f G Cro(RN) ^ (f(a)(0))agNN acts from E(n)(RN) into E(N^). If p is surjective, we will say that a version of Borel's extension theorem holds for the space E(q)(Rn) (for the original Borel's extension theorem see [7]). For minimal Beurling class (wn = nw, u is nonquasianalytic and w(2t) = O(u(t)) as t ^ to) Meise and Taylor [8] have shown that E(q)(Rn) admits a version of Borel's extension theorem if and only if w is

ro

strong, i. e. there exists a C > 0 such that J t-2u(yt) dt ^ Cu(y) + C for y ^ 0. The case of

1

normal Beurling class, when Q = {qnw }ro=i with qn ] q G (0, to) and w is a nonquasianalytic almost subadditive weight function, has been studied by the author in [4]. In this case, p : E(q)(Rn) ^ E(N is surjective iff w is slowly varying, i. e. ^lim ^^ = 1.

The main result of the present article is the following theorem.

Theorem. Let Q = {wn}ro=i G Wfnq. Each of the following two conditions is sufficient for E(q)(Rn ) to admit a version of Borel's extension theorem:

(I) for every n G N there exist m G N and C > 0 such that

ro

1| Un(I+tnII) dt < Wmdie + inII) + C for e + in G CN ;

—ro

(II) for every n G N there exist m G N and C > 0 such that

Wn(2t) < Wm(t) + C for t ^ 0

and

ro

4 / Unr^T dt < Um(y) + C for y > 0. n J t2 + 1 1

Suppose additionally that for each n G N there are m G N and C > 0 so that un(x + y) ^ um(x) + um(y) + C for x,y ^ 0. If Borel's extension theorem holds for E(q)(Rn ) for at least one N G N, then

(III) for every n G N there exist m G N and C > 0 such that

CO

2 f 72^ dt < 7m(y) + C for y > 0. n J t2 + 1 0

This theorem generalizes the results of [8] and [4] mentioned above. It should be also noted that (II) implies (I).

The paper has five sections. In Section 2 we get a criterion of surjectivity of p in terms of entire functions. In Section 3, using the method of Meise and Taylor [8], we obtain sufficient conditions on fi G W^9 under which E(q)(Rn) admits a version of Borel's extension theorem. Necessary conditions are derived in Section 4 by the method of [8] and [1]. The last section consists of two new examples of Beurling classes. We show that Borel's extension theorem holds for the first class and does not hold for the second one.

The author is grateful to professor Yu. F. Korobeinik for useful discussions.

2. Criterion in terms of entire functions

Let fi = be in Wf, and let the topology of E(n)(RN) (resp. E^) be given by the

system of seminorms (| ■ |Wn,n,N)neN (resp. by the normsystem (| ■ |a>n,w)neN).

For a weight function u and a number A G (0, to) we define the following space of entire functions

(CN) := {f G H(CN) : ||f:= jaup eA„fl/i|f+i(„z„) < '

where ||z|| = max{|zj| : 1 ^ j ^ N} for z = (z1;... ,zN) G CN. Obviously, Hw,a(Cn) is a Banach space with the norm || ■ . Next, for a weight sequence fi = {un}^=i G Wf we

put

H(n)(CN) := [J HWn,„(CN), H^ := [J H,n;o(CN).

n=1 n=1

Let H(q)(Cn) (resp. H^)) be equipped with the topology of ind HWn>n(CN) (resp. ind HWn;o(CN)). Note that H$) and H(Q)(RN) are (DFS)-spaces. By theorem 1 of [3], the Fourier-Laplace transform

F : ^ ^ £(z) = ^ (e-i<x'z>)

is a topological isomorphism from (E(q)(Rn))b onto H(^)(CN). As usual, we denote by the strong dual of a local convex space E. A description of (E^)) b is given by

Proposition 2.1. Let ea, a G Nn, be unit vectors in RnN, and fi = {un}^=1 be in Wf. Then the Fourier-Laplace transform

F : ^ ^ /x(z) = ^(e«)(-iz)a

aeNN

is a topological isomorphism from (E^) b onto H^).

zaN „ _ ^ it

Here za = zf1 ... for z = (z1,..., zw) G CN and a = (ab ..., aw) G N^.

< Since the proof is very similar to that of [4], we omit it. It should be only pointed out that the proof is based on the following property of Q = {wn}ro=i G Wf derived in Lemma 1 of [3]: if Bn is determined by the condition

Un(t) < Bnt for t ^ 1, (2)

and Cn is determined by (1), then

(s) - (s) < - log s + log(BneCn+1) for all s > 0. > Now we have a commutative diagram

N))b (« F I F

H(n)(CN) --- H()

where p' is the conjugate operator of p. It is easily checked that F o p' o F—1 is the identity mapping acting from into H(^)(CN). Our main result in this section is

Theorem 2.2. Let Q = {wn}ro=i G Wf. Then the following assertions are equivalent:

(i) a version of Borel's extension theorem holds for E(^)(Rn);

(ii) for each set B C H( contained and bounded in HUn;n(CN) for some n G N there exists an m G N such that B is contained and bounded in HUm;o(CN);

(iii) for each n G N there exist m G N and C > 0 such that

sup |f (Z)L < C sup llr |f(Z)L nn for all f G H(L; (3)

zecN eum(||z||) zgCiN en||1mz||+un(||z||) J (n) v ;

(iv) for each n G N there exist m G N and C > 0 so that

|f (z)| < en||/mz|l+u"(l|z|l) for all z G CN, f G H( (4)

imply

|f (z)| < CeUm(M) for all z G CN . (5)

< (i) ^ (ii): By the Surjectivity criterion 26.1 of [9], p maps E(^)(Rn) onto E( if and

only if for each bounded set A in (E(^)(Rn))b the set (p')—1 (A) is bounded in (E^)^ With the commutative diagram the first part of the theorem is proved.

(ii) ^ (iii): Fix any n G N and set Bn := {f G H(CN) : ||f ||un,n,N < 1}. Using (ii) with B = H() H Bn, we deduce that there exist m G N and C > 0 such that

IIsLmAN < C for all g G H^ H Bn. (6)

Let f G H() be fixed. If ||f ||Un,n)N = 0 or ||f ||Un,n)N = to, then (3) is trivially true. In case 0 < ||f |Ln n n < to we use (6) with g = llfn f-. Then we have

I|um,0,N ^ C ||f ||u„,n,N .

This means that (3) holds.

Implications (iii) ^ (iv) ^ (ii) are easily checked. >

3. Sufficient conditions

Throughout last three sections we suppose that Q = {wra}C=i G . We start by Proposition 3.1. Let Q = G . Then the space E(q)(Rn) is nonquasianalytic.

That is, there is a function f G E(^)(Rn)\{0} such that

f (a)(0) = 0 for all a G NN .

< First we choose 0 = to < ti < ... satisfying

CO

f Un(t) dt< -1 for each n G N. J t2 n3

tn

Then we introduce a function

'0 for t G [0,ti),

u(t) = {

nun(t) for t G [tn,tn+i).

Since we assume that un(t) ^ un+i(t) for t ^ 0, it follows that u is nondecreasing on [0, to). Next, un(t) = o(u(t)) as t ^ to and

tn+1

C

If dt =£ j'^ dt ^ <

1 n=1 + n=1

1 ^n

.

By Lemma 3.2 of [2] we find a cotinuous nondecreasing function a : [0, to) ^ [0, to) such that u(t) = o(a(t)), t ^ to; a(2t) < 4a(t) for all t ^ 0;

CO

fait) ,

dt < to.

1

Using Proposition 2.3 of [5] for the a and compact set K = {0}, we construct a function y G CC(RN) with the following properties:

y(x) = 1 for x G [—£,e]N; supp y C [—3e, 3e]N;

A^ := J \y(t)\dt< to.

rn

Here y(t) = f y(x) e-i(t'x> dx is a Fourier transformation of y.

rn

We wish now to show that y G E(^)(Rn ). By the Fourier inversion formula we have

\y(a)(x)\ ^ (21N / y'(t)(it)a ei(i'x> dt ^ yt)\HtyM dt <

^ T^V I \y(t)\ e"n("ty) dt • exp sup (\a\ log ||t|| - w„(||t||)) <

(2n)N J ÎSRN\{0}

< (2^/ \y(t)\e"n("i") dt • e^-n(|a|).

C

Since un(t) = o(u(t)) = o(o(a(t))) = o(a(t)) as t ^ to, there is an Mn > 0 such that

J |£(t)| (l|t|l) dt ^ |£(t)| dt = eMn A

Hence,

eMn a

|<^)(x)| < e^(|a|) for all a G <.

(2n)N

This means that ^ G E(h)(Rn). Setting f (x) = ^>(x) — 1 we finally obtain the result. >

To formulate the main result of the section we need some notation. For a nonquasianalytic function u we define the function Pa : CN —> R as follows:

Pa. (x + iy) = 1 J

w(||x + tyll) , „ mN

-— dt for x, y G RN.

n 7 t2 + 1

It should be noted that in earlier papers dealt with extension theorems (for instance, in [8] and [4]) the function Pa was considered for N = 1 only. As is well known, in case N = 1, Pa is harmonic in the open upper and lower half plane and it is continuous and subharmonic in the whole plane C. Moreover, u(|z|) ^ Pa(z) for z G C.

Theorem 3.2. Let fi = {wn}£°=1 G W™9. Suppose that (I) for each n G N there exist m G N and C > 0 such that

Pan (z) < Wm(||z||) + C for all Z G CN .

Then the operator p : E(h)(Rn) ^ is surjective.

< Let us show that condition (iv) of Theorem 2.2 holds. Fix any n G N. Assume that

N (H)

f G H(NH) satisfies (4). Then there are n/ G N and D/ > 0 such that

|f (z)| < D/ eanf (l|z|l) for z G CN . Given z = x + iy G CN with y = 0, we define the entire function

F : C ^ C, F(w) := fix + w ^ ) for w G C.

We can rewrite (7) as

Next, since

(7)

|F(w)| < Df eanf ("x"+|w|) for all w G C .

(8)

Im x + w

= | Im w |,

(4) implies that

|F(w)| ^ exp I n| Im w| + u

x+w

for w C .

(9)

CO

y

y

By the Phragmen-Lindelof principle (Theorem 6.5.4 in [6]) we find that for u G R, v G R\{0}

CO

|v| /" log IF(t)|

log|F(u + iv)I < M /" ( l0gt^ 2 dt + |v|d, n J (u — t)2 + v2

(10)

where

Using (8) we have

2 1/

d = lim sup--/log

r^C n r J 0

F (rei6)

sin 0 d0.

2 1 f 2 1 f

nr log F(re) sin0d0 < (logDf + Unf (H^H + r)) sin0d0 =

4 'logDf + Unf(INI + r)) for all r > 0.

■k \ r

Nonquasianalyticity of un/ gives us that

unf (r + ||x||) [ unf (r + ||x||) [ unf (s)

—--r—r— = —--~-ds ^ —-T.— ds ^ 0 as r ^ to .

r + ||x|| J s2 J s2

r+||x|| r+||x||

By the above this means that d ^ 0. Now, using (9) in (10) and (I), we have

, ,„, M |v| C Un(|x + t |2|H) log|F(u + „^ Uy (u — t)2 +^2 dt =

1 C Un(|x +(u + vt) A II) 1 C Un(H(x + u tv JU

dt = -

t2 + 1 W t2 + 1

—C

y ) , .■.. y w . / („ , , y ) , .■.. y

dt =

= P^J (x + u ) + iv-fT^) ^ Um( (x + u-^) + i^TT^ ) + C. n I yI I yI I yI I yI

Setting u = 0, v = I yI we get

log |f (x + iy)| = log |F(i|yH)| < um(|x + iy|) + C.

Thus,

|f (z)| < eC eWm(||z||) for all z G CN with Im z = 0 . By continuity this inequality holds for all z G CN. Theorem 3.2 is thus completely proved. > Corollary 3.3. Suppose that ^ = {un}CC=1 G Wn satisfies

(111) for each n G N there exist m G N and C > 0 such that Un(2t) < um(t) + C for all t ^ 0;

(112) for each n G N there exist m G N and C > 0 such that

CO

4 r Un(yt)

(II)

n J t2 + 1 i

dt ^ um(y) + C for all y ^ 0.

Then E(n)(RN) admits a version of Borel's extension theorem for all N £ N.

n

n

n

r

< Fix any n G N and find mi ^ n and Di > 0 such that

4 n

Wn (yt) t2 + 1

dt < wmi (y) + Di for y ^ 0.

(11)

Next, for the mi there exist m G N and D2 > 0 such that

(2t) < wm(i) + D2 for t ^ 0 . Combining (11) and (12), we have for all z = x + iy G CN

(12)

CO

1 / wn

P-n (z) = 1 f

Wn(IIx + tyN) dt < 2

Wn

NxN +

t2 + 1

< 2Wn(IMI +

< 2Wn(|x| +

+

n

CO

2 / w„

t2 + 1

dt <

IxN +

n

1

t2 + 1

dt <

+ 2Wmi (|x| +

< W.

mi 1

IIxII +

< wm(max{IIxI

+ Di <

+ Di < Wmi(2max{IxI, IIyII}) + Di <

+ Di + D2 < Wm (I z I) + C,

where C := Di + D2. This means that condition (I) of Theorem 3.2 holds. So p maps onto >

Remark. Let us explain how results of [8] and [4] for spaces of ultradifferentiable functions (UDF) of minimal and normal type can be derived from the present results. First recall that condition

(I') for each n G N there exist m G N and C > 0 such that

Po>n(z) < Wm(|z|) + C for all z G C

provides that Borel's extension theorem holds for the corresponding class ) of minimal

or normal type independently of the number N of variables.

In case of spaces of minimal type (see [8]), (IIi) means that w(2t) = O(w(t)), t ^ to. That was the general assumption of [8]. It is not hard to see that (II2) ^ (I') in this situation. Indeed, in the proof of Corollary 3.3 we just have proved that under assumption (II1), (II2) implies (I), and so (I'). Implication (I') ^ (II2) follows from

CO CO

4 r Wn(yt) dt < 4 r Wn(yt)

W t2 + 1 < n J t2 + 1 i 0

dt = 2P-n (iy) < 2wm(y) + 2C =

= 2mw(y) + 2C = W2m(y) + 2C for y ^ 0.

In case of spaces of normal type, in Lemmas 2.5 and 2.7 of [4] it was shown that (IIi) ^ (II2) and (IIi) ^ (I'). So (II) ^ (IIi) ^ (I').

4. Necessary conditions

Assume that Q = {wn}^=i G satisfies the additional condition (A) for each n G N there are m G N and An > 0 so that

w„(x + y) ^ wra(x) + wm(y) + An for x, y ^ 0.

(13)

C

Note that analogous assumption was also made in [4] for spaces of normal type (see Definition 6.1 of [4]). In case of spaces of minimal type, (13) is a simple consequence of the general assumption — (2t) = O(-(t)), t ^ to. We start by several lemmas.

Lemma 4.1. Suppose that ^ = {—n}C£Li £ Wn has property (A). For each n £ N there exists an m £ N such that for every £ > 0 we can find a C > 0 with

PWn(z) < £| Im z| + -m(| Re z|) + C for z £ C. (14)

< Fix any n £ N and find m £ N and An > 0 so that (13) holds. Next, for an arbitrary £ > 0 choose r > 0 with

CO

2 f -m(t) HjiT+idt<£.

r

We have for x ^ 0 and y > 0

CO CO

^ (x + iy) = y [ dt < 2y [ Um(x)+ +f + An dt = -m(x) + Im(y) + An,

n J t2 + y2 n J t2 + y2

-c 0

where Im(y) := — / -m(t)2 dt. If y ^ 1, then n J t2 + y2

CO CO

2 Г um(yt) 2 Г um(t)

Im(y) = - ,2 , -, dt dt =: Dm < TO .

n J t2 + 1 n J t2 + 1

0 0

If y > 1, then

r CO r/y CO

Im(y) = 2y dt + 2y /j^ dt < 2 ^ dt +2y dt < um(r) + ey.

n J t2 + y2 n J t2 + y2 n J t2 + 1 n J t2 + 1

0 r 0 r

Hence,

Im(y) ^ ey + um(r) + Dm for all y> 0,

and so,

P^n(x + iy) ^ um(x) + ey + C for x ^ 0, y > 0,

where C := um(r) + Dm + An. By continuity the preceding inequality holds also for y = 0. Since

PWn (x + iy) = PWn (x — iy) = PWn (—x + iУ), we finally obtain the result. >

Lemma 4.2. If fi = {un}C=1 G W™q has property (A), then for each n G N we can find an Rn > 0 such that for every R ^ Rn there exists an r > 0 for which

r

P"n (iR) « fjw+R dt. (15)

0

< Use of (1) gives that

CO r

p-„ (iR) < 2R / Wn+iR2 dt - log r+Cn=f /in+j! dt+Lr,R, (16) 00

where

CO

2R f Wn+^i) , _

Lr,R := — t2 + r2 dt - log R + C„

r

Put Rn := eCn+1. Given R ^ Rn choose r > 0 such that

CO

2R /Wn+i(t)

n J t2 + R2

dt < 1.

Then Lr,R < 1 - log Rn + Cn = 0. Use of this in (16) gives (15). >

In the next lemma we construct a special family of polynomials. Let us first introduce two new functions. For k G N and r > 0 we put

Lfe (t) for t G [0, r]

(t) := I .

I r(wfc)'_(r)log r + (r) for t G (r, to)

and

C

_ , _wk(|t|)

P-r (x + iy) : =

dt for y = 0

W (t - x)2 + y2 (x + iy € C)

— C

wr(|x|) for y = 0

P^r has the same properties that Pw, i. e. P^ is harmonic in the open upper and lower half plane and it is continuous and subharmonic in the whole plane.

Lemma 4.3. Suppose that ^ = {wn}C=i € satisfies (A). There exists a family

of polynomials {gR,n(Z) : n € N, R € [Rn, to)} of one variable ( € C with the following properties:

1) for each n € N and each R ^ Rn

9R,n(iR) ^ expP-„(iR); (17)

2) for each n € N there exist m € N and C > 0 such that

|gR,n(Z)l < C exp (m| Im (| + Wm(|ZI)) for all R € [Rn, to) and ( € C . (18)

< Given n € N we find an Rn > 0 according to Lemma 4.2. Next we fix any R ^ Rn and find r > 0 such that (15) holds. Then

r

P- (iR) < tI WtR) dt = ¥/!?+§ dt < P-n+i (iR). (19)

n J t2 + R2 n J t2 + R2 - -n+r 00

Applying Lemma 1 of [1] to the subharmonic function P^r+1 (z) and the point £ = iR, we construct an entire function gR,n (Z), Z G C, for which

gR,n(iR)=exp P^+1 (iR), (20)

r

|gR,n(Z)l < A(1 + |Z|2)2expP^+i(Z) for all Z G C. (21)

n + 1

Here A is an absolute constant and P^r i (Z) := sup |P^n+i. (Z + w) : |w| ^ l|. Combining (20) and (19), we immediately derive (17).

In order to show that (18) holds, we estimate P^r (Z). Since wn+i(t) ^ Wn+i(t) it follows

n + 1 +

that P^n+1 (Z) < P^n+1 (Z), and so, P^+1 (Z) < P^ (Z) for all Z G C. Next, by Lemma 4.1, there are k G N and Di > 0 such that

P^n+1 (z) < k| Im z| + (|z|) + Di for z G C .

In [3] it was proved (see inequality (5) of [3]) that

Wk(t + 1) < Wk (t) + e2 for all t ^ 0,

where Bk is determined by (2). Thus,

Pin+1 (Z) < P.Jn+1 (Z) < k| Im Z| + Wk(|Z| + 1) + k + Di < k| Im Z| + Wk(|Z|) + D,

where D := Bke2 + k + Di. It is easily checked that

(1 + |Z|2)2 < e4log(|z|+i) for Z G C.

Now we can continue (21) as follows

|gR,n(Z )| < A exp (k| Im Z | + 4log(|Z | + 1) + (|Z |) + D) .

Setting m = k + 4 and C = Aexp(Ck + Ck+i + Ck+2 + Ck+3 + D), we finally obtain (18).

Proof of the fact that gR,n are polynomials is the same as in Lemma 2 of [1], so we omit it. Our result is thus completely proved. > The main result of the section is

Theorem 4.4. Let Q = {wn}CC=i be a weight sequence in with property (A). If E(n)(RN) admits a version of Borel's extension theorem for at least one N G N, then (III) for each n G N there exist m G N and C > 0 so that

P^n (iy) < Wm(y) + C for all y ^ 0 . (22)

< First note that if p : ) ^ E^) is surjective for some N ^ 2, then it is also

surjective for N = 1.

Assume that there is an no G N such that for each m G N and each k G N there exists an Rm,k > Rn0 such that

P^n0 (iRm,k) > Wm(Rm,k) + k . (23)

Here Rn0 is determined by Lemma 4.2.

Let {gR,n(Z) : n G N, R G [Rn, to)} be a family of polynomials with properties of Lemma 4.3. Then there are ni G N and Di > 0 such that

|gr,no (Z)| < Di exp (ni| Im Z| + (|Z|)) for all Z G C .

1

Setting k := 7T~ gRm k no, we can rewrite the previous inequality as Di m-

|fm,k(Z)| < exp (ni| Im Z| + Wn1 (|Z|)) for all Z G C . (24)

Next, from (17) with n = no and (23) we have for all m,k G N

11 k

fm,k (iRm,k) = 77- 9Rmk ,no (iRm,k) ^77" exp (iRm,k) >77- exp um(Rm,k). (25)

Being polynomials fm,k are in H1^. From (24) and (25) it then follows that assertion (iv) of Theorem 2.2 is false. Thus, the operator p : E(q)(R) ^ is not surjective. This

contradiction proves the theorem. >

Remark. It should be noted that Theorem 4.4 gives us the corresponding necessary conditions of [8] and [4] for spaces of UDF of minimal and normal type. Moreover, in these two cases condition (II) of Corollary 3.3 is equivalent to condition (III) of the previuos theorem (see [4, 8]). This means that whole criteria of [8] and [4] can be derived from our present results.

5. Examples

One of the classical nonquasianalytic weight functions is w(t) = ta, 0 < a < 1. It was shown in [8] that space of UDF of minimal type defined by this function admits the analog of Borel's extension theorem. In contrast, the corresponding space of normal type does not admit this analog (see [4]). Let us consider Example 5.1. un(t) = tan, 0 < an | a ^ 1.

We should like to verify condition (II) of Corollary 3.3. It is easily seen that (II1) holds. Next, for n G N and y > 0 we have

те те те

4 Г dt = уап 4 Г dt ^ уап 4 у г„_2 dt = уап 4

W t2 + 1 у w t2 + п } У п(1 - an)'

1 1 1

Obviuosly, we can find a C > 0 such that

4

Уап —,-г < yan+1 + C for all у ^ 0.

n(1 - an)

This means that (II2) also holds. Thus the space E(q)(Rn) with Q = [ta"admits a version of Borel's theorem for all N £ N.

It is of particular interest that an could tend to 1, whereas w(t) = t is not a nonquasianalytic weight function.

Another well-known weight function is w(t) = loge(e+t), where в > 1. Recall (see [8] and [4]) that Borel's theorem does not hold for the corresponding spaces of UDF, both of minimal and normal type. Now we wish to consider a sequence of such functions.

Example 5.2. Un(t) = log/J(e+t), pn I p > 1

Without loss of generality we can assume that Pi < в +1. First note that Q = |wn}^°=i satisfies condition (IIi), and so, condition (A). Now, let us show that condition (III) of Theorem 4.4 does not hold. For у > 0 we have

тете те

^Ы = -! dt = У fd(lof + 1)) = y2 Pnf l0/+f + 1)__d^.

nKy> П 0 t2 + 1 n 0 loge™(e + yt) у ж 0 loge"+1(e + yt) e + yt

Put

t = y +yjy2 + 4(e - 1) ty := о .

Then t2 + 1 > e + yt for t > ty. Hence, for an arbitrary m G N we can write

C

P (. ) > 2 Pn / 1__= ^n__y =

-n (iy) > y n J log^n(e + yt) e + yt ^ - 1) log^n-i(e + yty)

ty

Pn log^m (e + y)

= Wm(y)

n(Pn - 1) log^n-i(e + yty)'

log^m (e + y)

Since > P > Pn — 1, the quotient -»—-- tends to to as y ^ to, and so, (III)

log^n (e + yty)

does not hold. By theorem 4.4 we derive that p : E(q)(Rn) ^ E^ is not surjective.

References

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2. Abanin A. V. Banach spaces of test functions in generalized approach of Beurling-Bjorck.—Vladikavkaz: Vladikavkaz Scientific Center, 2004.—P. 16-33.—In Russian.

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4. Abanina D. A. On Borel's theorem for spaces of ultradifferentiable functions of mean type // Results Math.—2003.—V. 44.—P. 195-213.

5. Abanina D. A. Absolutely representing systems of exponentials with imaginary exponents in ultrajet spaces of normal type and the extension of Whitney functions // Vladikavkaz. Math. J.—2005.—V. 7.— P. 3-15.—In Russian.

6. Boas R. P. Entire functions.—New York: Academic Press, 1954.—276 p.

7. Borel E. Sur quelques points de la theorie des fonctions // Ann. Sci. Ec. Norm. Super. 3 Ser.—1895.— V. 12.—P. 9-55.

8. Meise R., Taylor B. A. Whitney's extension theorem for ultradifferentiable functions of Beurling type // Ark. Mat.—1988.—V. 26.—P. 265-287.

9. Meise R., Vogt D. Einfiihrung in die Funktionalanalysis.—Braunschweig: Vieweg, 1992.—416 p.

Received 27 October, 2006

Abanina Daria Aleksandrovna

Rostov-on-Don, Southern Federal University,

Rostov-on-Don, 344090, RUSSIA;

Institute of Applied Mathematics and Informatics

Vladikavkaz Science Center of the RAS,

Vladikavkaz, 362040, RUSSIA

E-mail: abanina@math.rsu.ru