On some spaces of infinitely differentiable functions

ANNALES SOOIETATIS MATHEMATIĆAE POLONAE Series I: COMMENTATIONES MATHEMATIĆAE X I (1968)

J. M

(Poznań)

On some spaces of infinitely differentiable functions

J. Musielak introduced in [3] the spaces I)M of functions. The purpose of this paper is to investigate some further properties of spaces DM, in particular to prove the necessity-parts of some sufficiency theorems in [3].

Let M(u) be an even, continuous, convex, non-negative function vanishing only at и = 0 , u~lM(u) -> 0 as и -> 0 and u~xM(u) сю as и -> oo (cf. [2]). M(u) satisfies the condition (Д2) for small и if there exists positive numbers u 0 and Tc such that the inequality

M(2u) < hM(u) holds for every и < u0.

Let

— / M{(p(t))dt, where the integral is taken over the whole

w-dimensional space. Then

Z*M = {q> measurable:

(Ay) < oo for a certain X > 0}.

We define

D

— { 9 «f: Dp<p eL*M for every p},

here S is the space of all infinitely differentiable functions of n variables.

If the topology in DM is defined by the countable system of seminorms

\ \ D P <P\\m =

m f { e > 0:

< 1}, J)M becomes a jB 0 -space.

The following lemma is a specialization of the well-known theorem on the partition of unity (cf. [ 1 ]) such that the additional condition

is satisfied.

Dv coi are uniformly bounded for every p

L

1. Let e > 0 be given and let the sequence {r s a t i s f y the con­

dition

( ₁ . ₁ ) 0 , > 0 ,

R o czn ik i PTM — P ra ce M a tem a ty czn e XI.2

fi + l — Vi > €

20

3 0 6 J . M a g d z i a r z

for every i > 1. Moreover, let

A{ = 0 , A f E = {teBn: \t\ < r x}

and

A\ = {teEw: П-i < l<l < n_i + e}, A ^ e = {<ейи: П -1 + £ < |<| < r 4}

/or i ~ 2 , 3 , where e satisfies the condition ( 1 . 1 ) and / | = (tj + tl-j- + . . . + o l/2.

Then there exist functions оц defined in B n such that (i) щ е £ ,

(ii) a>i > 0 ,

(iii) denoting by 8 mi the support of щ , we have

8 W.

<=

w

1, гожего И* =

w

O O

(iy) £ coi = 1 ,

г=1

(v) D v coi are uniformly bounded for every p, i.e. there exist K v such that \Dpcoi\ ^ , K P.

Proof. Let

aa(x) =

for \x\ < a,

0 for \x\ > a,

where the number h is chosen so that f aa(x)dx = 1. Hence aa ^ 0 and Rn

( 1 . 2 ) aa (x) = aa(y) for \x\ = \y\-, moreover, aaeD (for the details of the proof, see [ 1 ]).

Let

—

w -d-i-j-i,

Q(i) = {te R n: r*_x + 0 < \t\ < г * + е — 0 }, where 0 < 0 < -|e, Gi = {teRn: г ^_!+0 — r\ < |t| < r*+ £—0 + rj}, where 0 < rj < 0 . Then both sequences of sets {Д } and are countable open coverings of the space B n and <= Gi a Gi cz Qi . How take an arbitrary i. If

№ = fx(£)aa(l~ £)d£,

R n

where % denotes the characteristic function of the set Gi, then we have /? e $. Since

f a j ( ) d i = l ,

Gi R n

we have 0 < P < 1 . Let a < d, where

0 < ó < m in[d(Qfi), Gi), d(Gt , £*)].

Then we have \t— || < d for teQ^y Hence £eGi and (1.3) f>(t) = f a a( t - ()ds = fa „ ( ()d s = 1 for

Gi R n

Moreover, we have \t— £\ > <5 for and hence aa{t— |) = 0.

Thus denoting b у 8p the support of /3 we have Bp <= Qi.

Further, let teB n be a point corresponding to te R n such that |£| = [/|, let JEJi(t) = {Я/ 6 -Й I Ti_\~\-0—f] <C 11 — x\ < f % -(-e — 0 -f" •

Then

p(t) = f a a(i— £)d£ = f aa(x)dx.

°i Щ&)

Applying (1.2) we can easily see that

j aa(x)dx = J aa(x)dx = J a a(t— £)d£,

Ei(i) Eid) Gi

hence

(1.4) /5(2) = 0CO for t,*e-ZT and |*[ = |t|.

In the same manner we construct the function (3 on each of the sets Qi with the same constants e, 0, rj, 6, a as above.

Thus we obtain a sequence of functions e $ such that (a) 0 < Pi(t) < 1 , te R n,

(b) (3{(t) = 1 (c) s fii <= Qi,

(d) let ie R n be a point corresponding to te R n such that if U A l e, then t e A f e, and if te A \ , then teAj and \t\— |t| = rj_1 — ri_1,

then

Ш = Ш , * = 1 , 2 , . . . , 1 = 1 , 2 , . . .

The condition (d) follows from formulas (1.4) and (1.3). We write and ан = р{/р.

1

Then a>i satisfy the lemma. The proofs of conditions (i), (ii), (iii) and (iv)

are trivial and will be omitted here. In the proof of (v) we shall apply

the following lemma:

3 0 8 J . M a g d z i a r z

I f f * S’ is a function of the radius, i.e.

f{x) = <p{r), where r = {х\ + х\ + • • • + #n)1/2, then for every p there exist constants ck (Tc = 1 , 2 ,

inequality

\v\

I-D7WI < 2 'Cil^ >(r)l

fc=l

.., |p|) such that the

is satisfied.

This lemma follows essentially from the formula

\V\ ^ ^ 4

q№ (r)

\ r r r /

fc=l

\v\

= 2 W t\ T '

where Wk(Ś) [к = 1 , 2 , ..., \p\) are polynomials (for the details of the proof see [4]).

From the condition (d), щ are functions of the radius, i.e., coi(x)

= iji(r). This implies

ipi

(1.5) l-O’WaOl < _ У ^ | ч»(г)| .

fc=l

Let te R n and \t\ = r, then there exists an i such that teQi. Hence, by the condition (d), we find teQ l such that \t\ — f and

nf>(r) = n f ]{r), ,(*) fc = 1 , 2 , . By the inequality (1.5) we have

ipi № 1

\Dpcoi(t)\ < ^ c k \ ^ ( r ) \ < Gkak = K *

&=i fc=i

Thus the Lemma 1 is proved.

The following theorem gives necessary conditions for the inclusion of spaces DM (see [3]).

T

2. I f D Ml <= D

2

then there exist constants Jc> 0 and u0> 0 such that the inequality

(2.1) M 2(Tcu) < M ^ u )

is satisfied for every 0 < и < u 0.

Proof. Let us suppose that for every number i there exists щ such that

(2.2) М 2{2~ъщ) > М ^ щ ) ,

(2.3) the sequence {[гЖ1(%)]-1} is increasing to infinity, i

(2.4) [(г + l) M ^ U i+ J]- 1 > a ^ [T c M x{uk) Y \

k

=l

where a = [@1/ri— l ] ~ n— 1 .

Let i 1 in r< = ( ^ [ f t j f 1 (« * )r 1) ■

fc=l

Then the sequence {r^} satisfies assumptions of the Lemma 1 with e = rx

= [Ж 1 ('и1)]_1/Г\ If the sets A \, A f eAi are defined as in lemma, this implies pAi = [гЖ 1(%)]“ 1 and p A ei+1 < pAi and then the function

oo

<p{t) = £ i =1

belongs to the space DMx. In fact, we have J Мг(ср{г)) dt = M 1(i2~iui) p A f e

A ~ e ^

Ai

j M 1(<p(t))dt= J M 1[i2~lUia)i(t) + { i+ l) 2 ~ {l+1)Ui+1a)i+1(t)]dt 2 We obtain easily

OO

J M x((p{t)) dt = f ’M^tpit)) ^ + ^ [ j Mi((p{t)) d t + j M x(cp{t)) dt j

R n A s- i=1 A T 8 a£.

г i+ l

OO

< j M x(<p{t)) dt-\-2 1 2i < 0 °'-

As

i=l

By lemma, for a fixed p , \JDpcoi\ < K. Thus we have J l-D’V «)l)< «

В» '

oo

i=0 Ae L J

^г +1

O O 00

< ^ 0 0 •

г=0 i= 0

Consequently, cpeDMl.

Let A be an arbitrary positive number, then there exists a number m such that nX ^ 1 for every n > m. Since

J* J f 2 (A<p(i)) dt = j M 2[Xi2~‘lUi(Oi(t)Jr X(i— l)2~^~1^ui_iO)i_1(t)]dt

3 1 0 J . M a g d z i a r z

and

we get

j M 2[A(p{t)]dt = f M 2[Ai2~iui]dt = М 2[Аг2~*щ]<иАГе,

AJ AT

>

W

f M 2[Acp{t)]dt > 2 V [M2(Ai2-iui)(iM1{ui))~1]

Rn <= 1

m— i oo

2 J ? [M2(Ai2-iui)(iM1(ui))~1\ + 2 ^ [ M 2(Ai2'iui)(iM1(ui))-1]

i= l

i=m

?»— 1 oo

2 У |Ж 2(Я*2-‘«()(*Ж1(%))-1] + 2 ^ ' - = oo

in contradiction to the fact that D Ml c DM2. Thus, the theorem is proved.

Theorems 3 and 4 yield necessary conditions for theorems 2.2(b) and 2.2(c) from [3].

T

3. I f

(3.1) DM = \cpe g: J M ( № pcp{t)) dt < oo for every p and A > 0j , nn

then M(u) satisfies the condition (Д2) for small u.

Proof. Suppose it is not true, i.e., for every % there exists an щ such that

(3.2) M(2ui) > 2 i М{щ),

(3.3) the sequence {[2г М ( щ ) У 1} is increasing to infinity, i

(3.4) [(^4-1) Ж (^ + 1)]-1 > a 2 [^(*Wft)rS

*=i where a = [(|) 1 /n- l ] - n- l .

i

Let fi = ( [2k M (u k)]~1f ln. Then the sequence {r*} satisfies the k =1

assumption of the Lemma 1 with в = гг — [2Ж(%)]~\ We consider the sets A \, Af% Ai defined in the lemma. Thus

juAi — [2г Ж (%)]-1 and y,Aei+1 ^ juAi.

OO

If <P — £ ui Mi, we have г =1

J M(cp(t)) dt = М{щ)[лА1е < М(щ)[2*М(щ)]~1 = 2~*

and

J M[(p{t))dt= J M^iCOi(t)-\-ui+1a)i+1(t)j dt ^ 2 г;

j

L*. t+i ■^i +1 this implies

OO OO

fM(<p(t))dt= j M(<p(t))dt+£ f M(<p(t))dt+ £ J M(<p{t))dt

R n

г=Х

A5 _{г + 1}

j M(<p(t)) dt + 2 2 2~*< °°- г=1

The proof of j M [Dv<p(t]) dt < oo is analogous. Consequently, <peDM.

R n

Since

J M{2(p(t)) dt = J Ж( 2 'М гС 0 г(^) + 2 %_ 1 С 0 г_1(^)) > J f (2%) /ь4*

^5 and

J M(2<p(t)j dt — J М(2щ) dt — M(2Ui)/łAi %

AT

we get

f M(2<p{t)) dt > 2 У M{2ui)(2iM (ui)}-1 > 2 У 2{М(щ) {2iM (ui))~1 = oo.

д 71 i=i Ći

Thus 2<p4DM — a contradiction. The theorem is proved.

T

4. I f D is dense in DM, then the function M ( и ) satisfies the condition (Д2) for small u.

Proof. By the assumption, for every <peDM there exists a sequence {<p*} of elements of D such that

%

<Pn -+q>.

Let p be fixed and let A be an arbitrary positive number. Choose a number e > 0. We find an n 0 such that

/ Jf ( 2 Л|Х>>> (<)-■»’>»„ (<)l) dt < e Rn

Moreover, denoting

/ M{2X\Dp<pne(t)\)dt = K

R n

we have

J M(K\Dpcp{t)\)dt < e + K <

R n

OO .

3 1 2 J . M a g d z i a r z

This implies X<peDM for every positive number X. Applying Theorem 3, we obtain the thesis.

The following theorem yields a necessary condition in order that the topology in DMi be stronger than the topology in В щ .

T

5. I f convergence in the space B Ml implies convergence in the space B M%, then there exist positive numbers Tc and u0 such that the con­

dition

(5.1) M 2(hu) < M x{u)

holds for every 0 < ^

Proof. If we consider the sequence of functions {<pk} where

OO

(pk — i% ' Ui coi 7

nM1

2 i=k

we obtain <pk -> 0 but <pk +*■ 0. The proof is analogous to the proof of the Theorem 2.

References

[1] A. F r ie d m a n , Generalized functions and partial differential equations, New York 1963.

[2] M. А. К р а с н о с е л ь с к и й , Я. Б. Р у т и ц к и й , Выпуклые функции и пространства Орлича, Москва 1958.

[3] J. M u sie la k , On some spaces and distributions (I), Studia Math. 21 (1962), pp. 195,202.

[4] — O pewnym twierdzeniu aproksymacyjnym dla funkcji nieskończenie róż- niczkowalnycli, Prace Mat. 7 (1962), pp. 63-69.

KATEDRA MATEMATYKI I UNIWERSYTETU A. MICKIEWICZA