On a modular space oî infinitely differentiable functions

A N N A L E S S O C I E T A T I S M A T H E M A T I C A L P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 ) R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O

S é r i a I : P E A G E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

J. Magdziarz (Poznan)

On a modular space oî infinitely differentiable functions

This paper contains generalizations of results contained in papers [1], [4] and [⁶]. In place of spaces D M considered in the above mentioned papers, there is considered a more general space De of infinitely differ­

entiable functions which is defined by means of a modular q. The problem of the dual of De considered in papers [⁶] and [7] in the special case of DM, is not investigated here.

1. First, we give the definition of a modular q, the space De and of convergence in De.

Let £ be the space of all real infinitely differentiable functions in the real n-dimensional space B n. Let an extended real-valued functional

q, called a modular, be defined on £, satisfying the following conditions:

(A.¹) ⁰ ^ q(<p)^ oo ; g(<p) = ⁰, if and only if, <p = ⁰,

(A.²) + +

for all a > 0 , ft > 0 such that a + /? = 1, (A.3) Q(<px) < q(pf) for \<px\ < \<p2\,

(A.4) if the support 8^ of a function pe £ is compact, then q{p) < oo, (A.5) QiPi + Pz) = QiVi) + Qipf) f or functions p x and p2 such that

S9i n S9i = 0 ,

(A.⁶) if Um p{ç £, then o(Hm cpi) <c lim

i —hOQ i —KX> OO

We shall say that a sequence {«pj, Pi e £ is ^-convergent to zero (we write cp{ 4» 0), if q{X(Pi) -> 0 as i -> ^{0 0} for some A > 0. By B Q we denote the subspace of £ which consists of all functions p e £ such that for every multiindex p = (px, p 2, . . . , p n) (p{^ ⁰) there exists a constant Лр > ⁰ satisfying the condition q(XpBpp) < ^{0 0}, where B p is the operator of differentation :

B p = d»i+.-+Pnidsc*i ...

In the space D e we define a sequence of pseudonorms IМ Г = sup ||1>>||((?), m = 0, 1 , 2, . .. ,

|p|<m

where \p\ + and

IMI(C) = in f j& :

O bviously, the topology defined in D e b y means of the base of neigh­

bourhoods of zero

V (m , s) = {<pe D q: \\y\\m < г }, m = 0 , 1 , 2 , . . . , e > 0, is equivalent to the topology defined in I)e b y means of the P-norm

M e

OQ

= У —Zj 2m

m=o

1М Г

1 + ¹МГ*

A sequence {99J , <p{e D Q w ill be called à-convergent to zero (we w rite ц\ Z 0) if B Qq>i ~> 0 fo r every p.

W e shall say that the m odular q satisfies condition (a) (we w rite pe(a)), if fo r every sequence {99J , D e such tha t <Pi Z 0 there exists a constant a > 0 satisfying the in eq u a lity |^(t)| < a for every t e Itn and every i.

N ow , we form ulate some lemmas needed in the sequel.

Lemma 1. Let <pe S. Then g(<p) < 00, if and only if,

00

(1.1) lim q(cp£ o)j) = 0,

k—xx> ' j ~ k

where the functions (Oj€ S, cOj of compact support, which give a partition of unity, are defined as in [4], Lemma 1.

г

P r o o f . L e t q { ( p ) < 00, and let <pt = Z Since Q (q > i) < q(<p) k ~1

and lim щ = 9о, we have

%— Ю О

9 ( < P ) < lim Q ( < P i ) < lim Q M < Q( <P) >

г—>00

i.e.

lim givi) = q(<p).

i-*OQ

L e t us choose an a rb itra ry e > 0. Then there exists an i Q such that

q(<P) ~~ Q(<Pi) < e f o T i ^ i 0 .

Infinitely differentiable functions 161

Hence

% ⁰⁰ oo m

Q(<p) > Q <pa>k + £ = e(<fr) + e \ 2 <P<»k),

k = l k = i + 2 ft=i+2

and so we get

00

в ( £ 9’CÜ*) ^ ~ e M < e for i0«

7 -__ V » л

k—i+2 This proves (¹.¹).

Now, let us suppose

00

q (⁹? JT* шг| < e for an e > ⁰ and a k0 > ¹ . i=*o .

Let a number a > ⁰ be chosen in such a manner that coko(t) = ¹ for \t\ = a.

Writing

A = {U |t| < a}, В = {t: |t| > a}, we obtain

00

tfjfce-i c -4 and £ (OjXB = Хв•

2! «V

i=l ' i=*o

Hence, one may construct non-negative functions cojti e ê such that l<l < j j + y } ’

OO

where j = ¹ ,² , . . . , i = ¹ ,² , . . . , and eow = ¹ for i = ¹ ,² , . . . j^{= 1}

Let us define two sequences of functions

i ⁰⁰

<pi,i = ( p j £ мл = у ^ wi,i'

3 = 1 j —i+l

It is easily observed that for each i,

A c S W s * v c B > 8n,i nfi4 < = 0 -

Since lim <phi = ухл and lim <p2ii = <pxB, we get lim {(plti + <piti) = <p.

%—Kx> i —x x ) i —>00

Hence we obtain

Q{<p) < Um[e(^M + 9?2,<)] = Jn^[e(<??i.î)+ Q{<P2,i)1 /co+l

i-kn

< Hm U<p JT o 3) -f q (<p JT1 a>i)l

ir~>00 j = X - ’ •

This proves the lemma.

1 1 ~ R o c z n i k i P T M — P r a c e M a t e m a t y c z n e X V I I

< OO.

Le m m a 2. I f ge(a), then every g-convergent to zero sequence { y j is convergent almost uniformly to zero.

Proof. Let <р{ 0. By the condition (a), there exists a sequence of positive numbers {ap} such that the inequality \JDp(pi{t)\ < ap holds for every t e R n and each i. Let p j = (plf p if ..., i b_i , îb + 1, pj+i, .. -, p n).

By the mean-value theorem,

П П

\ ^ П (1) - в ^ {(Г)\ = < «; у I*,— tj\,

3= 1 i = i

where a^, = max ay. Thus the functions Dp<Pi satisfy Lipschitz condi- i<i<n

tion, and so they are equicontinuous functions for each fixed p, uniformly bounded according to the assumption. Applying Arzelà theorem, the dia­

gonal method yields a subsequence {(pifc) such that B p<p{ -> <pp almost uni­

formly for every p separately. It is easily seen that (².¹) Dp(pik -* D pq) almost uniformly,

where <p — <Pq and tpe S. Next, we have g(X0<p) ^ Hm e(X0tpik) = ⁰, i.e.

г-wo

<p = ⁰. Prom this and from formula (².¹) it follows that (².²) <pifc -> ⁰ almost uniformly.

It is easily observed that the sequence {⁹⁹J is also almost uniformly con­

vergent to zero. Thus, we proved the lemma.

Le m m a 3. I f ge(a), then the space De is complete.

Proof. Let {<fi} be a Cauchy sequence in De, i.e. for each p and every constant Я > 0,

uniformly with respect to A. By the conditon (a), we have

(3.1) \DP<pk( t ) - D p<p0( t ) \ ^ a p

for every h. Now, we prove the function Dp<p0 to be bounded. Indeed, let g(XDp(p0) < со; then, by Lemma 1,

00

lim g{XBp<p0 У cok) = ⁰.

г-х»

00

Hence, by (a), there exists a constant > 0 such that \Dp<p0 со*] < fîp

k=*i

Infinitely differentiable functions 163

for each i, that is \Dpy 0\ < fip . From this and from (3.1) it follows that

\Dpy kI < ap + Pp ■= K p . Arguments analogous to those applied in the proof of Lemma ² show that for every p , Dp<pik -> L)py almost uniformly, where <p e S’. Consequently, applying Cauchy condition, we have for fixed p

elX(JPVl- I P v)l = g [lim Х Ц Р ^ - Г Р у , ) ]

< lim -.»*><)] < e

S —>OQ

for i > ix and for every constant Я > 0. Hence y t y. It is also easily seen that ye De, and the proof is finished.

Le m m a 4. Let д(ХрВ ру) < oo for every p and for a fixed sequence of positive numbers {Xp}. Let

oo supe(ApH »

£ < « > ? ) = ^ ,M<m

1 + sup д(ХрВ ру) ’ then the condition у{ Л- ⁰ is equivalent to the condition

(4.1) lim Q{{Xp},<Pi) = 0 .

%—WO

Proof. Let yt -Д- 0, i.e. Q{XpDpyi) -> 0, and let us choose an e > 0.

00 1

We take ^{& 0} so large that and we choose i0 in such a manner m—kfj+l ²

that e {XpDp(pi) < |e for \p\ ^ 7c0 âiiid. % ^ ( Then

* 0 o o

m = l m = k0+1

for i ^ i 0. Consequently, lim (>{{Xp},<Pi) = 0 . The converse implication

г —> o o

is obvious.

2. In order to formulate the fundamental theorems concerning spaces D e, we shall need some further notation and terminology.

A set 4 c De is called bounded in De, if there exists a sequence of positive numbers {Kp} such that the inequality \\Dpy\\^ < K p holds for every function ye A and for each p.

A modular is called non-stronger than a modular q2 (we write Pi ■*? Q2), if for every non-negative sequence of numbers {Xp} there exists a constant к > ⁰ such that for every function у е ё satisfying the condition 9г(ХрВ ру) ^ ¹ for every p, there holds the inequality Q%{ky) < ¹.

We shall say that a modular q satisfies the condition ( d 2) (we write

qç (d 2), if for every sequence of positive numbers {Ap} there exists a con­

stant a > ⁰ such that for every function <p satisfying the condition ^({Яр} , cp)

< a there holds the inequality q{2<p) < ¹.

Now, some important theorems concerning spaces De will be given.

Theorem 5. I f q e (a), then for every sequence , <p%e De such that the supports of (p{, Sv a A , where A is a compact set, the condition (p{ 0 is equivalent to the condition cp{ -> ⁰.

P roof. Let B p(pi -> 0 uniformly on a compact set A, and let JcQ be so large that A c S kQ , where coj are the same as in Lemma 1. Let e

j ~ l 3

satisfy the inequalities ⁰ < s < ¹, then there exists an i0 such that \Dp(Pi\

fco+i

< e for i ^ i 0, i.e. д(Л1)р(р^ ^ q{sX £ coj) < eM, where A is an arbitrary

j=1 dq

positive number. From the last inequality it follows that щ -> 0.

Converse implication follows from Lemma ², immediately.

oo ~

Theorem 6. Let <pe S. Then <peDQ, if and only if, £ <рю5 0, coj j*=k

being defined as in Lemma ¹. Moreover, Q{XDprp) < oo for every p and for every A > 0, if and only if,

(⁶.¹) / -4 ⁰.

i=*

Thus, the subspace D of functions from of compact support, is dense in De in the sense of g-convergence.

OO

Proof. Let g{XpB p(p) < oo. By Lemma 1, there holds lim g{hpDp cp^Wj)

ft—к» j = fc

= ⁰ for every p. Now, we prove that for every function y)€ ê such that д{Л\р) < oo for some A > ⁰, there holds

CO

(6.2)

j=k

Indeed, by [4], Lemma 1, there exists a sequence of numbers {jK^} such that |DP( J; wj) I < K p . Then

j=k

Г З ° ° Т Г 2 00 00 -, 00

s \wp _{L P} vI>p Œ_y=ft “')] = e _J [_{L P} k p * * Œ _{j= k i=ft-1} Ё a]_J < « h 2_J=ft-1 oj>]

Infinitely differentiable functions 16Ô

Hence, Lemma 1 implies

J *** К

= ⁰

and this proves (6.2). Now Leibniz formula gives

00 00

^ ² »<) = ² ’

j = k 0<|®|<¹Я j=k

where p = (px, . . . , p n), v = (vx, . . . , v n), p v = (px- v x, . . . , p n- v n).

Let

min Xe( ² fe ) •••(?»)) ' Then, by (⁶.²), we obtain

l i m g k l W , , V e [ V - » PV - » '(² a,<)] = 0 -

*~><50 i= * fc-*oo 0<|«|<|J3|

Now, let lim £>[ApI)p( <pu>j)] — ⁰; then

fc—^oo j—k

limg h pDp<p У aJ < lim q\àpDp L V о>,)1 = ⁰.

*~><30 j = k + l k - x x j*=fc

Hence, Lemma ¹ shows that (pe D e. Converse statement is proved analo­

gously.

Theorem 7. Let us consider the following conditions:

(A) i f a set A is bounded inD , then A is bounded in D e ;

Dei D z

(B) if <p4 -> ⁰, then <p{ ⁰; (C) gj -3 g2;

№) Я , <= Я 2-

Conditions (A), (B) and (C) are mutually equivalent. Condition (C) implies condition (D). Moreover, if qX€ (a) and q2€ («)> then all conditions (А), (В), (C) and (D) are mutually equivalent.

Proof. Let (A) be satisfied and let ç>t- -> 0. In particular, ||<^||(е1) -> 0;

hence there exists a subsequence {<pik} such that ll&<p*fcll(ei) -> 0. Let us arrange the systems p in a sequence {p8}, where p0 = (0, ⁰, .. ., 0). Since l|I>Pl⁹?fJ|(Ci) -> ⁰, from the sequence {⁹?^} one may extract a subsequence

{<fa } for which \\WPl(pik ||(ei) -> 0. Proceding further in the same manner, we get (applying the diagonal method) a subsequence {⁹⁹^} such that

||⁵Z>P⁹⁷^||(ei) < K p . By (A), there exists a sequence of numbers {K'p} satis­

fying the inequality (⁶²) ^ *,e’

(7.1) ||B > ,s||(e²) - >⁰. Hence it follows easily that ||-ZPV*||(e2) 0.

Now, let condition (B) be satisfied, and let us suppose that qx -3 ^(?2 does not hold, i.e. Qi{XpDp(pv) < 1 and Q2(v~2<pv)> 1 for every p, and for a sequence of positive numbers and a sequence of functions {<pv}, <pve ê.

Let y>v = v~2<pv and let X be an arbitrary positive number. Taking p fixed, we choose an index v0 such that v~1X < Xp for v0. Then

Q1{XDpipv) < — дг [— Dp<p\ < ~ Q i ixpDp<p^\ < —

v \ v ! v \ ! v

for v ^ v0, i.e. yjv — 0. Since QDei 2(yv) — ^{q2 (v} 2<pv) ^ 1, the condition

<pv -> 0 does not hold, a contradiction to the assumption. Thus, con­

dition (C) is satisfied.

Let qx^{- 3} q2 and let A be a bounded set in Hej, i.e. \ Kp l Dp <р\\^ < ¹ for every 99e A. Hence g1( K p 1B p (p) < 1, and so Q2(kcp) < 1, where fc is the constant in (0) chosen with respect to the sequence of numbers Xp

= K ~ x. Thus, we obtain IMI(e2) < 1/ft. Repeating the argument in case of the functions Dpcp in an analogous manner, we get condition (A). Hence, we proved the equivalency of conditions (A), (B) and (C).

The fact that (C) implies (D) is obvious.

Now, let qX€ (a), q2€ (a) and D c: D . Let T be the operation of

-D- — Dq2 _

embedding of D ex into DB2, and let (p{ Л ⁹⁹, T{(p{) = ⁹^ -> ⁹⁹. By Lemma 2,

<р{- р -> 0 and cpi—cp -> 0, both almost uniformly. Hence T(<p) = ⁹⁹, and the closed graph theorem implies condition (B).

Theorem 8. Let us consider the following conditions: (a) qe (A2),

(b) i f <р{ 4- ⁰, then <р{ Л ⁰,

(c) Dq = {⁹⁹e $ : q(XDv<p) < ⁰⁰ for every p and every X > 0}, (d) the space D is dense in Dq.

Then conditions (a), (b) are equivalent, conditions (c), (d) are equivalent, and condition (a) implies condition (c). Moreover, if Qe(a), then all condi­

tions (a), (b), (c), (d) are mutually equivalent.

Proof. Let Q e ( A 2) and ⁹⁹* -4 0. By Lemma 4, £({Я^}, ⁹?*) 0 for a sequence of numbers {Xp}. To this sequence we choose a number ax from the condition {Л2). Then there exists an index i x for which q{{Xp], ⁹⁹^) < аг

Infinitely differentiable functions 167

and the condition (A2) gives £>(²^ ) < ¹. Now, we choose to the sequence a number a² from the condition (zl2). Then there exists an index i2 such that £(-{АЯю}, 2epu) < a2, whence ^(2-2ср{ ) < 1, etc. In this manner we may define a subsequence {^-J- for which g(2K~12<pi ) < ¹ ; hence lim Q(2<p{.) = 0. An analogous argumentation enables us to extract

ft—>00 k

a subsequence {^ } from the sequence {ç>t- } for which lim Q{_{"iCa К} 2B p^-epi )_Wn

6 S —> o o 6

= ⁰, where p x is the first of all multiindices p t Ф (⁰, ⁰, ⁰) put in a sequence p x, p 2, ... Repeating the same argumentation and applying the diagonal method, one may finally define a subsequence {щ} such that lim Q(2Dp<pi}) = 0 for each p. It is easily seen that also lim д{ХВрер{^

l — >00 l — M O

Dq

= 0 for each p and for every Я > 0. But this proves that щ -> ⁰, and it is easily concluded that <р{ -> 0. Since the converse implication is obvious, the equivalence of (a) and (b) is proved.

Now, let us suppose (c) to be satisfied, and let ep e B e. We take a se­

quence of numbers {Kp} such that \BP oof < K p for every j, where ooi

к

are defined as in Lemma 1. Then \BP[ ooj)\ < K p for every k. Hence

3 = 1

we have for an arbitrary Я > 0

к ^{к o o o o}

Q ">)] = coj) X œi\ ^ 6 (ÀKp(P £ 0ii)'

j= 1 j = l j = k j= k

But, by condition (c) and Lemma ¹, the expression on the right-hand side of the last inequality tends to zero as Jc -> oo. Hence

к

(8.1) lim I\epBp ( У oo\ II(e) = 0

k-—юо j ^

for an arbitrary <pe B e. Applying (8.1) and the Leibniz formula we obtain easily that

к

limlLzWç.JT'e»,) ~D»<p\L = ⁰

к—> o o j= l

k D

i.e. (pk = £ 4>°>з Pi and 4>k € D. Hence В is dense in B Q. Conversely,

3 = 1

let us suppose condition (d), and let <peBQ.

We take a fixed p, and we choose an arbitrary positive Я. Then

к oo

q(XBp <p) < q [2XBpep oojj -|- q |2XBpep JT¹ o)jj .

3=1 3= k+ 1

Since В is dense in B e, there exists a sequence {ç>J, <pt e В such that <p{ De (p,

i.e. lim g [2ADpç>—2№ р<р{] — 0. Let г0 and &0 be tw o indices chosen in such a manner that

g[2,№ p<p — 2 № p<piQ] < 1 and S<pi(j n S = 0 . l-h + l 4 Then we have

k0 o (U )p rp) < {, [2W p<p 2 « ,] +

i^{- 1}

00 *0

\2Ш»<р g m/ + 2XJ)I,((p->pili)^ < O i] < 00.

j=k0+l /=1

Hence condition (c) is satisfied.

Next, let us suppose ge (Az) and <p* De, i.e. g(XpDp(p) < ^{0 0} for a se-

OO ~

quenee of numbers {Xp}. B y Theorem 6, <p £ Д - 0. W e choose a number j=k

a from the condition (z l2), corresponding to the sequence {Xp}. Then

OO 00

<?(■&>}> (p 2 œi) < a fo r Tc > fc0, and (z l2) implies q (2<p £ °>з) < 1- Hence

j —k j=k

q (2(p) < 00. The proof fo r functions D pq> and constants A of the form A = 2k is perform ed analogously. Thus, we proved (c).

F in a lly , le t ge (a), and le t condition (c) be satisfied. Le t {99J be ^-con­

vergent to zero, i.e. Umg(ApDpç>i) = 0 fo r a sequence of numbers {Ap}.

г—xx) oo

Since lim g ( A D ^ £ a)j) = 0 fo r an a rb itra ry A > 0, we obtain

5—X» j= S

OO

lim I (р{ £ coj||e = 0. Hence there exists a sequence of indices { s j fo r 8—*30 j = S

which

OO

(8.2) lim \\<pt £ a J I = 0.

t —>00 » .

j—»i

B y Lemma 2, the sequence {<p{} is u n iform ly convergent on every compact ri set. L e t rx be an a rb itra ry positive integer. Then the sequence {9oi £ coj}

j=i

ri is convergent in the sense of convergency in D. B y Theorem 5, lim ||<^ £ ^jWq

i-^00 j^{= 1} ri

= 0; hence there exists an index i x such that rx < and £ соу||г < 1.

. j=¹

r 2

L e t r z be an a rb itra ry positive integer such that r2 > . Then lim £ щ\\9 i—и» ^{/ = 1}

'2

= 0. Hence there exists an index i 2 such th at r2 < siz and ||^2 £ w,||ff

j=i

Infinitely differentiable functions 16»

< £, etc. Continuing this procedure, we may define two sequences of rm

indices {im} and { r m } such that s^{V - 1} < r m < sim and ||<pim £ o>,||e < ¹/m.

j=i Then

(8.3) lim

m—юо = ⁰.

Since lim e (Я0^г- ) = ⁰, there exists a subsequence {imQ fc} of the sequence of indices {гт } such that

Sj

l™ok

r~imQ,k

Let p19 p 2, ... be the sequence of all multiindices p Ф (⁰, 0 , . . . , 0). Then,, in particular, lim^(A¹DPl^ ) = 0. Hence there is a subsequence {»«,,*} of the sequence m0,kfor which

4mi,k

гИ(%, ^ el

’

etc. Continuing this procedure and applying the diagonal method we obtain a sequence {imkJ} such that

mk,k

« М ’ К м . И “ »)]

3~ Tmk,k

1 2*

for Jc^ кр ; it follows from the construction that the supports of functions on the left-hand side of this inequality are disjoint for different Jc. Let us define

OO

f = _fc^I₌₀ к * » ^У

3~ Гтк,к

Then

oo *m k,k

* = 0 ’ j=r™k,k

i.e. <pe D q. According to the assumption we obtain д ( № ру) < oo for each P and for an arbitrary A > 0. Hence we get

Sfm k,k

lim e [AZ>P (<Pi

k-xx> L v ^mk,k_3=r,I

m k,k

= 0

for an arbitrary Я > 0, i.e.

(8.4)

From (8.3), (8.4) and (^ ⁸.²) we conclude lim jj<pi _ k->oo _{_} Tfli» I» *K>K|| = ⁰, that is

■^n-> ⁰. This implies easily <p{ -> ⁰, and the proof is finished.

6. In the following, we shall construct some examples of spaces De. First, we give the definition of an M -function, and we prove a lemma useful in our further considerations. This lemma generalizes a lemma given in [¹].

A function M( t , u) = M( t x, t2, ..., tn, u) defined on Pn+1 will be called an M-function in the variable u, if

(B .l) for every u, the function M(t, u) is a continuous function of the variable t;

(B.2) M(t, u) > 0; if и = 0 , then M (t , u) = 0;

(B.3) M( t , u) = M(t, —u);

(B.4) M(t, аих-\-^и2) < aM(t, ux) (t, u2) for a, /? > 0, = 1;

(B.5) for every t, there holds

Let P. denote the set of all multiindices p = (px, ..., p n) such that p t = ⁰ for i = ik, where {i*.} are indices extracted from ¹ , ² , ..., n, p { ^{= 1} for the remaining i , and let \p\ = P i + p² + •••+!>«• Then we denote by dpM(t, и) the derivative dmM{t x, ..., tn, uj/dtf1 ... dtPn; dpM(t, u(t)) will stand for the derivative d]p]M( t x, . . . , t n, u)ldtfl ... dtPn calculated at the point и = u(tx, ..., tn).

Lemma 9. Let dpM ( t , и) be M-functions in the variable и and let у possess continuous derivatives Dpy in the set {t : \t\ > \y\) for all p e P . Then for every x such that \x\ ^ \y\ there holds the inequality

Proof. Let C{x, r) be one of the 2n n-dimensional closed cubes with a vertex at x and with sides of length r parallel to the coordinate axes, and let \x\ > \y\. Moreover, let J k be the projection of the cube C(x, r) on the axis xk\ we fix the cube C(x, r) in such a manner that

oo.

M ( x , y { x ) ) ^ J ? J dpM [ t ,2n~lP+QlI)Qy(t)]dt.

p + q*P \t\>\x\

Jk= xk < xk + r} îot xk > ⁰, J k = {tk: œk- r ^ t k ^ xk} for xk < ⁰,

Infinitely differentiable functions 171

where x = {хг, . .. , xn). Thus, C(x, r) = J x x J² X ... X J n. Let P k = { p e P : p = (рг, • • • j Pk—\ ? 0, 0)},P'k = { p'eP: p' = (0, .. ., 0 , p k, . . . , p n)}. Let us denote for every ¹ < к < n and every p

• • • ? ^я) • 5 J • • • ? ^n)]^*

' «^fc J»

Then the following inequality holds for every p'e P'k:

<9.1) £ £ ' [ > ] < У ( l + — ^ f +p'[ 2 * _ 1 _ lp + 2 li > V ] .

The proof of formula (9.1) will he performed by induction.

For к = 1 we have

= / др'Ж[£, ç?(<)]d<,

J x J n C ( x , r )

and formula (9.1) is true. Let us suppose (9.1) to be true for 1 < к < n.

The mean-value theorem gives

~ J дР M iXl ? • • • > xk~U • • ч tm ф{х1 > • • • > xk-1J be? • • • 1 tn)hdtk

r 4

= др Ш\ссг, . . ., xk_l76kJ tk+¹, .. . , tn, д)(хг, ..., хк_ Х1 0k, tkJrl1 ... ? ^n,) where 0* = 0*(®i, . . . ,xk_x, <*+¹, . . . , tn) e J* • Hence, we have for every p' еРк+1:

^fc+iL^] = J' ••• Ж [£tq,..., xk,tk+1, . . , , tn,<p(a?i,. . . , xkJ tk+1, . . . ,tn)]dt

J k+1 J n

x k

— J ... Ж [ж 1? .. . , be? • • • ? <р{х11 • • • JA;+1 Jn ek

..., , tk, ..., iw)]d^j-d£ + J' ... J' др Ж [#q, ..., x)k_l1 вк, ^ +1 ? • « •

•Ьс+l *bi

• • •, ^(^l? • • • ? xk—l ? • • • >

Denoting by p' multiindex, obtained from the multiindex p'e Pk+1 by putting ¹ in place of ⁰ at the fc-th place, we obtain

x k

^ f + l L ? 9] = J ' ” ’ J * • * *» x k - l ) ^kl * • •) • • •? x k - l ^ ^ k l • • “b

*^fc+l xn 6к

«= ?>(*!, X

We apply the following inequality (see [1])

d M[ x , f { t ) ] / d t ^ M[ æ, 2f{t)] + M\æ, df(t)ldt]

to the integrand in the second term on the right-hand side of the above inequality. We obtain

$ k + i t ? 5! ^

J

Jk Jn

+ J' • • • J* др Ж , . . . , , tk) ... , tnJ 2(p (#q , . . . , j , tky ..., ] dt -f-

+ J' • ' ‘ J * -З^Гя'! > • • • j i ? • • • j J • • • ? J'k—l > • • • > ^ra)l ^ ~h

4 ---J " ••• J * •••> ^k—11 •••> tnj %<P{X1> • •• ? ^ * - 1 ? • ••? ^ » ) ] ^

Г Jk Jn

= « r [ y + f1 + 7 ) г з д + ^ м • By (9.1), we get

+ / l + i \ У l1Jr l ] ^ - \ P + ^ S f +1>,[2k-1- lP+QlDQ(2(p)] +

\ r l p ^ p k\ r J ЛГ1 j -t \ f c - i - U H - a l

-f \ ( l + —I S i+P'[²fc~1“,2,+alDqcp'].

Pf ^ p k\ r f

Let p and q be arbitrary multiindices belonging to P*+1, and let p and q be obtained from p and q, respectively, by replacing ⁰ by ¹ at the fc-th place. Then

2 \ ( f c + i ) - i - i J H - a i

SS+i; , м < _ ^ f1 + - ) « f +3>'[2 ('t+i,- i - №+ii|j ) 39>]+

P+Q 'Pjc+l \ Г I

V~7 / 2\ ( ft+ 1) - l - l - P + 5 l

2 i + T ^$р+р' j-2(fc+x)-i-№ +ai j) « ç ,] 2^-

P+QtP/c '

1 \(*+l)-l-|2>+ff| - , „ „

Sp+p^ [2(&+1)_1_!2,+5,l Daç>]

Р + QtPk+J \ r I

+ ^ (1+7j

\Л l 1 \ ( * + l ) - I - № + e l

Z 1 + -

P+Q*Pk + 1 \ r

Thus, (9.1) is proved for ¹ < 7c < n

$ p + p ' |‘2(*+ i)-1-li,+el D 3Ç)].

Infinitely differentiable functions 173

Applying (9.1), we prove now the following inequality

(9.2) Ж(а?,⁹>(ж))<

У

Г

p+q7p \ r f c (x>r)

We have, by the mean-value theorem,

f • • • ? ®n- 1? bi? •: • ) ®п—1> %) №

Ж \хх, . .. , a?n_ i, 6n, cp(a?!, .. ., xn_1, ⁰n)], where вп = dn(xlf . .. , xn_j)e J n. Thus,

x n

M(x,<p{x)) = j у ^ М [ х г, ...,a?n_j, *n, ... , ®n. i , * » ) R +

en W

+ Ж [a?i, . . . , a?n_j, ⁶n, <p(xlf . .. , a?n_!, ⁰n)].

Denoting by p n a multiindex belonging to P which has 1 only at the w-th place, we get

Ж

74

(®,Ç>(®))< J арПЖ [Ж !, ...,Я ? П_ 1,#„,9?(®1, +

в п

xn

"b J " [*^i j • • • > i > > 2^9(а?!, . . . , #n—i , ^n) ] di n ~b

% xn

/ 0

Ж [а?!, . . . , x n_ i, i n, ^ (p (a?j ? • • • > i > ^n)J ~b en

H ~ Ж [a?i, . . . > a?n_ i , ifn, ç?(ajj, . . . , Jn

< S i d + ( i + y ) Si [²^ ] + s f M . By (9.1), we obtain the inequality

|¹ + -

p + q e P n ' Г

+ U +

£C»?(«))< ^{( 1} + 2 » - i - l P + f f l jr)<zj _ _ ç , j j _j_

7

)^

sp

]_\ »-l-№+el

+

/ p + q e P ,

q \n-l-|p+g|

2 ( i + -

âri>A r

p+q*p<

7

)]

Sp [2n- 1- ]p+Q]D q(2<p)] +

Sf+Pn [2n' 1~lP+q]