On Orlicz spaces with mixed norms (I)Inequalities for trigonometric polynomials *

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Seria I: PRACE MATEMATYCZNE X I (1968) ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)

Z. St y p i ń s k i (Poznań)

On Orlicz spaces with mixed norms (I)

Inequalities for trigonometric polynomials *

1. In the paper [4] there are given inequalities for the norm of an integral function and a trigonometric polynomial in the space IP. The purpose of the present paper is to prove analogus inequalities for trig

onometric polynomials of n variables in OrUcz spaces.

Let us denote by у (u) a convex, non-decreasing function defined and continuous for и > 0, <p(0) = 0, satisfying the following conditions:

(1) lim

«—>•0 (p(u)

и ⁰, lim ^<p{u) ^{= o o ,} (p{u) > 0 for и > 0 (2) cp(uv) < c<p{u) (p (-»), e > 1, 0, v ^ 0 (condition A').

Let f ( x x, . . . , xn) be a measurable function with period 2tu in each variable. Applying the notation of [1] we write

27T

(a) ll/f e , •• • j i ? 1 x k+1? • ■• • ) ^ra)||ę> = / l/(®i, •••, xn)\dxk 0

if (p{u) = и

(b) • ? ffijc—1 J * )i ? • •• ) **'w)||ę> := sup ess ¹ f { x 1, ..., xn)\

0<a5fc<27i:

if p('a) — °°

(e) № u •• •, ^{%k— 1 1} 1 ffik+ij •• * ?

C / l/Caq, ...,a?n)| \ d ,, s > 0: rp I ---Мхи < 1> otherwise.

o ' £

A function f ( x x, ..., xn) is said to belong to the space L*QM){n) if the following proceeding gives a finite value in each step. In the first step we calculate

* The results were presented April 14th 1966 in a Seminar of Dr. J. Musielak, Department of Mathematics, A. Mickiewicz University in Poznań.

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2 4 8 Z. S t y p i ń s k i

||/(*, x 2, .. . , %n)\\<p by means of the formulae (a), (b) or (c) depending on the definition of cp. Here, <0, 2ти> (j ~ 2 , 3 , n) are parameters.

In the second step we calculate j|/(*, •, x s, ..., xJIJ^ = ||||/(-, x 2, . . . , x n) |]„||^

again by means of one of the formulae (a), (b), (c). After n — 1 steps we obtain ||/(«, and in the nt~h step we calculate ||/(*, ..., •)!!,,

= ||ll/(*,---, • applying (a), (b) or (c). The number ||/( • , ..., -)IU obtained in this manner is denoted by Ц/Ц,, and is called a mixed norm:

(3) ll/ll, = l l / ( - ) l l , .

Thus, L*ot2n>(n) is the Orlicz space of functions / such that ||/||ф < o©.

Moreover, we denote by T the space of finite sequences, i.e. sequences containing a finite number of elements different from zero only. The norm of the element a = {ak} e l is defined as

(4) \\a\\Tip = in f j £ > 0: J ? <p < l j .

Similarly as in case of a function one may define a mixed norm of a sequence.

The following Jensen’s inequality will be of importance in our con

siderations :

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f f ( x ) d x \ j(p{f{x))dx --- 1 < ^{- A}---

mesA / mesA

We shall denote by T Vl...v (xlt ..., xn) a trigonometric polynomial of degrees vx, ..., vn with respect to variables x x, ..., xn, respectively.

We have

„ »■ ' n

1 П I 2 k SXS

(®) ^-Vx,...,Vn {XX ! •••! ®») = ^ --- Gkl---kn (i kl = ~ vl kn = ~ vn

Theorem 1.1 (a generalization of Bernstein’s inequality). Let Tn(x) be a trigonometric polynomial of degree n of one variable. There holds the following inequality:

(7) W T X ^ n W T X -

Proof. For the sake of completeness we give a simple proof of this inequality which is known in an even more general case (see [2]). By [6], Theorem 3.16, we have

2tt , 2tc

f V ^ ^ f <p(\Tn{x)\)dx.

o ' 0

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Let us choose a number e > 0 such that

Then

Hence

\Tn(x)\

dx < 1.

dx.

Corollary. Let T v (хг, . . . , xn) be a trigonometric polynomial of degrees v1} ..., vn with respect to variables x x, .. . , xn. Then —dTv is

dxk a trigonometric polynomial of degree vk with respect to the variable xk and

(8) dT*h <vn

dxk < vk IItVi

<p ^•vn^19»*

2. Theorem 2.1. Let T v{x) be a trigonometric polynomial of degree v1 and let us write h — 2n/v, xk = Ш (h = 0,1, ...), xk < xk < xk-\-rh, where r is a non-negative integer. Then

о) <р-лтт£т\ = ’Р-лттл^'Лт, « с а + г ^ м ,,

where <p_x(h) is the inverse of the function cp{h).

Proof. Let us write

In

Q{Ty) =

J

<p(\Tv(x)\)dx.

Then

xk+1

e{T.) = £ j <p{\T.m)dx = f?< p к як

\<P-i(h)Tv(h)\

Hence

Thus

for

\<P-iWTv(h)\

Cs

< р ~ л т щ \! * < с \\т х

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260 Z. S t y p i ń s k i

Next, applying triangle inequality for norms, substituting t — ruĄ-xk — rxk

— v and applying (2) and (5) we observe that

x k + r h

0 0 \\ J

11 x k

dt

xk+1

/

[Tvm<p-x(h)r\

du xk

inf ^{£ >}0

xk+1

/ (1Д0 / \iTv{v)]'<p_l {h)r\du 2 ' — ---;---

2_к

J — ;—

_{X k} ^{r ^ J}

q— e—

^{Г -}

But

J" ^<pj (\[Tv{v)]'\rh ^■IL ^'— ^I^{d u <}

\ s

2n

/ * (

\T'v{t)rh\

dt.

Hence

<P

^_1

W I

m \ł„ - \\T,\\f*^I

<

inf

je > 0: j

<p

j1--**— )<H <

l |

<

Ч.Г.Г

||T„||,.

Finally,

V - l W и г р \\x k ^ У - Л ^ 1 ) I n ^ 7 иX k u r n n l f e l , | | | fc

i » ; <

< (1 +2iTir)\\Tv\\ę .

Corollary. Theorem 2.1 remaim true if we replace the trigonometric polynomial T v by an arbitrary function f continuous in <0, 2tc).

Indeed, trigonometric polynomials are dense in the space L*$My.

Hence there exists a sequence of trigonometric polynomials {Tn(x)} such that \\Тп\\,р -> ||/||v. Moreover, we may suppose that \\Tn{xk)\\T<p -> H /M Iliy Consequently,

— — 1|/<®*)||г, = lim * = M - Ш х к)\\г < lim(l +2*r)||TJ|,.

C ■ę rtr^oo c * n-+oo

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Th e o r e m 2 . 2 . Let Tv%....Vn(xx, . . . y xn) . be a trigonometric polynomial of degrees rx, . . . , vn with respect to variables aq, ..., xn. Let us write

2n

hk = — (7c = О, 1, ..., n), %ik )= lh k (I = 0 , 1 , . . . ) , Vk

x {k) < x{k) < x\k)+rhk, where r is an integer. The following inequality holds:

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П

max

П

<P~i(hk)\\... \\TVl

k—1

,-(n)

<P

< [c{l+2nr)]n\\TVl....,J , . Proof. The variables и being fixed, the trigonometric polynomial Tv ....(aq — ux, .. . , xn — is a polynomial of degrees vx, ..., vn satisfying the equality

\\TVl,...,Vn{Xx Ux, . . . , Xn Un)\\Ty \\TVl....vn (®n • • • j МпШТу'

Hence it is sufficient to prove the inequality

■ n . - (1) ~ ( n )

П г - Л Щ - ц т . ,.... .„Igi . . . ] | > < [ с ( 1 + 2 т г г )Г ц г ,1.... „j i„.

fc=i

The proof will be performed by induction with respect to the number of variables. If n = 1, the inequality is already proved. Let us fix the variables xk+2, ..., xn and let us suppose the inequality is true for n = 1c.

Applying the previous Theorem and the Corollary we obtain [G (l+ 2ur)f\\TVl_ Vn( - , . . . , - , x k+1,...)\\4>

^ 1 1 || • • • II ^ (* ? • • • > * ? ^k+1 j • • -AO ^~(k I >, Hence

[^ (1 A )] ' 11 11-^ vb ...,vn ( * 1 • • • ) * ? & k+ 1 J • • • llę>||ę>

....

Theorem 2.3. Let Tt>b >Vn(x1, . . . , x n) be a trigonometric polynomial.

Then the following inequality holds:

4 ) ... . j , « <-nn y ...

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252 Z. S f c y p iń s k i

P ro o f by induction. If n — 1 we have 2л

/ cp(\Tn {xx)\)dx1 <

o ^ i 2rbxf ( \ T .l ( ^ ~ u 1)\).

h=i Hence

Thus

j

у n T ri(xx)cp_x{lf2r}i1)\

o ' C ‘ *i« i

\\TV1\

^{< G} p y gd1)

Let us remark that the above inequality remains valid for every function continuous in <0,2лг>.

Now, let us suppose the theorem is true for n — 1c. If we fix the variables xk+2, .. . , xn we obtain

R em ark 1. If x[k) — xjk) (lk = 0 ,1 , ...), then the coefficient 2r at the right-hand side of inequality (11) may be replaced by 1. It is

M 2 rhi

sufficient to take in the proof integrals f in place of f .

о 0

Е е mar к 2. Writing

=<!)

...))* = сПП ™ H - ||r ”.... ' • • nr;

if x f ] = (l = 0 ,1 , ...), x\kk] = x\kk] (lk = 0 ,1 , ...), we have

n

112;,...., J , < «Г .,....< е2п(1 + 2к)п ... ..II,•

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Theorem 2.4. I f <p{u) < y>(u) for и < we have

( 1 2 ) 1 И 1 т „ < I N l i y

n

(13) \\T'n ....J i , < C2n( l +2ТТ)” J 7 [?>_! (-^ ) , . 1 ( W ] 1

Proof. Let a?„ = (0, ..., ...) < {tx, ..., ...) = x. Applying mo-, notony of the norm (4) we get

\%I = Ы т у У - Л ! ) < INIz^V-iO-)- Hence

y J J A W y J - A U

4

^" ^\M^tJ ~ 4 \MtJ ^{= 1,} ^i-ev ll^llrr^ < INIrv •

Taking into account this inequality and inequalities (10) and (11) we obtain

П

.^<с”/

⁷

МйГ

^max ^IT^-Ł^v ^vn"T.^Ilf' ^—uⁿ

< C” ( l +2ru)

fj

|.Л , ( 2 - j ? _ ,(* * )] ‘ ||T ^ .... , J r .

Taking in the above results <p(u) — uv we obtain the respective results of [4].

R eferences

[1] J. 4 lb r y c h t, Teoria przestrzeni IP z mieszaną normą dla skończenie ad- dytywnych funkcji zbiorów, IJAM, Prace Wydziału Mat-Fiz-Chem, Poznań 1963.

[2] — A generalization of a Zygmund-Bernstein Theorem, Ann. Polon. Math.

2 (1955), pp. 64-66.

[3] 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.

[4] С. М. Н и к о л ь с к и й , Неравенства для целых функций конечной сте

пени и их применение в теории дифференцируемых функций многих переменных, Труды Мат. ин-та АН 38(1951), рр. 244-278.

[5] A. Z y g m u n d , Trigonometric series, Cambridge 1959, vol. II.