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On Hardy-Orlicz spaces, I

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ROCZNIK1 POLSKIEGO TOW ARZYSTW A M ATEMAT YCZNE GO Séria I: PRACE MATEMATYCZNE X V (1971)

A N N A L E S SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENT ATIO N ES MATHEMATICAE X V (1971)

esn iew ic z

ф

H p ( p

H p

p

(2)

and щ > 0 the following inequality holds :

(3)

O n H a r d y - O r l i c z s p a c e s , I 5

d

щ

9

e m m a

e m m a

<px

e m m a

e m m a

(4)

U ~ > C O

X

X

(**) Ф(х) — J p x(x)dx ( — o o < x < o o ) .

(5)

O n H a r d y - O r l i c z s p a c e s , I

X

0

и и и

— oo 0 0

{x+ y)l 2

— OO

Ж ( # + 3 / ) / 2 2/

— OO Ж (x+ y)/2

X у

£C/2

e m m a

и

(6)

2tc

tc

that M 9 is the greatest linear subset of the space L*9, which is contained

(7)

O n H a r d y - O r l i c z s p a c e s , 1

9

00 л oo v

m = 1 m = 1

h e o r e m

71=1

h e o r e m

n

71=1

h e o r e m

h e o r e m

(8)

3° if 1/iWI < \f2{t)\ for almost all te < 0 , 2 тг), then ll/J * ^ ||/2||*,4° Л (/Х 11/1С 4 II/IC<1; > » ( /) < 4 implies \\f\\l < 1 ,

3 ° i f W f n t 0 » M e n f n ^ 0 ,

y(u) ~ yi(us), where yj is a convex cp-function ([ 6 ] and [9]).

(9)

O n H a r d y - O r l i c z s p a c e s , I 11

h e o r e m

( o o x) ,

2tc

h e o r e m

h e o r e m

<

<

j t j ) < OO

h e o r e m

(10)

X

2rc 2n

0

we have

(11)

O n H a r d y - O r l i c z s p a c e s , I

E E

E

E

E

E

E

E

1 . 1 . 2 .

h e o r e m

(12)

2т:

2tu 2

2tz

4Ti

n { l —e)for 0 < о < 1,

(13)

O n H a r d y - O r l i c z s p a c e s , I 15

h e o r e m

E

E

X

(14)

h e o r e m

2rc 2tt

0

2тс 2tt

0

2k 2

0

7*—>1 —

271 27T

0 0

271 2 n

0

1 — z(l«l < !)•

(15)

O n H a r d y —O r l i c s s p a c e s , I 17

}

ф

1

r-> X— \ 1 V

— OO,

t

щ

J ^ F ^ é ) )

= J

< p ( l ) d t = 2тир( 1) .

= é = 1

R o czn ik i PTM — P r a c e M a tem a ty czn e XV 2

(16)

h e o r e m

e m m a

e m m a

0 < r < 1, are 27t-periodic.

(17)

O n H a r d y - O r l i c z s p a c e s , I 19

/av( F ) < o o .

O O OO

К с

с К 9 с

c

<=

c

h e o r e m

r - > l —

the interval <0, 2n). Thus, if we neglect the difference between isomorphic spaces, we may write Theorem 2 . 1.2 in the form

H 9 = N ' n L 9, H * 9 = N ' n L * 9, K 9 — N ' r\ M 9.

(18)

h e o r e m

h e o r e m

OO 2т: 27T

n = 2

n n

3

(19)

O n H a r d y - O r l i c z s p a c e s , 1 21

0 n — 3

e m m a

0

2

h e o r e m

OO

(20)

4

h e o r e m

OO

• v—X

OO

v— 1

h e o r e m

<Pz{u) < dq?1(u) for u ^ u0.

(21)

O n H a r d y - O r l i c s s p a c e s , 1

OO

v—l

h e o r e m

OO

h e o r e m

OO

v = l

OO

v = l

(22)

СО

Ш

со оо m

СО

СО

oo ^

positive integer m and for some constants d > 0 and > 0 there holds

(23)

O n H a r d y - O r l i c z s p a c e s , 1 2 5

h e o r e m

Proof. If cp2 -3 <Pi, then we deduce from Theorem 3.2.1 at once that

I F i c H * * 1 c= H * * 2 .

m —1

tc

i

w

t

tc

h e o r e m

JSf'

/ Р

then also IPi c H**2. By Theorem 3.2.2, we conclude cp2 -3 y x.

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