Isometric multiplication of Hardy-Orlicz spaces

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HARDY-ORLICZ SPACES. W. DEEB, R. KHALIL AND M, MARZUQ. For a modulus function , we define the Hardy-Orlicz space. H() . Two main questions ...
BULL. AUSTRAL. MATH. SOC.

46E30, 46E20

VOL. 34 (1986) 177-189

ISOMETRIC MULTIPLICATION OF HARDY-ORLICZ SPACES W. DEEB, R. KHALIL AND M, MARZUQ

For a modulus function

, we define the Hardy-Orlicz space

H() . Two main questions are discussed in this paper. when is a linear map m

: H($) -*• H($) , m (f) = g.f 3

y

First,

an isometry?

Second, when is

0. Let

A

and H(h)

Introduction.

be the open unit disc in

0 , the set of complex numbers,

be the space of analytic functions in

circle, then

A . If

T

is the unit

L (T) denotes the space of p-Lebesgue integrable functions,

0 < p S °° . The classical Hardy spaces will be denoted by

flP = ( / e H(A) : sup

[

\f(reLQ)

P

de < •» \,0 < p < » ,

0 CO

and

H

is the space of bounded analytic functions in

It is very well-known that every also denoted by

/ , in

closed subspace of

V(T)

f e IF

A .

has a radial limit function,

. Further, W

can be considered as a

L (T) , when equipped with the metric:

Received 31 October 1986.

Copyright Clearance Centre, Inc. Serial-fee code: ?A2.00 + 0.00. 177

0004-9727/86

178

W . D e e b , R. K h a l i l a n d M . M a r z u q

(j \f(x) - g(x)\P dx )P

p il

T T

d(f,g) = . j \f(x) - g(x)\P

dx

0

) is the Nevalinna class: r

r2v

*•

By N

n1°ij)0

.—j- is a

§.3§- , the function

is a modulus function.

Throughout the paper, we will assume that our modulus function satisfies the additional conditions that f|w|/l function whenever Let

u e H(A)

is a subharmonic

, and < j > is strictly increasing.

H (h) = {f e H(L)

: lim five)

exists a.e.}

. Thus H(L)

can be viewed as a space of functions on T . Now,

we define the Hardy-Orlicz space: 9

O

6

sup [ $\f(re* )\de = [ "$\f(e%%) \dQ >0 where

i j > is a modulus function.

>0 On H(i\>) we define a metric

r2-n

d(f,g)

9

With this metric,

fe J

= \\f - g\\,= '0

H($) is a topological vector space.

we are assuming that

|

U

4 C| I''

is an F-space, [ 2 ] .

i s

, noting that

Further, since

subharmonic for u e H(L)

If < f > is bounded, then

= ln(l + x?h 0 < p < 1 , then we write If2 = N+ c_N

g(ev*)\ds

N

, the space

H($) = H(b). If

for H(Q) . Clearly

ln(l + 3p) = i>1 ° ^Jx) , where

180

W. D e e b ,

t/Ax)

= ln(l

R. K h a l i l

and M. Marzuq

+ x),

In [ 2 ] , i t was shown t h a t

R c_ R($)

for a l l modulus functions

f2iT Throughout the paper, we write

| |/| | *

for

.

ie

1 $\f(e ) \dB >0

,

f e 2.

Multiplication on

Throughout this section, we will view defined on T . A function

g

R($) as a space of functions

defined on T

is called a (Schur)

multiplier of H($) if g.fe. H(§) for all f e R($) , where

(g.f)(x) = g(x).f(x). The set of all multipliers of H($) will be denoted by M(H(§)) . It is well known (and easy to prove) that we characterize

M(R(§))

M(If) = H

In this section

for a large class of modulus functions. First

we need the following

LEMMA 2.1. Let $ be a modulus function which satisfies: (i) for any f e H

there exists g e H($) such that

*(\g\> = I/I ; (ii) §(x).$(y) £ $(x.y) for all x £ 1 and y s 0 . If g e M(H($)), and f c H1 , then [ E = {9 Proof. ; \g(e7'eSince )\ > 1}1 e . H($) Then , it 2 iQ \2 \\g(e )

>E

By the f i r s t assumption on \f\

follows that g e R(§) .

| \f(eU) \de = \ *\g(eid) \ \f(eU) |de+f j\g(eiQ)

>0

=

f \g(e%Q) \) .f(e'l6)d6 < ».

$\h\



f \\g(ei6) >0

| \f(eiQ) \de .

>E^

E

,h(ei%)

such that

f , we have

\dS + $(1). \ \ h \ \ *

Hardy-Orlicz

since

Spaces

g e M(R($)).

D

THEOREM 2 . 2 .

Let

§

be a modulus function

satisfying

the conoo

ditions

in lemma Z.I. Proof.

-•tf

If

lim (x) = °° , then

Let g e M(R($)).

* I ^ C e ^ l + 3 . |/re7'e;ide 1 on E , by F a t o u ' s lemma we get -=f

f

lim ) = H1 and 1 1 / 1 1 ^ < A 1 1 / 1 1 ^ s j \ \ \ f \ \ 1 . f o r a l l

f e R(i>) , and some constants Proof. and Let

A, n e R .

(i) -y (ii) . Choose > s on

0 b} E(a,b) = {6 : a < \f\ < b] .

z±—L > r

on

184

W. D e e b , R. K h a l i l and M. Marzuq

Then

I If I 12 = \ \f(eiQ) \de + I \f(eiQ) \de + I \f(eiQ) |d9 E(a)

E(a,bl

E(b)

|

E(a,b) The function -"•—— -"•—— is continuous on [a,bl , consequently there exists

a e (a3b)

such t h a t

if(x) > t.a; on

'

> ™—— for a l l

Ca,i>] .

x e La^bl .

Hence

I t follows

I ll/ll,

where

X = max (—3 -r , — ) •

Similarly one can show that

| |/| |

^ X| |/| L . Thus (ii) is proved.

Consider the map f(eV ) = x e

(ii) -»• (i) .

From

1 1 / 1 1 - 2 = * * I I/I 1^ = *(x> •

i:L we

( )

g

, x > 0 . Then

et

x £ \§(x) < nx .

Hence

1 ^ A>(x) _ n

— S -"A

0 , then

f =

Z

d(afn,0)

for all

•* 0

/ e

for all

u. 0 V. e H($)

.

a e R .

Then

%1 0 < dia f,0) < E

u.\

IV

"V—~L

and

lim d(anf,0)

< X^ lim | \^

OO

Now, f o r

/ =

I

•1=1 sequence of functions

^

00

u. 8 V. , %

%

I

i=l

IIM.|| t

.

| | u . | | < » , %

*

define the



The Lebesgue dominated convergence theorem on the set on natural numbers with the counting measure implies: 00

lim dia. f , 0 ) n -Voo

S lim £ nH»

\\a u.\\,

.

| |u. | | .

Y

1=1

Y

lim I la M . I L • I \v • I M l * *

To prove (ii): let positive integer such that

fn e Ri$) 8 Ri$) , difn,O)+O . k > a . Then

.fn,O) < d(Kfn,O)

< k.difn,O) •* 0

Let

k

be a

Hardy-Orlicz

Spaces

189

This completes the proof of the theorem. Characterization of the (Schur) multipliers of

Q K(§1 8 K(§) would

be an interesting problem.

References [7]

J. Diestel and J. R. Uhl, Vector measures. (Math. Surveys, 15, 1977),

[2]

W. Deeb and M. Marzuq, "ff(J

spaces," Bull. Caned. Math. Soc.

(To appear). [3]

P. L. Duren, Theory of HP'spaces, (Pure and Applied Mathematics

[4]

M. Hasumi and L.A. Rubel, "Multiplication isometries of Hardy and

No. 38, Academic Press, New York and London 1970).

double Hardy spaces," Hokkaido Math. J. 10 (1981), 221-241. [5]

K. Hoffman, Banaah spaces of analytic functions, (Prentice Hall,

[6]

G. Kothe, Topological vector spaces, (Springer-Verlag, New York,

Inc. N.J. 1962).

1969). [7]

C. Romulus, Topological vector spaces, (Noord. Inter. Pub. Com.

[S]

G. Leibowits, Lectures on complex function algebras, (Scott.

Netherlands, 1977).

Forseman/Company, 1970). [9]

R. Lesniewicz, "On Hardy-Orlicz spaces. I," Corrmt. Math. Prace Mat, 15 (1971) 3-56.

C'0]

R. Lesniewicz, "On linear functionals in Hardy-Orlicz spaces. I," Studia Math. 46 (1973) 53-77.

Department of Mathematics Kuwait University KUWAIT.