) 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.