Internat. J. Math. & Math. Sci
Vol. 9 No. 3 (1986) 429-434
429
INCLUSIONS OF HARDY ORLICZ SPACES
ROSHDI KHALIL
Department of Mathematics
The University of Michigan
Ann Arbor, Michigan 48109 U.S.A.
(Received September 9, 1985 and in revised form February 24, 1986)
ABSTRACT.
Let
be a continuous positive increasing function defined on [0,) such
y) _< (x) + (y) and
0. The Hardy-Orlicz space generated by
(0)
is denoted by H(). In this paper, we prove that for
if H()
# 4,
H() as
sets, then H()
H() as topological vector spaces. Some other results are given.
that
(x
KEY WORDS AND PHRASES.
Modulus function, Orlicz spaces.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
30G99.
INTRODUCTION.
Let
(x)
[0,=)
:[0, =)
(y)
and
on
T.
if
fiIf(t) Idm(t)
by
L().
4(0)
be a continuous increasing function such that
Let
0.
T
be the unit circle, and
A complex valued measurable function
<
.
The space of all
f
defined on
m
T
(x
y)
be the Lebesgue measure
is called -integrable
T will
-integrable functions on
be denoted
by Orlicz, [8]. Subsequent papers were
L(). Examples of these papers are Cater, [4],
This space was first introduced
written to study different aspects of
Gramsch, [5] and Pallashke [9].
In [6] and [7], Lesniewicz introduced the so called Hardy-Orlicz spaces H() for
a given such function 9. The space H() was defined to be the space of all functions
such that
f e L()
is the radial limit of some function
f
unit disc and belongs to the Nevalinna class
N.
g
analytic in the open
The relation between different
H()-
In this paper, we show that the
spaces was studied by Deeb, Khalil and Marzug [3].
inclusion map between two H()-spaces is always continuous.
given.
It should be remarked that in the work of Lesniewicz,
is assumed to be a Q-convex function.
authors,
2.
PRELIMINARIES AND NOTATIONS.
A function ’[0,=) --> [0, )
[6], [7] and many other
In this paper it is not assumed so.
is called a modulus function if
is continuous and increasing
(i)
(ii)
(x)
(iii)
(x +y)
The functions
functions.
Some other results are
(x)
if and only if
0
p,
Further, if
0
bl
0
(y).
(x)
x
x
and
p
and
b2
(x)
in(1
x)
are examples of modulus
are modulus functions, then
l
2
and
bl
’}2
430
R. KHALIL
are modulus functions.
Izl
{z"
T
Let
is denoted by
@-
#
< i}.
Izl
{z"
i}, A
H+(A)
Let
H(4).
H +(4)
consider
Further,
is a modulus function if
The space of analytic functions on &
{f e H(4)" li
rX+
as a space of functions on
is.
T.
I
f (re @
exists a.e.@}
We will
For a given modulus function
we
define"
{f
H()
H (4)"
elf(re i @) Id0
sup
0 <r <I
If(e i@ )Ido
}.
H(40
[0,), d(f,g)
,If(e ie) g(e i0) Id defines
H(O), under which H(b) becomes a topological vector space. If one
assumes that Olul is subharmonic for u e I-I(4), then tt.) turns out to be complete
[21. For f e H(,), we write
x
0
p
i,
T If(e )ld@" If ,(x)
then H()
H p and for (x)
N
In(l + x), H()
{f e N"
in(l
Ifl) <},
T
d" l-t(qb)
The function
a metric on
p,
llfll
N
where
3.
f
is the Nevalinna class.
I- II
IS CONTINUOUS.
H()
In [2], it was shown that H
H() for all modulus functions
The authors
H
[3] were not able to show that the inclusion map
is
continuous.
H()
In this section we prove that
H
H() is continuous. Some other related
in
questions are discussed.
THEOREM 2.1.
exists.
Let
,
and
be two modulus functions such that
Then"
H() --H() if X
H() if
H()
H()
H() if
(i)
(ii)
(iii)
lira
(
x-
Let
PROOF. (i)
0
and
0
is finite
0
.
be finite.
Then there exists
[a2,
[b2,)
(*)
(**).
a
l,
b I, a 2, b
2
[0,) such
that
(x) _< al(X
(x) _< b l(x)
Let
f
for
x e
E(a2)
If(t) >_a2}.
,]f(e io)
{t e T
*lf(e )ld0
(a2)
_<
Hence
x
Set
H().
Ilfl[
for
a
EC(a2)
If flI
f e H()
Then
(a 2) <-
and
-
H()
H(). Similarly we sho H()
H(). Consequently,
H()
H(). Case (xi) and (iii) are proved similarly and details are omitted. This
ends the proof.
THEOREM 2.2.
Let
continuous.
PROOF.
(a)
b
Let
lira
X-Oo
O.
Then the inclusion map
From the proof of Theorem 2.1, there exists
for all f
H().
]lfl]
fn
(x)
0
in
H().
and consequently bounded in
Thus the sequence
H().
(fn)
a,b
0
I" H()
such that
tt()
is
IIfll
is bounded in the metric of
If possible let there exist a subsequence (f
n
H(,)
k
INCLUSIONS OF HARDY ORLICZ SPACES
II fnk I
such that
0.
a >
I]
Since
fnk I1
converges pointwise to the zero function.
f
that
(a)
n
With no loss of generality, we can assume
(x)
k
b-Ixl
+
has a subsequence which
Another application of the proof of Theorem 2.1, yields
a.e.
0
(fnk)
0
431
for all
x e [0,).
Ifnk(t)
,(a)
The sequence of functions
Hence
(t)
b
gnk
,(a)
f
gnk (t)dt
T
/
b
01fnk
converges a.e. to
,(a)
and
Consequently, by the generalized Lebesgue convergence theorem, [10], we have
lin 41fnk(t; Idt
Thus, the point
This is a contradiction.
(11
bounded sequence
fnll,)"
I: H(O)
Hence
zero.
lira
,,-
e(O,o),
lie
the sequence
n
This ends the proof.
H()
then
H()
converges to
as topological vector
X-o
spaces.
By Theorem 2.1, H(O)
H() as sets.
This ends the proof.
PROOF.
H()
is continuous.
0.
is the only limit point of the
0
w
Consequently, [11]
H()
If
COROLLARY 2.3.
limnk 41 fnk (t) Idt
Theorem 2.2 implies that I:
H()
is an isomorphism.
H() is called metrically bounded if
0. Clearly every metrically bounded map is contanuous.
The converse need not be true. However, for the inclusion map, we have the following:
A" H()
A linear map
for all
THEOREM 2.4.
f e H
for all
.
Since
But
I elI..
H
1,
a
n
Then there exists
X
such that
0
If film
If
f
get
be any modulus function.
I,
!1 fll,
have
Let
X
It is know, [2] (and easy to show) that H
H() for all modulus funcand
f e H
m, then using the argument in Theorem 2.1, we
PROOF.
tions
and some
f e H()
i
> 1.
Then there exists
there exists a natural nber
f[l_ llfll +
0
n
such that
K I fll
and
such that
a
I af[I1..
n+ 1-- a --n
fo an>, odu function
1.
Hence
It
follows that:
"f",
n+
X-
and consequently
II,,f,,,
xn+
Ilfll
This ends the proof.
TItEOREM 2.5.
Let
"
be a given modulus function such that
and topological bounded sets coincide in
for some
xllfll
X
0.
H(,), then
H
Ilfll xllfll
H().
for al
If metric
f e It(,),
R. KHALIL
432
1
H is an isomorphism of topological
Applying Corollary 2.3, I" H()
f
f
be not true on the unlt sphere of
vector spaces If possible let
__<
C H(),
such that
there exists
Then, for each n
H()
n
PROOF.
111 III
llf
fn
II
f
--= gn
Conslder the sequence
gn
By the assumption on bounded sets of H()
we
f
ll_-lll_>
llgn IIi
0 in H(). But
have, [12],
the continuity of the identity map I:
H()
H
for all n
This contradicts
Hence there exists
0
such that
< t
f
for all
Let
!1
11
seen that
K
is continuous.
Hence, from equation
II
f
Consequently,
{*)
]af
Il
f
+
2t
!
we can find
(*)
[0 )
Hence there exists
f
]]af
[0 )
K"
Consider the map
<
f E H()
Thus for every
Iaf
4.
f
f C H(),
K(a)
II
f
f C H(),
It can be easily
f
a
1]
]]tf
K(t)
a >
such that
such that
we get"
]I ! 2ak lf [
This end the proof
FURTHER RESULTS
The concept of metrically bounded linear operator was introduced in Section 3.
A linear map
I C (0,)
H()
H()
A:
such that
is called metrically bounded if there exists
[1_ III
]IAf
f
In general a continuous linear map need not
In this section we prove a result which is a generalizatlon
be metrically bounded
of Theorem 3.1 in [3].
THEOREM 4.1
be any two modules functlons.
and
Let
Then the following
are equivalent
(i)
(ii)
lit (x)
lit (x)
k+o (x)
x- (x)
PROOF.
in (0,)
H(), and the identity map
H()
(i)
is metrically bounded.
From the assumption in (i)
(ii)
(X)
(x]
Let
C (0,)
one can choose
such that
(x]
for some
c,
for some
r,s
f
(0,oo)
H()
>-
r
on
[0,a]
S
on
(b,)
Theorem 3.2 implies that
Consider the following sets:
H()
H(q)
a
and
b
INCLUSIONS OF HARDY ORLICZ SPACES
If(eil <a}
If(ei%l > b)
E(a) ={t-
0 <
E(b) ={t"
{t"
E(a,b)
433
fi
a_<
_< b}
Then-
E(a,b)
<
on
X
such that
f
]$
[$
!
]f(eit) Idt
7I f
+
(x)
(x) _< X(x)
If[ _< 8 ]f []$
Thus,
E(a,b)
[a,b], the continuity of (x)
the closed interval
> 0
,
f
--I]
r
E(b)
implies the existence of
Hence
(1 1)
m,,3
where
f g H($)
for all
I1
In a similar way one c show that
is metrically
Hence the identity map
H()
.bounded.
Conversely, (ii)
Asse
(i)
en
metrically bounded.
l] f
Hence
for
z
e
it
x g (0,)
proof.
(x)
---< (x)
ACKNOWLEDGEMENT.
< a
H()
and
]if ]]
f g H()
d
in
"
(0,)
H()
H()
is
such that
,
H()
Consider the function
f(z)
xz
Then
[If
Consequently
]if
< a
for all
!
H($)
there exists
[[
Since
}(x)
and
,8 g (0,)
[If
II
(x)
(i) then follows.
This end the
This work was done while the author was a visiting Professor
The author would also like to thank the
at the University of Michigan.
Department of Mathematics for their warm hospitality.
REFERENCES
5.
ALEKSANDROV, A.B. Essays On No Locally Convex Hardy Classes. Lecture Notes
In Math. 864(1-90).
DEEB, W. and RZUQ, M. H()-Spaces. To Appear In Bull. Canad. Math. Soc.
DEEB, W., KHALIL, R. and btARZUQ, M. Isometric Multiplication Of Hardy-Orlicz
Spaces. To Appear In ull. Aust. Math. Soc.
CATER, S. Continuous Linear Functionals On Certazn Topological Vector Spaces.
Pac. J. Math. 13(1963) 6S-71.
Math. Ann. 171(1967)
GRAMSCH, B. Die K1asse Der Metrischen L1nearen Raume L(
6.
61-78.
LESNIEWICZ, R. On Hardy-Orlicz Spaces I, Commt. Math. 15(1971) 3-56.
1.
2.
3.
4.
R. KHALIL
434
7.
LESNIEWICZ, R. On Linear Functionals In Hardy-Orlicz Spaces I, Stud. Math. 4_6(1973)
53-77.
ORLICZ, W. On Spaces Of -integrable Functions. Proc. Intern. Syrup. On Linear
Spaces. Hebrew Univ. Jerusalem, (1960) 357-365.
9. PALLASHKE, D. The Compact Endomorphisms Of The Metric Linear Spaces L().
_tudia Math. XLVII(1973).
I0. ROYDEN, H.L., Macmillan Company New York, 1963.
II. WILANSKY, A. Topology For Analysis, John Wiley, 1970.
12. WILANSKY, A. Modern Methods In Topological Vector Spaces. McGraw Hill. 1978.
8.