193
Internat. J. Math. & Math. Sci.
(1994) 193-196
VOL. 17 NO.
EXTENSIONS OF HARDY-LITTLEWOOD INEQUALITIES
ZENGJIAN LOU
Mathematics Department
Qufu Normal University
Qufu Shandong 273165
P. R. China
(Received December 30, 1992 and In revised form May 13, 1993)
ABSTRACT. For a function f 6- H’(B.)
Q) IfO<p<s,then
with f(0)
or-’( lOg 1)
0
f
we prove
"*-
f
Iip<,then
P
here
Rf is
the fractional derivative of
JO
f. These results generali the known cases
s
8,
KEY WORDS AND PHRASES. H’(B.) space,fractional derivative.
1991 AMS SUBJECT CLASSIFICATION CODES. 32A10.
1. INTRODUCTION.
Let C" denote the n-dimensional
vector space over
C. Let B. dehote the unit ball in C" with
.
boundary 3B,, and let o denote the rotation-invariant positive measure on 3B,. for which o(3B,.) 1.
We assume that fis holomorphic in B.. Let R#f(z)
]ala.be the fractional derivative of
f(z)
,
a.a(fl > 0).
For 0 < p,s,fl < oo, we
set
M(r,f)
/J If(r)
and
fll L.#
I
,If(rl l’-’lR#f(r) l’( lg 7)
r-’do(ldr
As usual ,for 0 < p < o, H(B.) denotes the space of holomorphic functions on B. for which
the means Mp(r,f) are bounded and the norm of f 6_ Ht(B,.) is defined by
Z. LOU
194
f II,
sup0<.<,M,(r,f).
Throughout this note,we assume that f 6_ H(B.) with f(0) 0.
In [1], Hardy Littlewood proved the following well--known theorem about H(BI).
THEOREM HL. If 0 < p 2 ,f 6_ /-P(B,), then
p<
f]0(1
, then
r)(r,fl)dr <
(.)
* implies f 6 H"(B,).
In this note, we generali these results to the unit ball B., with a new and short prof. That
is, we prove the following
If 2
If0<p,then
TOREM.
1
j
p<
@ I[
t
, then
f L.,
,
1
f
I n the Theorem 5y the ollowin
then
LEMMA. For 0 < p <
,
II/II ;
we have the
-1
’
II/II
{ollowin orollary, which extends Theorem (note that
COROLLARY.
0
,
<p
o
B., R’/()
r
then
1
I{ Z
p<
, then
2. PROOF OF THE MAIN RESULTS.
PROOF of the Theorem. Let 0 < p
Set/4’)
If(r)
IIf
;
then
aa./a()da()
-
=(llfll,.
f
>
f
s.
1
Assume without loss of generality that
we have
by Jensen’s.inequality, for each
R’f(r) I’/,(Dd())’/’
f(rD
j..I
If(r/Z) I’-’lRf(rD I’da(/;)
’
>
f II,
f f,-’(r,Rf)
r,
0,
EXTENSIONS OF HARDY-LITTLEWOOD INEQUALITIES
195
Therefore
fli ’-,f’f
* JoJ. If(r)lt-’lRPf(r)l’(log
fil T’ fll ;.,.,
rThe case p
log
r,
-)’P-’r-’d()dr
M;(r, Rf)dr
s is treated in a similar way to obtain,for each r,
f
f
tt-’Or-[joxlog 7)1
This completes the proof of the Theorem.
Now,we use the technique of [2] to give the prf of the lem.
B,, Rf() fr(), where ft() f(), R C B.
For
By the Hardy--Stein identity for one complex variable ([3]) we have
M(r’fr)
2ado
Ift(’)I’-’lr(
r
Integrating with resct to do(), using the Fubini theorem and the formula
see
[4, P. 15 ]),
we have
M,r,f,
P’IJ
]f()lt-’lR’f()l’(log
tting rl ,we obtain the Lemma.
ACKNOWLEDGEMENT. The author would like to thank the referee for a meticulous reading
of the manuscript, in particular for detecting an error in the first version of the Theorem and for
several helpful suggestions concerning presentation of the paper.
REFERENCES
[1] G. H. Hardy and J. E. Littlewood, Notes on the theory of series(XX), Generalizations of a theorem of Paley, Quart. J. Math., Oxford Ser., 8(1937), 161--171.
[2] Z. J. Lou, Generalizations of Littlewood and Paley inequalities, Internat. J. Math. & Math.
Sci., 16(1) (1993) ,205m207.
[3] P. Stein, On a theoem of Riesz, J. London Math. Soc. ,8(1933), 242--247
[4] W. Rudin, Function theory in the unit ball of C Springer--Verlag, New York, 1980.
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