Internat. J. Math. & Math. Sc.
VOL. 19 NO. 3 (1996) 611-612
611
FRACTIONAL DERIVATIVES OF HOLOMORPHIC FUNCTIONS ON
BOUNDED SYMMETRIC DOMAINS OF C"
ZENGJIAN LOU
Department of Mathematics
Qufu Normal University
Qufu Shandong, 273165
P R CHINA
(Received March 8, 1994 and in revised form January 13,1995)
_
ABSTRACT. Let f E H(Bn)
f[] denotes the /3th fractional derivative of f
A"q’(B,), we show that
_[_
<
s=
6, then f E As’t’O(B,), and [[f[[s,t,=,
(I) If/ <
n
--+l
(II) Iffl
(III) If
,
>--+ ,
o+1
+
a+l
C[[f[][[....,,q,o’
then
f E B(B,
and
[[f[[B --<
C[[f[][[
p,q,a
then f E A _o/____(B) especially If
where B. is the unit ball of C"
<- Cllf Ill
_-
I then
KEY WORDS AND PHRASES. Fractional derivative, Bergman space, Bloch space, Lipschitz space
1991 AMS SUBJECT CLASSIFICATION CODE. 32A10
Let f be a bounded symmetric domain in the complex vector space C’, o E f/, with BergmanSilov boundary b, F the group of holomorphic automorphisms of f and 1"0 its isotropy group. It is
known that [2 is circular and star-shaped with respect to o and b is circular. The group F0 is transitive
on b and b has a unique normalized F0-invariant measure a with a(b)= 1. Hua [2] constructed by
group representation theory a system {eke,} of homogeneous polynomials, k 0, 1,
v 1,
ink, mk= (n+kk-1), complete and orthogonal on f and orthonormal on b.
By H(f) we denote the class of all holomorphic functions on [2. Every f
expansion
f(z)
Z akvkv(Z)
akv
H(f) has a series
:
lim
f(r) @kv() da()
--,---,I
Jb
(0)
and the convergence is uniform on a compact subset of f.
where
k,v
Let
k=0 v=l
f H([2)
(0) and/ > 0. The/th fractional derivatives of f are defined,
r(k -+- 1 -F
fill(z)
with the expansion
respectively, by
f[#](z)
k,v
r(+l)
ak,@kv(Z)
F(k + 1 + fl)
It is known that f[t], f{a] E H([2) and
(1 p)#-lf[#](rp)dp.
f(r)
Let f E H(f). It will be said that f belongs to the Bergman spaces AP,q.’(f/), 0 < p, q < oo, a >
if
"’’
is finite, where
)’
(1 r)aMq(r, f)’dr
sup ( )M(, I),
O<r<l
p<oo
p
(1)
1
Z LOU
612
If(r)lqda()
Mq(r,f)
0
<q<
oo
and
M,,(r, f)
sup If(r()
(b
[1,3,5,6,7] for more on AP’q’(Q) For 0 < p < cx, let AP(f2)denote AP.P.() (see [10,12]), H(Q)
A’P’(Q) (see [9])
Let B denote the unit ball in C A nction f H(B) is called a Bloch nction, that is
f B(B), if([8,11])
Ilfll sup (1 -Izl)lffl(z)l <
see
denote
zB
,
the definition of Lipschitz space A (B) can be found in [4, 8 8]
For 0 < a <
In [10] and [12], Watanable and Stojan considered the problem If f’ AP(D)(D is the unit disc
of C), then q 9 such that f Aq(D) In this paper we consider and solve the same problem in
The main results of this paper are the following
THEOM 1. Let 0<p,q ,> -1,0<<5
f()
o(llf[]l[,..o(1-
then
,
f A.t.(a)
THEOM 2. Let 0 < p, q
a >
1, 0 < <
5, then f A’,t,O(B), and
(I) If <
+
same as above
,
and
o+l
+,
Ilfll.,.o
, fill
Ilfll,:,o_
if
f[a] AP’q’()
cllfta][[,.,o,
and
where
Ap.q.a(Bn)"
f
,where s,t are the
P,q,
(II) If:a+l + then f B(B) and tlfll c ft] ,,
o+1
(III) If> +,thenf
A (B) especially If=l, then
< c lft’) l
MA. (i) Theorem 2(I) (p q,
n
0,
1) eends the results of Watable’s d
Stoj’s (ii) Theorem (p ) extends the results of Shi’s ([9]) d Lou’s ([6,7])
REFERENCES
AHERN, P AND JEVTIC, M, Duality and multipliers for mixed norm spaces, Mtch. Math. J., 30
(1983), 53-63
[2] HUA, L K, Harmonic analysis of functions of several variables in the classical domains, Trans.
Amer. Math. Soc., 6 (1963).
[3] JEVTIC, M, Projection theorems, fractional derivatives and inclusion theorems for mixed-norm
spaces on the ball, Analysis, 9 (1989), 83-105
[4] KRANTZ, S.G, Functton Theory of Several Complex Vartables, John Wiley & Sons, New York,
1982
[5] LOU, Z J, A note on a conjecture of S Axler, J. Math. Res. & Exp., 11 (4) (1991), 629-630
(Chinese)
[6] LOU, Z J, Hardy-Littlewood type theorem on Bergman spaces, J. Math. Res. & Exp., 11 (3)
(1991) (Chinese)
[7] LOU, Z J, Hardy-Littlewood type theorem on Bergman spaces of several complex variables, J.
Qufu Normal Univ., (1992), 111-112 (Chinese)
[8] LOU, Z J., Characterizations of Bloch functions on the unit ball of C n, Kodat Math. J., 16 (1993),
74-78
[9] SHI, J H, Hardy-Littlewood theorems on bounded symmetric domains, Science m China, 4
(1988), 366-375
[10] STOJAN, D, Some new properties ofthe spaces AP, Mat. Vesmc, 5 18,33 (1981), 151-157
[11] TIMONEY, R M, Bloch functions in several complex variables, Bull. London. Math. Soc., 12
(1980), 241-267