Volume 8 (2007), Issue 3, Article 85, 8 pp.
COMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED BLOCH
SPACES OF POLYDISK
HAIYING LI, PEIDE LIU, AND MAOFA WANG
S CHOOL OF M ATHEMATICS AND S TATISTICS
W UHAN U NIVERSITY
W UHAN 430072, P.R. C HINA
tslhy2001@yahoo.com.cn
Received 23 March, 2007; accepted 26 June, 2007
Communicated by S.S. Dragomir
A BSTRACT. Let φ be a holomorphic self-map of the open unit polydisk U n in Cn and p, q > 0.
p
In this paper, the generally weighted Bloch spaces Blog
(U n ) are introduced, and the boundedp
q
ness and compactness of composition operator Cφ from Blog
(U n ) to Blog
(U n ) are investigated.
Key words and phrases: Holomorphic self-map; Composition operator; Bloch space; Generally weighted Bloch space.
2000 Mathematics Subject Classification. Primary 47B38.
1. I NTRODUCTION
Suppose that D is a domain in Cn and φ a holomorphic self-map of D. We denote by H(D)
the space of all holomorphic functions on D and define the composition operator Cφ on H(D)
by Cφ f = f ◦ φ.
The theory of composition operators on various classical spaces, such as Hardy and Bergman
spaces on the unit disk U in the finite complex plane C has been studied. However, the multivariable situation remains mysterious. It is well known in [3] and [5] that the restriction of
Cφ to Hardy or standard weighted Bergman spaces on U is always bounded by the Littlewood
subordination principle. At the same time, Cima, Stanton and Wogen confirmed in [1] that the
multivariable situation is much different from the classical case (i.e., the composition operators
on the Hardy space of holomorphic functions on the open unit ball of C2 as well as on many
other spaces of holomorphic functions over a domain of Cn can be unbounded, even when
n = 1 in [6]). Therefore, it would be of interest to pursue the function-theoretical or geometrical characterizations of those maps φ which induce bounded or compact composition operators.
In this paper, we will pursue the function-theoretic conditions of those holomorphic self-maps
φ of U n which induce bounded or compact composition operators from a generally weighted
p−Bloch space to a q−Bloch space with p, q > 0.
Supported by the Natural Science Foundation of China (No. 10671147, 10401027).
091-07
2
H AIYING L I , P EIDE L IU ,
AND
M AOFA WANG
For n ∈ N, we denote by U n the open unit polydisk in Cn :
U n = {z = (z1 , z2 , . . . , zn ) : |zj | < 1, j = 1, 2, . . . , n},
and
hz, wi =
n
X
zj wj ,
|z| =
p
hz, zi
j=1
for any z = (z1 , z2 , . . . , zn ), w = (w1 , w2 , . . . , wn ) in Cn . α = (α1 , α2 , . . . , αn ) is said to be
an n multi-index if αi ∈ N, written by α ∈ Nn . For α ∈ Nn , we write z α = z1α1 z2α2 · · · znαn
and zi0 = 1, 1 ≤ i ≤ n for convenience. For z, w ∈ Cn , we denote [z, w]j = z when j = 0,
[z, w]j = w when j = n, and
[z, w]j = (z1 , z2 , . . . , zn−j , wn−j+1 , . . . , wn )
when j ∈ {1, 2, . . . , n − 1}. Then [z, w]n−j = w when j = 1, and [z, w]n−j+1 = w when
j = n + 1, for j = 2, 3, . . . , n,
[z, w]n−j+1 = (z1 , z2 , . . . , zj−1 , wj , . . . , wn ).
For any a ∈ C and zj0 = (z1 , z2 , . . . , zj−1 , zj+1 , . . . , zn ), we write
(a, zj0 ) = (z1 , z2 , . . . , zj−1 , a, zj+1 , . . . , zn ).
0
Moreover, we adopt the notation (z [j ] )j 0 ∈N for an arbitrary subsequence of (z [j] )j∈N .
Recall that the Bloch space B(U n ) is the vector space of all f ∈ H(U n ) satisfying
b1 (f ) = sup Qf (z) < ∞,
z∈U n
where
|h∇f (z), ūi|
∂f
∂f
p
Qf (z) = sup
(z), . . . ,
(z)
,
∇f (z) =
∂z1
∂zn
H(z, u)
u∈Cn \{0}
and the Bergman metric H : U n × Cn → [0, ∞) on U n is
n
X
|uk |2
H(z, u) =
1 − |zk |2
k=1
(for example see [9], [15]). It is easy to verify that both |f (0)| + b1 (f ) and
n X
∂f
2
kf kB = |f (0)| + sup
(z)
∂zk (1 − |zk | )
z∈U n
k=1
n
are equivalent norms on B(U ). In [10], [12] and [8], some characterizations of the Bloch space
B(U n ) have been given.
In a recent paper [2], a generalized Bloch space has been introduced, the p−Bloch space:
for p > 0, a function f ∈ H(U n ) belongs to the p−Bloch space B p (U n ) if there is some
M ∈ [0, ∞) such that
n X
∂f
2 p
(z)
∀z ∈ U n .
∂zk (1 − |zk | ) ≤ M,
k=1
The references [13] to [7] studied these spaces and the operators in them.
Dana D. Clahane et al. in [2] proved the following two results:
Theorem A. Let φ be a holomorphic self-map of U n and p, q > 0. The following statements
are equivalent:
(a) Cφ is a bounded operator from B p (U n ) to B q (U n );
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 85, 8 pp.
http://jipam.vu.edu.au/
C OMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED B LOCH SPACES OF POLYDISK
(b) There is M ≥ 0 such that
n X
∂φl (1 − |zk |2 )q
(1.1)
∂zk (z) (1 − |φl (z)|2 )p ≤ M,
k, l=1
3
∀z ∈ U n .
Theorem B. Let φ be a holomorphic self-map of U n and p, q > 0. If condition (1.1) and
n X
∂φl (1 − |zk |2 )q
(1.2)
lim
∂zk (z) (1 − |φl (z)|2 )p = 0
φ(z)→∂U n
k, l=1
hold, then Cφ is a compact operator from B p (U n ) to B q (U n ).
p
Now we introduce the generally weighted Bloch space Blog
(U n ).
p
For p > 0, a function f ∈ H(U ) belongs to the generally weighted p−Bloch space Blog
(U n )
if there is some M ∈ [0, ∞) such that
n X
∂f
2
2 p
∀z ∈ U n .
∂zk (z) (1 − |zk | ) log 1 − |zk |2 ≤ M,
k=1
p
Its norm in Blog
(U n ) is defined by
p
kf kBlog
n X
∂f
2
(1 − |zk |2 )p log
= |f (0)| + sup
(z)
.
∂zk
1 − |zk |2
z∈U n
k=1
In this paper, we mainly characterize the boundedness and compactness of the composition
p
q
operators between Blog
(U n ) and Blog
(U n ), and extend some corresponding results in [2] and
[11] in several ways.
2. M AIN R ESULTS AND THEIR P ROOFS
First, we have the following lemma:
p
Lemma 2.1. Let f ∈ Blog
(U n ) and z ∈ U n , then:
p , when 0 < p < 1;
(a) |f (z)| ≤ 1 + (1−p)n log 2 kf kBlog
P
n
1
1
4
p
(b) |f (z)| ≤ 2 log
+ 2n log
k=1 log 1−|zk |2 kf kBlog , when p = 1;
2
2
Pn
2p−1
1
p
(c) |f (z)| ≤ n1 + (p−1)
k=1 (1−|zk |2 )p−1 kf kBlog , when p > 1.
log 2
p
p
Proof. Let p > 0, z ∈ U n , from the definition of k · kBlog
we have |f (0)| ≤ kf kBlog
and
(2.1)
p
p
kf kBlog
kf kBlog
∂f
≤
(z)
≤
∂zk (1 − |z |2 )p log 2
(1 − |zk |2 )p log 2
k
1−|zk |2
for every z ∈ U n and k ∈ {1, 2, . . . , n}. Notice that
f (z) − f (0) =
n
X
f ([0, z]n−k+1 ) − f ([0, z]n−k )
k=1
=
n
X
k=1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 85, 8 pp.
Z
zk
0
1
∂f ([0, (tzk , zk0 )]n−k+1 )
dt
∂zk
http://jipam.vu.edu.au/
4
H AIYING L I , P EIDE L IU ,
AND
M AOFA WANG
and then from the inequality (2.1), it follows that
Z 1 kf k p
n
X
Blog
|zk |
dt
|f (z)| ≤ |f (0)| +
2 p
log
2
(1
−
|tz
k| )
0
k=1
n Z |zk |
p
X
kf kBlog
1
p +
(2.2)
≤ kf kBlog
dt.
log 2 k=1 0
(1 − t2 )p
For p = 1, we have:
n Z
X
(2.3)
k=1
|zk |
0
n
n
X1
1
1 + |zk | X 1
4
dt
=
log
≤
log
.
(1 − t2 )p
2
1 − |zk |
2
1 − |zk |2
k=1
k=1
If p > 0 and p 6= 1, then
n Z |zk |
n Z |zk |
n
X
X
X
1
1
1 − (1 − |zk |)1−p
(2.4)
dt
≤
dt
=
.
(1 − t2 )p
(1 − t)p
1−p
k=1 0
k=1 0
k=1
Now for (a), from (2.4),
n Z
X
(2.5)
k=1
|zk |
0
1
1
dt ≤
.
2
p
(1 − t )
1−p
From (2.2) and (2.5), it follows that
|f (z)| ≤
n
1+
(1 − p) log 2
p .
kf kBlog
For (b), Since log 1−|z4 k |2 > log 4 = 2 log 2 for each k ∈ {1, 2, . . . , n}, then
n
X
4
1
1<
log
.
2n log 2 k=1
1 − |zk |2
(2.6)
Combining (2.2), (2.3) and (2.6) we get
X
n
1
1
4
p .
|f (z)| ≤
+
log
kf kBlog
2 log 2 2n log 2 k=1
1 − |zk |2
For (c), from (2.4) we have
n Z |zk |
n
n
X
X
X
1
1 − (1 − |zk |)p−1
2p−1
(2.7)
dt
≤
≤
.
2 )p
p−1
2 p−1
(1
−
t
(p
−
1)(1
−
|z
(p
−
1)(1
−
|z
k |)
k| )
0
k=1
k=1
k=1
By (2.2) and (2.7), we obtain
n
X
2p−1
1
p
|f (z)| ≤ kf k
+
kf kBlog
(p − 1) log 2 k=1 (1 − |zk |2 )p−1
X
n
1
2p−1
1
p .
≤
+
kf kBlog
n (p − 1) log 2 k=1 (1 − |zk |2 )p−1
p
Blog
Lemma 2.2. For p > 0, l ∈ {1, 2, . . . , n} and w ∈ U , the function fwl : U n → C,
Z zl
1
l
fw (z) =
2 dt
p
0 (1 − wt) log 1−wt
p
belongs to Blog
(U n ).
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 85, 8 pp.
http://jipam.vu.edu.au/
C OMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED B LOCH SPACES OF POLYDISK
5
Proof. Let k, l ∈ {1, 2, . . . , n} and w ∈ U , then
∂fwl
= 0,
∂zk
(2.8)
∀z ∈ U n , k 6= l
and
∂fwl
1
,
(z) =
2
∂zl
(1 − wzl )p log 1−wz
l
(2.9)
∀z ∈ U n .
An easy estimate shows that there is 0 < M < +∞ such that
2
(1 − |wz|)p log 1−|wz|
2
|1 − wz|p log |1−wz|
≤ M,
∀z, w ∈ U.
Therefore, by (2.8) and (2.9), we have
n X
∂fwl
2
l
(1 − |zk |2 )p log
|fw (0)| +
(z)
∂zk
1 − |zk |2
(1 −
=
≤
≤
k=1
|zl |2 )p
log 1−|z2 l |2
2
|1 − wzl |p | log 1−wz
|
l
(1 − |zl |2 )p log 1−|z2 l |2
2
(1 − |wzl |)p log 1−|wz
l|
·
2
(1 − |wzl |)p log 1−|wz
l|
2
|1 − wzl |p log |1−wz
l|
2p
· M < +∞
pe log 2
p
and thus {fwl : w ∈ U, l ∈ {1, 2, . . . , n}} ⊂ Blog
(U n ).
Theorem 2.3. Let φ be a holomorphic self-map of the open unit polydisk U n and p, q > 0, then
the following statements are equivalent:
p
q
(a) Cφ is a bounded operator from Blog
(U n ) and Blog
(U n );
(b) There is an M > 0 such that
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
(2.10)
≤ M,
∀z ∈ U n .
2
∂zk (z) · (1 − |φl (z)|2 )p · log
1−|φl (z)|2
k,l=1
Proof. Firstly, assume that (b) is true. By Lemma 2.1, there is a C > 0 such that for all
p
f ∈ Blog
(U n ),
p .
|f (φ(0))| ≤ Ckf kBlog
(2.11)
Then for all z ∈ U n ,
n X
∂(Cφ f ) 2
2 q
∂zk (z) (1 − |zk | ) log 1 − |zk |2
k=1
n X
∂f
2
(φ(z)) (1 − |φl (z)|2 )p log
≤
∂ξl
1 − |φl (z)|2
l=1
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
×
(z)
·
2
∂zk (1 − |φl (z)|2 )p log
1−|φl (z)|2
k=1
p ,
≤ M kf kBlog
and (a) is obtained.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 85, 8 pp.
http://jipam.vu.edu.au/
6
H AIYING L I , P EIDE L IU ,
AND
M AOFA WANG
Conversely, let l ∈ {1, 2, . . . , n}, if (a) is true, i.e. there is a C ≥ 0 such that
(2.12)
p ,
kCφ f k ≤ Ckf kBlog
p
∀f ∈ Blog
(U n ),
then, by Lemma 2.2 and (2.12), there is a Q > 0 such that
n
n X
l
X
∂φ
2
∂f
l
w
(φ(z)) ·
(z) (1 − |zk |2 )q log
≤ CQ,
2
∂ξ
∂z
1
−
|z
l
k
k|
k=1 l=1
∀w ∈ U, z ∈ U n .
Letting w = φ(z), and using (2.8) and (2.9), we have
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
≤ CQ.
2
∂zk (z) · (1 − |φl (z)|2 )p · log
2
1−|φ
(z)|
l
l,k=1
Lemma 2.4. Let φ : U n → U n be holomorphic and p, q > 0, then Cφ is compact from
p
q
p
Blog
(U n ) to Blog
(U n ) if and only if for any bounded sequence (fj )j∈N in Blog
(U n ), when fj → 0
q
uniformly on compacta in U n , then kCφ fj kBlog
→ 0 as j → ∞.
p
Proof. Assume that Cφ is compact and (fj )j∈N is a bounded sequence in Blog
(U n ) with fj → 0
n
uniformly on compacta in U . If the contrary is true, then there is a subsequence (fjm )m∈N and
q
a δ > 0 such that kCφ fjm kBlog
≥ δ for all m ∈ N. Due to the compactness of Cφ , we choose a
p
subsequence (fjml ◦ φ)l∈N of (Cφ fjm )m∈N = (fjm ◦ φ)m∈N and some g ∈ Blog
(U n ), such that
(2.13)
q
lim kfjml ◦ φ − gkBlog
= 0.
l→∞
Since Lemma 2.1 implies that for any compact subset K ⊂ U n , there is a Ck ≥ 0 such that
(2.14)
q ,
|fjml (φ(z)) − g(z)| ≤ Ck kfjml ◦ φ − gkBlog
∀l ∈ N, z ∈ K.
By (2.13), fjml ◦ φ − g → 0 uniformly on compact subset in U n . Since fjml φ(z) → 0 as l → ∞
for each z ∈ U n , and by (2.14), then g = 0; (2.13) shows
q
lim kCφ (fjml )kBlog
= 0,
l→∞
it gives a contradiction.
p
p
≤ M for all
Conversely, assume that (gj )j∈N is a sequence in Blog
(U n ) such that kgj kBlog
j ∈ N. Lemma 2.1 implies that if (gj )j∈N is uniformly bounded on any compact subset in
U n and normal by Montel’s theorem, then there is a subsequence (gjm )m∈N of (gj )j∈N which
∂g
∂g
converges uniformly on compacta in U n to some g ∈ H(U n ). It follows that ∂zjm
→ ∂z
l
l
p
uniformly on compacta in U n for each l ∈ {1, 2, . . . , n}. Thus g ∈ Blog
(U n ) with kgjm −
p
p
gkBlog
≤ M +kgkBlog
< ∞ and gjm −g converges to 0 on compacta in U n , so by the hypotheses,
q
p
q
gjm ◦φ → g◦φ in Blog
(U n ). Therefore Cφ is a compact operator from Blog
(U n ) to Blog
(U n ). p
q
q
Lemma 2.5. If for every f ∈ Blog
(U n ), Cφ f belongs to Blog
(U n ), then φα ∈ Blog
(U n ) for each
n-multi-index α.
p
Proof. As is well known, every polynomial pα : Cn → C defined by pα (z) = z α is in Blog
(U n ).
q
Thus, by the assumption Cφ (z α ) = φα ∈ Blog
(U n ).
Theorem 2.6. Suppose that p, q > 0, φ : U n → U n is a holomorphic self-map such that
q
φk ∈ Blog
(U n ) for each k ∈ {1, 2, . . . , n} and
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
·
(2.15)
lim n
(z)
·
= 0,
2
∂zk (1 − |φl (z)|2 )p log
φ(z)→∂U
2
1−|φl (z)|
k,l=1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 85, 8 pp.
http://jipam.vu.edu.au/
C OMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED B LOCH SPACES OF POLYDISK
7
p
q
then Cφ is a compact operator from Blog
(U n ) to Blog
(U n ).
p
Proof. Let (fj )j∈N be a sequence in Blog
(U n ) with fj → 0 uniformly on compacta in U n and
p
kfj kBlog
≤ C,
(2.16)
∀j ∈ N.
By Lemma 2.4, it suffices to show that
q
lim kCφ fj kBlog
= 0.
(2.17)
j→∞
q
Notice that if kφm kBlog
= 0 for all m ∈ {1, 2, . . . , n}, then φ = 0 and Cφ has finite rank.
q
Therefore, we can assume C > 0 and kφm kBlog
> 0 for some m ∈ {1, 2, . . . , n}. Now let
ε > 0, from (2.15), there is an r ∈ (0, 1) such that
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
ε
<
(2.18)
2
∂zk (z) · (1 − |φl (z)|2 )p · log
2C
1−|φl (z)|2
k,l=1
for all z ∈ U n satisfying d(φ(z), ∂U n ) < r. By using a subsequence and the chain rule for
derivatives, (2.16) and (2.18) guarantee that for all such z and j ∈ N,
n X
∂(Cφ f ) 2
2 q
∂zk (z) (1 − |zk | ) log 1 − |zk |2
k=1
n X
∂f
2
(1 − |φl (z)|2 )p log
≤
(φ(z))
∂ξl
1 − |φl (z)|2
l=1
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
×
2
∂zk (z) · (1 − |φl (z)|2 )p · log
1−|φl (z)|2
k=1
ε
ε
≤C·
= .
2C
2
To obtain the same estimate in the case d(φ(z), ∂U n ) ≥ r, let Er = {w : d(w, ∂U n ) ≥
r}. Since
Er is compact, by the hypothesis, (fj )j∈N and the sequences of partial derivatives
∂fj
∂zl
j∈N
converge to 0 uniformly on Er , respectively. Then
n X
∂(Cφ fj ) 2
2 q
∂zk (z) (1 − |zk | ) log 1 − |zk |2
k=1
X
n X
∂fj
n ∂φl 2
2 q
≤
∂ξl (φ(z)) ·
∂zk (z) · (1 − |zk | ) log 1 − |zk |2
k=1
k=1
n
X ∂fj
· kφl kB q
≤
(φ(z))
∂ξl
log
l=1
n
X
∂fj
ε
q
≤
sup (w) · kφl kBlog
≤
(as j → +∞).
2
w∈Er ∂ξl
l=1
q
Since {φ(0)} is compact, we have fj (φ(0)) → 0 as j → ∞, and kCφ fj kBlog
→ 0 as j → ∞,
p
q
n
n
thus Cφ is a compact operator from Blog (U ) to Blog (U ).
Theorem 2.7. Let φ be a holomorphic self-map of U n and p, q > 0. If conditions (2.10) and
p
q
(2.15) hold, then Cφ is a compact operator from Blog
(U n ) to Blog
(U n ).
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 85, 8 pp.
http://jipam.vu.edu.au/
8
H AIYING L I , P EIDE L IU ,
AND
M AOFA WANG
p
q
Proof. If (2.10) is true, then Cφ is bounded from Blog
(U n ) to Blog
(U n ) by Theorem 2.3, and
q
φk ∈ Blog (U n ) for each k ∈ {1, 2, . . . , n} by Lemma 2.5. The proof follows on applying (2.15)
and Theorem 2.6.
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