IJMMS 2004:71, 3941–3950
PII. S0161171204403548
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ESSENTIAL NORM OF WEIGHTED COMPOSITION
OPERATOR BETWEEN α-BLOCH SPACE AND
β-BLOCH SPACE IN POLYDISCS
LI SONGXIAO and ZHU XIANGLING
Received 29 March 2004
Let ϕ(z) = (ϕ1 (z), . . . , ϕn (z)) be a holomorphic self-map of Dn and ψ(z) a holomorphic
function on Dn , where Dn is the unit polydiscs of Cn . Let 0 < α, β < 1, we compute the
essential norm of a weighted composition operator ψCϕ between α-Bloch space Ꮾα (Dn )
and β-Bloch space Ꮾβ (Dn ).
2000 Mathematics Subject Classification: 47B35, 30H05.
1. Introduction. Let Dn be the unit polydiscs of Cn , the class of all holomorphic
functions with domain Dn will be denoted by H(Dn ). Let ϕ be a holomorphic self-map
of Dn , the composition operator Cϕ induced by ϕ is defined by (Cϕ f )(z) = f (ϕ(z))
for z in Dn and f ∈ H(Dn ). If, in addition, ψ is a holomorphic function defined on Dn ,
the weighted composition operators ψCϕ induced by ψ and ϕ is defined by ψCϕ (z) =
ψ(z)f (ϕ(z)) for z in Dn and f ∈ H(Dn ).
Let 0 < α < 1, a function f holomorphic in Dn is said to belong to the α-Bloch space
Ꮾα (Dn ) if
n ∂f
α
1 − zk 2 < +∞.
(z)
f α = f (0) + sup
n
∂z
z∈D
k=1
k
(1.1)
It is easy to show that Ꮾα (Dn ) is a Banach space with the norm ·α . These spaces are
called Lipschitz space Lipα (Dn ) by Zhou (see [6, 8]). It is easy to show that the usual
norm on Lip1−α (Dn ) defined by
f (z) − f (w)
f Lip = f (0) + sup
|z − w|1−α
z,w∈Dn
(1.2)
induces a Banach space structure on Lip1−α (Dn ). Clahane in [1] has shown that this
norm is equivalent to · α .
Essential norm formulas for composition operators are known in various settings.
Shapiro has given a formula for Cϕ e when Cϕ acting on the Hardy space H 2 (D)
in [5]; Montes-Rodríguez [4] has given the essential norm of composition operator on
the Bloch space in the unit disc; Donaway [2] has given upper and lower estimates
for Cϕ e when Cϕ maps the Bloch space, the Dirichlet space, or the Besov-p space to
itself; MacCluer and Zhao [3] have given an exact formula of essential norm of weighted
3942
L. SONGXIAO AND Z. XIANGLING
composition operator between the Bloch-type spaces in the unit disc, namely,
uCϕ = lim
e
sup u(z)ϕ (z) s→1 |ϕ(z)|>s
β
1 − |z|2
2 α .
1 − ϕ(z)
(1.3)
Here, ϕ is an analytic self-map of D and u is a fixed analytic function on D and 0 < α <
1, 0 < β < ∞; Zhou and Shi [7] have given the essential norm of composition operator
on the Bloch space in polydiscs, that is,
1
lim
n2 δ→0
sup
dist(ϕ(z),∂Dn )<δ
n 1 − z k 2
∂ϕl
2
∂z (z)
k
1 − ϕl (z)
k,l=1
≤ Cϕ e ≤ 2n2 lim
sup
δ→0 dist(ϕ(z),∂Dn )<δ
n 1 − zk 2
∂ϕl
2 .
∂z (z)
k
1 − ϕl (z)
k,l=1
(1.4)
Recently, Zhou [6] studied weighted composition operators between α-Bloch space
and β-Bloch space in polydiscs. He proved the following theorems.
Theorem 1.1. Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic self-map of Dn and ψ(z) a
holomorphic function of Dn , 0 < α, β < 1. Then, ψCϕ : Ꮾα (Dn ) → Ꮾβ (Dn ) is bounded if
and only if ψ ∈ Ꮾβ (Dn ) and
2 β
n 1 − z k ∂ϕl
−
ψ(z)
2 α = O(1) |z| → 1 .
∂z (z) k
1 − ϕl (z)
k,l=1
(1.5)
Theorem 1.2. Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic function of Dn and ψ(z) a
holomorphic function of Dn , 0 < α, β < 1. Then, ψCϕ : Ꮾα (Dn ) → Ꮾβ (Dn ) is compact if
and only if
(i) ψCϕ is bounded,
(ii)
n ∂ψ
β
1 − zk 2 = o(1),
(z)
∂z
k=1
k
ϕ(z) → ∂Dn ,
(1.6)
(iii)
2 β
n 1 − z k ∂ϕl
ψ(z)
2 α = o(1),
∂z (z) k
1 − ϕl (z)
k,l=1
ϕ(z) → ∂Dn .
(1.7)
It is reasonable to expect that the essential norm of ψCϕ : Ꮾα (Dn ) → Ꮾβ (Dn ) should
be given by the related lim sup expression. The following theorem is our main result.
ESSENTIAL NORM OF WEIGHTED COMPOSITION . . .
3943
Theorem 1.3. Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic self-map of Dn and ψ(z) a
holomorphic function of Dn , 0 < α, β < 1. Suppose the weighted composition operator
ψCϕ : Ꮾα (Dn ) → Ꮾβ (Dn ) is bounded, then,
1
lim
n δ→0
2 β
n 1 − z k ∂ϕl
sup
ψ(z)
2 α
∂z (z) k
dist(ϕ(z),∂Dn )<δ
1 − ϕl (z)
k,l=1
≤ ψCϕ e ≤ 2n lim
δ→0
2 β
n 1 − zk ∂ϕl
ψ(z)
sup
2 α .
∂z (z) k
dist(ϕ(z),∂Dn )<δ
1 − ϕl (z)
k,l=1
(1.8)
2. The proof of Theorem 1.3. In this section, we mainly give the proof of the main
theorem of this paper. We divided our proof into two parts.
The lower estimates. Since ψCϕ : Ꮾα (Dn ) → Ꮾβ (Dn ) is bounded, we have ψ ∈
Ꮾ (Dn ) by Theorem 1.1.
Note that for m ≥ 2,
β
α m
z = sup 1 − z1 2 mzm−1 = m
1 α
1
z∈Dn
2α
m − 1 + 2α
α m−1
m − 1 + 2α
(m−1)/2
,
(2.1)
where the maximum is attained at any point on the circle with radius
rm =
m−1
m − 1 + 2α
1/2
.
(2.2)
Hence, the sequence {z1m }m≥2 is bounded in Ꮾα (Dn ). Let fm = z1m /z1m α , then fm α =
1 and fm is bounded sequence and converges weakly to 0 in Ꮾα (Dn ). This follows since
a bounded sequence contained in Ꮾα (Dn ) which tends to 0 uniformly on compact subsets of Dn converges weakly to 0 in Ꮾα (Dn ). In particular, if K is any compact operator
from Ꮾα (Dn ) to Ꮾβ (Dn ), then limm→∞ Kfm β = 0.
For m ≥ 2, let
Am = z = z1 , z2 , . . . , zn ∈ Dn , rm ≤ z1 ≤ rm+1 ,
(2.3)
here, rm = ((m − 1)/(m − 1 + 2α))1/2 . Let g(x) = m(1 − x 2 )α x m−1 , then
α−1 g (x) = −mx m−2 1 − x 2
2αx 2 − (m − 1) 1 − x 2 ≤ 0
(2.4)
for x ∈ [((m − 1)/(m − 1 + 2α))1/2 , 1), that is, g(x) is a decreasing function for x ∈
[((m − 1)/(m − 1 + 2α))1/2 , 1). Therefore,
2
mr m−1 ∂fm α
m+1
1 − z1 2 = 1 − rm+1
min ∂z z m Am
1
1 α
(2.5)
(m−1)/2
m − 1 + 2α α
m2 + (2α − 1)m
=
= Cm .
m + 2α
m2 + (2α − 1)m − 2α
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L. SONGXIAO AND Z. XIANGLING
Note that Cm tends to 1 as m → ∞. Take any compact operator K from Ꮾα (Dn ) to
Ꮾβ (Dn ), we have
ψCϕ − K ≥ lim sup ψCϕ − K fm β
m→∞
≥ lim sup ψCϕ fm β − Kfm β
m→∞
= lim sup ψCϕ fm β
m→∞
≥ lim sup sup
m→∞
z∈D n
n ∂ ψfm ◦ ϕ
2 β
(z)
1 − z k ∂zk
k=1
n ∂ ψfm ◦ ϕ
β
1 − z k 2
(z)
∂zk
m→∞ ϕ(z)∈Am k=1
n n
2 β
∂ϕl
∂fm ∂ψ(z)
≥ lim sup sup
fm ϕ(z) +ψ(z)
(z) 1 − zk ϕ(z)
∂zk
∂wl
∂zk
m→∞ ϕ(z)∈A
≥ lim sup
sup
m k=1
l=1
∂ψ(z)
β
∂ϕ1
∂fm 1 − zk 2
ϕ(z)
+
ψ(z)
ϕ(z)
= lim sup sup
f
(z)
m
∂z
∂w1
∂zk
k
m→∞ ϕ(z)∈Am k=1
n
ψ(z) ∂fm ϕ(z) ∂ϕ1 (z)
≥ lim sup sup ∂w
∂z
1
k
m→∞ ϕ(z)∈Am k=1
n
2 β
1 − z k 2 α
×
2 α 1 − ϕ1 (z)
1 − ϕ1 (z)
−
n ∂ψ
β 1 − zk 2 fm ϕ(z) .
(z)
∂z
k
k=1
(2.6)
When 0 < α < 1, we know that ψ ∈ Ꮾβ (Dn ), so that
sup
lim sup
m→∞
n ∂ψ
2 β fm ϕ(z) ∂z (z) 1 − zk
ϕ(z)∈Am k=1
≤ ψβ lim sup
m→∞
k
sup
ϕ(z)∈Am
fm ϕ(z) m/2
m/(m + 2α)
= ψβ lim sup
z m m→∞
1
α
m/2
m/(m + 2α)
= ψβ lim sup α (m−1)/2
m→∞ m 2α/(m − 1 + 2α)
(m − 1)/(m − 1 + 2α)
= 0.
(2.7)
ESSENTIAL NORM OF WEIGHTED COMPOSITION . . .
3945
Therefore,
ψCϕ e
≥ lim sup
m→∞
sup
n
ψ(z) ∂fm ϕ(z) ∂ϕ1 (z)
∂w
∂z
ϕ(z)∈Am k=1
1
k
2 β
1 − z k 2 α
×
2 α 1 − ϕ1 (z)
1 − ϕ1 (z)
2 β
n
1 − z k ∂ϕ1
ψ(z)
≥ lim sup sup
2 α
∂z (z) k
m→∞ ϕ(z)∈Am k=1
1 − ϕ1 (z)
2 α
∂fm ϕ(z) × lim inf min 1 − ϕ1 (z)
ϕ(z)∈Am ∂w1
m
2 β
n
1 − z k ∂ϕ1
ψ(z)
≥ lim sup sup
2 α limminf Cm
∂z (z) k
m→∞ ϕ(z)∈Am k=1
1 − ϕ1 (z)
≥ lim sup
m→∞
sup
ϕ(z)∈Am
(2.8)
2 β
n
1 − z k ∂ϕ1
ψ(z)
2 α .
∂z (z) k
1 − ϕ1 (z)
k=1
So,
ψCϕ ≥ lim sup
e
m→∞
sup
ϕ(z)∈Am
2 β
n
1 − z k ∂ϕ1
ψ(z)
2 α .
∂z (z) k
1 − ϕ (z)
k=1
(2.9)
1
For l = 1, 2, . . . , n, define
al = lim
sup
δ→0 dist(ϕ(z),∂Dn )<δ
2 β
n
1 − z k ∂ϕl
ψ(z)
2 α .
∂z (z) k
1 − ϕl (z)
k=1
(2.10)
For any > 0, (2.10) shows that there exists δ0 , 0 < δ0 < 1, if dist(ϕ(z), ∂Dn ) < δ0 , then
2 β
n
1 − z k ∂ϕ1
ψ(z)
2 α > a1 − .
∂z (z) k
1 − ϕ1 (z)
k=1
(2.11)
Because rm → 1 (m → ∞), rm > 1 − δ0 for m being large enough. If ϕ(z) ∈ Am , then
rm ≤ |ϕ1 (z)| ≤ rm+1 , that is, 1 − rm+1 ≤ 1 − |ϕ1 (z)| ≤ 1 − rm , that is,
dist ϕ(z), ∂Dn ≤ dist ϕ1 (z), ∂D < δ0 .
(2.12)
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L. SONGXIAO AND Z. XIANGLING
By (2.9) and (2.11), we have
ψCϕ ≥ a1 − .
e
(2.13)
If we choose gm (z) = zlm /zlm , repeating similar argument as in the case l = 1, we have
ψCϕ ≥ al − e
(2.14)
for l = 2, . . . , n. Hence,
n
ψCϕ ≥ 1
al − e
n l=1


2 β
n
n
1 − z k ∂ϕl
1 

ψ(z)
sup
=
lim
2 α − 
∂z (z) n l=1 δ→0 dist(ϕ(z),∂Dn )<δ k=1
k
1 − ϕl (z)
1
= lim
n δ→0
sup
dist(ϕ(z),∂Dn )<δ
2 β
n
1 − z k ∂ϕl
ψ(z)
2 α − .
∂z (z) k
1 − ϕl (z)
k,l=1
(2.15)
Let → 0, we obtain the result.
The upper estimates. For this purpose, we define operator Km (m ≥ 2) as follows:
Km f (z) = f
m−1
z .
m
(2.16)
the method of [7], we can show that Km has the following properties:
Km is compact operator from Ꮾα (Dn ) to Ꮾα (Dn ),
limm→∞ supf α ≤1 supz∈Dn |(I − Km )f (z)| = 0,
for any f ∈ Ꮾα (Dn ), (I − Km )f → 0 uniformly on compact subsets of Dn , hence,
for l = 1, 2, . . . , n, ∂(I − Km )f /∂wl → 0 uniformly on compact subsets of Dn ,
(d) I − Km ≤ 2.
The details of parts (b) and (c) are left to the reader, we will show the details for part
(a) and (d).
First, we show the details of part (a). In fact, let E = {((m − 1)/m)z, z ∈ Dn }, then
E is a compact subset of Dn . For any sequence {fj } ⊂ Ꮾα (Dn ), there exists a subsequence {fjs } of {fj } converging uniformly to f ∈ Ꮾα (Dn ) on compact subsets of
Dn and f α ≤ M. It is obvious that {∂fjs /∂zi }, i = 1, 2, . . . ,n, converges uniformly to
{∂f /∂zi } on compact subsets of Dn . So, for large enough s, w ∈ E and l = 1, 2, . . . , n,
Using
(a)
(b)
(c)
∂ fjs − f
< .
(w)
∂w
l
(2.17)
ESSENTIAL NORM OF WEIGHTED COMPOSITION . . .
3947
Hence,
Km fj − Km f s
α
m−1 m
−
1
z
−
f
z
=
f
j
s
m
m
α
n
∂ fjs − f
α
(m
−
1)/m
z 1 − z k 2
≤ sup
n
∂z
z∈D
k
k=1
n n α
∂ fjs − f
(m − 1)/m z m − 1 1 − z k 2
≤ sup
n
∂w
m
z∈D
(2.18)
l
k=1 l=1
n n ∂ fjs − f
(m − 1)/m z m−1
m
∂w
≤ sup
z∈Dn k=1 l=1
m−1
m
w∈E
≤ sup
l
n
k=1
∂ fjs − f
(w)
→ 0
∂w
l
l=1
n for s → ∞. These show that Km is a compact operator.
Next, we show the details of part (d). In fact, for any f ∈ Ꮾα (Dn ), we have
n ∂ I − Km f
α
I − Km f ≤ sup
1 − zk 2
(z)
α
n
∂z
z∈D
= sup
k=1
n
z∈Dn k=1
≤ sup
z∈Dn
n
k=1
+ 1−
k
∂f
2 α
1 ∂f
1
z (z)
−
1
−
1
−
1 − z k ∂z
m ∂z
m
k
k
∂f
2 α
∂z (z) 1 − zk
1
m
(2.19)
k
sup
z∈Dn
2 α
n ∂f
1
1 − 1 − 1 zk z
1
−
∂z
m
m
k
k=1
≤ f α + f α = 2f α .
Since each Km is compact operator from Ꮾα (Dn ) to Ꮾα (Dn ), so is ψCϕ Km . Hence,
ψCϕ ≤ ψCϕ − ψCϕ Km = ψCϕ I − Km = sup ψCϕ I − Km f .
e
β
(2.20)
f α ≤1
We bound the last expression from above by
sup ψ(0) I − Km f ϕ(0) f α ≤1
(2.21)
+ sup
sup
n
β
ψ(z) ∂ I − Km f ϕ(z) ∂ϕl (z) 1 − zk 2
∂w
∂z
(2.22)
+ sup
sup
n β
∂ψ
I − Km f ϕ(z) 1 − zk 2 .
(z)
∂z
(2.23)
f α ≤1 z∈Dn k,l=1
f α ≤1 z∈Dn k=1
l
k
k
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L. SONGXIAO AND Z. XIANGLING
By the property (b), we know that the supremum in (2.21) can be made arbitrarily small
as m → ∞. Since ψ(z) ∈ Ꮾβ (Dn ),
sup
n ∂ψ
β
1 − zk 2 < ∞.
(z)
∂z
z∈Dn k=1
k
(2.24)
Property (b) also ensures that the supremum in (2.23) tends to 0 as m → ∞. Now, we
need only consider the term
sup
sup
n
β
ψ(z) ∂ I − Km f ϕ(z) ∂ϕl (z) 1 − zk 2 .
∂w
∂z
f α ≤1 z∈Dn k,l=1
l
k
(2.25)
For arbitrary 0 < δ < 1, we define
G1 = z ∈ D n : dist ϕ(z), ∂Dn < δ ,
G2 = z ∈ D n : dist ϕ(z), ∂Dn ≥ δ .
(2.26)
Here, G2 is compact subset of Cn . We consider
sup
n
β
ψ(z) ∂ I − Km f ϕ(z) ∂ϕl (z) 1 − zk 2
∂w
∂z
sup
f α ≤1 z∈D n k,l=1
l
k
n
β
ψ(z) ∂ I − Km f ϕ(z) ∂ϕl (z) 1 − zk 2
sup
∂w
∂z
≤ sup
f α ≤1 z∈G1 k,l=1
l
k
β
ψ(z) ∂ I − Km f ϕ(z) ∂ϕl (z) 1 − zk 2
+ sup sup
∂wl
∂zk
f α ≤1 z∈G2 k,l=1
(2.27)
n
= I + II.
First, we write I as follows:
2 β n
1 − z k ∂ϕl
ψ(z)
(z)
I = sup sup
2 α ∂z
k
f α ≤1 z∈G1 k,l=1
1 − ϕl (z)
2 α
∂ I − Km f ϕ(z) ×
1 − ϕl (z)
∂w
(2.28)
l
and observe that this is bounded above by
2 β
n
1 − z k ∂ϕl
ψ(z)
nI − Km sup
2 α .
∂z (z) k
z∈G1 k,l=1
1 − ϕl (z)
(2.29)
ESSENTIAL NORM OF WEIGHTED COMPOSITION . . .
3949
Then, by property (d), we know that
2 β
n
1 − z k ∂ϕl
I ≤ 2n sup
(z) ψ(z) 2 α .
∂zk
z∈G1 k,l=1
1 − ϕ (z)
(2.30)
l
Next, we prove that limm→∞ II = 0. Since ψCϕ is bounded from Ꮾα (Dn ) to Ꮾβ (Dn ),
by Theorem 1.1 we have
2 β
n 1 − z k ∂ϕl
−
ψ(z)
2 α < ∞ |z| → 1 .
∂z (z) k
1 − ϕl (z)
k,l=1
(2.31)
Thus,
n ∂ϕl
2 β
< ∞.
sup ψ(z)
∂z (z) 1 − zk
z∈G2
k,l=1
(2.32)
k
Then, using property (c), we have
lim II = lim
m→∞
sup
n
∂ I − Km f 2 β
∂ϕl
ϕ(z)
sup
(z)
= 0.
ψ(z) 1 − z k ∂w
∂z
m→∞ f α ≤1 z∈G2 k,l=1
l
k
(2.33)
Combining the estimates for (2.21), (2.22), and (2.23) as m → ∞, we get
ψCϕ ≤ 2n lim
e
sup
δ→0 dist(ϕ(z),∂D n )<δ
2 β
n
1 − z k ψ(z) ∂ϕl (z) 2 α .
∂z
k
1 − ϕ (z)
k,l=1
(2.34)
l
Acknowledgments. The authors thank the referee for several helpful comments
and suggestions for improvements. The first author also thanks professor Zehua Zhou
for many help. The first author is supported in part by the National Natural Science
Foundation of China (no. 10371051) and the NSF of the Zhejiang province of China (no.
102025).
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3950
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Li Songxiao: Department of Mathematics, Jiaying University, Meizhou 514015, Guangdong,
China
E-mail address: lsx@mail.zjxu.edu.cn
Zhu Xiangling: Department of Mathematics, Shantou University, Shantou 515063, Guangdong,
China
E-mail address: g_xlzhu@stu.edu.cn