Weighted Composition Operators from H^∞ to the Bloch Space on the Polydisc

• Published in: Abstract and Applied Analysis Vol. 2007; pp. 280 - 292
• Main Authors: Li, Songxiao, Stevic, Stevo
• Format: Journal Article
• Language: English
• Published: Hindawi Limiteds 2007; Hindawi Publishing Corporation; Hindawi Limited
• ISSN: 1085-3375; 1687-0409
• DOI: 10.1155/2007/48478

Let
D
n
be the unit polydisc of
ℂ
n
,
ϕ
(
z
)
=
(
ϕ
1
(
z
)
,
…
,
ϕ
n
(
z
)
)
be a holomorphic self-map of
D
n
, and
ψ
(
z
)
a holomorphic function on
D
n
. Let
H
(
D
n
)
denote the space of all holomorphic functions with domain
D
n
,
H
∞
(
D
n
)
the space of all bounded holomorphic functions on
D
n
, and
B
(
D
n
)
the Bloch space, that is,
B
(
D
n
)
=
{
f
∈
H
(
D
n
)
|
‖
f
‖
B
=
|
f
(
0
)
|
+
sup
z
∈
D
n
∑
k
=
1
n
|
(
∂
f
/
∂
z
k
)
(
z
)
|
(
1
−
|
z
k
|
2
)
<
+
∞
}
. We give necessary and sufficient conditions for the weighted composition operator
ψ
C
ϕ
induced by
ϕ
(
z
)
and
ψ
(
z
)
to be bounded and compact from
H
∞
(
D
n
)
to the Bloch space
B
(
D
n
)
.