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THE ESSENTIAL NORMS OF COMPOSITION OPERATORS
BETWEEN GENERALIZED BLOCH SPACES IN THE POLYDISC
AND THEIR APPLICATIONS
ZEHUA ZHOU AND YAN LIU
Received 27 December 2005; Revised 26 June 2006; Accepted 22 July 2006
Let U
n
be the unit polydisc of C
n
and φ = (φ
1
, ,φ
n
) a holomorphic self-map of U
n
.
Ꮾ
p
(U
n
), Ꮾ
p
0
(U
n
), and Ꮾ
p
0
∗
(U
n
) denote the p-Bloch space, little p-Bloch space, and little
star p-Bloch space in the unit polydisc U
n
, respectively, where p, q>0. This paper gives
the estimates of the essential nor ms of bounded composition operators C
φ
induced by
φ between Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
)) and Ꮾ
q
(U
n
)(Ꮾ
q
0
(U
n
)orᏮ
q
0
∗
(U
n
)). As their
applications, some necessary and sufficient conditions for the (bounded) composition
operators C
φ
to be compact from Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
)) into Ꮾ
q
(U
n
)(Ꮾ
q
0
(U
n
)
or Ꮾ
q
0
∗
(U
n
)) are obtained.
Copyright © 2006 Z. Zhou and Y. Liu. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The class of all holomorphic functions with domain Ω will be denoted by H(Ω), where
Ω is a bounded homogeneous domain in
C
n
.Letφ be a holomorphic self-map of Ω,the
composition operator C
φ
induced by φ is defined by
C
φ
f
(z) = f
φ(z)
, (1.1)
for z in Ω and f
∈ H(Ω).
Let K(z,z)betheBergmankernelfunctionofΩ,theBergmanmetricH
z
(u,u)inΩ is
defined by
H
z
(u,u) =
1
2
n
j,k=1
∂
2
logK(z,z)
∂z
j
∂z
k
u
j
u
k
, (1.2)
where z
∈ Ω and u =(u
1
, ,u
n
) ∈ C
n
.
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 90742, Pages 1–22
DOI 10.1155/JIA/2006/90742
2 Essential norm of composition operators
Following Timoney [5], we say that f
∈ H(Ω)isintheBlochspaceᏮ(Ω)if
f
Ꮾ(Ω)
= sup
z∈Ω
Q
f
(z) < ∞, (1.3)
where
Q
f
(z) =sup
f (z)u
H
1/2
z
(u,u)
: u
∈ C
n
−{0}
, (1.4)
and
f (z) =(∂f(z)/∂z
1
, ,∂f(z)/∂z
n
), f (z)u =
n
l
=1
(∂f(z)/∂z
l
)u
l
.
ThelittleBlochspaceᏮ
0
(Ω) is the closure in the Banach space Ꮾ(Ω)ofthepolynomial
functions.
Let ∂Ω denote the boundary of Ω. Following Timoney [6], for Ω
= B
n
the unit ball of
C
n
, Ꮾ
0
(B
n
) ={f ∈ Ꮾ(B
n
):Q
f
(z) → 0, as z →∂B
n
};forΩ = Ᏸ the bounded symmetric
domain other than the ball B
n
, {f ∈ Ꮾ(Ᏸ):Q
f
(z) → 0, as z → ∂Ᏸ} is the set of constant
functions on Ᏸ.SoifᏰ is a bounded symmetric domain other than the ball, we denote
the Ꮾ
0∗
(Ᏸ) ={f ∈ Ꮾ(Ᏸ):Q
f
(z) → 0, as z → ∂
∗
Ᏸ} and call it little star Bloch space;
here ∂
∗
Ᏸ means the distinguished boundary of Ᏸ. The unit ball is the only bounded
symmetric domain Ᏸ with the property that ∂
∗
Ᏸ = ∂Ᏸ.
Let U
n
be the unit polydisc of C
n
.Timoney[5] shows that f ∈Ꮾ(U
n
)ifandonlyif
f
1
=
f (0)
+sup
z∈U
n
n
k=1
∂f
∂z
k
(z)
1 −
z
k
2
< +∞, (1.5)
where f
∈ H(U
n
).
This definition was the starting point for introducing the p-Bloch spaces.
Let p>0, a function f
∈ H(U
n
)issaidtobelongtothep-Bloch space Ꮾ
p
(U
n
)if
f
p
=
f (0)
+sup
z∈U
n
n
k=1
∂f
∂z
k
(z)
1 −
z
k
2
p
< +∞. (1.6)
It is an easy exercise to show that Ꮾ
p
(U
n
) is a Banach space with the norm ·
p
for p ≥1; and for 0 <p<1, Ꮾ
p
(U
n
) is a nonlocally convex topological vector space and
d( f ,g)
=f −g
p
p
is a complete metric for it. Its proof idea is basic, we refer the reader
to see the proof of Proposition 3.1 or the statement corresponding the Bloch-type space
for the unit ball in [13].
Just like Timoney [6], if
lim
z→∂U
n
n
k=1
∂f
∂z
k
(z)
1 −
z
k
2
p
= 0, (1.7)
it is easy to show that f must be a constant. Indeed, for fixed z
1
∈ U,(∂f/∂z
1
)(z)(1 −
|
z
1
|
2
)
p
is a holomorphic function in z
= (z
2
, ,z
n
) ∈ U
n−1
.Ifz → ∂U
n
,thenz
→ ∂U
n−1
,
which implies that
lim
z
→∂U
n−1
∂f
∂z
1
(z)
1 −
z
1
2
p
= 0. (1.8)
Z. Zhou and Y. Liu 3
Hence, (∂f/∂z
1
)(z)(1 −|z
1
|
2
)
p
≡ 0foreveryz
∈ ∂U
n−1
,andforeachz
1
∈ U, and con-
sequently (∂f/∂z
1
)(z) =0foreveryz ∈U
n
. Similarly, we can obtain that (∂f/∂z
j
)(z) =0
for every z
j
∈ U
n
and each j ∈{2, ,n}; therefore f ≡ const.
So, there is no sense to introduce the corresponding little p-Bloch space in this way.
We will say that the little p-Bloch space Ꮾ
p
0
(U
n
) is the cl osure of the polynomials in the
p-Bloch space. If f
∈ H(U
n
)and
sup
z∈∂
∗
U
n
n
k=1
∂f
∂z
k
(z)
1 −
z
k
2
p
= 0, (1.9)
we say f belongs to little star p-Bloch space Ꮾ
p
0
∗
(U
n
). Using the same methods as that
of [6, Theorem 4.15], we can show that Ꮾ
p
0
(U
n
)isapropersubspaceofᏮ
p
0
∗
(U
n
)and
Ꮾ
p
0
∗
(U
n
) is a nonseparable closed subspace of Ꮾ
p
(U
n
).
For the unit disc U
⊂ C, Madigan and Matheson [1]provedthatC
φ
is always bounded
on Ꮾ(U)andboundedonᏮ
0
(U)ifandonlyifφ ∈ Ꮾ
0
(U). They also gave the sufficient
and necessary conditions that C
φ
is compact on Ꮾ(U)orᏮ
0
(U).
The analogues of these facts for the unit polydisc and classical symmetric domains
were obtained by Zhou and Shi in [8–10]. The y had already shown that C
φ
is always
bounded on the Bloch space of these domains, and also gave some sufficient and necessary
conditions for C
φ
to be compact on those spaces. For the results on the unit ball, we refer
the reader to see [4, 12].
We recall that the essential norm of a continuous linear operator T is the distance from
T to the compact operators, that is,
T
e
= inf
T −K : K is compact
. (1.10)
Notice that
T
e
= 0ifandonlyifT is compact, so that estimates on T
e
lead to condi-
tions for T to be compact.
As we have known that C
φ
is always bounded on the Bloch space in the unit disc and
polydisc, in [2], Montes-Rodriguez gave the exact essential norm of a composition oper-
ator on the Bloch space in the disc and obtained a different proof for the corresponding
compactness results in [1]. After that, Zhou and Shi generalized Alsonso’s result to the
polydisc in [11].
In [7], Zhou stated and proved the corresponding compactness characterization for
Ꮾ
p
(U
n
)for0<p<1, however, C
φ
is not always bounded, and the test functions used
in [7] are only suitable for handling the case 0 <p<1. It is therefore natural to won-
der what results can be proven about boundedness and compactness of C
φ
on p-Bloch
spaces for an arbitrary positive number p or, more generally, between possibly different
p-andq-Bloch spaces of multivariable domains. In this paper, we answer these questions
completely for U
n
with essential norm approach, we give some estimates of the essen-
tial norms of bounded composition operators C
φ
between Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
))
and Ꮾ
q
(U
n
)(Ꮾ
q
0
(U
n
)orᏮ
q
0
∗
(U
n
)). Further, we apply t hese results to obtain some nec-
essary and sufficient conditions for the composition operators C
φ
to be compact from
Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
)) into Ꮾ
q
(U
n
)(Ꮾ
q
0
(U
n
)orᏮ
q
0
∗
(U
n
)). The fundamental
4 Essential norm of composition operators
ideas of the proof are those used by Shapiro [3] to obtain the essential norm of a com-
position operator on Hilbert spaces of analytic functions (Hardy and weighted Bergman
spaces) in terms of natural counting functions associated with φ. This paper generalizes
the results on the Bloch space for the unit disc in [2] and the unit polydisc in [11].
Throughout the remainder of this paper C will denote a positive constant, the exact
value of which will vary from one appearance to the next.
Our main results are the following.
Theorem 1.1. Let φ
= (φ
1
,φ
2
, ,φ
n
) be a holomorphic self-map of U
n
and C
φ
e
the
essent ial norm of a bounded composition operator C
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
) or Ꮾ
p
0
∗
(U
n
)) →
Ꮾ
q
(U
n
)(Ꮾ
q
0
(U
n
) or Ꮾ
q
0
∗
(U
n
)), then
1
n
lim
δ→0
sup
dist(φ(z),∂U
n
)<δ
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
≤
C
φ
e
≤ 2lim
δ→0
sup
dist(φ(z),∂U
n
)<δ
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
.
(1.11)
By Theorem 1.1 and the fact that C
φ
: Ꮾ
p
(U
n
)(orᏮ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
)) → Ꮾ
q
(U
n
)
(or Ꮾ
q
0
(U
n
)orᏮ
q
0
∗
(U
n
)) is compact if and only if C
φ
e
= 0, we obtain Theorem 1.2 at
once.
Theorem 1.2. Let φ
= (φ
1
, ,φ
n
) be a holomorphic self-map of U
n
. Then the bounded
composition operator C
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
) or Ꮾ
p
0
∗
(U
n
)) → Ꮾ
q
(U
n
)(Ꮾ
q
0
(U
n
) or Ꮾ
q
0
∗
(U
n
))
is compact if and only if for any ε>0, there exists a δ with 0 <δ<1, such that
sup
dist(φ(z),∂U
n
)<δ
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
<ε. (1.12)
Remark 1.3. When n
= 1, p = q = 1, on Ꮾ(U)weobtain[1, Theorem 2]. Since ∂U =
∂
∗
U, Ꮾ
0
(U) = Ꮾ
0∗
(U), we can also obtain [1,Theorem1].
Remark 1.4. When n>1, p
= q = 1, C
φ
is always bounded on Ꮾ(U
n
), so we can obtain
the corresponding results in [8, 11].
The remainder of the present paper is assembled as follows: in Section 2,westatesome
lemmas for the proof of Theorem 1.1. In terms of mapping properties of symbol φ,Lem-
mas 2.3, 2.4,and2.6 will give some conditions for C
φ
to be bounded between possibly
different p-andq-Bloch spaces, “little” or “little star” p-andq-Bloch spaces, the methods
used are different from that of [7], since the test functions used in [7]areonlysuitable
for handling the p-Blochspaceforthecase0<p<1, not others. In Section 3,wegive
the proof of Theorem 1.1.InSection 4, as applications of Theorems 1.1 and 1.2,wegive
some corollaries for C
φ
to be compact on those spaces.
Z. Zhou and Y. Liu 5
2. Some lemmas
In order to p rove Theorem 1.1, we need some lemmas.
Lemma 2.1. Let f
∈ Ꮾ
p
(U
n
), then
(1) if 0
≤ p<1, then |f (z)|≤|f (0)|+(n/(1 −p))f
p
;
(2) if p
= 1, then |f (z)|≤(1 + 1/nln2)(
n
k
=1
ln(2/(1 −|z
k
|
2
)))f
p
;
(3) if p>1, then
|f (z)|≤(1/n +2
p−1
/(p −1))
n
k
=1
(1/(1 −|z
k
|
2
)
p−1
)f
p
.
Proof. This Lemma can be easily obtained by some integral estimates, so we omit the
detail.
Lemma 2.2. For p>0,set
f
w
(z) =
z
l
0
dt
(1 −wt)
p
, (2.1)
where w
∈ U. Then f ∈ Ꮾ
p
0
(U
n
) ⊂ Ꮾ
p
0
∗
(U
n
) ⊂ Ꮾ
p
(U
n
).
Proof. Since
∂f
w
∂z
l
=
1 −wz
l
−p
,
∂f
w
∂z
i
= 0, i = l, (2.2)
it follows that
f (0)
+
n
k=1
∂f
w
∂z
k
(z)
1 −
z
k
2
p
=
1 −
z
l
2
p
1 −wz
l
p
≤
1+
z
l
p
≤ 2
p
. (2.3)
Hence f
w
∈ Ꮾ
p
(U
n
).
Now we prove that f
w
∈ Ꮾ
p
0
(U
n
). Using the asymptotic formula
(1
−wt)
−p
=
+∞
k=0
p(p +1)···(p + k −1)
k!
(
w)
k
t
k
, (2.4)
we obtain
f
w
(z) =
+∞
k=0
p(p +1)···(p + k −1)
k!
(
w)
k
z
l
0
t
k
dt. (2.5)
Denoting P
n
(z) =
n
k
=0
(p(p +1)···(p +k −1)/k!)(w)
k
z
l
0
t
k
dt,itiseasytoseethat
∂
f
w
−P
n
∂z
l
≤
+∞
k=n+1
p(p +1)···(p + k −1)
k!
|w|
k
−→ 0, as n −→ ∞ . (2.6)
6 Essential norm of composition operators
Thus
f
w
−P
n
p
=
f
w
(0) −P
n
(0)
+sup
z∈U
n
∂
f
w
−P
n
∂z
l
1 −
z
l
2
p
≤ sup
z∈U
n
∂
f
w
−P
n
∂z
l
−→
0,
(2.7)
which shows that f
w
∈ Ꮾ
p
0
(U
n
). So f ∈Ꮾ
p
0
(U
n
) ⊂ Ꮾ
p
0
∗
(U
n
) ⊂ Ꮾ
p
(U
n
).
Lemma 2.3. Let φ = (φ
1
, ,φ
n
) be a holomorphic self-map of U
n
, p, q>0. Then C
φ
:
Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
) or Ꮾ
p
0
∗
(U
n
)) →Ꮾ
q
(U
n
) is bounded if and only if there exists a constant
C such that
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
≤ C, (2.8)
for all z
∈ U
n
.
Proof. First a ssume that condition (2.8)holdsandlet f
∈ Ꮾ
p
(U
n
). By Lemma 2.1,we
know the evaluation at φ(0) is a bounded linear functional on Ꮾ
p
(U
n
), so |f (φ(0))|≤
Cf
p
.
On the other hand we have
n
k=1
∂
C
φ
f (z)
∂z
k
1 −
z
k
2
q
=
n
k=1
n
l=1
∂f
∂φ
l
φ(z)
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
≤
n
k,l=1
∂f
∂φ
l
φ(z)
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
≤
n
l=1
∂f
∂φ
l
φ(z)
1 −
φ
l
(z)
2
p
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
≤f
p
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
≤ Cf
p
.
(2.9)
So C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
)isbounded.
For the converse, assume that C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
) is bounded, with
C
φ
f
q
≤ Cf
p
(2.10)
for all f
∈ Ꮾ
p
(U
n
).
For fixed l (1
≤ l ≤ n), we will make use of a family of test functions {f
w
: w ∈ C, |w| <
1
} defined in Lemma 2.2.
Z. Zhou and Y. Liu 7
Since
f
w
∈ Ꮾ
p
0
U
n
⊂
Ꮾ
p
0
∗
U
n
⊂
Ꮾ
p
U
n
, (2.11)
it follows from (2.10)thatforz
∈ U
n
,
n
k=1
n
l=1
∂f
w
φ(z)
∂φ
l
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
≤ C. (2.12)
Let w
= φ
l
(z). Then
n
k=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
≤ C. (2.13)
The results are stated above for Ꮾ
p
(U
n
), but they also hold with minor modifications
for Ꮾ
p
0
(U
n
)andᏮ
p
0
∗
(U
n
). Now the proof of Lemma 2.3 is completed.
Lemma 2.4. Let φ = (φ
1
,φ
2
, ,φ
n
) be a holomorphic self-map of U
n
. Then C
φ
: Ꮾ
p
0
∗
(U
n
)(Ꮾ
p
0
(U
n
)) →Ꮾ
q
0
∗
(U
n
) is bounded if and only if φ
l
∈ Ꮾ
q
0
∗
(U
n
) for every l = 1,2, ,n
and (2.8) holds.
Proof. If C
φ
: Ꮾ
p
0
∗
(U
n
)(Ꮾ
p
0
(U
n
)) → Ꮾ
q
0
∗
(U
n
) is bounded, it is clear that, for every l =
1,2, ,n, f
l
(z) =z
l
∈ Ꮾ
p
0
(U
n
) ⊂ Ꮾ
q
0
∗
(U
n
), so C
φ
f
l
= φ
l
∈ Ꮾ
q
0
∗
(U
n
). Furthermore, (2.12)
holds by Lemma 2.3.
Inordertoprovetheconverse,wefirstprovethatifφ
l
∈ Ꮾ
q
0
∗
(U
n
), for every l =
1,2, ,n,then f ◦φ ∈Ꮾ
q
0
∗
(U
n
)forany f ∈Ꮾ
p
0
∗
(U
n
).
Without loss of generality, we prove this result when n
= 2.
For any sequence
{z
j
= (z
j
1
,z
j
2
)}⊂U
n
with z
j
→ ∂
∗
U
n
as j →∞,then
z
j
1
−→
1,
z
j
2
−→
1. (2.14)
Since
|φ
1
(z
j
)| < 1and|φ
2
(z
j
)| < 1, there exists a subsequence {z
j
s
} in {z
j
} such that
φ
1
z
j
s
−→
ρ
1
,
φ
2
z
j
s
−→
ρ
2
, (2.15)
as s
→∞.
8 Essential norm of composition operators
It is clear that 0
≤ ρ
1
, ρ
2
≤ 1. Then for k = 1,2, we have
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
≤
∂f
∂w
1
φ
z
j
s
∂φ
1
∂z
k
z
j
s
1 −
z
j
s
k
2
q
+
∂f
∂w
2
φ
z
j
s
∂φ
2
∂z
k
z
j
s
1 −
z
j
s
k
2
q
=
∂f
∂w
1
φ
z
j
s
1 −
φ
1
z
j
s
2
p
∂φ
1
∂z
k
z
j
s
1 −
z
j
s
k
2
q
1 −
φ
1
z
j
s
2
p
+
∂f
∂w
2
φ
z
j
s
1 −
φ
2
z
j
s
2
p
∂φ
2
∂z
k
z
j
s
1 −
z
j
s
k
2
q
1 −
φ
2
z
j
s
2
p
.
(2.16)
Now we prove the left-hand side of (2.16)
→ 0ass →∞according to four cases.
Case 1. If ρ
1
< 1andρ
2
< 1, there exist r
1
and r
2
such that ρ
1
<r
1
< 1andρ
2
<r
2
< 1, so
as j is large enough,
|φ
1
(z
j
s
)|≤r
1
and |φ
2
(z
j
s
)|≤r
2
.
Since φ
1
,φ
2
∈ Ꮾ
q
0
∗
(U
n
), by (2.16), we get
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
≤f
p
1
1 −r
2
1
p
∂φ
1
∂z
k
z
j
s
1 −
z
j
s
k
2
q
+ f
p
1
1 −r
2
2
p
∂φ
2
∂z
k
z
j
s
1 −
z
j
s
k
2
q
−→ 0
(2.17)
as s
→∞.
Case 2. If ρ
1
= 1andρ
2
= 1, then φ(z
j
s
) → ∂
∗
U
n
,by(2.8) and, since f ∈ Ꮾ
p
0
∗
(U
n
), (2.16)
yields that
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
≤ C
∂f
∂w
1
φ
z
j
s
1 −
φ
1
z
j
s
2
p
+ C
∂f
∂w
2
φ
z
j
s
1 −
φ
2
z
j
s
2
p
−→ 0
(2.18)
as s
→∞.
Z. Zhou and Y. Liu 9
Case 3. If ρ
1
< 1andρ
2
= 1, similarly to Case 1,wecanprovethat
∂f
∂w
1
φ
z
j
s
1 −
φ
1
z
j
s
2
p
∂φ
1
∂z
k
z
j
s
1 −
z
j
s
k
2
q
1 −
φ
1
z
j
s
2
p
≤f
p
1
1 −r
2
1
p
∂φ
1
∂z
k
z
j
s
1 −
z
j
s
k
2
q
1 −
φ
1
z
j
s
2
p
−→ 0
(2.19)
as s
→∞.
On the other hand, for fixed s,letw
j
s
2
= φ
2
(z
j
s
). Then |w
j
s
2
| < 1. Denote
F
w
1
=
∂f
∂w
2
w
1
,w
j
s
2
. (2.20)
It is clear that F(w
1
) is holomorphic on |w
1
| < 1. Choosing R
j
s
→ 1withr
1
≤ R
j
s
< 1.
|φ
1
(z
j
s
)|≤r
1
,so
F
φ
1
z
j
s
≤
max
|w
1
|≤r
1
F
w
1
≤
max
|w
1
|≤R
j
s
F
w
1
=
max
|w
1
|=R
j
s
F
w
1
=
F
w
j
s
1
,
(2.21)
where w
j
s
1
is a point of modulus R
j
s
where maximum of F(w
1
) is attained. This means
that
|(∂f/∂w
2
)(φ
1
(z
j
s
),φ
2
(z
j
s
))|≤|(∂f/∂w
2
)(w
j
s
1
,w
j
s
2
)|.Since|w
j
s
1
|→1, |w
j
s
2
|→ρ
2
= 1
and f
∈ Ꮾ
p
0
∗
(U
n
),
∂f
∂w
2
w
j
s
1
,w
j
s
2
1 −
w
j
s
2
2
p
−→ 0 (2.22)
as s
→∞,soby(2.8),
∂f
∂w
2
φ
z
j
s
1 −
φ
2
z
j
s
2
p
∂φ
2
∂z
k
z
j
s
1 −
z
j
s
k
2
q
1 −
φ
2
z
j
s
2
p
≤ C
∂f
∂w
2
w
j
s
1
,w
j
s
2
1 −
w
j
s
2
2
p
−→ 0
(2.23)
as s
→∞.
By (2.19)and(2.23), (2.16)yields
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
−→ 0, (2.24)
as s
→∞.
Case 4. If ρ
1
= 1andρ
2
< 1, similarly to Case 3,wecanprove
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
−→ 0, (2.25)
as s
→∞.
10 Essential norm of composition operators
Combining Cases 1, 2, 3,and4, we know there exists a subsequence
{z
j
s
} in {z
j
} such
that
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
−→ 0, (2.26)
as s
→∞for k =1,2. We claim that
∂( f ◦φ)
∂z
k
z
j
1 −
z
j
k
2
q
−→ 0, (2.27)
as j
→∞. In fact, if it fails, then there exists a subsequence {z
j
s
} such that
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
−→ ε>0 (2.28)
for k
= 1 or 2. But from the above discussion, we can find a subsequence in {z
j
s
};westill
write
{z
j
s
} with
∂( f ◦φ)
∂z
k
z
j
s
1 −
z
j
s
k
2
q
−→ 0, (2.29)
it contradicts with (2.28).
So for any sequence
{z
j
}⊂U
n
with z
j
→ ∂
∗
U
n
as j →∞,wehave
∂( f ◦φ)
∂z
k
z
j
1 −
z
j
k
2
q
−→ 0 (2.30)
for k
= 1,2. By (2.8)andLemma 2.3, it is clear that f ◦φ ∈Ꮾ
q
(U
n
), so f ◦φ ∈ Ꮾ
q
0
∗
(U
n
).
For any f
∈ Ꮾ
p
0
(U
n
). Since Ꮾ
p
0
(U
n
) ⊂ Ꮾ
p
0
∗
(U
n
), then f ◦φ ∈ Ꮾ
q
0
∗
(U
n
).
By closed graph theorem, we know that
C
φ
: Ꮾ
p
0
∗
U
n
Ꮾ
p
0
U
n
−→
Ꮾ
q
0
∗
U
n
(2.31)
is bounded. This ends the proof of Lemma 2.4.
Remark 2.5. For the case C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
0
∗
(U
n
), the necessity also holds, but we cannot
guarantee that the sufficiency holds because we cannot be sure that C
φ
f ∈ Ꮾ
q
0
∗
(U
n
)for
all f
∈ Ꮾ
p
(U
n
).
Lemma 2.6. Let φ
= (φ
1
,φ
2
, ,φ
n
) be a holomorphic self-map of U
n
. Then
C
φ
: Ꮾ
p
0
U
n
−→
Ꮾ
q
0
U
n
(2.32)
is bounded if and only if φ
γ
∈ Ꮾ
q
0
(U
n
) for every multiindex γ,and(2.8)holds.
Proof (sufficiency). From (2.8)andbyLemma 2.3 we k now that C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
)
is bounded, in particular
C
φ
f
q
≤
C
φ
Ꮾ
p
(U
n
)→Ꮾ
q
(U
n
)
f
p
, ∀f ∈ Ꮾ
p
0
U
n
. (2.33)
Z. Zhou and Y. Liu 11
The boundedness of C
φ
: Ꮾ
p
0
(U
n
) → Ꮾ
q
0
(U
n
) directly follows, if we prove C
φ
f ∈ Ꮾ
q
0
(U
n
)
whenever f
∈ Ꮾ
p
0
(U
n
). So, let f ∈ Ꮾ
p
0
(U
n
). By the definition of Ꮾ
p
0
(U
n
) it follows that
for every ε>0 there is a polynomial p
ε
such that f − p
ε
p
<ε.Hence
C
φ
f −C
φ
p
ε
q
≤
C
φ
Ꮾ
p
(U
n
)→Ꮾ
q
(U
n
)
f − p
ε
p
<ε
C
φ
Ꮾ
p
(U
n
)→Ꮾ
q
(U
n
)
. (2.34)
Since φ
γ
∈ Ꮾ
q
0
(U
n
) for every multiindex γ,weobtainC
φ
p
ε
∈ Ꮾ
q
0
(U
n
). From this and
(2.34) the result follows.
If C
φ
: Ꮾ
p
0
(U
n
) → Ꮾ
q
0
(U
n
)isbounded,then(2.8) can be proved as in Lemma 2.3, since
the test functions appearing there belong to Ꮾ
p
0
(U
n
). Since the polynomials z
γ
∈ Ꮾ
p
0
(U
n
)
for every multiindex γ,wegetC
φ
z
γ
∈ Ꮾ
q
0
(U
n
), as desired.
Remark 2.7. For the case C
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
∗
(U
n
)) → Ꮾ
q
0
(U
n
), in analogy to Remark 2.5,
the necessity also holds, but we cannot guarantee that the sufficiency holds.
Lemma 2.8. If
{f
k
} is a bounded sequence in Ꮾ
p
(U
n
),thenthereexistsasubsequence{f
k
l
}
of {f
k
} which converges uniformly on compact subsets of U
n
to a holomorphic function f ∈
Ꮾ
p
(U
n
).
Proof. Let
{f
k
} be a bounded sequence in Ꮾ
p
(U
n
)withf
k
p
≤ C.ByLemma 2.1, {f
j
}
is uniformly bounded on compact subsets of U
n
and hence normal by Montel’s theorem.
So we may extract a subsequence
{f
j
k
} which converges uniformly on compact subsets of
U
n
to a holomorphic function f . It follows that ∂f
j
k
/∂z
l
→ ∂f/∂z
l
for each l ∈{1,2, ,n},
so
n
l=1
∂f
∂z
l
1 −
z
l
2
p
= lim
k→∞
n
l=1
∂f
j
k
∂z
l
1 −
z
l
2
p
≤ sup
k
f
j
k
p
≤ C, (2.35)
which implies f
∈ Ꮾ
p
(U
n
). The Lemma is proved.
Lemma 2.9. Let Ω be a domain in C
n
, f ∈ H(Ω). If a compact set K and its ne ighborhood
G satisfy K
⊂ G ⊂G ⊂Ω and ρ =dist(K,∂G) > 0, then
sup
z∈K
∂f
∂z
j
(z)
≤
√
n
ρ
sup
z∈G
f (z)
. (2.36)
Proof. For any a
∈ K, the polydisc
P
a
=
z
1
, ,z
n
∈ C
n
:
z
j
−a
j
<
ρ
√
n
, j
= 1, ,n
(2.37)
is contained in G. By Cauchy’s inequality,
∂f
∂z
j
(a)
≤
√
n
ρ
sup
z∈∂
∗
P
a
f (z)
≤
√
n
ρ
sup
z∈G
f (z)
. (2.38)
Taking the supremum for a over K gives the desired inequality.
12 Essential norm of composition operators
3. The proof of Theorem 1.1
NowweturntotheproofofTheorem 1.1. In the following, we are dealing with the case
for C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
), but if we note that the test functions f
m
introduced below be-
long to Ꮾ
p
0
(U
n
) ⊂Ꮾ
p
0∗
(U
n
) ⊂Ꮾ
p
(U
n
), the results in Theorem 1.1 also hold w ith minor
modifications for the other cases.
We begin by proving the lower estimate. It is clear that
{m
p−1
z
m
1
}⊂Ꮾ
p
0
(U
n
) ⊂
Ꮾ
0∗
(U
n
) ⊂ Ꮾ(U
n
)form = 1,2, , and this sequence converges to zero uniformly on
compact subsets of the unit polydisc U
n
.Furthermore
m
p−1
z
m
1
p
= sup
z∈U
n
1 −
z
1
2
p
m
p
z
1
m−1
. (3.1)
Let p(x)
= m
p
(1 −x
2
)
p
x
m−1
,then
p
(x) =−m
p
x
m−2
1 −x
2
p−1
(2p + m −1)x
2
−(m −1)
, (3.2)
so
p
(x) ≤0forx ∈
(m −1)/(2p + m −1),1
,
p
(x) ≥0forx ∈
0,
(m −1)/(2p + m −1)
.
(3.3)
That is, p(x) is a decreasing function for x
∈ [
(m −1)/(2p + m −1),1] and p(x)isan
increasing function for x
∈ [0,
(m −1)/(2p + m −1)]. Hence
max
x∈[0,1]
p(x) = p
m −1
2p + m −1
. (3.4)
It follows from (3.1)that
m
p−1
z
m
1
p
= p
m −1
2p + m −1
=
2p
2p + m −1
p
m
p
m −1
2p + m −1
(m−1)/2
−→
2p
e
p
,
(3.5)
as m
→∞.
Therefore, the sequence
{m
p−1
z
m
1
}
m≥2
is bounded away from zero. Now we consider
the normalized sequence
{f
m
= m
p−1
z
m
1
/m
p−1
z
m
1
p
} which also tends to zero uniformly
on compact subsets of U
n
.Foreachm ≥ 2, we define
A
m
=
z =
z
1
, ,z
n
∈
U
n
: r
m
≤
z
1
≤
r
m+1
, (3.6)
Z. Zhou and Y. Liu 13
where r
m
=
(m −1)/(2p + m −1). So
min
A
m
n
l=1
∂f
m
∂z
l
(z)
1 −
z
l
2
p
=
min
A
m
∂f
m
∂z
1
1 −
z
1
2
p
=
1 −r
2
m+1
p
m
p
r
m−1
m+1
m
p−1
z
m
1
p
=
2p + m −1
2p + m
m(2p + m −1)
(m −1)(2p + m)
((m−1)/2)
= c
m
.
(3.7)
It is easy to show that c
m
tendsto1asm →∞. For the moment fix any compact opera-
tor K : Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
). The uniform convergence on compact subsets of the sequence
{f
m
} to zero and the compactness of K imply that Kf
m
q
→ 0. It is easy to show that if
a bounded sequence that is contained in Ꮾ
p
0
∗
(U
n
) converges uniformly on compact sub-
sets of U
n
, then it also converges weakly to zero in Ꮾ
p
0
∗
(U
n
)aswellasinᏮ
p
(U
n
). Since
f
m
p
= 1, we have
C
φ
−K
≥
limsup
m
C
φ
−K
f
m
q
≥ limsup
m
C
φ
f
m
q
−
Kf
m
q
=
limsup
m
C
φ
f
m
q
≥ limsup
m
sup
z∈U
n
n
k=1
∂
f
m
◦φ
∂z
k
(z)
1 −
z
k
2
q
=
limsup
m
sup
z∈U
n
n
k=1
∂f
m
∂w
1
φ(z)
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
= limsup
m
sup
z∈U
n
n
k=1
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
1
(z)
2
p
∂f
m
∂w
1
φ(z)
1 −
φ
1
(z)
2
p
≥ limsup
m
sup
φ(z)∈A
m
n
k=1
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
1
(z)
2
p
∂f
m
∂w
1
φ(z)
1 −
φ
1
(z)
2
p
≥ limsup
m
sup
φ(z)∈A
m
n
k=1
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
1
(z)
2
p
×liminf
m
min
φ(z)∈A
m
∂f
m
∂w
1
φ(z)
1 −
φ
1
(z)
2
p
14 Essential norm of composition operators
≥ limsup
m
sup
φ(z)∈A
m
n
k=1
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
1
(z)
2
p
liminf
m
c
m
≥ limsup
m
sup
φ(z)∈A
m
n
k=1
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
1
(z)
2
p
.
(3.8)
So
C
φ
e
= inf
C
φ
−K
: K is compact
≥
limsup
m
sup
φ(z)∈A
m
n
k=1
∂φ
1
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
1
(z)
2
p
.
(3.9)
For each l
= 1,2, ,n,define
a
l
= lim
δ→0
sup
dist(φ(z),∂U
n
)<δ
n
k=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
. (3.10)
For any ε>0, (3.10) shows that there exists a δ
0
with 0 <δ
0
< 1, such that
n
k=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
>a
l
−ε, (3.11)
whenever dist(φ(z),∂U
n
) <δ
0
and l =1,2, ,n.
Since r
m
→ 1asm →∞,wemaychoosem large enough so that r
m
> 1 −δ
0
.Ifφ(z) ∈
A
m
, r
m
≤|φ
1
(z)|≤r
m+1
,so1−r
m+1
< 1 −|φ
1
(z)| < 1 −r
m
<δ
0
; hence dist(φ
1
(z), ∂U) <
δ
0
. There exists w
1
with |w
1
|=1 such that dist(φ
1
(z), w
1
) = dist(φ
1
(z), ∂U) <δ
0
.
Let w
= (w
1
,φ
2
(z), , φ
n
(z)) ∈∂U
n
.Then
dist
φ(z),∂U
n
≤
dist
φ(z),w
=
dist
φ
1
(z), w
1
<δ
0
. (3.12)
By (3.11), (3.9) implies that
C
φ
e
≥ a
1
−ε. (3.13)
Similarly, if we choose g
m
(z) =m
p−1
z
m
l
/m
p−1
z
m
l
,wehave
C
φ
e
≥ a
l
−ε, (3.14)
Z. Zhou and Y. Liu 15
for every l
= 2 ,n.So
C
φ
e
≥
1
n
n
l=1
a
l
−ε
=
1
n
n
l=1
⎛
⎜
⎝
lim
δ→0
sup
dist(φ(z),∂U
n
)<δ
n
k=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
−ε
⎞
⎟
⎠
≥
1
n
lim
δ→0
sup
dist(φ(z),∂U
n
)<δ
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
−ε.
(3.15)
Let ε
→ 0, the low estimate follows.
To obtain the upper estimate we first prove the following proposition.
Proposition 3.1. Let φ
= (φ
1
, ,φ
n
) be a holomorphic self-map of U
n
.Thenform ≥ 2,the
operator K
m
on H(U
n
) defined by K
m
f (z) = f (((m −1)/m)z) has the following properties.
For each f
∈ H(U
n
),
(i) K
m
f ∈ Ꮾ
p
0
(U
n
) ⊂ Ꮾ
p
0
∗
(U
n
) ⊂ Ꮾ
p
(U
n
);
(ii) if C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
) is bounded, then C
φ
K
m
f ∈ Ꮾ
q
(U
n
);
(iii) for fixed m,theoperatorK
m
is compact on Ꮾ
p
(U
n
);
(iv) if C
φ
: Ꮾ
p
(U
n
) → Ꮾ
q
(U
n
) is bounded, then C
φ
K
m
f ∈ Ꮾ
q
(U
n
) is compact;
(v)
I −K
m
≤2;
(vi) (I
−K
m
) f converges to zero uniformly on compacta in U
n
.
Proof. (i) Let f
∈ H(U
n
), r
m
= (m −1)/m,and f
m
(z) =K
m
f (z) = f (r
m
z). First note that
f
m
p
=
f (0)
+sup
z∈U
n
n
k=1
r
m
∂f
∂z
k
r
m
z
1 −
z
k
2
p
≤
f (0)
+sup
z∈U
n
n
k=1
∂f
∂z
k
r
m
z
1 −
r
m
z
k
2
p
≤f
p
.
(3.16)
On the other hand, f
m
∈ H((1/r
m
)U
n
), and observe that (2/(1 + r
m
))U
n
⊂ (1/r
m
)U
n
which implies that for fixed m, corresponding to each j =1,2, , there is a polynomial
P
(j)
m
such that
sup
z∈(2/(1+r
m
))U
n
f
m
(z) −P
(j)
m
(z)
<
1 −r
m
2
1
j
. (3.17)
Let K
= U
n
, G = (2/(1 + r
m
))U
n
, Ω = (1/r
m
)U
n
,thenK ⊂ G ⊂ G ⊂ Ω and ρ =
dist(K,∂G) =(1 − r
m
)/(1 + r
m
) > 0, so for all w ∈ U
n
, k ∈{1, ,n},itfollowsfrom
16 Essential norm of composition operators
Lemma 2.9 that
∂
f
m
−P
(j)
m
∂w
k
(w)
≤
sup
w∈K
∂
f
m
−P
(j)
m
∂w
k
(w)
≤
√
n
1+r
m
1 −r
m
sup
w∈G
f
m
(w) −P
(j)
m
(w)
≤
√
n
1+r
m
1 −r
m
1 −r
2
m
1
j
≤ 4
√
n
1
j
.
(3.18)
Therefore
n
k=1
∂
f
m
−P
(j)
m
∂w
k
(w)
1 −
w
k
2
p
≤ 4n
√
n
1
j
−→ 0 (3.19)
as j
→∞, that is,
f
m
−P
(j)
m
Ꮾ
p
=
f
m
(0) −P
(j)
m
(0)
+sup
w∈U
n
n
k=1
∂
f
m
−P
(j)
m
∂w
k
(w)
1 −
w
k
p
p
−→ 0.
(3.20)
P
(j)
m
(w) ∈ Ꮾ
p
0
(U
n
) implies that f
m
∈ Ꮾ
p
0
(U
n
).
(ii) follows immediately from (i).
(iii) For any sequence
{f
j
}⊂Ꮾ
p
(U
n
)withf
j
p
≤ M, by (i), {K
m
f
j
}∈Ꮾ
p
0
(U
n
). By
Lemma 2.8, there is a subsequence
{f
j
s
} of {f
j
} which converges uniformly on compact
subsets of U
n
to a holomorphic function f ∈ Ꮾ
p
(U
n
)andf
p
≤ M. The sequence
{∂f
j
s
/∂z
i
}, i =1,2, ,n, also converges uniformly on compact subsets of U
n
to the holo-
morphic function ∂f/∂z
i
.Soass is large enough, for any w ∈E ={((m −1)/m)z : z ∈
U
n
}⊂U
n
,
∂
f
j
s
− f
∂w
l
(w)
<ε, (3.21)
for every l
= 1,2, ,n.So
K
m
f
j
s
−K
m
f
p
=
f
j
s
m −1
m
z
−
f
m −1
m
z
p
= sup
z∈U
n
n
k=1
∂
f
j
s
− f
(m −1)/m
z
∂z
k
1 −
z
k
2
p
+
f
j
s
(0) − f (0)
Z. Zhou and Y. Liu 17
≤ sup
z∈U
n
n
k=1
n
l=1
∂
f
j
s
− f
∂w
l
m −1
m
z
m −1
m
+
f
j
s
(0) − f (0)
≤ nsup
w∈E
m −1
m
n
l=1
∂
f
j
s
− f
∂w
l
(w)
+
f
j
s
(0) − f (0)
−→
0,
(3.22)
as s
→∞. This shows that {K
m
f
j
s
}converges to g =K
m
f ∈ Ꮾ
p
0
(U
n
)⊂Ꮾ
p
0
∗
(U
n
)⊂Ꮾ
p
(U
n
).
So K
m
is compact on Ꮾ
p
(U
n
).
(iv) follows immediately from (i) and (iii).
(v) follows from the fact that for any f
∈ Ꮾ
p
(U
n
), (I −K
m
) f (0) = 0, so
I −K
m
f
p
= sup
z∈U
n
n
k=1
∂
I −K
m
f
∂z
k
(z)
1 −
z
k
2
p
= sup
z∈U
n
n
k=1
∂f
∂z
k
(z) −
1 −
1
m
∂f
∂z
k
1 −
1
m
z
1 −
z
k
2
p
≤ sup
z∈U
n
n
k=1
∂f
∂z
k
(z)
1 −
z
k
2
p
+
1 −
1
m
sup
z∈U
n
n
k=1
∂f
∂z
k
1 −
1
m
z
1 −
1 −
1
m
z
k
2
p
≤f
p
+ f
p
= 2f
p
,
(3.23)
so
I −K
m
≤2.
(vi) For any compact subset E
⊂ U
n
, there exists r,0<r<1suchthatE ⊂ rU
n
⊂
rU
n
⊂ U
n
.Forallz ∈E,
I −K
m
f (z)
=
f (z) − f
m
(z)
=
f (z) − f
r
m
z
≤
n
k=1
1
r
m
∂f
∂w
k
(tz)
dt.
(3.24)
For t
∈ [r
m
,1] and z ∈ E,wehave|tz
k
|=t|z
k
|≤|z
k
| <r, tz ∈ rU
n
, so there exists M>0
such that
|(∂f/∂w
k
)(tz)|≤M for all t ∈[r
m
,1] and z ∈ E.Thus
I −K
m
f (z)
≤
nM
1 −r
m
−→
0 (3.25)
as m
→∞, proving the results in Theorem 1.1.
18 Essential norm of composition operators
Let us now return to the proof of the upper estimate. For convenience, we remove the
subscript p from
f
p
,
C
φ
e
≤
C
φ
−C
φ
K
m
=
C
φ
I −K
m
=
sup
f =1
C
φ
I −K
m
f
q
= sup
f =1
sup
z∈U
n
n
k=1
∂
I −K
m
( f ◦φ)
∂z
k
1 −
z
k
2
q
+
I −K
m
f
φ(0)
≤
sup
f =1
sup
z∈U
n
n
k=1
n
l=1
∂
I −K
m
f
∂w
l
φ(z)
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
+sup
f =1
f
φ(0)
−
f
m −1
m
φ(0)
≤
sup
f =1
sup
z∈U
n
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
∂
I −K
m
f
∂w
l
φ(z)
1 −
φ
l
(z)
2
p
+sup
f =1
f
φ(0)
−
f
m −1
m
φ(0)
.
(3.26)
Fix δ>0, let G
1
={z ∈ U
n
: dist(φ(z),∂U
n
) <δ}, G
2
={z ∈ U
n
: dist(φ(z),∂U
n
) ≥ δ},
G
={w ∈ U
n
: dist(w,∂U
n
) ≥ δ}, and observe that G is a compact subset of C
n
.
Then by Lemmas 2.3, 2.4,and2.6,andbyProposition 3.1,wededuce
C
φ
e
≤ sup
f =1
sup
z∈G
1
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
∂
I −K
m
f
∂w
l
φ(z)
1 −
φ
l
(z)
2
q
+ C sup
f =1
sup
z∈G
2
n
l=1
1 −
φ
l
(z)
2
p
∂
I −K
m
f
∂w
l
φ(z)
+sup
f =1
f
φ(0)
−
f
m −1
m
φ(0)
≤
I −K
m
sup
z∈G
1
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
+ C sup
f =1
sup
z∈G
2
n
l=1
1 −
φ
l
(z)
2
p
∂
I −K
m
f
∂w
l
φ(z)
+sup
f =1
f
φ(0)
−
f
m −1
m
φ(0)
Z. Zhou and Y. Liu 19
≤ 2sup
z∈G
1
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
+ C sup
f =1
sup
z∈G
2
n
l=1
1 −
φ
l
(z)
2
p
∂
I −K
m
f
∂w
l
φ(z)
+sup
f =1
f
φ(0)
−
f
m −1
m
φ(0)
.
(3.27)
Denoting the second term and third term of the right-hand side of (3.27)byI
1
and I
2
,
then Theorem 1.1 is proved if we can prove
lim
m→∞
I
1
= 0, lim
m→∞
I
2
= 0. (3.28)
To do this, let z
∈ G
2
and w =φ(z) ∈G.Then
I
1
≤ C sup
f =1
sup
w∈G
n
l=1
1 −
w
l
2
p
∂f
∂w
l
(w) −
1 −
1
m
∂f
∂w
l
1 −
1
m
w
≤
C sup
f =1
sup
w∈G
n
l=1
1 −
w
l
2
p
∂f
∂w
l
(w) −
∂f
∂w
l
1 −
1
m
w
+
C
m
sup
f =1
sup
w∈G
n
l=1
1 −
w
l
2
p
∂f
∂w
l
1 −
1
m
w
≤
C sup
f =1
sup
w∈G
n
l=1
1 −
w
l
2
p
∂f
∂w
l
(w) −
∂f
∂w
l
1 −
1
m
w
+
C
m
.
(3.29)
Letting w
= (w
1
,w
2
, ,w
n−1
,w
n
), for m largeenough,wehave
∂f
∂w
l
(w) −
∂f
∂w
l
1 −
1
m
w
≤
n
j=1
∂f
∂w
l
1 −
1
m
w
1
, ,
1 −
1
m
w
j−1
,w
j
, ,w
n
−
∂f
∂w
l
1 −
1
m
w
1
, ,
1 −
1
m
w
j
,w
j+1
, ,w
n
=
n
j=1
w
j
(1−(1/m))w
j
∂
2
f
∂w
l
∂w
j
1 −
1
m
w
1
, ,
1 −
1
m
w
j−1
,ζ,w
j+1
, ,w
n
dζ
≤
1
m
n
j=1
sup
w∈G
∂
2
f
∂w
l
∂w
j
(w)
.
(3.30)
Denote G
3
by the set {w ∈ U
n
: dist(w,∂U
n
) >δ/2}.ThenG ⊂G
3
⊂ G
3
⊂ U
n
.
20 Essential norm of composition operators
Since dist(G,∂G
3
) = δ/2, then by Lemma 2.9,(3.30)gives
∂f
∂w
l
(w) −
∂f
∂w
l
1 −
1
m
w
≤
2n
√
n
mδ
max
z∈G
3
∂f
∂w
l
(w)
. (3.31)
On the other hand, on the unit ball of Ꮾ
p
(U
n
), we have
sup
z∈G
3
1 −
w
l
2
p
∂f
∂w
l
(w)
=
sup
dist(w,∂U
n
)>δ/2
1 −
w
l
2
p
∂f
∂w
l
(w)
≤
f
p
= 1, (3.32)
namely,
sup
z∈G
3
∂f
∂w
l
(w)
≤
1
1 −(δ/2)
2
p
=
4
p
(4 −δ
2
)
p
. (3.33)
Combining (3.29), (3.31), and (3.33)), it follows that
I
1
≤
2n
√
nC
mδ
4
p
4 −δ
2
p
+
C
m
(3.34)
and lim
m→∞
I
1
= 0.
Now we can prove lim
m→∞
I
2
= 0. In fact,
f
φ(0)
−
f
m −1
m
φ(0)
=
1
(m
−1)/m
df
tφ(0)
dt
dt
=
n
l=1
1
(m
−1)/m
φ
l
(0)
∂f
∂ζ
l
tφ(0)
dt.
(3.35)
By Lemma 2.1, it follows that for any compact subset K
⊂ U
n
, |f (z)|≤C
K
f
p
= C
K
.
Let K
={z ∈ U
n
: |z
i
|≤|φ
i
(0)|, i =1, ,n},So
f
φ(0)
−
f
m −1
m
φ(0)
≤
n
l=1
φ
l
(0)
1
(m
−1)/m
C
K
dt ≤ nC
K
1 −
m −1
m
=
nC
K
m
,
(3.36)
so I
2
≤ nC
K
/m → 0. Thus letting first m →∞and then δ →0in(3.27), we get the upper
estimate of
C
φ
e
:
C
φ
e
≤ 2lim
δ→0
sup
dist(φ(z),∂U
n
)<δ
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
. (3.37)
Now the proof of Theorem 1.1 is finished.
Z. Zhou and Y. Liu 21
4. Some corollaries
The following three corollaries follow from Theorem 1.2.
Corollary 4.1. Let φ
= (φ
1
, ,φ
n
) be a holomorphic self-map of U
n
. Then C
φ
:
Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
) or Ꮾ
p
0
∗
(U
n
)) → Ꮾ
q
(U
n
) is compact if and only if
n
k,l=1
∂φ
l
∂z
k
(z)
1 −
z
k
2
q
1 −
φ
l
(z)
2
p
≤ C (4.1)
for all z
∈ U
n
and (1.12)holds.
Proof. By Lemma 2.3,weknowC
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
))→Ꮾ
q
(U
n
) is bounded.
It follows from Theorem 1.2 that C
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
))→ Ꮾ
q
(U
n
)iscom-
pact.
Conversely, if C
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
)) → Ꮾ
q
(U
n
)iscompact,itisclearthat
C
φ
: Ꮾ
p
(U
n
)(Ꮾ
p
0
(U
n
)orᏮ
p
0
∗
(U
n
)) → Ꮾ
q
(U
n
) is bounded, by Theorem 1.2,(1.12)holds.
Corollary 4.2. Let φ = (φ
1
, ,φ
n
) be a holomorphic self-map of U
n
. Then C
φ
:
Ꮾ
p
0
∗
(U
n
)(Ꮾ
p
0
(U
n
)) → Ꮾ
q
0
∗
(U
n
) is compact if and only if φ
l
∈ Ꮾ
q
0
∗
(U
n
) for every l = 1,
2, ,n and (1.12)holds.
The proof follows from Lemma 2.4.
Corollary 4.3. Let φ
=(φ
1
, ,φ
n
) be a holomorphic self-map of U
n
. Then C
φ
: Ꮾ
p
0
(U
n
) →
Ꮾ
q
0
(U
n
) is compact if and only if φ
l
∈ Ꮾ
q
0
(U
n
) for every l = 1,2, ,n and (1.12)holds.
The proof follows from Lemma 2.6.
Acknowledgments
The authors would like to thank the editor and referee(s) for helpful comments on the
manuscript. The first author is supported in part by the National Natural Science Foun-
dation of China (Grants no. 10671141 and no. 10371091) and LiuHui Center for Applied
Mathematics, Nankai University & Tianjin University.
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Zehua Zhou: Department of Mathematics, Tianjin University, Tianjin 300072, China
E-mail address:
Yan Liu: Department of Mathematics, Tianjin University, Tianjin 300072, China
E-mail address:
