Essential commutation of composition and integral-type operators

0.15 MB·English

by  Ce-Zhong Tong

ESSENTIAL COMMUTATION OF COMPOSITION AND
INTEGRAL-TYPE OPERATORS
1
1 CE-ZHONGTONGANDZE-HUAZHOU∗
0
2 Abstract. Let H∞ represent the bounded holomorphic function algebra on
n theunitdiskDinthecomplexplane. TheBlochspaceandlittleBlochspace
a in D are denoted by B and B0 respectively. In this paper, we investigate the
J essential commutation of composition and integral-type operators from H∞
2 orBtoBorB0. Correspondingresultsfortheboundedness andcompactness
2 havealsobeenobtained.

]
A
F 1. Introduction
.
h Let D be the unit disk in the complex plane. Denote by H(D) the class of all
at holomorphic functions on D, and S(D) the collection of all the holomorphic self
m mappings of D. Every ϕ ∈ S(D) induces the composition operator C defined by
ϕ
[ Cϕf =f ◦ϕ for f ∈H(D).
Let g ∈H(D), the Volterra operator Jg is defined by
2
z
v J f(z)= f(ζ)g′(ζ)dζ, z ∈D, f ∈H(D);
8 g Z
0
3
and another integral operator I is defined by
3 g
0 z
. Igf(z)= f′(ζ)g(ζ)dζ, z ∈D, f ∈H(D).
1 Z
0
0
Use the notation H∞(D) to represent bounded holomorphic function algebra
1
1 with k·k∞ as its supreme norm. The Bloch space B consists of all f ∈H(D) such
: that
v
kfk :=sup(1−|z|2)|f′(z)|<∞,
i ∗
X z∈D
then k·k is a complete semi-norm on B, which is Mo¨bius invariant. Bloch space
r ∗
a is a Banach space under the norm
kfk=|f(0)|+kfk .
∗
The little Bloch space is denoted by B consisting of f ∈B for which
0
lim (1−|z|2)|f′(z)|=0.
|z|→1
Composition and integral-type operators are studied quite well in some fields,
just like that all composition operators are bounded naturally on H∞ or B. We
2000 Mathematics Subject Classification. Primary: 47B38; Secondary: 47B33,
47B35,46E15,32A36.
Key words and phrases. composition operator; integral-type operator; Bloch space; essential
commutation.
∗ Corresponding author. Supported in part by the National Natural Science Foundation of
China(GrandNos. 10971153, 10671141).
1
2 C.Z.TONGANDZ.H.ZHOU
refer to the books [2, 8] and some recent papers [11, 12, 13] to learn much more
on this subject. The boundedness and compactness of Volterra operators on Q
p
were characterizedin [10]. The same problems for the product of composition and
Volterra operators on all kinds of function spaces on the disk of C have also been
discussed, see some recent papers [4, 5, 6].
Based on these results, we consider the composition operator C : X → X,
ϕ
and the integral-type operator J (or I ) : X → B where X represents H∞ or B.
g g
Generally, it is clear that C J 6= J C (or C I 6= I C ), but it is interesting to
ϕ g g ϕ ϕ g g ϕ
study when
C J =J C mod K (or C I =I C mod K)? (1.1)
ϕ g g ϕ ϕ g g ϕ
Here K represents the collection of all compact operators from X to B.
If (1.1) holds for any ϕ ∈ S(D) and g ∈ H(D), we call C and J or I to be
ϕ g g
essentially commutative.
Since all the composition operators are bounded on X, it is nature to ask:
Main Questions: what properties does a non-constant g have, while (1.1)
holds true for every ϕ∈S(D) ?
In this paper, we dedicate to answer those questions. Even on the disk of C, some
properties are not easily managed.
2. Preliminaries
Before the discussion of our main results, we need some preliminary notations
and propositions, and we just state them without proving.
Schwarz derivative. Let ϕ ∈ S(D), denote the Schwarz derivative by ϕ#(z) =
1−|z|2 ϕ′(z). From Schwarz Lemma we know that |ϕ#(z)| ≤ 1, and the equality
1−|ϕ(z)|2
holds if and only if ϕ is an automorphism of the unit disk.
The following criterion for compactness follows from standard arguments which
is similar to the method of Proposition3.11 in [2]. Hence we omit the details. The
lemma is also true when we substitute I by J .
g g
Lemma 2.1. Suppose that ϕ∈S(D) and g ∈H(D). Then C I −I C is compact
ϕ g g ϕ
from X to B if and only if for any bounded sequence {f }, k =1,2,... in X which
k
converges to zero uniformly on compact subsets of D, k(C I −I C )f k → 0 as
ϕ g g ϕ k ∗
k →∞.
Recall that the notation CN represents the N dimensional complex Euclidean
space. Denote the unit ball of CN by B . The following Lemma will be used in
N
Section 4, which was first presented by Berndtsson in [1].
Lemma 2.2. Let {x } be a sequence in the ball B satisfying
i N
|Φ (x )|≥d>0 for any k. (2.1)
xj k
j:Yj6=k
Then there exists a number M = M(d) < ∞ and a sequence of functions h ∈
k
H∞(B ) such that
N
(a) h (x )=δ ; (b) |h (z)|≤M for |z|<1. (2.2)
k j kj k
Xk
(The symbol δ is equal to 1 if k =j and 0 otherwise.)
kj
Next lemma was proved by Carl Toews in [9].
ESSENTIAL COMMUTATION 3
Lemma 2.3. Let {z } ⊂ B be a sequence with |z | → 1 as n → ∞. Then for
n N n
any given d∈(0,1) there is a subsequence such that {x }:={z } satisfies (2.1).
i ni
From this lemma, there is always a subsequence which satisfies (2.1) for every
sequence converging to the boundary of the ball, and Lemma 2.2 holds for this
subsequence. What we need is just the case in one dimension.
To get some simple consequences of our main problems, we will consider the
situations on little Bloch settings. Next lemma is as usual, see [7].
Lemma 2.4. A closed set K in B is compact if and only if it is bounded and
0
satisfies
lim sup(1−|z|2)|f′(z)|=0.
|z|→1f∈K
TheboundednessofoperatorsJ andI ,fromX totheBlochspace,arerequired
g g
in the following contents, and several characterizations are listed here before our
discussion.
Lemma 2.5. Suppose that g ∈H(D). Then
(Corollary 4 in [4]): J :H∞ →B is bounded if and only if g ∈B;
g
(Theorem 3 in [6]): J :B →B is bounded if and only if
g
2
sup(1−|z|2)|g′(z)|ln <∞;
z∈D 1−|z|2
(Corollary 1 in [4] and Theorem 14 in [6]): I : X → B is bounded if
g
and only if g ∈H∞.
3. Essential commutation of C and I
ϕ g
FirstweconsiderC I −I C asawholeoperatorfromtheBlochspacetoitself.
ϕ g g ϕ
Theorem 3.1. Suppose that ϕ ∈ S(D) and g ∈ H(D). Then C I − I C is
ϕ g g ϕ
bounded on the Bloch space if and only if
sup|ϕ#(z)|·|g(ϕ(z))−g(z)|<∞. (3.1)
z∈D
Proof. Suppose (3.1) holds, we will show that C I −I C is bounded on B. For
ϕ g g ϕ
any f ∈B,
z
((C I −I C )f)(z) = C f′(ζ)g(ζ)dζ −I f(ϕ(z))
ϕ g g ϕ ϕ g
Z
0
ϕ(z) z
= f′(ζ)g(ζ)dζ − (f ◦ϕ)′(ζ)g(ζ)dζ.
Z Z
0 0
k(C I −I C )fk = sup(1−|z|2)|ϕ′(z)f′(ϕ(z))g(ϕ(z))−ϕ′(z)f′(ϕ(z))g(z)|
ϕ g g ϕ ∗
z∈D
= sup(1−|z|2)|ϕ′(z)f′(ϕ(z))(g(ϕ(z))−g(z))|
z∈D
= sup|ϕ#(z)||g(ϕ(z))−g(z)|(1−|ϕ(z)|2)|f′(ϕ(z))|
z∈D
≤ sup|ϕ#(z)||g(ϕ(z))−g(z)|·kfk .
∗
z∈D
From which we obtain that C I −I C is bounded by (3.1).
ϕ g g ϕ
4 C.Z.TONGANDZ.H.ZHOU
NowsupposethatC I −I C isboundedonB,thenthereisaconstantC such
ϕ g g ϕ
that
k(C I −I C )fk ≤k(C I −I C )fk<C
ϕ g g ϕ ∗ ϕ g g ϕ
for kfk ≤ 1. We will prove condition (3.1). Suppose not, there exists a sequence
{w } in D such that
n
lim |ϕ#(w )||g(ϕ(w ))−g(w )|=∞.
n n n
n→∞
Let α (z) = ϕ(wn)−z , and α′ (z) = − 1−|ϕ(wn)|2 for n = 1,2,.... It is easy to
n 1−ϕ(wn)z n (1−ϕ(wn)z)2
check that kα k =1.
n ∗
C > k(C I −I C )α k
ϕ g g ϕ n ∗
= sup|ϕ#(z)||g(ϕ(z))−g(z)|·(1−|ϕ(z)|2)|α′ (ϕ(z))|
n
z∈D
1−|ϕ(w )|2
≥ (1−|ϕ(w )|2) n ·|ϕ#(w )||g(ϕ(w ))−g(w )|
n n n n
|(1−ϕ(w )ϕ(w ))2|
n n
= |ϕ#(w )||g(ϕ(w ))−g(w )|→∞.
n n n
That is impossible. So (3.1) holds. (cid:3)
When we investigate essential commutation of C and I , we need add the
ϕ g
conditiong ∈H∞ to ensure the boundedness ofI on the Bloch space,see Lemma
g
2.5.
Theorem 3.2. Suppose that ϕ ∈ S(D) and g ∈ H∞(D). Then C and I are
ϕ g
essentially commutative on B if and only if
lim |ϕ#(z)||g(ϕ(z))−g(z)|=0. (3.2)
|ϕ(z)|→1
Proof. Sufficiency. Suppose (3.2) holds. Notice that g ∈ H∞ and |ϕ#| ≤ 1, we
have
sup|ϕ#(z)|·|g(ϕ(z))−g(z)|<∞.
z∈D
From which we conclude that C I − I C is a bounded operator by Theorem
ϕ g g ϕ
3.1. For anybounded sequence {f } inB convergingto zerouniformly oncompact
k
subsets of D with kf k≤M. From (3.2), it follows that for any small ε>0, there
k
exists a δ >0 such that
ε
|ϕ#(z)||g(ϕ(z))−g(z)|<
M
with ϕ(z)∈D\(1−δ)D.
k(C I −I C )f k = sup|ϕ#(z)||g(ϕ(z))−g(z)|(1−|ϕ(z)|2)|f′(ϕ(z))|
ϕ g g ϕ k ∗ k
z∈D
≤ (A)+(B), (3.3)
where
(A)= sup |ϕ#(z)||g(ϕ(z))−g(z)|(1−|ϕ(z)|2)|f′(ϕ(z))|,
k
ϕ(z)∈(1−δ)D
and
(B)= sup |ϕ#(z)||g(ϕ(z))−g(z)|(1−|ϕ(z)|2)|f′(ϕ(z))|.
k
ϕ(z)∈D\(1−δ)D
ESSENTIAL COMMUTATION 5
(A)<ε for sufficient large number k since (1−δ)D⊂⊂D, and
ε ε
(B)< ·sup(1−|ϕ(z)|2)|f′(ϕ(z))|= kfk <ε.
M z∈D k M ∗
Thus C I −I C is compact on B.
ϕ g g ϕ
Necessity. If C I −I C is compact on B, certainly C I −I C is a bounded
ϕ g g ϕ ϕ g g ϕ
operator, then
sup|ϕ#(z)|·|g(ϕ(z))−g(z)|<∞
z∈D
by Theorem 3.1. Suppose (3.2) is wrong, then there exists a sequence {w } in D
n
with |ϕ(w )|→1 and
n
|ϕ#(w )||g(ϕ(w ))−g(w )|>ǫ>0. (3.4)
n n n
Let
1−|ϕ(w )|2
h (z)= n . (3.5)
n
1−ϕ(w )z
n
A simple computation shows that
ϕ(w )
h′ (z)=(1−|ϕ(w )|2) n .
n n
(1−ϕ(w )z)2
n
So kh k ≤ 1 and {h } converges to zero uniformly on compact subsets of D. By
n ∗ n
Lemma 2.2, it follows that k(C I −I C )h k →0. On the other hand, by (3.4),
ϕ g g ϕ n ∗
we have
k(C I −I C )h k
ϕ g g ϕ n ∗
= sup|ϕ#(z)||g(ϕ(z))−g(z)|·(1−|ϕ(z)|2)|h′ (ϕ(z))|
n
z∈D
|ϕ(w )|(1−|ϕ(w )|2)
= sup|ϕ#(z)||g(ϕ(z))−g(z)|·(1−|ϕ(z)|2) n n
z∈D |1−ϕ(wn)ϕ(z)|2
≥ |ϕ#(w )||g(ϕ(w ))−g(w )||ϕ(w )|>ǫ
n n n n
whenn→∞. Andwefindacontradiction. So(3.2)holdswhenC andI commute
ϕ g
essentially. (cid:3)
Corollary 3.3. Let ϕ ∈ S(D) and g ∈ H(D), then C I −I C : H∞ → B is
ϕ g g ϕ
bounded if and only if (3.1) holds; C I −I C :H∞ →B is compact if and only
ϕ g g ϕ
if (3.2) holds.
Proof. To prove this corollary, we only adjust (3.5), in the proof of Theorem 3.2,
as
1−|ϕ(w )|2 ϕ(w )−z
f (z)= n · n .
n
1−ϕ(w )z 1−ϕ(w )z
n n
The other proofs are similar to that of Theorems 3.1 and 3.2. We omit the details.
(cid:3)
Using Lemma 2.4, we can easily get the following corollary on little Bloch set-
tings. The method is as usual and we omit its proof.
Corollary 3.4. Let ϕ∈S(D) and g ∈H(D), and notation X represents either the
space H∞, B or B . The following three statements are equivalent:
0
(a): C I −I C :X →B is bounded;
ϕ g g ϕ 0
(b): C I −I C :X →B is compact;
ϕ g g ϕ 0
6 C.Z.TONGANDZ.H.ZHOU
(c): lim |ϕ#(z)||g(ϕ(z))−g(z)|=0.
|z|→1
Now we consider the problem raised in the first section:
• Is there any non-constant holomorphic function g on D satisfying
(C :B →B)I =I (C :X →X) mod K (3.6)
ϕ g g ϕ
for every analytic self-mapping ϕ of D?
The answer is negative, see the following theorem.
Theorem 3.5. Equation (3.6) holds for every self-mapping ϕ of the unit disk if
and only if g is a constant.
Proof. Sufficiency is obvious. For clarify the necessity, just consider ϕ as any au-
tomorphism in condition (3.2). Then maximum modular theorem implies that g
must be a constant. (cid:3)
4. Essential commutation of C and J
ϕ g
First, we consider the case for X =H∞ in the main question.
Theorem 4.1. Suppose that ϕ∈S(D) and g ∈H(D). Then
(1) C J −J C is bounded from H∞ to B if and only if
ϕ g g ϕ
sup(1−|z|2)|(g◦ϕ)′(z)−g′(z)|<∞; (4.1)
z∈D
(2) C J −J C is compact from H∞ to B if and only if (4.1) holds and
ϕ g g ϕ
lim (1−|z|2)|(g◦ϕ)′(z)−g′(z)|=0. (4.2)
|ϕ(z)|→1
Proof. Being similar with the proofs of Theorems 3.1 and 3.2, we just need some
modifications.
k(C J −J C )fk
ϕ g g ϕ ∗
= ϕ(z)f(ζ)g′(ζ)dζ − zf(ϕ(ζ))g′(ζ)dζ
0 0
= s(cid:13)(cid:13)uRp(1−|z|2)|f(ϕ(z))gR′(ϕ(z))ϕ(z)−f(ϕ(cid:13)(cid:13)(∗z))g′(z)| (4.3)
(cid:13)z∈D (cid:13)
= sup(1−|z|2)|(g◦ϕ)′(z)−g′(z)||f(ϕ(z))|.
z∈D
Sufficiency of the two items in the theorem is obvious from the last formula in
(4.3).
Necessity of boundedness can be proved by computing test functions
ϕ(w )−z
n
f (z)=1− ,
n
1−ϕ(w )z
n
where sequence {w } makes the equation (4.1) wrong.
n
Necessity of compactness. Suppose not, we can find a sequence {ϕ(w )} con-
n
verging to the boundary of D and ǫ >0 such that
0
lim (1−|w |2)|(g◦ϕ)′(w )−g′(w )|>ǫ . (4.4)
n n n 0
n→∞
Further we may assume that {ϕ(w )} is interpolating. Then there exist functions
n
{h } in H∞ for {ϕ(w )} such that
n n
1 n=k,
h (ϕ(w ))= (4.5)
n k (cid:26) 0 n6=k.
ESSENTIAL COMMUTATION 7
and
|h (z)|≤M <∞, (4.6)
n
Xn
by Lemma 2.3 and 2.2, or see [3]. Equation (4.6) guarantee that {h } is bounded
n
in H∞ and converges to zero on compact subsets of D.
k(C J −J C )h k
ϕ g g ϕ n ∗
= sup(1−|z|2)|(g◦ϕ′(z)−g′(z)||h (ϕ(z))|
n
z∈D
≥ (1−|w |2)|(g◦ϕ′(w )−g′(w )||h (ϕ(w ))|
n n n n n
= (1−|w |2)|(g◦ϕ)′(w )−g′(w )|.
n n n
The last equation follows by (4.5). Let n→∞, we find contradiction by (4.4). (cid:3)
Corollary 4.2. Let ϕ∈S(D) and g ∈H(D). Next three conditions are equivalent:
(a): C J −J C :H∞ →B is bounded;
ϕ g g ϕ 0
(b): C J −J C :H∞ →B is compact;
ϕ g g ϕ 0
(c): lim (1−|z|2)|(g◦ϕ)′(z)−g′(z)|=0.
|z|→1
ForcompositionoperatorandJ ,theresultofourmainquestionturnsouttobe
g
interesting. Notethatg ∈B impliesthatsup(1−|z|2)|(g◦ϕ)′(z)−g′(z)|<∞,thus
z∈D
C J −J C is a boundedoperatorfromH∞ toB. AndC andJ :H∞ →B are
ϕ g g ϕ ϕ g
essential commutative if and only if (4.2) holds. Further, we are going to answer
the main question in next corollary.
Corollary 4.3. Let g ∈B , then for every analytic self-mapping ϕ, we have
0
(C :B →B)J =J (C :H∞ →H∞) mod (K). (4.7)
ϕ g g ϕ
Proof. Since g is in the little Bloch space, (1−|z|2)|g′(z)| tends to 0 whenever z
tends to the boundary of the disk. So we have that
lim (1−|z|2)|g′(ϕ(z))ϕ′(z)−g′(z)|
|ϕ(z)|→1
≤ lim |ϕ#(z)|(1−|ϕ(z)|2)|g′(ϕ(z))|+ lim (1−|z|2)|g′(z)|.
|ϕ(z)|→1 |ϕ(z)|→1
Conditions g ∈B and|ϕ#|≤1 imply that C J −J C is compact fromH∞ to B
ϕ g g ϕ
for every self-mapping ϕ. (cid:3)
Being different with the situation of C and I , there are great many of non-
ϕ g
constantfunctions g’s, indeedall the little Blochfunctions, making J to commute
g
withallthecompositionoperatorsessentially. Naturally,wearegoingtoask: Does
J commuting essentially with all the composition operators imply that g is in the
g
little Bloch space? The answer is positive!
Lemma 4.4. If g is a Bloch function on the unit disk with the property that, for
any rotation τ(z)=eitz, g◦τ −g is in the little Bloch space, then g itself must be
in the little Bloch space.
Proof. Conditiong ∈B isnecessarytoensurethatJ :H∞ →B isbounded. Since
g
g◦τ −g is in the little Bloch space, we have
lim (1−|z|2) g′(eitz)eit−g′(z) =0. (4.8)
|z|→1
(cid:12) (cid:12)
(cid:12) (cid:12)
8 C.Z.TONGANDZ.H.ZHOU
∞ ∞
Suppose that g(z)= a zn, then g′(z)= na zn−1. Integrating with respect
n n
nP=0 nP=1
to t from 0 to 2π, and some calculations show that
2π
lim (1−|z|2) g′(eitz)eit−g′(z) dt
Z0 |z|→1 (cid:12) (cid:12)
2π(cid:12) (cid:12)
≥ lim (1−|z|2) g′(eitz)eit−g′(z)dt
|z|→1 (cid:12)(cid:12)Z0 (cid:12)(cid:12)
= lim (1−|z|2)(cid:12)(cid:12) 2π ∞ (eitz)n−1eit−n(cid:12)(cid:12)a zn−1 dt
(cid:12) n (cid:12)
|z|→1 (cid:12)(cid:12)Z0 nX=1(cid:0) (cid:1) (cid:12)(cid:12)
= lim (1−|z|2)(cid:12)(cid:12) ∞ na zn−1 2π(eint−1)dt (cid:12)(cid:12)
(cid:12) n (cid:12)
|z|→1 (cid:12)(cid:12)nX=1 Z0 (cid:12)(cid:12)
= 2π· lim (1−|(cid:12)(cid:12)z|2)|g′(z)|, (cid:12)(cid:12)
|z|→1
where the inequality in the second line holds true by the dominated convergence
theorem. Thus we get g ∈B from (4.8). (cid:3)
0
Theorem 4.5. Equation (4.7) holds for every ϕ if and only if g ∈B .
0
Proof. Sufficiency is stated in Corollary 4.3. To prove necessity, just consider the
rotation of the disk. That is to say put ϕ(z) = τ(z) = eitz in the condition (4.2),
and we have g ∈B by Lemma 4.4. (cid:3)
0
Next several propositions are concerning C and J from B to itself.
ϕ g
Proposition 4.6. Assume that ϕ ∈ S(D) and g ∈ H(D). Then C J −J C is
ϕ g g ϕ
bounded from B to itself if and only if
2
sup(1−|z|2)|(g◦ϕ)′(z)−g′(z)|ln <∞. (4.9)
z∈D 1−|ϕ(z)|2
Proof. Sufficiency can be clarified by some regular computations and inequalities.
Necessity will be proved by choosing text function
2
f (z)=ln .
w
1−wz
(cid:3)
Proposition 4.7. Assume that ϕ ∈ S(D) and g ∈ H(D). Then the following
statements are equivalent:
(a): C J −J C :B →B is compact and (4.9) holds;
ϕ g g ϕ
(b): C J −J C :B →B is compact;
ϕ g g ϕ 0 0
(c): C J −J C :B →B is weakly compact;
ϕ g g ϕ 0 0
(d): Condition (4.9) holds and
2
lim (1−|z|2)|(g◦ϕ)′(z)−g′(z)|ln =0; (4.10)
|ϕ(z)|→1 1−|ϕ(z)|2
(e): C J −J C :B →B is compact;
ϕ g g ϕ 0
(f): C J −J C :B →B is bounded;
ϕ g g ϕ 0
ESSENTIAL COMMUTATION 9
The compactness of operators C J and J C are characterized in Theorem 4
ϕ g g ϕ
andTheorem11 in [6]. The proofs are similar with Theorem4 in [6], thus we omit
it.
Tocontinueourdiscussion,weneedanupperboundonthemodularofϕ(z)(see
Corollary 2.40 in [2]).
Lemma 4.8. If ϕ∈S(D), then |ϕ(z)|≤ |z|+|ϕ(0)| .
1+|z||ϕ(0)|
Now we are ready to consider our main question for C and J , where both of
ϕ g
them are operators from B to itself.
Theorem 4.9. Assume g ∈ H(D) such that J is bounded on the Bloch space.
g
Then
(C :B →B)J =J (C :B →B) mod (K) (4.11)
ϕ g g ϕ
for every analytic self-mapping ϕ of unit disk, if and only if
2
lim (1−|z|2)|g′(z)|ln =0. (4.12)
|z|→1 1−|z|2
Proof. FollowingthemethodintheproofofLemma4.4,thenecessitycanbeproved
similarly.
Now suppose (4.12) holds. We have that
2
(1−|z|2)|(g◦ϕ)′(z)−g′(z)|ln
1−|ϕ(z)|2
2
≤ |ϕ#(z)|(1−|ϕ(z)|2)|g′(ϕ(z)|ln
1−|ϕ(z)|2
2 ln2−ln(1−|ϕ(z)|2)
+(1−|z|2)|g′(z)|ln · .
1−|z|2 ln2−ln(1−|z|2)
Just consider those z’s tending to ∂D with |ϕ(z)|→1.
ln2−ln(1−|ϕ(z)|2)
lim
|z|→1− ln2−ln(1−|z|2)
−ln(1−|ϕ(z)|)
= lim
|z|→1− −ln(1−|z|)
−ln(1− |z|+|ϕ(0)| )
1+|z||ϕ(0)|
≤ lim
|z|→1− −ln(1−|z|)
ln(1−|z|)(1−|ϕ(0)|)−ln(1+|z||ϕ(0)|)
= lim
|z|→1− ln(1−|z|)
− 1 − |ϕ(0)|
1−|z| 1+|z||ϕ(0)|
= lim =1
|z|→1− − 1
1−|z|
where Lemma 4.8 and L’Hospital Law are utilized. Thus (4.10) holds for every
ϕ∈S(D) by (4.12). The proof is finished by Proposition 4.7. (cid:3)
Acknowledgements. The work receivedgreathelp from ProfessorKehe Zhu. We
would like to take this opportunity to express our appreciation to him.
10 C.Z.TONGANDZ.H.ZHOU
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Departmentof Mathematics
TianjinUniversity
Tianjin300072
P.R. China.
E-mail address: [email protected]
Departmentof Mathematics
TianjinUniversity
Tianjin300072
P.R. China.
E-mail address: [email protected]