Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 214762, 14 pages
doi:10.1155/2010/214762
Research Article
On an Integral-Type Operator Acting between
Bloch-Type Spaces on the Unit Ball
Stevo Stević1 and Sei-Ichiro Ueki2
1
2
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
Faculty of Engineering, Ibaraki University, Hitachi 316-8511, Japan
Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs
Received 30 January 2010; Accepted 18 March 2010
Academic Editor: Narcisa C. Apreutesei
Copyright q 2010 S. Stević and S.-I. Ueki. 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.
Let B denote the open unit ball of Cn . For a holomorphic self-map ϕ of B and a holomorg
phic function g in B with g0 0, we define the following integral-type operator: Iϕ fz 1
Rfϕtzgtzdt/t, z ∈ B. Here Rf denotes the radial derivative of a holomorphic function
0
f in B. We study the boundedness and compactness of the operator between Bloch-type spaces
Bω and Bμ , where ω is a normal weight function and μ is a weight function. Also we consider the
operator between the little Bloch-type spaces Bω,0 and Bμ,0 .
1. Introduction
Let B denote the open unit ball of the n-dimensional complex vector space Cn and HB the
space of all holomorphic functions on B. For f ∈ HB with the Taylor expansion fz γ
|γ|≥0 aγ z , let
Rfz γ aγ zγ
|γ|≥0
1.1
be the radial derivative of f, where γ γ1 , . . . , γn is a multi-index, |γ| γ1 · · · γn , and
zγ z1 γ1 . . . zn γn . It is well known that
Rfz n
j1
where ∇ is the usual gradient on Cn .
zj
∂f
z ∇fz, z ,
∂zj
1.2
2
Abstract and Applied Analysis
Let ϕ be a holomorphic self-map of B and g ∈ HB with g0 0. Then ϕ and g define
g
an operator Iϕ on HB as follows:
g
Iϕ fz
1
Rf ϕtz gtz
0
dt
,
t
f ∈ HB, z ∈ B.
1.3
g
The following important formula involving R and Iϕ f was proved, for example, in 1
g
R Iϕ f z Rf ϕz gz,
z ∈ B.
1.4
g
Motivated by papers 2, 3, operators Iϕ were introduced by the first author of the
present paper and Zhu in 1, 4–6, where its boundedness and compactness from the α-Bloch
space, the Zygmund space, the mixed-norm space, and the generalized weighted Bergman
space into the Bloch-type space on the unit ball are studied. In our previous work 7, we
g
studied the boundedness and compactness of Iϕ acting between weighted-type spaces. For
n
related operators on C see, for example, 8–21 and the references therein.
Let ω be a strictly positive continuous function on B weight. If ωz ω|z| for every
z ∈ B, we call it radial weight. A weight ω is called normal 9, 22 if it is radial and there are a
and b, 0 < a < b < ∞ such that ωr/1 − ra is decreasing on 0, 1, ωr/1 − rb is increasing
on 0, 1,
ωr
a 0,
r → 1 1 − r
lim
lim
r →1
ωr
1 − rb
∞.
1.5
A radial weight ω is called typical if it is nonincreasing with respect to |z| and ωz → 0
as |z| → 1− . If ω is normal, then by the monotonicity of ωr/1 − ra , for 0 ≤ r1 < r < 1, we
have that
ωr 1 − ra
ωr
a ωr1 < ωr1 ,
a < 1 − r
1 − r
1 − r1 a
1.6
that is, ω is decreasing on 0, 1. On the other hand, from the first equality in 1.5, we have
that for any ε > 0, there is a δ > 0 such that
0 < ωr < ε1 − ra ,
1.7
δ < r < 1,
which implies limr → 1− ωr 0. Hence every normal weight ω is also typical.
For a weight ω, the associated weight ω 23 is defined by
ωz :
1
,
sup fz : f ∈ Hω∞ , f Hω∞ ≤ 1
z ∈ B.
1.8
Abstract and Applied Analysis
3
Here Hω∞ denotes the weighted-type space consisting of all f ∈ HB with
f Hω∞
sup ωzfz < ∞
z∈B
1.9
see, e.g., 23, 24. Associated weights assist us in studying of weighted-type spaces of
holomorphic functions. It is known that associated weights are also continuous, 0 < ω ≤ ω,
∞
and for each z ∈ B, we can find an fz ∈ Hω∞ , fz Hω∞ ≤ 1 such that fz z 1/ωz. Let Hω,0
be the little weighted-type space, that is, the space of all f ∈ HB such that ωz|fz| → 0 as
∞ for the compact open
|z| → 1− . If ω is typical, then the unit ball BHω∞ is the closure of BHω,0
topology. Hence we have
ωz 1
∞ sup fz : f ∈ Hω,0
, f Hω∞ ≤ 1
1.10
∞ such that f z 1/ωz. A weight ω is
and so for each z ∈ B, we can choose an fz ∈ BHω,0
z
called essential if it satisfies that ω ≤ Cω for some positive constant C. By the arguments in
25, we see that a normal weight function is also essential. For some examples of essential
weights, see, for example, 25. Related results can also be found in 22, 26.
The Bloch-type space Bω is the space of all holomorphic functions f on B such that
bω f sup ωzRfz < ∞,
z∈B
1.11
where ω is a weight see, e.g., 20. The little Bloch-type space Bω,0 consists of all f ∈ HB
such that
lim − ωzRfz 0.
1.12
|z| → 1
Both spaces Bω and Bω,0 are Banach spaces with the norm
f f0 bω f ,
Bω
1.13
and Bω,0 is a closed subspace of Bω . When ωr 1 − r 2 , the space Bω is a classical Bloch
space.
The purpose of this paper is to characterize the boundedness and compactness of the
g
g
operators Iϕ : Bω → Bμ and Iϕ : Bω,0 → Bμ,0 .
Throughout this paper, we assume that ϕ is a holomorphic self-map of B and g ∈ HB
with g0 0. Furthermore, some constants are denoted by C; they are positive and may
differ from one occurrence to the other. The notation a b means that there exists a positive
constant C such that a ≤ Cb. Moreover, if both a b and b a hold, then one says that a b.
2. Auxiliary Results
Here we formulate and prove some auxiliary results which are used in the proofs of the main
ones.
4
Abstract and Applied Analysis
The following lemma was proved in 20, Theorem 2.1.
Lemma 2.1. Let ω be a normal weight function and f ∈ HB. Then f ∈ Bω if and only if
supz∈B ωz|∇fz| < ∞ and it holds that
f Bω
f0 sup ωz∇fz.
z∈B
2.1
Moreover, f ∈ Bω,0 if and only if lim|z| → 1− ωz|∇fz| 0.
As an application of Lemma 2.1, we have the following result.
Lemma 2.2. Let ω be a normal weight function and f ∈ Bω . Then f ∈ Bω,0 if and only if it holds
that limr → 1− fr − fBω 0, where fr z frz.
Proof. Take an f ∈ Bω,0 . For a fixed ε > 0, by Lemma 2.1, we can choose a δ0 ∈ 0, 1 such that
ε
ωz∇fz <
2
2.2
for any z ∈ B \ δ0 2 B. Since ∂fr /∂zj z r∂f/∂zj rz for j ∈ {1, . . . , n}, r ∈ 0, 1, and
z ∈ B, we have
fr − f Bω
sup ωzr∇frz − ∇fz
z∈B
≤ sup ωzr∇frz − ∇fz
z∈B\δ0 B
2.3
sup ωzr∇frz − ∇fz.
z∈δ0 B
Since
max r∇frz − ∇fz −→ 0,
|z|≤δ0
as r −→ 1− ,
2.4
we see that the second term in 2.3 converges to 0 as r → 1− .
If r ∈ δ0 , 1 and z ∈ B \ δ0 B, then by 2.2 we have
ε
ωrz∇frz < .
2
2.5
By 1.6 we have that ωz ≤ ωrz for r, |z| ∈ 0, 1.
Hence we have
sup ωzr∇frz − ∇fz ≤ sup ωrz∇frz sup ωz∇fz < ε,
z∈B\δ0 B
z∈B\δ0 B
z∈B\δ0 B
for all r ∈ δ0 , 1. This proves that limr → 1− fr − fBω 0 whenever f ∈ Bω,0 .
2.6
Abstract and Applied Analysis
5
Conversely, the normality of ω implies that for any ε > 0 we have
ωz∇frz ≤ ε1 − |z|a sup ∇fw −→ 0,
|w|≤r
as |z| −→ 1− ,
2.7
so that fr ∈ Bω,0 for any r ∈ 0, 1. On the other hand, by the assumption limr → 1− fr − fBω 0, we have that for every ε > 0 there is an r1 ∈ 0, 1 such that
fr − f < ε,
Bω
2.8
for r ∈ r1 , 1.
By letting |z| → 1− in the following inequality, which easily follows from Lemma 2.1:
ωz∇fz ωz∇frz fr − f Bω ,
2.9
then using 2.7 and 2.8, we get f ∈ Bω,0 , as claimed.
Corollary 2.3. Let ω be a normal weight function. Then the set of all holomorphic polynomials is
dense in Bω,0 .
j
j
Proof. For the homogeneous expansion f ∞
F
k0 Fk of an f ∈ Bω,0 , we set P k0 k
for each j ∈ N. Since P j → f uniformly on compact subsets of B as j → ∞, we see that
j
RPr → Rfr uniformly on B for any r ∈ 0, 1. Moreover, we have
j
Pr − f Bω
j
≤ Pr − fr Bω
fr − f Bω
j
≤ sup ωz sup R Pr z − R fr z fr − f Bω .
2.10
z∈B
z∈rB
Combining this with Lemma 2.2, we get the desired result.
The following lemma can be found in 1, Lemma 3. Its proof is similar to the proof of
the corresponding one-dimensional result in 27, for the case of the little Bloch space B1−r,0 .
Hence we omit the proof.
Lemma 2.4. A closed subset K in Bω,0 is compact if and only if it is bounded and
lim − sup ωzRfz 0.
|z| → 1 f∈K
2.11
The following lemma is very useful for estimating the norm of the Bloch-type space.
Lemma 2.5. Assume that m is a positive integer and ω is normal. Then for every f ∈ HB,
sup ωzfz
z∈B
f0 sup 1 − |z|m ωzRm fz.
z∈B
Proof. For the details of the proof, we can refer 9 or 28.
2.12
6
Abstract and Applied Analysis
g
3. The Boundedness of Operator Iϕ
g
g
In this section we consider the boundedness of the operator Iϕ : Bω → Bμ or Iϕ : Bω,0 → Bμ,0 .
Theorem 3.1. Let ω be a normal weight function and μ a weight function. Then the following
conditions are equivalent:
g
a Iϕ : Bω → Bμ is bounded;
g
b Iϕ : Bω,0 → Bμ is bounded;
c ϕ and g satisfy
sup
μzgzϕz
ω ϕz
z∈B
< ∞.
3.1
g
Moreover, if Iϕ : Bω → Bμ is bounded, then
g
Iϕ Bω → Bμ
sup
z∈B
μzgzϕz
ω ϕz
.
3.2
Proof. The implication a ⇒ b is clear, so we only prove b ⇒ c and c ⇒ a.
g
b ⇒ c: assume that Iϕ : Bω,0 → Bμ is bounded and fix z ∈ B. We may assume that
∞
such that hw Hω∞ ≤ 1 and hw w 1/ωw.
ϕz /
0. For w : ϕz, there exists hw ∈ Hω,0
We define the function fw as follows:
fw v 1
0
hw tv
tv, w dt
,
|w| t
v ∈ B.
3.3
Since Rfw v hw v v, w /|w|, we see that fw ∈ Bω,0 and fw Bω ≤ 1. Hence, by 1.4,
we have
g
Iϕ Bω → Bμ
g ≥ Iϕ fw Bμ
μzgzϕz
,
≥ μz gz Rfw ϕz ω ϕz
3.4
and so condition 3.1 is true.
c ⇒ a: we assume 3.1 and take an f ∈ Bω . Since ω is an essential weight due to
its normality, 1.4 gives
g
μzR Iϕ f z μzgzRf ϕz ω ϕz ∇f ϕz ≤ μzgzϕz
ω ϕz
μzgzϕz
sup ωw∇fw,
ω ϕz
w∈B
3.5
Abstract and Applied Analysis
7
for any z ∈ B. By Lemma 2.1, we have supw∈B ωw|∇fw|
g Iϕ f Bμ
fBω , and so we obtain
μzgzϕz f .
sup
Bω
ω ϕz
z∈B
3.6
g
This implies that Iϕ : Bω → Bμ is bounded. The relation 3.2 follows from 3.4 and 3.6.
This completes the proof.
Theorem 3.2. Let ω be a normal weight function and μ a weight function. Then the following
conditions are equivalent:
g
a Iϕ : Bω,0 → Bμ,0 is bounded;
b ϕ and g satisfy
lim − μzgzϕz 0,
sup
|z| → 1
μzgzϕz
z∈B
ω ϕz
< ∞.
3.7
Proof. a ⇒ b: as in the proof of Theorem 3.1, for fixed z ∈ B and w ϕz, we see that ϕ
and g satisfy the condition
sup
μzgzϕz
ω ϕz
z∈B
< ∞.
3.8
On the other hand, since the normality of ω implies that the function πj z :
zj 1 ≤ j ≤ n belongs to Bω,0 , we obtain that μz|gz| |ϕj z| → 0 for each j, and so
μz|gz||ϕz| → 0 as |z| → 1− .
g
b ⇒ a: the assumption lim|z| → 1− μz|gz||ϕz| 0 shows that Iϕ p ∈ Bμ,0 for any
polynomial p. For each f ∈ Bω,0 , by Corollary 2.3, we can choose a sequence of polynomials
{pj }j∈N such that f − pj Bω → 0 as j → ∞. Furthermore, the assumption
sup
μzgzϕz
ω ϕz
z∈B
<∞
3.9
g
shows that Iϕ : Bω,0 → Bμ is bounded by Theorem 3.1. Thus we obtain
g g
0 ≤ Iϕ f − Iϕ pj Bμ
g
g
g
≤ Iϕ Bω → Bμ
f − pj −→ 0
Bω
as j −→ ∞ .
3.10
g
Since Iϕ f ∈ Bμ , {Iϕ pj }j∈N ⊂ Bμ,0 , and Bμ,0 is closed in Bμ , we have Iϕ f ∈ Bμ,0 for any f ∈
g
g
Bω,0 . Hence Iϕ Bω,0 ⊆ Bμ,0 which means that Iϕ : Bω,0 → Bμ,0 is bounded. The proof is
accomplished.
The following corollary is an immediate consequence of Theorems 3.1 and 3.2.
g
Corollary 3.3. Let ω be a normal weight function and μ a weight function. Then Iϕ : Bω,0 → Bμ,0
g
is bounded if and only if lim|z| → 1− μz|gz||ϕz| 0 and Iϕ : Bω → Bμ is bounded.
8
Abstract and Applied Analysis
g
4. The Compactness of Operator Iϕ
g
g
In this section we characterize the compactness of Iϕ : Bω → Bμ or Iϕ : Bω,0 → Bμ,0 . To do
this, we need the following standard lemma see, e.g., 13, Lemma 3.
g
Lemma 4.1. Let ω and μ be weight functions. Suppose that the operator Iϕ : Bω or Bω,0 → Bμ is
g
bounded. Then Iϕ : Bω or Bω,0 → Bμ is compact if and only if for every bounded sequence {fj }j∈N
g
in Bω or Bω,0 which converges to 0 uniformly on compact subsets of B, Iϕ fj Bμ → 0 as j → ∞.
Theorem 4.2. Let ω and μ be weight functions. Suppose that ϕ is a holomorphic self-map of B such
g
g
that ϕ∞ < 1 and the operator Iϕ : Bω → Bμ is bounded. Then Iϕ : Bω → Bμ is compact. Here
ϕ∞ denotes the supremum supz∈B |ϕz|.
Proof. Since ϕ∞ < 1, we see that |ϕz| ≤ r for some r ∈ 0, 1 and any z ∈ B. From the proof
g
of Theorem 3.1, we see that the boundedness of Iϕ : Bω → Bμ implies
M : sup
μzgzϕz
ω ϕz
z∈B
< ∞.
4.1
Thus we obtain that
sup μzgzϕz ≤ M sup ωw < ∞.
z∈B
w∈rB
4.2
Take a bounded sequence {fj }j∈N in Bω such that fj → 0 uniformly on compact subsets of B
as j → ∞. By 1.4, we have
g Iϕ fj Bμ
sup μzgzRfj ϕz z∈B
≤ sup μzgzϕz∇fj ϕz z∈B
4.3
≤ M sup ωw sup ∇fj w.
w∈rB
w∈rB
Since ∂fj /∂zk 1 ≤ k ≤ n also converges to 0 uniformly on rB as j → ∞, 4.2 and 4.3 show
g
g
that Iϕ fj Bμ → 0 as j → ∞. From Lemma 4.1, it follows that Iϕ : Bω → Bμ is compact, and
so we get the assertions.
Lemma 4.3. Suppose that ω is a weight function. Then there exists a sequence {fk }k∈N in the closed
unit ball of Bω such that fk → 0 uniformly on compact subsets of B as k → ∞.
Proof. Let {wk }k∈N ⊂ B with |wk | → 1− as k → ∞. For each wk , there exists hk : hwk ∈ Hω∞
such that hk Hω∞ ≤ 1 and hk wk 1/ωwk . We define fk as follows:
fk z 1
0
hk tz
tz, wk
|wk |
1/1−|wk |
dt
,
t
z ∈ B.
4.4
Abstract and Applied Analysis
9
Since fk 0 0 and |Rfk z| ≤ |hk z|, we have {fk }k∈N ⊂ Bω and fk Bω ≤ 1 for each k ∈ N.
For any compact subset K of B, we can choose an r ∈ 0, 1 such that K ⊂ rB. Hence we
obtain that for any z ∈ K
fk z ≤
1
0
|hk tz|t|wk |/1−|wk | dt ≤ hk Hω∞
1
0
1 |wk |/1−|wk |
1
t
dt ≤ max
1 − |wk |.
ωtz
ωw
w∈rB
4.5
From the above inequality, it follows that fk converges to 0 uniformly on compact subsets of
B as k → ∞. This completes the proof.
∞
Remark 4.4. If we assume that ω is typical in Lemma 4.3, then we can choose hk ∈ Hω,0
. In
this case, hence, we see that fk belongs to Bω,0 for each k ∈ N.
Theorem 4.5. Let ω be a normal weight function and μ a weight function. Suppose that the operator
g
Iϕ : Bω → Bμ is bounded and ϕ∞ 1. Then the following conditions are equivalent:
g
a Iϕ : Bω → Bμ is compact;
g
b Iϕ : Bω,0 → Bμ is compact;
c ϕ and g satisfy
lim
μzgzϕz
|ϕz| → 1−
ω ϕz
0.
4.6
Proof. a ⇒ b: this implication is obvious.
b ⇒ c: take a sequence {zk }k∈N in B with |ϕzk | → 1− as k → ∞ and put wk ϕzk for each k. Then, by Remark 4.4 after Lemma 4.3, there exists a sequence {fk }k∈N in Bω,0
such that supk∈N fk Bω ≤ 1 and fk → 0 uniformly on compact subsets of B as k → ∞. By
g
g
Lemma 4.1, the compactness of Iϕ : Bω,0 → Bμ implies that Iϕ fk Bμ → 0 as k → ∞.
g
On the other hand, 1.4 gives RIϕ fk z Rfk ϕzgz, and so we have
g Iϕ fk Bμ
≥ μzk Rfk ϕzk gzk ≥ μzk Rfk ϕzk gzk ϕzk .
4.7
From the construction 4.4 of fk , we obtain
Rfk ϕzk ϕzk 1/1−|ϕzk |
ω ϕzk ,
4.8
for each k ∈ N. Combining this with 4.7, we have
g Iϕ fk Bμ
μzk gzk ϕzk ϕzk 1/1−|ϕzk | .
≥
ω ϕzk 4.9
10
Abstract and Applied Analysis
Letting k → ∞, we have
lim
μzk gzk ϕzk ω ϕzk k→∞
0,
4.10
for any sequence {zk }k∈N with |ϕzk | → 1− . This proves that 4.6 is true.
c ⇒ a: we will prove the following estimate:
g
Iϕ e
lim sup
|ϕz| → 1−
g
μzgzϕz
ω ϕz
4.11
.
g
Here Iϕ e denotes the essential norm of Iϕ : Bω → Bμ , namely,
g
g
Iϕ inf Iϕ K
Bω → Bμ
e
| K : Bω −→ Bμ is compact .
4.12
Now we take a sequence {rl }l∈N ⊂ 0, 1 which increasingly converges to 1 and put
g
Irl ϕ fz 1
Rf rl ϕtz gtz
0
dt
.
t
4.13
g
Since rl ϕ∞ ≤ rl < 1, Theorem 4.2 implies that Irl ϕ : Bω → Bμ is compact for each l ∈ N. For
any f ∈ Bω with fBω ≤ 1, from 1.4 it follows that
g
g
Iϕ f − Irl ϕ f Bω
sup μzgzRf ϕz − Rf rl ϕz z∈B
≤
sup μzgzRf ϕz − Rf rl ϕz
R<|ϕz|<1
sup μzgzRf ϕz − Rf rl ϕz
|ϕz|≤R
4.14
,
for some fixed R ∈ 0, 1. The essentiality of ω and Lemma 2.1 give
μzgzϕz
μzgzRf ϕz ≤
ω ϕz ∇f ϕz ω ϕz
μzgzϕz
sup ωw∇fw
ω ϕz
w∈B
μzgzϕz
.
ω ϕz
4.15
Abstract and Applied Analysis
11
Similarly, we also have
μzgzRf rl ϕz μzgzϕz
ω rl ϕz
4.16
,
for each l ∈ N. The normality of ω implies that
ω rl ϕz
1 − rl ϕz
a
≥
ω ϕz
1 − ϕz
a,
4.17
a.
4.18
for each l ∈ N and some a > 0, and so by the essentiality,
ω rl ϕz
1 − rl ϕz
a
ω ϕz
1 − ϕz
Thus 4.16 and 4.18 give
μzgzRf rl ϕz μzgzϕz
ω ϕz
4.19
,
for each l ∈ N. By 4.15 and 4.19, we obtain
sup
R<|ϕz|<1
μzgzRf ϕz − Rf rl ϕz sup
R<|ϕz|<1
μzgzϕz
ω ϕz
.
4.20
When |ϕz| ≤ R, by using the mean value theorem, we have
μzgzRf ϕz − Rf rl ϕz ≤ 1 − rl μzgzϕz sup ∇ Rf w
|w|≤R
≤
1 − rl
1
max
sup μzgzϕz
1 − R w∈RB ωw z∈B
× sup ωw1 − |w|∇ Rf w.
4.21
w∈B
Since ωw1 − |w| is also normal, by Lemmas 2.1 and 2.5, we have
sup ωw1 − |w|∇ Rf w
sup ωw1 − |w|R2 fw
w∈B
w∈B
sup ωwRfw.
w∈B
4.22
12
Abstract and Applied Analysis
Hence we obtain
sup μzgzRf ϕz − Rf rl ϕz |ϕz|≤R
1 − rl
1
max
sup μzgzϕz.
1 − R w∈rB ωw z∈B
4.23
g
Since the boundedness of Iϕ : Bω → Bμ implies supz∈B μz|gz||ϕz| < ∞, letting l → ∞
in the above inequality, we have
sup sup μzgzRf ϕz − Rf rl ϕz −→ 0.
f Bω ≤1 |ϕz|≤R
4.24
By using 4.14, 4.20, and 4.24 and letting R → 1− , we obtain the desired estimate
g
Iϕ lim sup
|ϕz| → 1−
e
μzgzϕz
ω ϕz
g
.
4.25
g
So if condition 4.6 is true, then Iϕ e 0, which means that Iϕ : Bω → Bμ is compact. Our
proof is accomplished.
Theorem 4.6. Let ω be a normal weight function and μ a weight function. Suppose that the operator
g
g
Iϕ : Bω,0 → Bμ,0 is bounded. Then Iϕ : Bω,0 → Bμ,0 is compact if and only if
lim
μzgzϕz
|z| → 1−
ω ϕz
0.
4.26
Proof. Suppose that 4.26 holds. For any f ∈ Bω,0 , by Lemma 2.1 and 1.4, we have
g
μzR Iϕ f z μzRf ϕz gz
≤ μzgzϕz∇f ϕz μzgzϕz
ω ϕz ∇f ϕz ω ϕz
μzgzϕz f .
Bω
ω ϕz
4.27
Combining this with 4.26, we obtain
g
lim − sup μzR Iϕ f z 0.
|z| → 1
f Bω ≤1
g
Hence it follows from Lemma 2.4 that Iϕ : Bω,0 → Bμ,0 is compact.
4.28
Abstract and Applied Analysis
13
g
Conversely, we assume that Iϕ : Bω,0 → Bμ,0 is compact. By Theorem 3.2, we see that
lim − μzgzϕz 0.
4.29
|z| → 1
Thus this implies 4.26 if ϕ∞ < 1.
Now assume ϕ∞ 1. We claim that
lim sup
|z| → 1−
μzgzϕz
ω ϕz
lim sup
|ϕz| → 1−
μzgzϕz
ω ϕz
.
4.30
.
4.31
Further assume that {zk }k∈N is a sequence in B such that
lim sup
|z| → 1−
μzgzϕz
ω ϕz
lim
μzk gzk ϕzk k→∞
ω ϕzk If supk∈N |ϕzk | < 1, then from this and 4.29 we have that both limits in 4.30 are equal to
zero. If supk∈N |ϕzk | 1, then there is a subsequence {ϕzkl }l∈N such that |ϕzkl | → 1− as
l → ∞. Hence we have
lim sup
μzgzϕz
|z| → 1−
ω ϕz
lim
l→∞
μzkl gzkl ϕzkl ω ϕzkl ≤ lim sup
|ϕz| → 1−
μzgzϕz
ω ϕz
,
4.32
and so 4.30 holds.
g
Since Iϕ : Bω,0 → Bμ is also compact, by Theorem 4.5, we see that the second limit in
4.30 is equal to zero, so that 4.26 holds. This completes the proof.
Acknowledgment
This work was supported by JSPS Grant-in-Aid for Young Scientists Start-up: no. 20840004
and Ibaraki University.
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