Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 952831, 21 pages
doi:10.1155/2009/952831
Research Article
Weighted Composition Operators and
Integral-Type Operators between Weighted
Hardy 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 2 May 2009; Revised 3 June 2009; Accepted 8 June 2009
Recommended by Leonid Berezansky
We study the boundedness and compactness of the weighted composition operators as well as
integral-type operators between weighted Hardy spaces on the unit ball.
Copyright q 2009 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.
1. Introduction
Let B denote the open unit ball of the n-dimensional complex vector space Cn , ∂B its
boundary, and let HB denote the space of all holomorphic functions on B. For 0 < p < ∞
p
and α ≥ 0 we define the weighted Hardy space Hα B as follows:
p
Hα B
f ∈ HB : sup 1 − r
p
α
|frζ| dσζ < ∞ ,
1.1
∂B
0<r<1
where dσ is the normalized Lebesgue measure on ∂B see, also 1, as well as 2, for an
equivalent definition of the space. Note that for α 0 the weighted Hardy space becomes
the Hardy space H p B. We define the norm · Hαp on this space as follows:
p
f p sup 1 − rα
H
α
0<r<1
frζp dσζ.
∂B
1.2
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p
With this norm Hα B is a Banach space when 1 ≤ p < ∞. For a related space on the
unit polydisk; see 3. In this paper, we investigate two types of operators acting between
weighted Hardy spaces.
Let ϕ be a holomorphic self-map of B and u ∈ HB. Then ϕ and u induce a weighted
composition operator uCϕ on HB which is defined by uCϕ f uf ◦ ϕ. This type of operators
has been studied on various spaces of holomorphic functions in Cn , by many authors; see, for
example, 4, recent papers 5–17, and the references therein.
Let g ∈ HD and ϕ be a holomorphic self-map of the open unit disk D in the complex
plane. Products of integral and composition operators on HD were introduced by S. Li and
S. Stević in a private communication see 18–21, as well as papers 22 and 23 for closely
related operators as follows:
Cϕ Jg fz Jg Cϕ fz ϕz
0z
fζg ζdζ,
1.3
f ϕζ g ζdζ.
1.4
0
In 24 the first author of this paper has extended the operator in 1.4 in the unit ball
settings as follows see also 25, 26. Assume g ∈ HB, g0 0, and ϕ is a holomorphic
self-map of B, then we define an operator on the unit ball as follows:
g Pϕ f z 1
dt
f ϕtz gtz ,
t
0
f ∈ HB,
z ∈ B.
1.5
If n 1, then g ∈ HD and g0 0, so that gz zg0 z, for some g0 ∈ HD. By the
change of variable ζ tz, it follows that
g
Pϕ fz 1
dt
f ϕtz tzg0 tz t
0
z
f ϕζ g0 ζdζ.
1.6
0
Thus the operator 1.5 is a natural extension of operator Jg Cϕ in 1.4. For related operators
see 27–33 as well as the references therein.
In this paper we study the boundedness and compactness of the weighted composition
g
operators as well as the integral-type operator Pϕ , between different weighted Hardy spaces
on the unit ball.
Throughout this paper, constants are denoted by C, they are positive and may differ
from one occurrence to the other. The notation a b means that there is a positive constant C
such that a ≤ Cb. Moreover, if both a b and b a hold, then one says that a b.
2. Weighted Composition Operators
This section is devoted to studying weighted composition operators between weighted
Hardy spaces. Weighted composition operators between different Hardy spaces on the unit
ball were previously studied in 15, 34, while the composition operators on the unit ball were
studied in 35, 36. For the case of the unit disk see also 37.
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Before we formulate the main results in this section we quote several auxiliary results
which will be used in the proofs of these ones.
Lemma 2.1. Let 0 < p < ∞ and α ≥ 0. Suppose that u ∈ HB and ϕ is a holomorphic self-map of B.
Then for each f ∈ HB
uCϕ f p ≤ lim inf uCϕ fR p ,
H
H
−
R→1
α
α
2.1
where uCϕ fR z uzfRϕz.
Proof. Fix r ∈ 0, 1. Fatou’s lemma shows that
1 − rα
∂B
p
|urζf ϕrζ | dσζ ≤ 1 − rα lim inf
−
R→1
lim inf
1 − rα
−
R→1
p
|urζf Rϕrζ | dσζ
∂B
urζf Rϕrζ p dσζ
2.2
∂B
≤ lim inf
uCϕ fR H p .
−
R→1
α
Hence we have the desired inequality.
Recall that an f ∈ HB has the homogeneous expansion
fz ∞ c γ zγ ,
k0 |γ |k
2.3
where γ γ1 , . . . , γn is a multi-index, |γ| γ1 · · · γn and zγ z1 γ1 · · · zn γn . For the
homogeneous expansion of f and any integer j ≥ 1, let
Rj fz ∞ c γ zγ ,
k0 |γ |k
2.4
and Kj I − Rj where If f is the identity operator. Note that Kj is compact operator on
p
Hα B for each j ∈ N.
Lemma 2.2. If 1 < p < ∞, then Rj converges to 0 pointwise in the Hardy space H p B as j → ∞.
Proof. See 34, Corollary 3.4 .
Lemma 2.2 and the uniform boundedness principle show that {Rj } is an uniformly
bounded sequence in H p B.
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The following lemma is proved similar to 4, Lemma 3.16. We omit its proof.
p
q
Lemma 2.3. If uCϕ is bounded from Hα B into Hβ B, then
uCϕ e,H p B → H q B ≤ lim infuCϕ Rj H p B → H q B ,
α
j →∞
β
α
β
2.5
where · e,Hαp B → H q B and · Hαp B → H q B denote the essential norm and the operator norm,
β
β
respectively.
Lemma 2.4. Let 0 < p ≤ q < ∞. Suppose that μ is a positive Borel measure on B which satisfies
μBζ, t ≤ C1 tqn/p
ζ ∈ ∂B, t > 0,
2.6
for some positive constant C1 . Then there exists a positive constant C2 which depends only on p, q,
and the dimension n such that
q
B
|f|q dμ ≤ C1 C2 fH p ,
2.7
for any f ∈ H p B. Here Bζ, t {z ∈ B : |1 − z, ζ| < t}.
Proof. See 38, page 13, Theorem or 34, Lemma 2.1 .
Let 0 < q < ∞. For each r ∈ 0, 1, a holomorphic self-map ϕ of B and u ∈ HB, we
define a positive Borel measure μru,ϕ on B by
μru,ϕ E
ϕr −1
E
|ur |q dσ,
2.8
for all Borel sets E of B. By the change of variables formula from measure theory, we can
verify
B
g
dμru,ϕ
|urζ|q g ◦ ϕ rζ dσζ,
2.9
∂B
for each nonnegative measurable function g in B.
Theorem 2.5. Let 0 < p ≤ q < ∞ and α, β ≥ 0. Suppose that u ∈ HB and ϕ is a holomorphic
p
q
self-map of B. Then uCϕ : Hα B → Hβ B is bounded if and only if
sup sup 1 − r
w∈B 0<r<1
β
|urζ|
∂B
q
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ < ∞.
2.10
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Proof. For w ∈ B we put
fw z αn/p
1 − |w|2
2.11
.
1 − z, w2
p
Then we see that fw ∈ Hα B and moreover supw∈B fw Hαp ≤ C. By a straightforward
calculation, we have
q
uCϕ fw H q sup 1 − r
β
β
|urζ|
∂B
0<r<1
p
q
1 − |w|2
1 − ϕrζ, w 2
qαn/p
2.12
dσζ,
q
for all w ∈ B. Hence if uCϕ : Hα B → Hβ B is bounded, then u and ϕ satisfy the condition
sup sup 1 − r
β
|urζ|
q
∂B
w∈B 0<r<1
1 − |w|2
1 − ϕrζ, w 2
qαn/p
q
dσζ ≤ CuCϕ H p B → H q B < ∞.
α
β
2.13
Next we assume
M : sup sup 1 − r
q
β
|urζ|
∂B
w∈B 0<r<1
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ < ∞.
2.14
Fix r ∈ 0, 1 and R ∈ 0, 1, respectively. For ζ ∈ ∂B and t, 0 < t ≤ tR 1 − R, we put
w 1 − tζ and wR 1 − tR ζ. Since the function fw z, which is defined by 2.11 for this
w, satisfies
|fw z|q > 4−qαn/p t−qn/p 1 − R−qα/p
2.15
for all z ∈ Bζ, t, we have
μru,ϕ Bζ, t
tqn/p
≤4
qαn/p
1 − R
qα/p
Bζ,t
|fw z|q dμru,ϕ z
≤4
qαn/p
1 − R
qα/p
|urζ|
q
∂B
≤ 4qαn/p
1 − Rqα/p
1 − rβ
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ
2.16
M.
By the same argument, the function fwR z gives the following estimate:
μru,ϕ Bζ, 2tR ≤ 9qαn/p
1 − Rqα/p
1 − rβ
MtR qn/p .
2.17
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Now we need to prove that there exists a positive constant C such that
μru,ϕ Bζ, t ≤ C
1 − Rqα/p
1 − rβ
2.18
Mtqn/p ,
for all ζ ∈ ∂B and t > 0. By the estimate 2.16, we see that the inequality 2.18 is true for all
t ∈ 0, tR . Thus we assume t > tR . By the same argument as in 36, pages 241-242, proof of
Theorem 1.1 , we see that the inequality 2.17 shows that there exists a positive constant Cn
which depends only on the dimension n such that
μru,ϕ Bζ, t ≤ Cn
t
tR
n
9qαn/p
Cn 9qαn/p
≤ Cn 9qαn/p
1 − Rqα/p
1 − r
1 − Rqα/p
1 − r
β
1 − Rqα/p
1 − rβ
β
MtR qn/p
Mtn tR q/p−1n
2.19
Mtqn/p .
Hence for C max{4qαn/p , Cn 9qαn/p }, we have the inequality in 2.18.
p
For f ∈ Hα B the dilate function fR belongs to the ball algebra, and so fR is in the
Hardy space H p B. Hence Lemma 2.4 gives
B
|fR z|q dμru,ϕ z ≤ C C
1 − Rqα/p
1 − r
β
q
MfR H p ,
2.20
for some positive constant C and all R ∈ 0, 1. This implies that
1 − r
q/p
uCϕ fR rζq dσζ ≤ C CM 1 − Rα
fRζp dσζ
,
β
∂B
2.21
∂B
and so we have
q
q
uCϕ fR H q ≤ C CMfH p ,
2.22
α
β
for all R ∈ 0, 1. By Lemma 2.1 we have
q
q
uCϕ fH q ≤ C CfH p
α
β
× sup sup 1 − r
w∈B 0<r<1
This completes the proof.
β
q
|urζ|
∂B
1 − |w|2
1 − ϕrζ, w 2
qαn/p
2.23
dσζ.
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The following proposition is proved in a standard way; see, for example, the proofs of
the corresponding results in 4, 32, 33, 39. Hence we omit its proof.
Proposition 2.6. Let 0 < p, q < ∞ and α, β ≥ 0. Suppose that u ∈ HB and ϕ is a holomorphic selfp
q
p
q
map of B which induce the bounded operator uCϕ : Hα B → Hβ B. Then uCϕ : Hα B → Hβ B
p
is compact if and only if for every bounded sequence {fj }j∈N in Hα B which converges to 0 uniformly
q
on compact subsets of B, {uCϕ fj }j∈N converges to 0 in Hβ B.
In the proof of Theorem 2.8, we need the following lemma.
Lemma 2.7. Let 1 < p < ∞, α ≥ 0, and fw be the family of test functions defined in 2.11. Then
p
fw → 0 weakly in Hα B as |w| → 1− .
p
Proof. The family {fw }w∈B is bounded in Hα B and fw → 0 uniformly on compact subsets
p
of B as |w| → 1− . By the definitions of the space Hα B and the norm · Hαp , we see that
p
p
Hα B is a subspace of the weighted Bergman space Aα B and
fApα ≤ C α, p, n fHαp
p
f ∈ Hα B ,
2.24
for some positive constant Cα, p, n which depends on α, p, and n. This inequality implies
p
that the family {fw }w∈B is also bounded in Aα B. Note also that the family converges to 0
p
uniformly on compact subsets of B as |w| → 1− . Hence fw → 0 weakly in Aα B as |w| → 1− .
p
In order to prove that fw → 0 weakly in Hα B as |w| → 1− , we take an arbitrary
p
bounded linear functional Λ on Hα B. By the Hahn-Banach theorem, Λ can be extended to
w Λfw for all w ∈ B. Since fw → 0
on Apα B so that Λf
a bounded linear functional Λ
p
−
w → 0 as |w| → 1− , and so fw → 0
weakly in Aα B as |w| → 1 , we have Λfw Λf
p
−
weakly in Hα B as |w| → 1 .
Theorem 2.8. Let 1 < p ≤ q < ∞ and α, β ≥ 0. Suppose that u ∈ HB and ϕ is a holomorphic
p
q
self-map of B such that uCϕ : Hα B → Hβ B is bounded. Then the qth power of the essential norm
uCϕ e,H p B → H q B is comparable to
α
β
lim sup sup 1 − r
β
|w| → 1− 0<r<1
p
|urζ|
∂B
q
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ.
2.25
q
Hence uCϕ : Hα B → Hβ B is compact if and only if
lim sup 1 − r
|w| → 1− 0<r<1
β
|urζ|
∂B
q
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ 0.
2.26
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Proof. To prove a lower estimate
q
uCϕ e,H p B → H q B ≥ lim sup sup 1 − r
α
β
|urζ|
|w| → 1− 0<r<1
β
∂B
q
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ,
2.27
p
we consider the test functions fw defined in 2.11. The family {fw }w∈B is bounded in Hα B,
say by L, and fw → 0 uniformly on compact subsets of B as |w| → 1− . Thus by Lemma 2.7
p
we have that fw → 0 weakly in Hα B as |w| → 1− , so that Kfw H q → 0 as |w| → 1− for
p
β
q
every compact operator K : Hα B → Hβ B. Hence
LuCϕ − KHαp B → H q B ≥ lim sup uCϕ − K fw H q
β
β
|w| → 1−
2.28
≥ lim supuCϕ fw H q .
β
|w| → 1−
q
This inequality and 2.12 give the lower estimate for uCϕ e,H p B → H q B .
α
β
p
Next we prove an upper estimate. Take f ∈ Hα B with fHαp ≤ 1. Fix ε > 0 and put
M1 : lim sup sup 1 − r
q
β
|w| → 1− 0<r<1
|urζ|
∂B
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ.
2.29
dσζ < M1 ε,
2.30
Then we can choose R0 ∈ 0, 1 such that
sup 1 − r
0<r<1
β
|urζ|
∂B
q
1 − |w|2
1 − ϕrζ, w 2
qαn/p
for w ∈ B with |w| > R0 . Fix r ∈ 0, 1 and R ∈ R0 , 1. By the same argument as in the proof
of inequality 2.20 in Theorem 2.5, we obtain that
q
1 − Rqα/p
q
M1 εRj fR H p ,
Rj f R z dμru,ϕ z ≤ C
β
1 − r
B
2.31
where the positive constant C is independent of r, R and a positive integer j. Since fR is in
the ball algebra, Lemma 2.2 gives
q
q
q
q
Rj fR H p Rj fR H p ≤ supRj H p B → H p B fR H p .
j≥1
2.32
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Combining this with 2.31, we have
q/p
q
fR ζp dσζ
|uCϕ Rj fR rζ| dσζ ≤ C M1 ε 1 − Rα
1 − rβ
∂B
2.33
q
q
uCϕ Rj fR H q ≤ C M1 εfH p .
2.34
∂B
q
≤ C M1 εfH p ,
α
and so we have
α
β
Letting R → 1− , by Lemma 2.1, we obtain
q
q
uCϕ Rj fH q ≤ C M1 εfH p .
2.35
α
β
Since ε > 0 is arbitrary, this estimate and Lemma 2.3 imply
q
q
uCϕ e,H p B → H q B ≤ lim infuCϕ Rj H p B → H q B
α
j →∞
β
α
β
≤ C lim sup sup 1 − r
|w| → 1− 0<r<1
q
β
|urζ|
∂B
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ,
2.36
which completes the proof.
Remark 2.9. In the above proof, we used Lemma 2.2. This lemma required the assumption 1 <
p < ∞. Hence we cannot have an upper estimate for uCϕ e,H p B → H q B in the case 0 < p ≤ 1.
α
p
β
q
However, Proposition 2.6 shows that the compactness of uCϕ : Hα B → Hβ B 0 < p ≤ q <
∞ is equivalent to
lim sup 1 − r
|w| → 1− 0<r<1
q
β
|urζ|
∂B
1 − |w|2
1 − ϕrζ, w 2
qαn/p
dσζ 0.
2.37
3. Integral-Type Operators
g
Here we study the boundedness and compactness of the integral-type operators Pϕ between
weighted Hardy spaces on the unit ball.
For f ∈ HB with the Taylor expansion fz |γ|≥0 aγ zγ , let Rfz |γ|≥0 |γ|aγ zγ
be the radial derivative of f.
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The following lemma was proved in 24 see also 25.
Lemma 3.1. Assume that ϕ is a holomorphic self-map of B, g ∈ HB and g0 0. Then for every
f ∈ HB one holds
g R Pϕ f z f ϕz gz.
3.1
A positive continuous function ω on the interval 0, 1 is called normal 40 if there is
a δ ∈ 0, 1 and a and b, 0 < a < b such that
ωr
ωr
a is decreasing on δ, 1 and lim
a 0,
r → 1 1 − r
1 − r
ωr
1 − rb
is increasing on δ, 1 and lim
ωr
r → 1 1
3.2
∞.
− rb
If it is said that a function ω : B → 0, ∞ is normal, it is also assume that it is radial.
Lemma 3.2. Assume that 0 < q ≤ ∞, m is a positive integer and ω is normal. Then for every
f ∈ HB
sup ωrMq f, r f0 sup 1 − rm ωrMq Rm f, r ,
0<r<1
3.3
0<r<1
where
Mq f, r 1/q
|frζ|q dσζ
,
M∞ f, r supfrζ.
∂B
3.4
ζ∈∂B
Proof. The proof of the lemma in the case 1 ≤ q ≤ ∞ can be found in 27, Theorem 2. However,
due to an overlook, the proof for the case q ∈ 0, 1 has a gap. Hence we will give a correct
proof here in the case.
We may assume that f0 0, otherwise we can consider the functions hz fz −
f0. Also we may assume that δ 0, to avoid some minor technical difficulties.
By 27, Lemma 1, for each fixed q ∈ 0, 1, there is a positive constant C depending
only on q and the dimension n such that
C
Mq f, r ≤
r
r
1/q
q
r − tq−1 Mq Rf, t dt
0
for every r ∈ 0, 1 and f ∈ HB such that f0 0.
,
3.5
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From 3.5 and the fact that ω is normal, we have
1
sup ωrMq f, r ≤ C sup ωr
r
0≤r<1
0≤r<1
r
0
≤ C sup 1 − ra
0≤r<1
≤ C sup 1 − r
1/q
q
r − tq−1 Mq Rf, t dt
1
r
a1
0≤r<1
C sup 1 − ra
r
1/q
r − tq−1
0
ωq t
q
Mq Rf, t dt
1 − taq
r
1/q
r − tq−1
aqq dt
0 1 − t
r
1
0≤t<1
1/q
1 − uq−1
aqq du
0 1 − ur
0≤r<1
sup 1 − tωtMq Rf, t
sup 1 − tωtMq Rf, t .
0≤t<1
3.6
By 40, page 291, Lemma 6 there exists a positive constant C such that
1
1 − uq−1
C
,
aqq du ≤
1 − raq
0 1 − ur
3.7
for every r ∈ 0, 1. Combining this with 3.6, we have
1/q
1
sup ωrMq f, r ≤ C sup 1 − r
sup 1 − tωtMq Rf, t
aq
1 − r
0≤r<1
0≤r<1
0≤t<1
Csup 1 − tωtMq Rf, t .
a
3.8
0≤t<1
The reverse inequality is proved by the following inequality:
1r
1 − rMq Rf, r ≤ CMq f,
2
3.9
and the fact that ωr ω1 r/2 for ω normal see 27. Hence, we obtain the result for
the case m 1.
For m ≥ 2 it should be only noticed that 1 − rm ωr is still normal, that Rm f0 0
for every integer m ≥ 1, and use the method of induction.
Theorem 3.3. Let 0 < p ≤ q < ∞ and α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ is a
g
p
q
holomorphic self-map of B. Then Pϕ : Hα B → Hβ B is bounded if and only if
sup sup 1 − r
w∈B 0<r<1
βq
qαn/p
1 − |w|2
grζq dσζ < ∞.
1 − ϕrζ, w 2
∂B
3.10
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p
Proof. Take f ∈ Hα B with fHαp ≤ 1. Since the function 1 − rβ/q for β > 0 and 0 < q < ∞ is
normal, Lemma 3.2 gives
g
g g
sup 1 − rβ/q Mq Pϕ f, r Pϕ f0 sup 1 − rβ/q1 Mq R Pϕ f , r .
0<r<1
3.11
0<r<1
g
g
The assumption g0 0 implies Pϕ f0 0, and Lemma 3.1 shows RPϕ f gCϕ f. Hence
we obtain
g
q
q
sup 1 − rβ Mq Pϕ f, r sup 1 − rβq Mq gCϕ f, r ,
0<r<1
g
3.12
0<r<1
q
q
g
p
and so we obtain Pϕ fH q gCϕ fH q . This implies that the boundedness of Pϕ : Hα B →
β
q
βq
p
q
Hβ B is equivalent to the boundedness of gCϕ : Hα B → Hβq B. So Theorem 2.5 shows
that the condition
sup sup 1 − r
βq
w∈B 0<r<1
qαn/p
2
q
1
−
|w|
grζ dσζ < ∞
1 − ϕrζ, w 2
∂B
g
p
3.13
q
is a necessary and sufficient condition for the boundedness of Pϕ : Hα B → Hβ B. This
completes the proof.
The next proposition is proved similar to Proposition 2.6.
Proposition 3.4. Let 0 < p, q < ∞, and α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ
g
p
q
is a holomorphic self-map of B which induce the bounded operator Pϕ : Hα B → Hβ B. Then
g
p
q
p
Pϕ : Hα B → Hβ B is compact if and only if for every bounded sequence {fj }j∈N in Hα B which
g
q
converges to 0 uniformly on compact subsets of B, {Pϕ fj }j∈N converges to 0 in Hβ B.
Theorem 3.5. Let 0 < p ≤ q < ∞ and α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ
g
p
q
is a holomorphic self-map of B which induce the bounded operator Pϕ : Hα B → Hβ B. Then
g
p
q
Pϕ : Hα B → Hβ B is compact if and only if
lim sup 1 − r
|w| → 1− 0<r<1
βq
qαn/p
2
q
1
−
|w|
grζ dσζ 0.
1 − ϕrζ, w 2
∂B
3.14
p
Proof. First we assume that condition 3.14 holds. Take a bounded sequence {fj }j∈N ⊂ Hα B
which converges to 0 uniformly on compact subsets of B. Theorem 2.8 and the remark in
p
q
Section 2 show that gCϕ : Hα B → Hβq B is compact. Thus Proposition 2.6 implies that
lim gCϕ fj H q 0.
j →∞
βq
3.15
Discrete Dynamics in Nature and Society
q
g
13
q
q
g
From 3.15 and since Pϕ fj H q gCϕ fj H q , we have that Pϕ fj H q → 0 as j → ∞. By
β
g
β
βq
p
q
Proposition 3.4, we see that Pϕ : Hα B → Hβ B is compact.
To prove the necessity of the condition in 3.14, we consider the family of test
functions fw which is defined by 2.11. Hence we have
q
g
q
Pϕ fw H q gCϕ fw H q
β
βq
sup 1 − r
βq
0<r<1
qαn/p
1 − |w|2
grζq dσζ,
1 − ϕrζ, w 2
∂B
3.16
p
for all w ∈ B. Since {fw }w∈B is a bounded sequence in Hα B and fw → 0 uniformly on
g
compact subsets of B as |w| → 1− , the compactness of Pϕ and Proposition 3.4 show that
q
g
Pϕ fw H q → 0 as |w| → 1− . This fact along with 3.16 implies the condition in 3.14,
β
finishing the proof of the theorem.
Theorem 3.6. Let 1 < p ≤ q < ∞ and α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ is a
g
p
q
holomorphic self-map of B which induce the bounded operator Pϕ : Hα B → Hβ B. Then the qth
g
power of the essential norm of Pϕ is comparable to
lim sup sup 1 − r
βq
|w| → 1− 0<r<1
qαn/p
1 − |w|2
grζq dσζ.
1 − ϕrζ, w 2
∂B
3.17
p
Proof. To prove a lower estimate, we take an arbitrary compact operator K : Hα B →
q
Hβ B. Since Lemma 2.7 implies that the family of functions fw defined by 2.11 converges
p
to 0 weakly in Hα B as |w| → 1− , we obtain
g
g
g
CPϕ − KH p B → H q B ≥ lim sup Pϕ fw H q − Kfw H q ≥ lim supPϕ fw H q .
α
|w| → 1−
β
β
β
|w| → 1−
β
3.18
Combining this with 3.16, we have
g q
CPϕ e,H p B → H q B
α
β
≥ lim sup sup 1 − r
|w| → 1− 0<r<1
βq
qαn/p
2
q
1
−
|w|
grζ dσζ,
1 − ϕrζ, w 2
∂B
3.19
which is a lower estimate.
By some modification of Lemma 2.3 and the application of Lemmas 3.1 and 3.2, we get
g q
g
q
Pϕ e,H p B → H q B ≤ lim inf sup Pϕ Rj fH q
α
β
j → ∞ f p ≤1
H
β
α
q
≤ C lim inf sup gCϕ Rj fH q .
j → ∞ f p ≤1
H
α
βq
3.20
14
Discrete Dynamics in Nature and Society
As in the proof of Theorem 2.8, we obtain
q
lim inf sup gCϕ Rj fH q ≤ C lim sup sup 1 − rβq
j → ∞ f p ≤1
H
|w| → 1− 0<r<1
βq
α
qαn/p
q
1 − |w|2
grζ
×
dσζ,
1 − ϕrζ, w 2
∂B
3.21
g q
and so we have an upper estimate for Pϕ e,H p B → H q B .
α
β
g
4. The Case Pϕ : Hα∞ B → Hβ∞ B
When p ∞ and α > 0, we define the weighted-type space Hα∞ B as follows:
Hα∞ B
α
f ∈ HB : sup 1 − r M∞ f, r < ∞ .
4.1
0<r<1
It is easy to see that f ∈ Hα∞ B if and only if supz∈B 1 − |z|α |fz| < ∞, so we define the
norm fHα∞ on Hα∞ B by this supremum.
∞
B defined by
Furthermore we consider the subspace Hα,0
∞
Hα,0
B f ∈ HB : lim− 1 − rα M∞ f, r 0 .
r →1
4.2
Theorem 4.1. Let α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ is a holomorphic self-map
g
∞
B → Hβ∞ B is bounded if and only if
of B. Then Pϕ : Hα∞ B or Hα,0
1 − |z|β1 gz
sup
α < ∞.
1 − ϕz
z∈B
g
In this case, the operator norm Pϕ H ∞ B
α
∞
or Hα,0
B → Hβ∞ B
4.3
is comparable to the above supremum.
Proof. By the definition of the space Hα∞ B, f ∈ Hα∞ B satisfies the growth condition
fw ≤ 1 − |w|−α f
Hα∞
w ∈ B,
4.4
so it follows from Lemma 3.1 and Lemma 3.2 that
g
Pϕ fH ∞
β
for every f ∈ Hα∞ B.
sup1 − |z|
z∈B
1 − |z|β1 gz
gCϕ fz ≤ fHα∞ sup α ,
1 − ϕz
z∈B
β1 4.5
Discrete Dynamics in Nature and Society
15
Hence we obtain
g
Pϕ H ∞ B or H ∞ B → H ∞ B
α
α,0
β
1 − |z|β1 gz
≤ Csup
1 − |ϕz|α
z∈B
.
4.6
Now we prove the reverse inequality. For w ∈ B, we put
1
.
1 − z, wα
fw z 4.7
∞
Note that fw ∈ Hα,0
B for each w ∈ B and moreover supw∈B fw Hα∞ ≤ 1.
When ϕz /
0, we have
g
g
Pϕ H ∞ B → H ∞ B ≥ Pϕ ftϕz/|ϕz| α,0
β
Hβ∞
≥ 1 − |z|β1 gzf
1 − |z|β1 gz
α ,
1 − tϕz
sup1 − |w|β1 gCϕ ftϕz/|ϕz| w
w∈B
tϕz/|ϕz|
ϕz 4.8
for all t ∈ 0, 1. Letting t → 1− in 4.8, we have
g
Pϕ H ∞ B → H ∞ B
α,0
β
1 − |z|β1 gz
≥C
α .
1 − ϕz
4.9
∞
B we obtain
For the constant function 1 ∈ Hα,0
g
Pϕ 1H q sup1 − |w|β1 gCϕ 1w ≥ 1 − |z|β1 gz.
β
4.10
w∈B
Inequality 4.10 shows that the estimate in 4.9 also holds when ϕz 0.
Hence, from 4.9 we obtain
g
Pϕ H ∞ B → H ∞ B
α,0
β
1 − |z|β1 gz
≥ Csup
α ,
1 − ϕz
z∈B
4.11
which along with the obvious inequality
g
g
Pϕ H ∞ B → H ∞ B ≥ Pϕ H ∞ B → H ∞ B
α
β
α,0
β
4.12
completes the proof of the theorem.
g
∞
For the compactness of Pϕ : Hα∞ B or Hα,0
B → Hβ∞ B, we can also prove the
following proposition which is similar to Proposition 2.6.
16
Discrete Dynamics in Nature and Society
Proposition 4.2. Let α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ is a holomorphic
g
∞
B → Hβ∞ B. Then
self-map of B which induce the bounded operator Pϕ : Hα∞ B or Hα,0
g
∞
Pϕ : Hα∞ B or Hα,0
B → Hβ∞ B is compact if and only if for every bounded sequence {fj }j∈N
g
∞
in Hα∞ B or Hα,0
B which converges to 0 uniformly on compact subsets of B, {Pϕ fj }j∈N
converges to 0 in Hβ∞ B.
Theorem 4.3. Let α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ is a holomorphic selfg
∞
map of B such that Pϕ : Hα∞ B or Hα,0
B → Hβ∞ B is bounded. Then the essential norm
g
Pϕ e,H ∞ B or
α
∞
Hα,0
B → Hβ∞ B
is comparable to
1 − |z|β1 gz
lim sup
α .
|ϕz| → 1− 1 − ϕz 4.13
g
∞
In particular, Pϕ : Hα∞ B or Hα,0
B → Hβ∞ B is compact if and only if
1 − |z|β1 gz
lim
α 0.
|ϕz| → 1− 1 − ϕz
4.14
Proof. First we consider the family {fw }w∈B where
fw z 1 − |w|
1 − z, wα1
4.15
.
∞
We can easily check that fw ∈ Hα,0
B, fw Hα∞ ≤ 1 for all w ∈ B and fw → 0 uniformly on
compact subsets of B as |w| → 1− . Hence 40, page 296, Theorem 2 implies that fw → 0
∞
B as |w| → 1− .
weakly in Hα,0
If ϕ∞ < 1, then as in the proof of 26, Theorem 3 it can be seen that the operator
g
∞
B → Hβ∞ B is compact, so that
Pϕ : Hα∞ B or Hα,0
g
Pϕ e,H ∞ B or
α
∞
Hα,0
B → Hβ∞ B
0.
4.16
On the other hand, the limit in 4.13 is vacuously equal to zero, from which the result follows
in this case. If ϕ∞ 1, then take a sequence {ϕzj }j∈N in B with |ϕzj | → 1 as j → ∞
∞
B
and put Fj z fϕzj z for each j ∈ N. Then {Fj }j∈N is a bounded sequence in Hα,0
Discrete Dynamics in Nature and Society
17
∞
B, as j → ∞. Hence for every compact operator
and {Fj }j∈N converges to 0 weakly in Hα,0
∞
B → Hβ∞ B we have KFj H ∞ → 0 as j → ∞. So we have
K : Hα,0
β
g
g
Pϕ − KH ∞ B → H ∞ B ≥ lim supPϕ Fj − KFj H ∞
α,0
j →∞
β
β
g
≥ lim supPϕ Fj H ∞
j →∞
β
lim sup sup 1 − |w|β1 gw Fj ϕw j →∞
w∈B
4.17
β1 ≥ lim sup 1 − zj g zj Fj ϕ zj j →∞
1 − |zj |β1 g zj ≥ α1 lim sup
α ,
2
1 − ϕ zj j →∞
1
∞
for all compact operators K : Hα,0
B → Hβ∞ B. Taking the infimum over the set of all
∞
B → Hβ∞ B, we obtain
compact operators K : Hα,0
β1 g zj 1 − zj ≥ C lim sup α .
1 − ϕ zj j →∞
g
Pϕ e,H ∞ B → H ∞ B
α,0
β
g
4.18
g
Combining this with the estimate Pϕ e,H ∞ B → H ∞ B ≥ Pϕ e,H ∞ B → H ∞ B , we have
α
g
Pϕ e,H ∞ B → H ∞ B
α
β
β
α,0
β
1 − |z|β1 gz
≥ Clim sup
α .
|ϕz| → 1− 1 − ϕz 4.19
Next we prove an upper estimate. Assume that {rl }l∈N ⊂ 0, 1 is a sequence which
increasingly converges to 1. For this {rl }l∈N , we define the operators defined by
g
Prl ϕ fz 1
dt
.
gtzf rl ϕtz
t
0
4.20
g
As in the proof of 26, Theorem 3, Proposition 4.2 shows that Prl ϕ : Hα∞ B → Hβ∞ B is
compact for each l ∈ N.
Put
1 − |z|β1 gz
M2 : lim sup α ,
|ϕz| → 1− 1 − ϕz
4.21
18
Discrete Dynamics in Nature and Society
and fix ε > 0. Then we can choose R ∈ 0, 1 such that
1 − |z|β1 gz
α < M2 ε,
1 − ϕz
4.22
if R < |ϕz| < 1. Take f ∈ Hα∞ B with fHα∞ ≤ 1 and an integer l ∈ N. By Lemma 3.1 and
Lemma 3.2, we have
g g
g
g
Pϕ f − Prl ϕ fH ∞ sup1 − |z|β1 R Pϕ f z − R Prl ϕ f z
β
z∈B
sup1 − |z|β1 gzf ϕz − f rl ϕz z∈B
sup 1 − |z|β1 gzf ϕz − f rl ϕz |ϕz|≤R
sup 1 − |z|β1 gzf ϕz − f rl ϕz .
R<|ϕz|<1
4.23
By using the mean value theorem and the asymptotic relation
sup1 − |z|α1 ∇fz sup1 − |z|α1 Rfz,
z∈B
4.24
z∈B
we obtain
sup f ϕz − f rl ϕz ≤ sup 1 − rl ϕz sup ∇fw
|w|≤R
|ϕz|≤R
|ϕz|≤R
≤
1 − rl R
1 − R
α1
sup1 − |w|α1 ∇fw
w∈B
1 − rl R
sup1 − |w|
1 − Rα1 w∈B
1 − rl R
1 − Rα1
Rfw
α1 4.25
fHα∞ .
g
g
Since the boundedness of Pϕ : Hα∞ B → Hβ∞ B implies that Pϕ 1 ∈ Hβ∞ B, we see
sup1 − |z|β1 gz < ∞,
4.26
z∈B
and so we have
1 − rl R
sup 1 − |z|β1 gzf ϕz − f rl ϕz ≤ C
sup1 − |z|β1 gz.
α1
1 − R
z∈B
fHα∞ ≤1 |ϕz|≤R
4.27
sup
Discrete Dynamics in Nature and Society
19
On the other hand, the monotonicity of M∞ f, r shows
f rl ϕz ≤ 1 − ϕz −α fr l
Hα∞
−α
≤ 1 − ϕz fHα∞ .
4.28
Thus we have
sup
sup 1 − |z|
1 − |z|β1 gz
gz f ϕz − f rl ϕz ≤ 2 sup
α
1 − ϕz
R<|ϕz|<1
β1 fH ∞ ≤1 R<|ϕz|<1
α
≤ 2M2 ε.
4.29
g
From 4.23, 4.27, 4.29, and the compactness of Prl ϕ , we obtain
g
g
g
Pϕ e,H ∞ B → H ∞ B ≤ Pϕ − Prl ϕ H ∞ B → H ∞ B
α
α
β
≤C
1 − rl R
1 − R
α1
β
sup1 − |z|β1 gz 2M2 ε.
4.30
z∈B
Letting l → ∞ and ε → 0, we have
g
Pϕ e,H ∞ B → H ∞ B
α
β
1 − |z|β1 gz
≤ 2lim sup α .
|ϕz| → 1− 1 − ϕz
4.31
This completes the proof.
g
∞
∞
∞
When Pϕ : Hα∞ B or Hα,0
B → Hβ,0
B is bounded, we see that g ∈ Hβ1,0
B. By
a standard argument as in the proof of 26, Corollary 3, we have
lim sup
|z| → 1−
1 − |z|β1 gz
1 − |ϕz|α
1 − |z|β1 gz
lim sup α ,
|ϕz| → 1− 1 − ϕz
4.32
and so
g
Pϕ e,H ∞ Bor H ∞ B → H ∞ B
α
α,0
β,0
1 − |z|β1 gz
lim sup α .
1 − ϕz
|z| → 1−
4.33
g
Hence we obtain the following characterization for the compactness of the operator Pϕ :
∞
∞
B → Hβ,0
B.
Hα∞ B or Hα,0
20
Discrete Dynamics in Nature and Society
Corollary 4.4. Let α, β > 0. Suppose that g ∈ HB with g0 0 and ϕ is a holomorphic self-map of
g
g
∞
∞
∞
B → Hβ,0
B is bounded. Then Pϕ : Hα∞ B or Hα,0
B →
B such that Pϕ : Hα∞ B or Hα,0
∞
Hβ,0 B is compact if and only if
1 − |z|β1 gz
lim α 0.
|z| → 1−
1 − ϕz
4.34
Acknowledgment
The second author of this paper is supported by Grant-in-Aid for Young Scientists Start-up;
no.20840004.
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