Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 698038, 16 pages
doi:10.1155/2011/698038
Research Article
Integral-Type Operators from Bloch-Type Spaces to
QK Spaces
Stevo Stević1 and Ajay K. Sharma2
1
2
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, India
Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs
Received 16 May 2011; Accepted 29 June 2011
Academic Editor: Narcisa C. Apreutesei
Copyright q 2011 S. Stević and A. K. Sharma. 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.
z
n
The boundedness and compactness of the integral-type operator Iϕ,g fz 0 f n ϕζgζdζ,
where n ∈ N0 , ϕ is a holomorphic self-map of the unit disk D, and g is a holomorphic function on
D, from α-Bloch spaces to QK spaces are characterized.
1. Introduction
Let D be the open unit disk in the complex plane, ∂D be its boundary, Dw, r be disk centered
at w of radius r, and let HD be the class of all holomorphic functions on D. Let
ηa z a−z
,
1 − az
a ∈ D,
1.1
be the involutive Möbius transformation which interchanges points 0 and a. If X is a Banach
space, then by BX we will denote the closed unit ball in X.
The α-Bloch space, Bα D Bα , α > 0, consists of all f ∈ HD such that
α sup 1 − |z|2 f z < ∞.
z∈D
1.2
2
Abstract and Applied Analysis
The little α-Bloch space Bα0 D Bα0 consists of all functions f holomorphic on D such that
α
lim|z| → 1 1 − |z|2 |f z| 0. The norm on Bα is defined by
α f α f0 sup 1 − |z|2 f z.
B
z∈D
1.3
With this norm, Bα is a Banach space, and the little α-Bloch space Bα0 is a closed subspace of
the α-Bloch space. Note that B1 B is the usual Bloch space.
Given a nonnegative Lebesgue measurable function K on 0, 1 the space QK consists
of those f ∈ HD for which
2
bQ
f sup
K
a∈D
D
2 f z K 1 − ηa z2 dmz < ∞,
1.4
where dmz 1/πdx dy 1/πr dr dθ is the normalized area measure on D 1. It is
known that bQK is a seminorm on QK which is Möbius invariant, that is,
bQK f ◦ η bQK f ,
η ∈ AutD,
1.5
where AutD is the group of all automorphisms of the unit disk D. It is a Banach space with
the norm defined by
f f0 bQ f .
K
QK
1.6
The space QK,0 consists of all f ∈ HD such that
lim
|a| → 1
D
2 f z K 1 − ηa z2 dmz 0.
1.7
It is known that QK,0 is a closed subspace of QK . For classical Q spaces, see 2.
It is clear that each QK contains all constant functions. If QK consists of just constant
functions, we say that it is trivial. QK is nontrivial if and only if
1
sup
t∈0, 1
K1 − r
0
1 − t2
1 − tr 2 3
r dr < ∞.
1.8
Throughout this paper, we assume that condition 1.8 is satisfied, so that the space QK is
nontrivial. An important tool in the study of QK spaces is the auxiliary function λK defined
by
Kst
,
0<t≤1 Kt
λK s sup
0 < s < ∞,
1.9
where the domain of K is extended to 0, ∞ by setting Kt K1 when t > 1. The next two
conditions play important role in the study of QK spaces.
Abstract and Applied Analysis
3
a There is a constant C > 0 such that for all t > 0
K2t ≤ CKt.
1.10
b The auxiliary function λK satisfies the following condition:
1
0
λK s
ds < ∞.
s
1.11
Let Ω0, ∞ denote the class of all nondecreasing continuous functions on 0, ∞
satisfying conditions 1.8, 1.10, and 1.11.
A positive Borel measure μ on D is called a K-Carleson measure 3 if
K
sup
SI
I
1 − |z|
dμz < ∞,
|I|
1.12
where the supermum is taken over all subarcs I ⊂ ∂D, |I| is the length of I, and SI is the
Carleson box defined by
SI z
∈I .
z : 1 − |I| < |z| < 1,
|z|
1.13
A positive Borel measure μ is called a vanishing K-Carleson measure if
lim
K
|I| → 0
SI
1 − |z|
dμz 0.
|I|
1.14
We also need the following results of Wulan and Zhu in 3, in which QK spaces are
characterized in terms of K-Carleson measures.
Theorem 1.1. Let K ∈ Ω0, ∞. Then a positive Borel measure μ on D is a K-Carleson measure if
and only if
sup
a∈D
D
2 K 1 − ηa z dμz < ∞.
1.15
Also, μ is a vanishing K-Carleson measure if and only if
lim
|a| → 1
D
2 K 1 − ηa z dμz 0.
1.16
From Theorem 1.1 and the definition of the spaces QK and QK,0 , we see that when
K ∈ Ω0, ∞, then f ∈ QK if and only if the measure dμf |f z|2 dmz is a K-Carleson
measure, while f ∈ QK,0 if and only if this measure is a vanishing K-Carleson measure.
4
Abstract and Applied Analysis
Let ϕ ∈ SD be the family of all holomorphic self-maps of D, g ∈ HD, and n ∈ N0 .
We define an integral-type operator as follows:
n
Iϕ,g fz z
f n ϕζ gζdζ,
z ∈ D.
1.17
0
Operator 1.17 extends several operators which has been introduced and studied recently
see, e.g., 4–9. For related operators in n-dimensional case, see, for example, 10–19. For
some classical operators see, for example, 20, 21 and the related references therein. For other
product-type operators, see 22 and the references therein.
Motivated by 23, 24 see also 25–29, we characterize when ϕ and g induce
bounded and/or compact operators in 1.17 from α-Bloch to QK spaces.
Throughout this paper, constants are denoted by C; they are positive and not
necessarily the same at each occurrence. The notation A B means that there is a positive
constant C such that B/C ≤ A ≤ CB.
2. Auxiliary Results
Here, we quote several lemmas which will be used in the proofs of the main results in this
paper. The following lemma is folklore see, e.g., 30.
Lemma 2.1. For any f ∈ HD and z ∈ D, the following inequalities hold
⎧ f α ,
⎪
if 0 < α < 1,
⎪
B
⎪
⎪
⎪
⎪
e
⎪
⎨f Bα ln
, if α 1,
fz ≤ C
1 − |z|2
⎪
f α
⎪
⎪
B
⎪
if α > 1,
⎪
,
⎪
⎪
⎩ 1 − |z|2 α−1
2.1
α supw∈Dz,1−|z|/2 1 − |w|2 f w
n f z ≤ C
αn−1
1 − |z|2
f α
B
≤ C
αn−1 , if n ∈ N.
2
1 − |z|
2.2
The next lemma is obtained in 31, 32.
Lemma 2.2. Let α > 0. Then there are two functions f1 , f2 ∈ Bα such that
f z f z ≥ 2
1
C
1 − |z|2
α ,
z ∈ D.
2.3
Abstract and Applied Analysis
5
Also, if α /
1, then there are two functions f3 , f4 ∈ Bα and C > 0, such that
f3 z f4 z ≥ C
1 − |z|
2
α−1 ,
z ∈ D.
2.4
The next Schwartz-type lemma 33 is proved in a standard way, so we omit the proof.
n
Lemma 2.3. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N0 . Then Iϕ,g :
Bα or Bα0 → QK is compact if and only if for any bounded sequence fm m∈N in Bα converging
n
to zero on compacts of D, we have limm → ∞ Iϕ,g fm Q 0.
K
n
Lemma 2.4. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N0 . Then Iϕ,g : Bα0 →
QK or QK,0 is weakly compact if and only if it is compact.
n ∗
n
Proof. By a known theorem Iϕ,g : Bα0 → QK or QK,0 is weakly compact if and only if Iϕ,g :
∗
∗
or QK,0
→ Bα0 ∗ is weakly compact. Since Bα0 ∗ ∼
QK
A1 the Bergman space and A1
n ∗
∗
∗
has the Schur property, it follows that it is equivalent to Iϕ,g : QK
or QK,0
→ Bα0 ∗ , is
n
compact, which is equivalent to Iϕ,g : Bα0 → QK or QK,0 , is compact, as claimed.
n
Lemma 2.5. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N0 . Then Iϕ,g : Bα0 → QK,0
n
is compact if and only if Iϕ,g : Bα → QK,0 is bounded.
n
Proof. By Lemma 2.4, Iϕ,g : Bα0 → QK,0 is compact if and only if it is weakly compact, which,
n ∗∗
by Gantmacher’s theorem 34, is equivalent to Iϕ,g Bα0 ∗∗ ⊆ QK,0 . Since Bα0 ∗∗ Bα
n ∗∗
and by a standard duality argument Iϕ,g n
n
Iϕ,g on Bα , this can be written as Iϕ,g Bα ⊆ QK,0 ,
which by the closed graph theorem is equivalent to
n
Iϕ,g
: Bα → QK,0 is bounded.
For a ∈ D, set
Φϕ,g,K a D
2 21−α−n
2 2 K 1 − ηa z gz 1 − ϕz
dmz.
2.5
Lemma 2.6. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N0 . If Φϕ,g,K is finite at some
point a ∈ D, then it is continuous on D.
Proof. We follow the lines of Lemma 2.3 in 24. From the elementary inequality
2
1 − ηa z
4
1 − |a|1 − |a1 |
≤
,
2 ≤
4
−
− |a1 |
1
|a|1
1 − ηa1 z
a, a1 , z ∈ D,
2.6
and since K is nondecreasing and satisfies 1.10, we easily get
2 2 K 1 − ηa1 z ≤ Clog2 4/1−|a|1−|a1 |1 K 1 − ηa z .
2.7
6
Abstract and Applied Analysis
From 2.7 and since Φϕ,g,K a is finite, it follows that Φϕ,g,K is finite at each point a1 ∈ D. Let
a ∈ D be fixed, and let al l∈N ⊂ D be a sequence converging to a.
We have
Φϕ,g,K a − Φϕ,g,K al ≤
gz2 K 1 − ηa z2 − K 1 − ηa z2 l
dmz.
2 2αn−1
D
1 − ϕz
2.8
From 2.6, we have that for l such that 1 − |al | ≥ 1 − |a|/2, say l ≥ l0 , holds
2
1 − ηal z ≤
8
1 − |a|2
2 1 − ηa z ,
2.9
and consequently for l ≥ l0 , it holds
2 2 2 2
K 1 − ηa z − K 1 − ηal z ≤ 1 Clog2 8/1−|a| 1 K 1 − ηa z .
2.10
From this and since Φϕ,g,K is finite at a, by the Lebesgue dominated convergence
theorem, we get that the integral in 2.8 converges to zero as l → ∞ which implies that
Φϕ,g,K al → Φϕ,g,K a as l → ∞, from which the lemma follows.
n
3. Boundedness and Compactness of Iϕ,g : Bα or Bα0 → QK or QK,0 n
In this section, we characterize the boundedness and compactness of the operators Iϕ,g :
Bα or Bα0 → QK or QK,0 . Let
2 21−α−n
2 dμϕ,g,n,α z gz 1 − ϕz
dmz.
3.1
Theorem 3.1. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N, or n 0 and α > 1.
Then the following statements are equivalent.
n
a Iϕ,g : Bα → QK is bounded.
n
b Iϕ,g : Bα0 → QK is bounded.
21−α−n
c M : supa∈D D K1 − |ηa z|2 |gz|2 1 − |ϕz|2 dmz < ∞.
d dμϕ,g,n,α z is a K-Carleson measure.
n
Moreover, if Iϕ,g : Bα → QK is bounded, then the next asymptotic relations hold
n Iϕ,g Bα → QK
n Iϕ,g Bα0 → QK
M1/2 .
3.2
Abstract and Applied Analysis
7
Proof. By Theorem 1.1, it is clear that c and d are equivalent.
n
c ⇒ a. Let f ∈ BBα . First note that Iϕ,g f0 0 for each f ∈ HB and n ∈ N0 . From
this and by Lemma 2.1, we have
n 2
Iϕ,g f QK
sup
2 n 2
Iϕ,g f z K 1 − ηa z dmz
a∈D
D
a∈D
D
2 2 2 sup f n ϕz gz K 1 − ηa z dmz
2
≤ Cf Bα sup
a∈D
D
3.3
2 21−α−n
2 2 K 1 − ηa z gz 1 − ϕz
dmz,
n
from which the boundedness of Iϕ,g : Bα → QK follows, and moreover
n Iϕ,g Bα → QK
≤ CM1/2 .
3.4
a ⇒ b. This implication is obvious.
b ⇒ c. By Lemma 2.2, if n ∈ N, there are two functions f1 , f2 ∈ Bα such that 2.3
holds, and if n 0 and α > 1 such that 2.4 holds. Let
h1 z f1 z −
n−1 f k 0
1
k1
k!
k
z ,
h2 z f2 z −
n−1 f k 0
2
k1
k!
zk .
3.5
It is known see 30 that for each f ∈ HD and n ∈ N, we have
1 − |z|2
n−1
αn−1 α n k f z f 0 1 − |z|2 f z.
3.6
k1
k
k
From this, Lemma 2.2, and since h1 0 h2 0 0, k 0, 1, . . . , n − 1, we have that
there is a δ > 0 such that
−αn−1 n n C 1 − |z|2
≤ h1 z h2 z,
for |z| > δ.
3.7
Now note that for any f ∈ Bα , the functions fr z frz, r ∈ 0, 1 belong to Bα , and
moreover, sup0<r<1 fr Bα ≤ fBα .
8
Abstract and Applied Analysis
n
Applying 3.7, using an elementary inequality, the boundedness of Iϕ,g : Bα0 → QK ,
and the last inequality, we obtain
|rϕz|>δ
2 21−α−n
2 2 r 2n K 1 − ηa z gz 1 − r ϕz
dmz
≤C
C
D
D
2 n 2 2 n 2
r 2n K 1 − ηa z gz h1 rϕz h2 rϕz dmz
2
2 n
K 1 − ηa z Iϕ,g h1 r z dmz
C
3.8
2
2 n
K 1 − ηa z Iϕ,g h2 r z dmz
D
2
n ≤ Iϕ,g α
B0 → QK
h1 2Bα h2 2Bα .
Letting r → 1 in 3.8 and using the monotone convergence theorem, we get
|ϕz|>δ
2
2 21−α−n
2 2 n K 1 − ηa z gz 1 − ϕz
dmz ≤ CIϕ,g α
B0 → QK
.
3.9
n
On the other hand, for f0 z zn /n! ∈ Bα0 , we get Iϕ,g f0 ∈ QK which implies
2
2
n f0 α
I
ϕ,g
B
α
2 21−α−n
2
2 B0 → QK
gz 1 − ϕz
sup
K 1 − ηa z
dmz ≤
.
2αn−1
α∈D |ϕz|≤δ
1 − δ2 3.10
n
From 3.9 and 3.10, c follows. Moreover we get M1/2 ≤ CIϕ,g Bα → Q . From this, 3.4
n
n
0
K
and since Iϕ,g Bα → Q ≤ Iϕ,g Bα → Q the asymptotic relations in 3.2 follow, finishing the
K
K
0
proof of the theorem.
Theorem 3.2. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N, or n 0 and α > 1. Let
n
Iϕ,g : Bα → QK be bounded. Then the following statements are equivalent.
n
a Iϕ,g : Bα → QK is compact.
n
b Iϕ,g : Bα0 → QK is compact.
n
c Iϕ,g : Bα0 → QK is weakly compact.
d supa∈D D K1 − |ηa z|2 |gz|2 dmz < ∞, and
lim sup
r → 1 a∈D
|ϕz|>r
2 21−α−n
2 2 K 1 − ηa z gz 1 − ϕz
dmz 0.
3.11
Abstract and Applied Analysis
9
Proof. By Lemma 2.4, we have that b is equivalent to c.
d ⇒ a. Let fl l∈N be a bounded sequence in Bα , say by L, converging to zero
n
uniformly on compacts of D. Then fl also converges to zero uniformly on compacts of D.
From 3.11 we have that for every ε > 0 there is an r1 ∈ 0, 1 such that for r ∈ r1 , 1
sup
a∈D
|ϕz|>r
2 21−α−n
2 2 K 1 − ηa z gz 1 − ϕz
dmz < ε.
3.12
Therefore, by Lemma 2.1 and 3.12, we have that for r ∈ r1 , 1
n 2
Iϕ,g fl QK
|ϕz|≤r
|ϕz|>r
2
2 2 n fl ϕz K 1 − ηa z gz dmz
2
2 2
n < sup fl ϕz K 1 − ηa z gz dmz CL2 ε.
D
|ϕz|≤r
3.13
n
Letting l → ∞ in 3.13, using the first condition in d and sup|w|≤r |fl w| → 0 as l → ∞,
n
n
it follows that liml → ∞ Iϕ,g fl Q 0. Thus, by Lemma 2.3, Iϕ,g : Bα → QK is compact.
K
a ⇒ b. The implication is trivial since Bα0 ⊂ Bα .
b ⇒ d. By choosing fz zn /n! ∈ Bα0 , n ∈ N0 , we have that the first condition
in d holds. Let fl z zl /l, l ∈ N. It is easy to see that fl l∈N is a bounded sequence in
Bα0 converging to zero uniformly on compacts of D. Hence, by Lemma 2.3, it follows that
n
Iϕ,g fl QK → 0 as l → ∞. Thus, for every ε > 0, there is an l0 ∈ N, l0 > n such that for l ≥ l0
⎛
⎞2
n−1
2l−n 2
2 ⎝
⎠
sup ϕz
K 1 − ηa z gz dmz < ε.
l−j
a∈D
j1
D
3.14
From 3.14 we have that for each r ∈ 0, 1 and l ≥ l0
⎛
r 2l−n ⎝
⎞2
⎠
sup
l−j
n−1
a∈D
j1
Hence, for r ∈ n−1
j1 l0
−1/l0 −n
− j
sup
a∈D
|ϕz|>r
|ϕz|>r
2
2 K 1 − ηa z gz dmz < ε.
3.15
, 1, we have that
2
2 K 1 − ηa z gz dmz < ε.
3.16
10
Abstract and Applied Analysis
Let f ∈ BBα0 , and let ft z ftz, 0 < t < 1. Then sup0<t<1 ft Bα ≤ fBα , ft ∈ Bα0 ,
t ∈ 0, 1, and ft → f uniformly on compact subsets of D as t → 1. The compactness of
n
Iϕ,g : Bα0 → QK implies
n
n limIϕ,g ft − Iϕ,g f t→1
QK
0.
3.17
Hence, for every ε > 0, there is a t ∈ 0, 1 such that
2
2 2 n sup ft ϕz − f n ϕz K 1 − ηa z gz dmz < ε.
D
a∈D
3.18
−1/l0 −n
From this and 3.16, we have that for such t and each r ∈ n−1
, 1
j1 l0 − j
sup
a∈D
|ϕz|>r
2
2 2 n ϕz K 1 − ηa z gz dmz
f
≤ 2 sup
|ϕz|>r
a∈D
2
2 2 n ft ϕz − f n ϕz K 1 − ηa z gz dmz
2 sup
|ϕz|>r
a∈D
2
2 2 n ft ϕz K 1 − ηa z gz dmz
3.19
n 2
< 2ε 1 ft .
∞
From 3.19 we conclude that for every f ∈ BBα0 , there is a δ0 ∈ 0, 1 and δ0 δ0 f, ε such
that for r ∈ δ0 , 1
sup
a∈D
|ϕz|>r
2
2 2 n ϕz K 1 − ηa z gz dmz < ε.
f
3.20
n
Since Iϕ,g : Bα0 → QK is compact, we have that for every ε > 0 there is a finite collection
of functions f1 , f2 , . . . , fk ∈ BBα0 such that, for each f ∈ BBα0 , there is a j ∈ {1, . . . , k}, such that
2
2 2 n sup f n ϕz − fj ϕz K 1 − ηa z gz dmz < ε.
a∈D
D
3.21
1 and
On the other hand, from 3.20, it follows that if δ : max1≤j≤k δj fj , ε, then for r ∈ δ,
all j ∈ {1, . . . , k}, we have
sup
a∈D
|ϕz|>r
2
2 2 n fj ϕz K 1 − ηa z gz dmz < ε.
3.22
Abstract and Applied Analysis
11
1 and every f ∈ BBα
From 3.21 and 3.22, we have that for r ∈ δ,
0
sup
a∈D
|ϕz|>r
2
2 2 n ϕz K 1 − ηa z gz dmz < 4ε.
f
3.23
If we apply 3.23 to the delays of the functions in 3.5 which are normalized so that they
belong to BBα and then use the monotone convergence theorem, we easily get that for r >
where δ is chosen as in 3.7
max{δ, δ}
sup
a∈D
|ϕz|>r
2 21−α−n
2 2 K 1 − ηa z gz 1 − ϕz
dmz < Cε,
3.24
from which 3.11 follows, as desired.
Theorem 3.3. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD and n ∈ N, or n 0 and α > 1. Then
the next statements are equivalent.
n
a Iϕ,g : Bα → QK,0 is bounded.
n
b Iϕ,g : Bα → QK,0 is compact.
n
c Iϕ,g : Bα0 → QK,0 is compact.
n
d Iϕ,g : Bα0 → QK,0 is weakly compact.
21−α−n
e lim|a| → 1 D K1 − |ηa z|2 |gz|2 1 − |ϕz|2 dmz 0.
f dμϕ,g,n,α z is a vanishing K-Carleson measure.
Proof. By Theorem 1.1, e and f are equivalent; by Lemma 2.4, c is equivalent to d,
while, by Lemma 2.5, a is equivalent to c. Also b obviously implies a.
a ⇒ e Let h1 and h2 be as in 3.5. Then from 3.7 and an elementary inequality,
we get
|ϕz|>δ
2 21−α−n 2 gz2 dmz
1 − ϕz
K 1 − ηa z
≤C
D
2 n 2
K 1 − ηa z Iϕ,g h1 z dmz
C
3.25
2 n 2
K 1 − ηa z Iϕ,g h2 z dmz.
D
n
n
For f0 z zn /n! ∈ Bα , we get Iϕ,g f0 ∈ QK,0 . From this and since Iϕ,g hj ∈ QK,0 , j 1, 2, by
letting |a| → 1, we get that e holds.
e ⇒ b. We have that for every ε > 0 there is a δ ∈ 0, 1 so that for |a| > δ
Φϕ,g,K a < ε.
3.26
12
Abstract and Applied Analysis
On the other hand, by Lemma 2.6, Φϕ,g,K is continuous on |a| ≤ δ, so uniformly
bounded therein, which along with 3.26 gives the boundedness of Φϕ,g,K on D. Hence, by
n
Theorem 3.1, Iϕ,g : Bα → QK is bounded. By Lemma 2.1, we have
n 2 K 1 − ηa z2 dmz
I
lim sup
f
z
ϕ,g
|a| → 1
f Bα ≤1 D
2
≤ C sup f Bα lim Φϕ,g,K a C lim Φϕ,g,K a 0,
|a| → 1
|a| → 1
f Bα ≤1
3.27
n
so Iϕ,g : Bα → QK,0 is bounded.
Now assume that fl l∈N is a bounded sequence in Bα , say by L, converging to zero
n
uniformly on compacta of D as l → ∞. To show that the operator Iϕ,g : Bα → QK,0 is
n
compact, it is enough to prove that there is a subsequence flk k∈N of fl l∈N such that Iϕ,g flk
converges in QK,0 as k → ∞. By Lemma 2.1 and Montel’s theorem, it follows that there is a
subsequence, which we may denote again by fl l∈N converging uniformly on compacta of D
n
n
to an f ∈ Bα , such that fBα ≤ L. Since Iϕ,g Bα ⊆ QK,0 , then clearly Iϕ,g f ∈ QK,0 . We show
that
n
n lim Iϕ,g fl − Iϕ,g f l→∞
QK
0.
3.28
From 3.26, Lemma 2.1, and some simple calculation, we obtain
sup
δ<|a|<1
2 n
n
Iϕ,g fl z − Iϕ,g
K 1 − ηa z2 dmz < 4CL2 ε.
fz
D
3.29
For a ∈ D and t ∈ 0, 1, let
Ψt a D\tD
2 21−α−n
2 2 K 1 − ηa z gz 1 − ϕz
dmz.
3.30
Lemma 2.6 essentially shows that Ψt is continuous on D. Hence, for each a ∈ D, there is a
ta ∈ r, 1 such that Ψta a < ε/2. Moreover, there is a neighborhood Oa of a such that,
for every b ∈ Oa, Ψta b < ε. From this and since the set |a| ≤ δ is compact, it follows that
there is a t0 ∈ 0, 1 such that Ψt0 a < ε when |a| ≤ δ. This along with Lemma 2.1 implies that
sup
|a|≤δ
D\t0 D
2 n
n
Iϕ,g fl z − Iϕ,g
K 1 − ηa z2 dmz
fz
2
≤ Cfl − f Bα sup Ψt0 a < 4CL2 ε.
|a|≤δ
3.31
Abstract and Applied Analysis
13
n
By the Weierstrass theorem fl → f n uniformly on compacta as l → ∞, from which
along with 2.2 and since ϕt0 D is compact, for r supw∈ϕt0 D |w|, it follows that
sup
t0 D
|a|≤δ
2 n
n
Iϕ,g fl z − Iϕ,g
K 1 − ηa z2 dmz
fz
n 2
≤ C sup fl − f
z sup Φϕ,g,K a −→ 0,
|z|≤r
|a|≤δ
3.32
as l −→ ∞.
n
From 3.29–3.32 and since Iϕ,g f0 0 for each f ∈ HD, we easily get 3.28, from
which b follows, finishing the proof of this theorem.
Theorem 3.4. Let α > 0, K ∈ Ω0, ∞, ϕ ∈ SD, g ∈ HD, and n ∈ N, or n 0 and α > 1.
Then the following statements are equivalent.
n
a Iϕ,g : Bα0 → QK,0 is bounded,
21−α−n
dmz < ∞, and
b supa∈D D |gz|2 K1 − |ηa z|2 1 − |ϕz|2 lim
|a| → 1
D
gz2 K 1 − ηa z2 dmz 0.
3.33
n
Proof. Suppose b holds and f ∈ Bα0 . Then by Theorem 3.1, Iϕ,g : Bα0 → QK is bounded. We
n
show Iϕ,g f ∈ QK,0 , for every f ∈ Bα0 . Since f ∈ Bα0 , we have that, for every ε > 0, there is an
r ∈ 0, 1 such that see, e.g., the idea in 35, Lemma 2.4
2 2αn−1
n ϕz 1 − ϕz2
f
<ε
for ϕz > r.
3.34
Thus,
sup
a∈D
|ϕz|>r
2 n
Iϕ,g fz K 1 − ηa z2 dmz
< ε sup
D
a∈D
2 21−α−n 2 gz2 dmz.
1 − ϕz
K 1 − ηa z
3.35
We also have
lim
|a| → 1
|ϕz|≤r
≤C
≤C
2 n
Iϕ,g fz K 1 − ηa z2 dmz
2
f α
B
1 −
lim
r 2 2αn−1 |a| → 1
2
f α
B
1 − r 2 |ϕz|≤r
lim
2αn−1 |a| → 1
D
2
2 K 1 − ηa z gz dmz
2
2 K 1 − ηa z gz dmz 0.
3.36
14
Abstract and Applied Analysis
n
n
Combining 3.35 and 3.36, we get Iϕ,g f ∈ QK,0 . Hence, Iϕ,g : Bα0 → QK,0 is bounded.
n
n
Conversely, if Iϕ,g : Bα0 → QK,0 is bounded, then Iϕ,g : Bα0 → QK is bounded too. Thus,
n
by Theorem 3.1, we get the first condition in b. For f0 z zn /n! ∈ Bα0 , we get Iϕ,g f0 ∈ QK,0 ,
which is equivalent to 3.33, finishing the proof of the theorem.
0
If n 0, we simply denote the operator Iϕ,g by Iϕ,g .
Theorem 3.5. Let α ∈ 0, 1, K ∈ Ω0, ∞, ϕ ∈ SD, and g ∈ HD. Then the following
statements are equivalent.
a Iϕ,g : Bα → QK is bounded.
b Iϕ,g : Bα0 → QK is bounded.
c M1 : supa∈D D K1 − |ηa z|2 |gz|2 dmz < ∞.
d dμ1 z |gz|2 dmz is a K-Carleson measure.
e Iϕ,g : Bα → QK is compact.
f Iϕ,g : Bα0 → QK is compact.
g Iϕ,g : Bα0 → QK is weakly compact.
Moreover, if Iϕ,g : Bα → QK is bounded, then the next asymptotic relations hold
Iϕ,g α
Iϕ,g Bα → QK M11/2 .
B → QK
0
3.37
Proof. The proof of the equivalence of statements a–d of this theorem is similar to the
proof of Theorem 3.1; moreover, the implication b ⇒ c is much simpler since it follows by
using the test function f0 z ≡ 1. That c is equivalent to e–g is proved similarly as in
Theorem 3.2, by using the well-known fact that if a bounded sequence fl l∈N in Bα , α ∈ 0, 1
converges to zero uniformly on compacts of D, then it converges to zero uniformly on the
whole D. The details are omitted.
The proof of the next theorem is similar to the proofs of Theorems 3.3 and 3.4 and will
be omitted.
Theorem 3.6. Let α ∈ 0, 1, K ∈ Ω0, ∞, ϕ ∈ SD, and g ∈ HD. Then the following
statements are equivalent.
a Iϕ,g : Bα0 → QK,0 is bounded.
b Iϕ,g : Bα → QK,0 is bounded.
c Iϕ,g : Bα → QK,0 is compact.
d Iϕ,g : Bα0 → QK,0 is compact.
e Iϕ,g : Bα0 → QK,0 is weakly compact.
f lim|a| → 1 D K1 − |ηa z|2 |gz|2 dmz 0.
g dμ1 z |gz|2 dmz is a vanishing K-Carleson measure.
Abstract and Applied Analysis
15
Acknowledgment
This work is partially supported by the National Board of Higher Mathematics
NBHM/DAE, India Grant no. 48/4/2009/R&D-II/426 and by the Serbian Ministry of
Science Projects III41025 and III44006.
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