# Hankel Operators on the WeightedL

Authors: Hong Rae Cho; Jeong Wan Seo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weightedL
P-Bergman spaces with exponential type weights.


---

## Body

## 1. Introduction

Let𝔻 be the unit disc in the complex plane ℂ and d
A
(
z
) the area measure on 𝔻, and denote by H
(
𝔻
) the space of all analytic functions in 𝔻. Let φ
∈
C
2
(
𝔻
) with Δ
φ
>
0. For 0
<
p
≤
∞, the weighted Bergman space A
φ
p is the space of functions f
∈
H
(
𝔻
) such that
(1)
∥
f
∥
p
,
φ
p
=
∫
𝔻
‍
|
f
(
z
)
e
-
φ
(
z
)
|
p
d
A
(
z
)
<
∞
for
0
<
p
<
∞
,
∥
f
∥
∞
,
φ
=
sup
⁡
z
∈
𝔻
|
f
(
z
)
|
e
-
φ
(
z
)
.
Note that A
φ
p is the closed subspace of L
φ
p
:
=
L
p
(
𝔻
,
e
-
p
φ
d
A
) consisting of analytic functions. Since the space A
φ
2 is a reproducing kernel Hilbert space, for each z
∈
𝔻, there are functions K
z
∈
A
φ
2 with f
(
z
)
=
〈
f
,
K
z
〉, where 〈
·
,
·
〉 is the usual inner product in L
φ
2. The orthogonal projection from L
φ
2 to A
φ
2 is given by
(2)
P
f
(
z
)
=
∫
𝔻
‍
f
(
w
)
K
(
w
,
z
)
e
2
φ
d
A
(
w
)
,
where K
(
w
,
z
)
=
K
z
(
w
)
¯.Givenσ
∈
C
1
(
𝔻
) so that there exists a dense subset 𝒟 of A
φ
2 with σ
f
∈
L
φ
2 for f
∈
𝒟, the big Hankel operator H
σ with symbol σ is densely defined by
(3)
H
σ
f
=
σ
f
-
P
(
σ
f
)
,
f
∈
𝒟
,
where P is the orthogonal projection of L
φ
2 onto A
φ
2.We write∂
¯
=
∂
/
∂
z
¯. Then the ∂
¯-equation can be written by
(4)
∂
¯
u
=
f
.
For f
∈
A
φ
2, we look for a solution v
∈
L
φ
2 of minimal L
φ
2-norm. Notice that the solution of minimal norm is the one that is orthogonal to the kernel of ∂
¯ on L
φ
2; that is, v
⊥
A
φ
2. Then, if u
∈
L
φ
2 solves (4), we get
(5)
v
=
(
I
-
P
)
u
.
The linear operator N
:
A
φ
2
→
L
φ
2 given by
(6)
N
(
f
)
=
v
is called the canonical solution operator to ∂
¯ on A
φ
2.For anyf
∈
A
φ
2, obviously ∂
¯
(
z
¯
f
)
=
f and
(7)
N
(
f
)
=
(
I
-
P
)
(
z
¯
f
)
=
H
z
¯
f
.
That is, the canonical solution operator coincides with the big Hankel operator acting on A
φ
2 with symbol z
¯. Motivated by this fact, we now consider Hankel operators with conjugate analytic symbols on A
φ
2. For f
,
g
∈
A
φ
2, we do not necessarily have g
¯
f
∈
L
φ
2. Let 𝒟
:
=
Span
⁡
{
K
z
:
z
∈
𝔻
}. Then 𝒟 is dense in A
φ
2. For symbol g
∈
A
φ
2 such that
(8)
g
¯
K
z
∈
L
φ
2
∀
z
∈
𝔻
,
we consider the densely defined big Hankel operator on A
φ
2 given by
(9)
H
g
¯
f
=
(
I
-
P
)
(
g
¯
f
)
.A positive functionτ on 𝔻 is said to belong to the class ℰ if it satisfies the following three properties.(a)
There exists a constantC
1
>
0 such that
(10)
τ
(
z
)
≤
C
1
(
1
-
|
z
|
)
,
for
z
∈
𝔻
.
(b)
There exists a constantC
2
>
0 such that
(11)
|
τ
(
z
)
-
τ
(
w
)
|
≤
C
2
|
z
-
w
|
,
for
z
,
w
∈
𝔻
.
(c)
For eachm
≥
1, there are constants b
m
>
0 and 0
<
t
m
<
1
/
m such that
(12)
τ
(
z
)
≤
τ
(
w
)
+
t
m
|
z
-
w
|
,
for
|
z
-
w
|
>
b
m
τ
(
w
)
.In this paper, we characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weightedL
p-Bergman spaces with exponential type weights as follows.Theorem 1.
Let1
≤
p
<
∞ and g
∈
A
φ
p. Let φ
∈
C
2
(
𝔻
) with Δ
φ
>
0, and the function τ
(
z
)
=
(
Δ
φ
(
z
)
)
-
1
/
2 is in the class ℰ. Then H
g
¯ extends to a bounded linear operator on A
φ
p if and only if
(13)
sup
⁡
z
∈
𝔻
τ
(
z
)
|
g
′
(
z
)
|
<
∞
.Theorem 2.
Let1
≤
p
<
∞ and g
∈
A
φ
p. Let φ
∈
C
2
(
𝔻
) with Δ
φ
>
0, and the function τ
(
z
)
=
(
Δ
φ
(
z
)
)
-
1
/
2 is in the class ℰ. Then H
g
¯ extends to a compact linear operator on A
φ
p if and only if
(14)
lim
⁡
|
z
|
→
1
-
τ
(
z
)
|
g
′
(
z
)
|
=
0
.In [1], Luecking firstly proved the same results in the context of the ordinary L
2-Bergman spaces. For L
2-Bergman spaces with exponential type weights, the same results were proved in [2–4]. Moreover, Schatten-class Hankel operators are also indicated in their papers.The expressionf
≲
g means that there is a constant C independent of the relevant variables such that f
≤
C
g, and f
≈
g means that f
≲
g and g
≲
f.


## 2. Preliminaries

From now on we assume thatφ
∈
C
2
(
𝔻
), Δ
φ
>
0, and the function τ
(
z
)
=
(
Δ
φ
(
z
)
)
-
1
/
2 is in the class ℰ. The following notations will be frequently used:
(15)
m
τ
=
min
⁡
(
1
,
C
1
-
1
,
C
2
-
1
)
4
,
where C
1 and C
2 are the constants in the conditions (a) and (b) in Section 1 and
(16)
d
τ
(
z
,
w
)
=
|
z
-
w
|
min
⁡
[
τ
(
z
)
,
τ
(
w
)
]
.Lemma 3 (see [5]).
For eachM
≥
1, there exists a constant C
>
0 (depending on M) such that for z
,
w
∈
𝔻, one has
(17)
|
K
(
w
,
z
)
|
≤
C
e
φ
(
z
)
+
φ
(
w
)
τ
(
z
)
τ
(
w
)
min
⁡
[
1
,
1
d
τ
(
z
,
w
)
]
M
.By using the upper estimates forK
(
w
,
z
) in Lemma 3, Arroussi and Pau [5] proved that the orthogonal projection P projects L
φ
p boundedly onto A
φ
p for 1
≤
p
≤
∞.Lemma 4 (see [6]).
Let0
<
ρ
≤
m
τ and w
∈
𝔻. Then,
(18)
3
4
τ
(
w
)
≤
τ
(
z
)
≤
5
4
τ
(
w
)
,
z
∈
D
ρ
(
w
)
.By using the third Green formula, we get the following two approximation results.Lemma 5 (see [7]).
For smallK
>
0, there exists A
=
A
(
K
)
>
0 such that
(19)
sup
⁡
w
∈
D
K
(
z
)
|
φ
(
w
)
-
φ
(
z
)
-
h
z
(
w
)
|
≤
A
,
where h
z is a harmonic function in D
K
(
z
) with h
z
(
z
)
=
0.Lemma 6 (see [7]).
For smallK
>
0, one has the estimate
(20)
|
∂
φ
∂
w
(
w
)
-
∂
h
z
∂
w
(
w
)
|
≲
1
τ
(
w
)
f
o
r
w
∈
D
K
(
z
)
,
where h
z is defined in Lemma 5.The following is a certain submean value property of|
f
(
z
)
|
e
-
φ
(
z
). We follow the proof of ([8], Lemma 19).Lemma 7.
Let0
<
p
<
∞. For any small r
>
0, there exists C
=
C
(
r
)
>
0 such that for any f
∈
A
φ
p and z
∈
𝔻
(a)
|
f
(
z
)
|
e
-
φ
(
z
)
≤
C
(
(
1
/
τ
(
z
)
2
)
∫
D
r
(
z
)
‍
|
f
|
p
e
-
p
φ
d
A
)
1
/
p;
(b)
|
∇
(
|
f
|
e
-
φ
)
(
z
)
|
≤
C
(
1
/
τ
(
z
)
)
(
(
1
/
τ
(
z
)
2
)
∫
D
r
(
z
)
‍
|
f
|
p
e
-
p
φ
d
A
)
1
/
p, provided f
(
z
)
≠
0.Proof.
(a) By Lemma5, there exists some constant A
=
A
(
K
)
>
0 such that
(21)
sup
⁡
w
∈
D
K
(
z
)
|
φ
(
w
)
-
φ
(
z
)
-
h
z
(
w
)
|
≤
A
,
for z
∈
𝔻. Since h
z is harmonic, there is an analytic function Φ
z on D
K
(
z
) such that Φ
z
(
z
)
=
0 and Re
Φ
z
=
h
z on D
K
(
z
). Thus we have |
e
Φ
z
|
=
e
h
z. Hence by the submean value property together with Lemma 5, we get
(22)
|
f
(
z
)
|
p
=
|
f
(
z
)
e
-
(
1
/
p
)
Φ
z
(
z
)
|
p
≤
1
(
π
K
τ
(
z
)
)
2
∫
D
K
(
z
)
‍
|
f
e
-
(
1
/
p
)
Φ
z
|
p
d
A
=
1
(
π
K
τ
(
z
)
)
2
∫
D
K
(
z
)
‍
|
f
e
-
(
1
/
p
)
h
z
|
p
d
A
≲
e
φ
(
z
)
τ
(
z
)
2
∫
D
K
(
z
)
‍
|
f
e
-
(
1
/
p
)
φ
|
p
d
A
.
(b) We begin as follows:(23)
|
∇
(
|
f
|
e
-
φ
)
|
=
|
1
2
1
|
f
|
f
′
f
¯
e
-
φ
-
|
f
|
e
-
φ
∂
φ
∂
w
|
≲
|
f
′
e
-
φ
-
2
f
e
-
φ
∂
φ
∂
w
|
≤
|
f
′
e
-
φ
-
2
f
e
-
φ
∂
h
z
∂
w
|
+
|
2
f
e
-
φ
∂
h
z
∂
w
-
2
f
e
-
φ
∂
φ
∂
w
|
≲
|
f
′
-
2
f
∂
h
z
∂
w
|
e
-
φ
+
|
f
|
e
-
φ
|
∂
h
z
∂
w
-
∂
φ
∂
w
|
.
Since h
z is harmonic, there is an analytic function Φ
z
∈
H
(
D
K
(
z
)
) such that
(24)
Φ
z
(
z
)
=
0
,
Φ
z
′
=
2
∂
h
z
∂
w
,
|
e
Φ
z
|
=
e
h
z
.
Note that
(25)
|
∇
(
f
e
-
Φ
z
)
(
z
)
|
=
|
f
′
(
z
)
-
f
(
z
)
Φ
z
′
(
z
)
|
=
|
f
′
(
z
)
-
2
f
(
z
)
∂
h
z
∂
w
(
z
)
|
.
On the other hand,
(26)
|
∇
(
f
e
-
Φ
z
)
(
z
)
|
≲
|
∫
|
z
-
ζ
|
=
τ
(
z
)
‍
f
(
ζ
)
e
-
Φ
z
(
ζ
)
(
z
-
ζ
)
2
d
ζ
|
≲
1
τ
(
z
)
2
∫
|
z
-
ζ
|
=
τ
(
z
)
‍
|
f
(
ζ
)
|
e
-
h
z
(
ζ
)
|
d
ζ
|
.
For |
z
-
ζ
|
=
τ
(
z
), we have
(27)
|
f
(
ζ
)
|
e
-
h
z
(
ζ
)
≲
|
f
(
ζ
)
|
e
-
φ
(
ζ
)
+
φ
(
z
)
≲
(
1
τ
(
z
)
2
∫
D
K
(
z
)
‍
|
f
|
p
e
-
p
φ
d
A
)
1
/
p
e
φ
(
z
)
.
Hence we have
(28)
|
∇
(
f
e
-
Φ
z
)
(
z
)
|
≲
1
τ
(
z
)
(
1
τ
(
z
)
2
∫
D
K
(
z
)
‍
|
f
|
p
e
-
p
φ
d
A
)
1
/
p
e
φ
(
z
)
.
Thus
(29)
|
∇
(
|
f
|
e
-
φ
)
(
z
)
|
≲
1
τ
(
z
)
(
1
τ
(
z
)
2
∫
D
K
(
z
)
‍
|
f
|
p
e
-
p
φ
d
A
)
1
/
p
.Despite that the next result was proved in [4], we give the proof of different method by using Lemma 7.Proposition 8.
There is anr
>
0 independent of z such that
(30)
|
K
(
w
,
z
)
|
≳
e
φ
(
z
)
+
φ
(
w
)
τ
(
z
)
τ
(
w
)
,
w
∈
D
r
(
z
)
.Proof.
Letr
>
0. By (b) of Lemma 7, we have
(31)
|
|
K
(
w
,
z
)
|
e
-
φ
(
w
)
-
|
K
(
z
,
z
)
|
e
-
φ
(
z
)
|
≲
|
w
-
z
|
τ
(
z
)
(
1
τ
(
z
)
2
∫
D
r
(
z
)
‍
|
K
(
ζ
,
z
)
|
2
e
-
2
φ
(
ζ
)
d
A
(
ζ
)
)
1
/
2
≤
r
τ
(
z
)
K
(
z
,
z
)
1
/
2
,
w
∈
D
r
(
z
)
.
Hence it follows that
(32)
|
K
(
w
,
z
)
|
e
-
φ
(
w
)
≳
K
(
z
,
z
)
e
-
φ
(
z
)
-
r
τ
(
z
)
K
(
z
,
z
)
1
/
2
,
|
K
(
w
,
z
)
|
e
-
φ
(
w
)
≳
K
(
z
,
z
)
e
-
φ
(
z
)
-
r
τ
(
z
)
w
∈
D
r
(
z
)
.
Note that by Lemma3 and (a) of Lemma 7, we have
(33)
K
(
z
,
z
)
≈
τ
(
z
)
-
2
e
2
φ
(
z
)
,
z
∈
𝔻
.
Thus we have
(34)
|
K
(
w
,
z
)
|
e
-
φ
(
w
)
≳
τ
(
z
)
-
2
e
φ
(
z
)
-
r
τ
(
z
)
-
2
e
φ
(
z
)
≥
(
1
-
r
)
τ
(
z
)
-
2
e
φ
(
z
)
,
(
z
)
-
2
e
φ
(
z
)
w
∈
D
r
(
z
)
,
if we choose small r
>
0.


## 3. Hankel Operators onA
φ
p

For the proof of boundedness of Hankel operator onL
2-Bergman spaces with exponential type weights in [2–4], they used Hörmander's L
2-estimates for ∂
¯. However, for L
p-Bergman spaces, we need the following L
p-estimates for ∂
¯.Theorem 9 (see [6]).
Letφ
∈
C
2
(
𝔻
) with Δ
φ
>
0. Suppose that the function τ
(
z
)
=
(
Δ
φ
(
z
)
)
-
1
/
2 satisfies conditions (a) and (b) in Section 1. Let 1
≤
p
≤
∞. Then there is a solution u to the equation ∂
¯
u
=
f such that
(35)
∫
𝔻
‍
|
u
e
-
φ
|
p
d
A
(
z
)
≲
∫
𝔻
‍
|
f
e
-
φ
Δ
φ
|
p
d
A
(
z
)
,
provided the right hand side integral is finite.Letg be an analytic function in 𝔻 satisfying the condition (8). Let
(36)
k
ζ
(
z
)
=
K
(
ζ
,
z
)
K
(
ζ
,
ζ
)
,
ζ
,
z
∈
𝔻
.
By the reproducing formula in A
φ
2 we get
(37)
H
g
¯
k
ζ
(
z
)
=
(
g
(
z
)
¯
-
g
(
ζ
)
¯
)
k
ζ
(
z
)
,
ζ
,
z
∈
𝔻
.Lemma 10.
Let0
<
p
<
∞. Then
(38)
∥
k
z
∥
p
,
φ
p
≈
τ
(
z
)
2
-
p
.Proof.
Consider(39)
∥
k
z
∥
p
,
φ
p
=
∫
|
z
-
w
|
≤
δ
0
τ
(
z
)
‍
+
∫
|
z
-
w
|
>
δ
0
τ
(
z
)
‍
|
k
z
(
w
)
|
p
e
p
φ
(
w
)
d
A
(
w
)
.
First,
(40)
∫
|
z
-
w
|
≤
δ
0
τ
(
z
)
‍
|
k
z
(
w
)
|
p
e
p
φ
(
w
)
d
A
(
w
)
≈
∫
|
z
-
w
|
≤
δ
0
τ
(
z
)
‍
d
A
(
w
)
τ
(
w
)
p
≈
τ
(
z
)
2
-
p
.
Now,
(41)
∫
|
z
-
w
|
>
δ
0
τ
(
z
)
|
k
z
(
w
)
|
p
e
p
φ
(
w
)
d
A
(
w
)
‍
≲
∫
|
z
-
w
|
>
δ
0
τ
(
z
)
‍
1
τ
(
w
)
p
(
min
⁡
[
τ
(
z
)
,
τ
(
w
)
]
|
z
-
w
|
)
p
M
d
A
(
w
)
.
We take a large constant M
>
1 so that p
M
>
2. Then
(42)
∫
|
z
-
w
|
>
δ
0
τ
(
z
)
1
τ
(
w
)
p
(
min
⁡
[
τ
(
z
)
,
τ
(
w
)
]
|
z
-
w
|
)
p
M
d
A
(
w
)
‍
≲
τ
(
z
)
p
M
-
p
∫
δ
0
τ
(
z
)
<
|
z
-
w
|
<
2
‍
1
|
z
-
w
|
p
M
d
A
(
w
)
≲
τ
(
z
)
p
M
-
p
∫
δ
0
τ
(
z
)
2
‍
1
r
p
M
-
1
d
r
≲
τ
(
z
)
2
-
p
.
Thus we get the result.Theorem 11.
Let1
≤
p
<
∞. Let g
∈
A
φ
p. Then H
g
¯ extends to a bounded linear operator on A
φ
p if and only if
(43)
sup
⁡
z
∈
𝔻
τ
(
z
)
|
g
′
(
z
)
|
<
∞
.Proof.
Assume first that(44)
sup
⁡
z
∈
𝔻
τ
(
z
)
|
g
′
(
z
)
|
<
∞
.
By Theorem 9, there is a solution u of the equation ∂
¯
u
=
f
∂
¯
g
¯ such that
(45)
∥
u
∥
p
,
φ
≤
C
∥
τ
f
∂
¯
g
¯
∥
p
,
φ
.
Since H
g
¯
f is the minimal L
φ
2-norm solution of the ∂
¯-equation, we have H
g
¯
f
=
(
I
-
P
)
u. In [5], Arroussi and Pau proved that the orthogonal projection P projects L
φ
p boundedly onto A
φ
p for 1
≤
p
≤
∞. Thus we have
(46)
∥
H
g
¯
f
∥
p
,
φ
=
∥
(
I
-
P
)
u
∥
p
,
φ
≲
∥
u
∥
p
,
φ
.
By (45) and (46), we have
(47)
∥
H
g
¯
f
∥
p
,
φ
≲
∥
τ
f
∂
¯
g
¯
∥
p
,
φ
≤
sup
⁡
z
∈
𝔻
τ
(
z
)
|
g
′
(
z
)
|
∥
f
∥
p
,
φ
,
which shows that H
g
¯ can be extended to a bounded linear operator on A
φ
p.
Conversely, assume thatH
g
¯ is bounded on A
φ
p. Then we have
(48)
∥
H
g
-
(
k
z
∥
k
z
∥
p
,
φ
)
∥
p
,
φ
<
M
for
z
∈
𝔻
.
Using Proposition 8 and Lemma 3, there exists r
>
0 such that
(49)
|
k
ζ
(
z
)
|
=
|
K
(
ζ
,
z
)
|
K
(
ζ
,
ζ
)
≳
e
φ
(
z
)
τ
(
z
)
on
⁡
z
∈
D
r
(
ζ
)
.
Hence we have
(50)
M
p
>
1
∥
k
z
∥
p
,
φ
p
∥
H
g
¯
k
ζ
∥
p
,
φ
p
≈
τ
(
z
)
p
-
2
∫
𝔻
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
|
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
≥
τ
(
z
)
p
-
2
∫
D
r
(
ζ
)
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
|
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
≳
1
τ
(
z
)
2
∫
D
r
(
ζ
)
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
d
A
(
z
)
.
Since g is an analytic function in 𝔻, by the Cauchy estimates applied to g
ζ
(
z
)
:
=
g
(
z
)
-
g
(
ζ
), we can now conclude
(51)
τ
(
ζ
)
p
|
g
′
(
ζ
)
|
p
≲
1
τ
(
ζ
)
2
∫
D
r
(
ζ
)
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
d
A
(
z
)
≲
M
p
,
ζ
∈
𝔻
.
Thus we get the result.Lemma 12.
Let0
<
p
<
∞. Then
(52)
k
z
∥
k
z
∥
p
,
φ
⟶
0
u
n
i
f
o
r
m
l
y
o
n
c
o
m
p
a
c
t
s
u
b
s
e
t
s
,
|
z
|
⟶
1
-
.Proof.
LetK be a compact subset of 𝔻. We choose a large constant M so that M
-
(
2
/
p
)
-
1
>
0. Then we have for w
∈
K
(53)
|
k
z
(
w
)
|
∥
k
z
∥
p
,
φ
≲
τ
(
z
)
(
2
/
p
)
-
1
e
φ
(
w
)
τ
(
w
)
min
⁡
[
τ
(
z
)
,
τ
(
w
)
]
M
|
z
-
w
|
≲
τ
(
z
)
M
-
(
2
/
p
)
-
2
1
dist
⁡
(
K
,
z
)
sup
⁡
w
∈
K
e
φ
(
w
)
≲
(
1
-
|
z
|
)
M
-
(
2
/
p
)
-
2
1
dist
⁡
(
K
,
z
)
sup
⁡
w
∈
K
e
φ
(
w
)
⟶
0
,
|
z
|
⟶
1
-
,
where dist
⁡
(
K
,
z
)
=
min
⁡
{
|
z
-
w
|
:
w
∈
K
}.Theorem 13.
Let1
≤
p
<
∞. Let g
∈
A
φ
p. Then H
g
¯ extends to a compact linear operator on A
φ
p if and only if
(54)
lim
⁡
|
z
|
→
1
-
τ
(
z
)
|
g
′
(
z
)
|
=
0
.Proof.
Suppose now thatH
g
¯ is compact on A
φ
p. Then by Riesz-Tamarkin compactness theorem, we have
(55)
lim
⁡
r
→
1
-
1
∥
k
ζ
∥
p
,
φ
p
∫
r
<
|
z
|
<
1
‍
|
H
g
¯
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
=
0
,
uniformly in ζ
∈
𝔻. Now, by Lemma 12,
(56)
1
∥
k
ζ
∥
p
,
φ
p
∫
|
z
|
≤
r
‍
|
H
g
¯
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
≤
sup
⁡
|
z
|
≤
r
(
|
k
ζ
(
z
)
|
∥
k
ζ
∥
p
,
φ
)
p
×
∫
|
z
|
≤
r
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
⟶
0
,
as |
ζ
|
→
1
-. Thus we have
(57)
∥
H
g
-
(
k
z
∥
k
z
∥
p
,
φ
)
∥
p
,
φ
=
1
∥
k
ζ
∥
p
,
φ
∥
H
g
-
k
ζ
∥
p
,
φ
⟶
0
,
|
ζ
|
⟶
1
-
.
We chooseρ
>
0 so that
(58)
|
k
ζ
(
z
)
|
=
|
K
(
ζ
,
z
)
|
K
(
ζ
,
ζ
)
≳
e
φ
(
z
)
τ
(
z
)
on
⁡
z
∈
D
r
(
ζ
)
.
Then(59)
1
∥
k
ζ
∥
p
,
φ
p
∫
𝔻
‍
|
H
g
¯
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
≳
1
τ
(
z
)
2
∫
D
r
(
ζ
)
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
d
A
(
z
)
≳
τ
(
ζ
)
p
|
g
′
(
ζ
)
|
p
.
This implies that
(60)
lim
⁡
|
z
|
→
1
-
τ
(
z
)
|
g
′
(
z
)
|
=
0
.
For|
ζ
|
>
r
+
ρ, the inclusion D
ρ
(
ζ
)
⊂
{
r
<
|
z
|
<
1
} holds, and
(61)
1
∥
k
ζ
∥
p
,
φ
p
∫
r
<
|
z
|
<
1
‍
|
H
g
¯
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
=
1
∥
k
ζ
∥
p
,
φ
p
∫
r
<
|
z
|
<
1
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
|
k
ζ
(
z
)
|
p
e
-
p
φ
(
z
)
d
A
(
z
)
≳
1
τ
(
z
)
2
∫
D
r
(
ζ
)
‍
|
g
(
z
)
-
g
(
ζ
)
|
p
d
A
(
z
)
≳
(
τ
(
ζ
)
|
g
′
(
ζ
)
|
)
p
.
This implies that
(62)
lim
⁡
|
z
|
→
1
-
τ
(
z
)
|
g
′
(
z
)
|
=
0
.
Assume now that(63)
lim
⁡
|
z
|
→
1
-
τ
(
z
)
|
g
′
(
z
)
|
=
0
.
It is enough to show that for any sequence {
f
n
} that is bounded in norm and converges uniformly to zero on compact subsets, we have ∥
H
g
¯
f
n
∥
p
,
φ
→
0 as n
→
∞. As in relation (46), we have
(64)
∥
H
g
¯
f
n
∥
p
,
φ
p
≲
∫
𝔻
‍
|
g
′
|
p
τ
p
|
f
n
|
p
e
-
p
φ
d
A
.
Now(65)
∫
𝔻
‍
|
g
′
|
p
τ
p
|
f
n
|
p
e
-
p
φ
d
A
=
∫
|
z
|
≤
r
‍
+
∫
r
<
|
z
|
<
1
‍
|
g
′
|
p
τ
p
|
f
n
|
p
e
-
p
φ
d
A
.
Since {
f
n
} converges uniformly to zero on compact subsets,
(66)
∫
|
z
|
≤
r
‍
|
g
′
|
p
τ
p
|
f
n
|
p
e
-
p
φ
d
A
≤
sup
⁡
|
z
|
≤
r
|
f
n
|
p
∫
|
z
|
≤
r
‍
|
g
′
|
p
τ
p
e
-
p
φ
d
A
⟶
0
,
n
⟶
∞
.
Now
(67)
∫
r
<
|
z
|
<
1
‍
|
g
′
|
p
τ
p
|
f
n
|
p
e
-
p
φ
d
A
≤
sup
⁡
r
<
|
z
|
<
1
|
g
′
|
p
τ
p
∫
𝔻
|
f
n
|
p
e
-
p
φ
d
A
⟶
0
,
r
⟶
1
-
.
Hence H
g
¯ is compact.



---

*Source: 304867-2014-02-12.xml*