Complex Anal. Oper. Theory
DOI 10.1007/s11785-014-0404-0
Complex Analysis
and Operator Theory
Characterizations of the Dirichlet-Type Space
Xiaosong Liu · Gerardo R. Chacón · Zengjian Lou
Received: 19 January 2014 / Accepted: 10 July 2014
© Springer Basel 2014
Abstract Some characterizations of the so-called Dirichlet-type spaces D(μ) are
given. First we characterize D(μ) by means of the derivative free integral and of the
mean oscillation in the Bergman metric. We then obtain a characterization for D(μ)
that makes use of high-order derivative. Finally, as the main result of this article, we
establish a decomposition theorem of D(μ).
Keywords
Dirichlet-type spaces · Characterizations · Decomposition theorem
Mathematics Subject Classification
Primary 30D45; Secondary 30D50
Communicated by Vladimir Bolotnikov.
This work was supported by NNSF of China (Grant No. 11171203, 11201280), NSF of Guangdong
Province (Grant No. 10151503101000025, S2011010004511, S2011040004131), and was partially
supported by Pontificia Universidad Javeriana (Research Proyect No. 5568).
X. Liu · Z. Lou (B)
Department of Mathematics, Shantou University, Shantou 515063, China
e-mail: [email protected]
Present Address:
X. Liu
Department of Mathematics, Jiaying University, Meizhou 514015, China
e-mail: [email protected]
G. R. Chacón
Departamento de Matematicas, Pontificia Universidad Javeriana,
Cra. 7 No. 43-82, Bogotá, Colombia
e-mail: [email protected]
X. Liu et al.
1 Introduction
Let D be the unit disk of complex plane C and H (D) the space of analytic functions
on D. Given a positive Borel measure μ defined on the boundary of the unit disc ∂D
and let Pμ be the positive harmonic function defined on the unit disc D by
Pμ (z) =
1 − |z|2 dμ(t)
.
|eit − z|2 2π
2π
0
The Dirichlet-t ype space D(μ) is defined as the space of all analytic functions
on D such that
| f (z)|2 Pμ (z)d A(z) < ∞,
D
where d A(z) = π1 d xd y denotes the normalized area Lebesgue measure. It was shown
in [10] that the space D(μ) is contained as a subset in the Hardy space H 2 . A norm
of D(μ) can be defined as
f 2D(μ)
=:
f 2H 2
+
D
| f (z)|2 Pμ (z)d A(z).
If μ = 0, let D(μ) = H 2 . Notice that if μ is the arc-length Lebesgue measure on
∂D, then the Dirichlet-type space D(μ) coincides with the classical Dirichlet space
D. Moreover D(μ) is a Hilbert space with the inner product given by
f, gD = f, g H 2 +
D
f (z)g (z)Pμ (z)d A(z).
For a positive finite Borel measure μ on ∂D, we consider a family of functions
Pμr (z) =
r 2 (1 − |z|2 )
dμ(ζ ), z ∈ D, r ∈ (0, 1).
2
∂ D |ζ − r z|
It is well-known [16] that Pμr (z) is a subharmonic function and
lim Pμr (z) = Pμ (z).
r →1−
Following [10], we define the local Dirichlet integral of f at λ ∈ ∂D as
Dλ ( f ) =
1
2π
0
2π
f (eit ) − f (λ) 2
dt.
eit − λ
If μ is a positive finite Borel measure on ∂D, we have a representation of the norm of
f ∈ D(μ) as a consequence of the following formula showed in [10, Proposition 2.2]
Characterizations of the Dirichlet-Type Space
∂D
Dζ ( f )dμ(ζ ) =
D
| f (z)|2 Pμ (z)d A(z).
Give a finite and positive Borel measure ν on D, we say that ν is a μ-Carleson
measure if there exists a constant C independent of f such that for all f ∈ D(μ) ([2,5])
| f (z)|2 dν(z) ≤ C f 2D(μ) .
D
The Dirichlet-t ype space D(μ) was introduced by Richter in [10]. It has been
studied extensively. The study of Carleson formula for the local Dirichlet integral,
multiplication and cyclic vectors of D(μ) can be found in [11], on reproducing kernels
and extremal functions in [15], on Nevanlinna-Pick property in [16], on Carleson
measure and interpolation problems in [2,3,5,7], on Toeplitz operators in [6] and
composition operators in [4].
The aim of this article is to devote a further study of the space D(μ). Firstly,
we characterize the space D(μ) by means of the derivative-free integral and of the
mean oscillation in the Bergman metric. We also give a characterization that makes
use of high-order derivative. Finally, as the main result of this article, we establish a
decomposition theorem of D(μ).
The article is organized as follows. In Sect. 2, we show two derivative-free characterizations of D(μ) whereas a further characterization based on higher-order derivative
is given Sect. 3. The main theorem of the article-decomposition theorem of D(μ) is
in Sect. 4.
Throughout the article, we will denote by C a positive constant that may differ
from line to line and C is independent of the functions and variables in the inequality.
The notation F ≈ G means that there exist positive constants C1 and C2 such that
C1 F ≤ G ≤ C2 F.
2 Derivative-Free Integral Characterization
In this section, we characterize the Dirichlet-t ype space D(μ) by means of
derivative-free integral. Similar characterizations for other function spaces can be
found in [1,18].
p
The weighted Bergman space Aα is defined as the space of all analytic functions
on D such that
| f (z)| p d Aα (z) < ∞
D
where d Aα (z) = (1 − |z|2 )α . When p = 2 and α = 0, Aα is the Bergman space A2 .
Let d(z, w) denote the Bergman metric between two points z, w ∈ D:
p
d(z, w) = log
1 + |ϕz (w)|
,
1 − |ϕz (w)|
z, w ∈ D,
z−w
where ϕz (w) = 1−z̄w
. For z ∈ D and R > 0, we denote by B(z, R) = {w ∈ D :
d(z, w) < R} the Bergman ball at z with radius R and by |B(z, R)| the area of B(z, R).
X. Liu et al.
If R > 0 is fixed, then it is well-known that |B(z, R)| ≈ (1 − |z|2 )2 as |z| → 1−
(see, for example, Section 4.2 of [21]). Given a function f ∈ L 2 (D, d A), we define
the mean oscillation of f as
M O f (z) =
D
| f ◦ ϕz (w) − f (z)|2 d A(w)
1
2
.
For 0 < r < 1, let
fˆr (z) =
1
|B(z, r )|
B(z,r )
f (w)d A(w)
denote the average of f over the Bergman ball B(z, r ). The mean oscillation of f at
z in the Bergman metric is defined by
M Or f (z) =
1
|B(z, r )|
B(z,r )
| f (w) − fˆr (z)|2 d A(w)
1
2
.
It is easy to check that for z ∈ D, we have
(M Or f (z))2 = | f |r2 (z) − | fˆr (z)|2
1
=
| f (u) − f (v)|2 d A(u)d A(v).
|B(z, r )|2 B(z,r ) B(z,r )
In order to prove our theorems, we need the following two lemmas. The first one
can be found in [9, Lemma 3.5] (see also [20, Lemma 1]).
Lemma 2.1 Suppose that η, ζ, z ∈ D. Let s > −1, r, t > 0 and t < s + 2 < r .
Then
(1 − |η|2 )s
C
d A(η) ≤
.
r |1 − η̄ζ |t
2
r
−s−2
|1
−
η̄z|
(1 − |z| )
|1 − ζ̄ z|t
D
Lemma 2.2 Let s > −2 and p > s + 3. Then
(1 − |w|2 ) p−s−2 (1 − |z|2 )s
Pμ (z)d A(z) ≤ C Pμ (w).
|1 − w̄z| p
D
Proof From Lemma 2.1 and Fubini’s theorem, we have
(1 − |w|2 ) p−s−2 (1 − |z|2 )s
Pμr (z)d A(z)
|1 − w̄z| p
D
(1 − |w|2 ) p−s−2 (1 − |z|2 )s
r 2 (1 − |z|2 )
=
dμ(ζ )d A(z)
p
2
|1 − w̄z|
D
∂ D |ζ − r z|
r 2 (1 − |w|2 )
C
≤C
dμ(ζ ) = C Pμr (w) ≤
.
2
|ζ
−
r
w|
1
−
|w|2
∂D
Characterizations of the Dirichlet-Type Space
Letting r → 1− and using Fatou Lemma imply
(1 − |w|2 ) p−s−2 (1 − |z|2 )s
Pμ (z)d A(z)
|1 − w̄z| p
D
(1 − |w|2 ) p−s−2 (1 − |z|2 )s
=
limr →1− Pμr (z)d A(z)
|1 − w̄z| p
D
(1 − |w|2 ) p−s−2 (1 − |z|2 )s
≤ limr →1−
Pμr (z)d A(z)
|1 − w̄z| p
D
≤ Climr →1− Pμr (w) = C Pμ (w).
This finishes the proof.
We are ready to establish the main theorem of this section.
Theorem 2.3 Suppose σ, τ > −1. Then f ∈ D(μ) if and only if
| f (z) − f (w)|2
P (z)d Aσ (z)d Aτ (w) < ∞.
4+σ +τ μ
D D |1 − z̄w|
Proof Suppose first that σ = τ . We may assume that σ > τ . For z, w ∈ D, we have
(1 − |w|2 )σ (1 − |z|2 )σ
(1 − |w|2 )σ (1 − |z|2 )τ (1 − |z|2 )σ −τ
=
|1 − z̄w|4+2σ
|1 − z̄w|4+σ +τ
|1 − z̄w|σ −τ
2
σ
(1 − |w| ) (1 − |z|2 )τ
≤ 2σ −τ
|1 − z̄w|4+σ +τ
and
(1 − |w|2 )σ (1 − |z|2 )τ
(1 − |w|2 )σ (1 − |z|2 )τ (1 − |z|2 )σ −τ
=
|1 − z̄w|4+σ +τ
|1 − z̄w|4+2τ
|1 − z̄w|σ −τ
2
σ
(1 − |w| ) (1 − |z|2 )τ
≤ 2σ −τ
.
|1 − z̄w|4+2τ
So,
2τ −σ
(1 − |w|2 )σ (1 − |z|2 )σ
(1 − |w|2 )σ (1 − |z|2 )τ
≤
4+2σ
|1 − z̄w|
|1 − z̄w|4+σ +τ
(1 − |w|2 )σ (1 − |z|2 )τ
≤ 2σ −τ
.
|1 − z̄w|4+2τ
Consequently, the case σ = τ can be obtained from the case σ = τ .
In what follows, we may assume that σ = τ . It is well-known ([22, Theorem 4.27])
that for any F ∈ H (D)
2
|F(w) − F(0)| d Aσ (w) ≈
|F (w)|2 (1 − |w|2 )2 Aσ (w).
(2.1)
D
D
X. Liu et al.
From Lemma 2.2 and (2.1), we get
| f (z) − f (w)|2
Pμ (z)d Aσ (z)d Aσ (w)
|1 − z̄w|4+2σ
D D
Pμ (z)
| f (ϕz (w)) − f (ϕz (0))|2 d Aσ (w)
d A(z)
=
(1
−
|z|2 )2+σ
D D
Pμ (z)
| f (ϕz (w)) |2 (1 − |w|2 )2 d Aσ (w)
d Aσ (z)
≈
(1 − |z|2 )2+σ
D D
(1 − |w|2 )σ +2 (1 − |z|2 )σ
≈
| f (w)|2
d A(w)Pμ (z)d A(z)
|1 − z̄w|4+2σ
D D
≤C
| f (w)|2 Pμ (w)d A(w).
I( f ) =
D
Conversely, for any f ∈ H (D), from the following estimates (cf. [22, Charpter 4])
C
| f (z)| ≤
(1 − |z|2 )2+σ
2
B(z,r )
| f (w)|2 d Aσ (w)
and
(1 − |w|2 )2
1
≈
, w ∈ B(z, r ).
|1 − z w̄|4+σ
(1 − |z|2 )2+σ
Combining these with the estimate of I ( f ) yields
1 − |w|2 2
f (w)
d Aσ (w)Pμ (z)d Aσ (z)
|1 − z̄w|2+σ
D B(z,r )
1
≈
| f (w)|2 d Aσ (w)Pμ (z)d A(z)
2 )2+σ
(1
−
|z|
D
B(z,r )
≥
| f (z)|2 Pμ (z)d A(z).
I( f ) ≥
D
The theorem is proved.
Now, we give a characterization of D(μ) in terms of the mean oscillation in the
Bergman metric.
Theorem 2.4 Let f ∈ A2 , 0 < r < 1 and dτ (z) = d A(z)/(1 − |z|2 )2 on D. Then
the following statements are equivalent:
(1) f ∈ D(μ);
(2)
D
2
M O f (z) Pμ (z)dτ (z) < ∞;
Characterizations of the Dirichlet-Type Space
(3)
D
2
M Or f (z) Pμ (z)dτ (z) < ∞.
Proof (1) ⇒ (2). For f ∈ A2 , from [21, Section 7.1], we have
2
2π M O f (z) =
D
| f (w) − f (z)|2
(1 − |z|2 )2
d A(w).
|1 − z̄w|4
Thus,
2
| f (z) − f (w)|2
P
(z)d
A(z)d
A(w)
≈
M O f (z) Pμ (z)dτ (z).
μ
4
|1 − z̄w|
D D
D
Theorem 2.3 yields the desired result.
(2) ⇒ (3). Its proof follows from the fact that ([21, Theorem 7.1.6])
M Or f (z) ≤ M O f (z).
(3) ⇒ (1). Since
(1 − |z|2 )| f (z)| ≤ M Or f (z),
(see [18, p.292]), we have
D
| f (z)|2 Pμ (z)d A(z) =
≤
D
D
(1 − |z|2 )2 | f (z)|2 Pμ (z)dτ (z)
2
M Or f (z) dτ (z).
The proof is finished.
3 Higher Order Derivative Characterization
In this section, we show a further characterization of the D(μ) space by means of
higher order derivatives. For this, we need to show the boundedness of certain integral
operator by making use of Schur’s lemma.
Let (X, μ) be a measure space. For f ∈ L p (X, dμ), we define the integral operator
H (x, y) f (y)dμ(y),
T f (x) =
X
where H is a non-negative and measurable function on X × X .
X. Liu et al.
Lemma 3.1 ([21, Corollary 3.2.3]) Assume that μ is a σ -finite measure. If there exists
a positive and measurable function h on X and a positive constant C > 0 such that
H (x, y)h(y)dμ(y) ≤ Ch(x)
X
for almost all x ∈ X and
H (x, y)h(x)dμ(x) ≤ Ch(y)
X
for almost all y ∈ X , then the integral operator T is bounded on L 2 (X, dμ). Furthermore, the norm of T on L 2 (X, dμ) does not exceed the constant C.
Lemma 3.2 ([21, Lemma 4.2.2]) Suppose t > −1. If s > 0, then there exists a
constant C such that
(1 − |w|2 )t
C
d A(w) ≤
2+s+t
(1 − |z|2 )s
D |1 − z w̄|
for all z ∈ D. If s < 0, then there exists a constant C such that
(1 − |w|2 )t
d A(w) ≤ C
2+s+t
D |1 − z w̄|
for all z ∈ D.
Given μ a finite Borel positive measure on ∂D, define the measure ν on D as
dν(z) = Pμ (z)d A(z)
and the integral operator T by
T f (z) =
D
H (z, w) f (w)dν(z),
f ∈ L 2 (D, dν),
(3.1)
where
H (z, w) =
(1 − |z|2 )n (1 − |w|2 )α
|1 − z w̄|2+n+α Pμ (w)
is a positive integral kernel and α a sufficiently large constant.
Also, consider integral operator S define as
S f (z) =
D
L(z, w) f (w)dν(w),
f ∈ L 2 (D, dν),
(3.2)
Characterizations of the Dirichlet-Type Space
where
L(z, w) =
(1 − |w|2 )α
.
|1 − z w̄|2+α Pμ (w)
(3.3)
Again, α is a sufficiently large constant.
We now show that the operators T and S are bounded on L 2 (D, dν). As a consequence, we will give the announced characterization of D(μ) in terms of higher order
derivatives.
Theorem 3.3 The operator T defined in (3.1) is bounded on L 2 (D, dν) for α sufficiently large.
Proof Fix constants σ and α such that
σ < n, α > σ + 1, α + σ > −1.
We apply Lemma 3.1 for the test function
h(z) = (1 − |z|2 )σ ,
z ∈ D.
Since α + σ > −1 and n − σ > 0, using Lemma 3.2 to conclude that there exists a
constant C > 0, such that
D
H (z, w)h(w)dν(w) = (1 − |z|2 )n
≤ Ch(z)
(1 − |w|2 )α+σ
d A(w)
2+(α+σ )+(n−σ )
D |1 − z w̄|
for all z ∈ D.
Next, for any w ∈ D, applying Lemma 2.2, we get
D
(1 − |w|2 )α−σ (1 − |z|2 )n+σ Pμ (z)
(1 − |w|2 )σ
d A(z)
Pμ (w)
|1 − z w̄|2+n+α
D
≤ Ch(w).
H (z, w)h(z)dν(z) =
As a consequence of Lemma 3.1, the proof of the theorem is complete.
Theorem 3.4 The operator S defined in (3.2) is bounded on L 2 (D, dν) for α sufficiently large.
Proof For 0 < < 1 and α > − + 1, we consider the function
h(z) = (1 − |z|2 )− ,
z ∈ D.
Again, we will apply Lemma 3.1 to show the boundedness of S on L 2 (D, dν).
X. Liu et al.
In fact, for any z ∈ D , from Lemma 3.2, we have
D
(1 − |w|2 )α−
d A(w)
2+(α−)+
D |1 − z w̄|
≤ Ch(z).
L(z, w)h(w)dν(w) =
For any w ∈ D, using Lemma 2.2 again, we obtain
D
(1 − |w|2 )α+ (1 − |z|2 )− Pμ (z)
(1 − |w|2 )−
d A(z)
Pμ (w)
|1 − z w̄|2+α
D
≤ Ch(w).
L(z, w)h(z)dν(z) =
Hence, the boundedness of S on L 2 (D, dν) follows.
Theorem 3.5 Let n be any non-negative integer. Then f ∈ D(μ) if and only if
D
| f (n+1) (z)|2 (1 − |z|2 )2n Pμ (z)d A(z) < ∞.
(3.4)
Proof Suppose that f ∈ D(μ), then it has the following integral representation:
f (z) = (α + 1)
f (w)(1 − |w|2 )α
d A(w), z ∈ D, α > 1.
(1 − z w̄)2+α
D
Differentiating under the integral sign n times and multiplying the result by (1−|z|2 )n ,
we have
(1 − |z|2 )n f (n+1) (z) = C
(1 − |z|2 )n (1 − |w|2 )α w̄ n f (w)
d A(w),
(1 − z w̄)2+α+n
D
where C is a positive constant depending only on α and n. In particular,
2 n (n+1)
(1 − |z| )
f
(z) ≤ C
D
H (z, w)| f (w)|dν(w).
From Theorem 3.3, we obtain
D
(1 − |z|2 )2n | f (n+1) (z)|2 dν(z) ≤ C
D
| f (z)|2 dν(z).
Conversely, integrating n-times both sides of the following representation (see, for
example, [8, Corollary 1.5] or [19, Corollary 8]),
Characterizations of the Dirichlet-Type Space
f (n+1) (z) = (n + α + 1)
f (n+1) (w)(1 − |w|2 )n (1 − |w|2 )α
d A(w),
(1 − z w̄)2+n+α
D
we get
f (z) =
h(z, w) f (n+1) (w)(1 − |w|2 )n (1 − |w|2 )α
d A(w),
(1 − z w̄)2+α
D
where h(z, w) is a bounded function in z and w. In particular, we have
| f (z)| ≤ C
D
L(z, w)| f (n+1) (w)|(1 − |w|2 )n dν(w)
From Theorem 3.4, we obtain
D
| f (z)|2 dν(z) ≤ C
D
(n+1) 2
f
(z) (1 − |z|2 )2n dν(z)
and the desired result follows.
4 Decomposition Theorem
Decomposition theorems in different function spaces such as Bergman spaces, Bloch
spaces, Dirichlet spaces, B M O A space and Q p spaces have been established and
proved their usefulness in several articles. See, for example, [12–14,17]. In this section,
we show the decomposition theorem for the Dirichlet − t ype space D(μ). To prove
the theorem, we need some notations and lemmas.
We say that a sequence of points z j ∈ D ( j = 1, 2, . . .) is η-separated, if there
exists η > 0 such that
inf d(z j , z k ) ≥ η.
j=k
On the other hand, we say that {z j }∞
j=1 is η-dense if
D=
∞
B(z j , η).
j=1
Lemma 4.1 ([22, Lemmas 4.7]) For any η ∈ (0, 1), there exist an η2 -separated and
η-dense sequence {z j }∞
j=1 and Lebesgue measurable sets D j ( j = 1, 2, . . .) such that:
(1) B(z j , η4 ) ⊂ D j ⊂ B(z j , η);
D j = ∅, if i = j;
(2) Di ∩ (3) D = ∞
j=1 D j .
X. Liu et al.
Lemma 4.2 ([22, Lemmas 4.10]) For any η ∈ (0, 1) and positive integer N , there
exist an η2 -separated and η-dense sequence {z j }∞
j=1 ⊂ D such that any z ∈ D lies in
at most N of the sets B(z j , 2η) ( j = 1, 2, . . .).
The following lemma can be found in [12,13].
Lemma 4.3 If z 0 ∈ D and η ≤ 1, there exists a constant C > 0, independent of η
and z 0 , such that
|kw (z) − kw (z 0 )| ≤ Cη|kw (z)|
for all w ∈ D and z ∈ B(z 0 , η), where
kw (z) =
(1 − |z|2 )b−1
,
(1 − z̄w)b+1
b > 0.
Lemma 4.4 Let μ be a positive finite Borel measure on ∂D. For η ∈ (0, 1), let
{z j }∞
j=1 be an η-separated sequence. If z ∈ B(z j , η), j = 1, 2, . . . , then there exist
two positive constants C1 and C2 such that
C1 Pμ (z j ) ≤ Pμ (z) ≤ C2 Pμ (z j ), j = 1, 2, . . . .
Proof Let z ∈ B(z j , η), j = 1, 2, . . ., and r ∈ (0, 1). It is easy to check that there
exists a constant C > 0, independent of the sequence {z j }∞
j=1 and η such that
|1 − r ζ̄ z| ≤ C|1 − r ζ̄ z j |,
ζ ∈ ∂D.
Also, by Lemma 4.3.4 in [21], there is a constant C > 0 independent of {z j }∞
j=1 and
η such that, for z ∈ B(z j , η)
1 − |z j |2 ≤ C(1 − |z|2 ).
Therefore,
r 2 (1 − |z j |2 )
r 2 (1 − |z|2 )
≤
C
.
|1 − r ζ̄ z j |2
|1 − r ζ̄ z|2
Integrating on ∂D with respect to μ and letting r → 1− , we have
Pμ (z j ) ≤ C Pμ (z).
The other inequality follows in a similar way.
Lemma 4.5 Let f ∈ D(μ), 0 < η < 1 and
{z j }∞
j=1
be an η-separated. Then
∞
2 2
(1 − |z j |) | f (z j )| Pμ (z j ) ≤ C
| f (z)|2 Pμ (z)d A(z).
j=1
D
Characterizations of the Dirichlet-Type Space
Proof For any f ∈ H (D), we have ([22, Proposition 4.13])
C
| f (z j )|2 ≤
| f (w)|2 d A(w).
|B(z j , η)| B(z j ,η)
Note that |B(z j , η)| ≈ (1 − |z j |2 )2 , from Lemma 4.4, we obtain
∞
(1 − |z j |)2 | f (z j )|2 Pμ (z j ) ≤ C
j=1
∞ j=1
≤C
D
B(z j ,η)
| f (w)|2 Pμ (w)d A(w)
| f (w)|2 Pμ (w)d A(w).
Now we are ready to prove the following result.
Theorem 4.6 (Decomposition Theorem) Let μ be a nonnegative Borel measure on
∂D and b > 2. Then there exists a d-separated sequence {z j }∞
j=1 in D such that the
following are true.
(1) If f ∈ D(μ), then there exists a sequence {λ j }∞
j=1 in C such that
f (z) = f (0) +
∞
λ j (1 − |z j | )
2 b
j=1
1
−1
(1 − z j z)b
(4.1)
and
∞
|λ j |2 Pμ (z j ) ≤ C f 2D(μ) .
j=1
∞
2
(2) If a sequence {λ j }∞
j=1 |λ j | Pμ (z)δz j is a μ-Carleson
j=1 ⊂ C satisfies that
measure, then the series defined in (4.1) converges in D(μ) and
f 2D(μ)
≤C
∞
|λ j |2 Pμ (z j ).
(4.2)
j=1
Proof For part (1), recall that an equivalent norm for the Dirichlet type spaces D(μ)
is given by (see, for example, [4, Lemma 2.3])
| f (z)|2 Pμ (z)d A(z).
f 2D(μ) ≈ | f (0)|2 +
D
If we define the space D0 (μ) := { f ∈ D(μ) : f (0) = 0} with the norm
f D0 (μ) =
D
| f (z)|2 Pμ (z)d A(z)
1
2
,
X. Liu et al.
then f − f (0) ∈ D0 (μ) for f ∈ D(μ). Moreover, the space D(μ) can be written as
D(μ) = D0 (μ) ⊕ C.
For b > 2, assume that f ∈ D0 (μ). Then f ∈ H 2 and f ∈ A21 ⊂ A2b−1 . By the
reproducing formula of the Bergman space, we have
(1 − |w|2 )b−1 b
f (w)d A(w).
f (z) =
π D (1 − w̄z)b+1
η
∞
Since {z j }∞
j=1 is 2 -separated and η-dense, then there exists a disjoint partition {D j } j=1
of D
f (z) =
∞ (1 − |w|2 )b−1 b
f (w)d A(w).
b+1
π
D j (1 − w̄z)
j=1
Define the linear operator A on D0 (μ) by
A( f )(z) =
∞
(1 − |z j |2 )b−1
b
f (z j )|D j |
.
π
z¯j (1 − z¯j z)b
j=1
We will show first that
f − A( f ) D0 (μ) ≤ Cη f D0 (μ) .
Notice that
∞ f (z) − A( f ) (z) ≤ b
| f (w)||k z (w) − k z (z j )|d A(w)
π
Dj
+
j=1
∞ b
π
j=1
| f (w) − f (z j )||k z (z j )|d A(w)
Dj
= I1 + I2 .
Using Lemma 4.3 implies
| f (w)||k z (w)|d A(w).
I1 ≤ Cη
D
From [17, p.394], we have
I2 ≤ Cη
D
| f (w)||k z (w)|d A(w).
(4.3)
Characterizations of the Dirichlet-Type Space
Applying Theorem 3.4 yields
2
| f (z)− A( f ) (z)|2 Pμ (z)d A(z) ≤ Cη
|k z (w)|| f (w)|d A(w) Pμ (z)d A(z)
D
D
D
≤ Cη
| f (z)|2 Pμ (z)d A(z).
D
Thus, inequality (4.2) holds.
Now, define the operator A : D(μ) → D0 (μ) as
∞
(1 − |z j |2 )b−1
1
1
A( f − f (0))(z) :=
f (z j )|D j |
−1 .
π
zj
(1 − z j z)b
j=1
In other words, A is the operator A followed by the projection into the space D0 (μ).
Consider the operator B : D(μ) → D(μ) defined by
B=
A
0
0
1
Then, inequality (4.3) gives
(I − B) f 2D(μ) = f − A( f − f (0)) − f (0)2D(μ)
= f − A( f − f (0)) − A( f − f (0))(0) − f (0)2D(μ)
= ( f − f (0)) − A( f − f (0))2D0 (μ)
≤ Cη f − f (0) D0 (μ)
≤ Cη f D(μ) ,
where I stands for the identity operator acting on D(μ). Taking η > 0 small enough,
we have the invertibility of the operator B. Its bounded inverse is defined by
B −1 = (I − (I − B))−1 =
∞
(I − B)n .
n=0
We have constructed an approximation operator B with bounded inverse. For any
f ∈ D(μ), we can write
f (z) = B B −1 f (z)
= AA−1 ( f − f (0))(z) + f (0)
∞
(1 − |z j |2 )b−1
1
b −1
=
(A ( f − f (0))) (z j )|D j |
− 1 + f (0)
π
zj
(1 − z j z)b
j=1
= f (0) +
∞
j=1
λ j (1 − |z j |2 )b
1
−
1
,
(1 − z j z)b
X. Liu et al.
where
λj =
b(A−1 ( f − f (0))) (z j )|D j |
.
π z j (1 − |z j |2 )
We now show that
∞
|λ j |2 Pμ (z j ) < C f 2D(μ) .
(4.4)
j=1
By Lemma 4.5, we have
∞
|λ j |2 Pμ (z j ) ≤ C
j=1
≤C
∞
j=1
∞
|D j |2
|A−1 ( f − f (0)) (z j )|2 Pμ (z j )
(1 − |z j |2 )2
(1 − |z j |2 )2 |A−1 ( f − f (0)) (z j )|2 Pμ (z j )
j=1
≤C
D
|A−1 ( f − f (0)) (z)|2 Pμ (z)d A(z)
≤ CA−1 ( f − f (0))2D0 (μ) ≤ C f 2D(μ) .
Thus (4.4) is proved.
2
Next we show part (2). Suppose that ∞
j=1 |λ j | Pμ (z)δz j is a μ-Carleson measure.
Then there exists a constant C > 0 such that
∞
|λ j |2 | f (z j )|2 Pμ (z j ) ≤ C f 2D(μ) ,
f ∈ D(μ).
j=1
In particular, if f ≡ 1, we have that
∞
|λ j |2 Pμ (z j ) ≤ C.
(4.5)
j=1
It is sufficient to show that f ∈ D(μ) for f defined as in Eq. (4.1). In this case,
f (w) = b
∞
j=1
λ j z¯j
(1 − |z j |2 )b
.
(1 − z¯j w)b+1
We know that there exists a positive constant (cf. [17])
η
Cj =
2
π( eeη −1
+1 )
η
2b
η
4e
2
2
b
1 − ( eeη −1
+1 ) |z j | 1 − ( (eη +1)2 )
,
j = 1, 2, . . . ,
Characterizations of the Dirichlet-Type Space
such that
B(z j , η4 )
|B(z j , η4 )| (1 − |z j |2 )b−1
(1 − |z|2 )b−1
d
A(z)
=
.
(1 − z̄w)b+1
Cj
(1 − z¯j w)b+1
Therefore, for b > 2,
1 − |z j |2
(1 − |z|2 )b−1
η (z)d A(z)
λ j z¯j C j
χ
f (w) = b
η
|B(z j , 4 )| D (1 − z̄w)b+1 B(z j , 4 )
j=1
∞
1 − |z j |2
(1 − |z|2 )b−1 η (z) d A(z).
χ
λ
z
¯
C
=b
j
j
j
η
B(z
,
)
j 4
b+1
|B(z j , 4 )|
D (1 − z̄w)
∞
j=1
Consequently, if
∞
2
1 − |z j |2
λ j z¯j C j
η χ B(z j , η4 ) (z) Pμ (z)d A(z) < ∞,
|B(z
,
)|
j 4
D
(4.6)
j=1
then by Theorem 3.4
D
| f (z)|2 Pμ (z)d A(z) ≤ C
∞
2
1 − |z j |2
η (z) Pμ (z)d A(z).
χ
λ j z¯j C j
|B(z j , η4 )| B(z j , 4 )
D
j=1
(4.7)
Hence f ∈ D(μ). It remains to show that the inequality (4.6) holds.
In fact, since {B(z j , η4 )}∞
j=1 is a set of disjoint Bergman discs, then the sequence
is
bounded.
Finally,
from (4.5) and Lemmas 4.4, we have
{C j }∞
j=1
∞
2
1 − |z j |2
λ j z¯j C j
η χ B(z j , η4 ) (z) Pμ (z)d A(z)
|B(z
,
)|
j 4
D
j=1
≤C
≤C
≤C
∞
D j=1
∞ |λ j |2
(1 − |z j |2 )2
η Pμ (z)d A(z)
χ
|B(z j , η4 )|2 B(z j , 4 )
|λ j |2
η
j=1 B(z j , 4 )
∞
|λ j |2 Pμ (z j )
j=1
(1 − |z j |2 )2
Pμ (z)d A(z)
|B(z j , η4 )|2
≤ C.
Combining this with (4.7) we get (4.2). This finishes the proof.
X. Liu et al.
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