# Characterizations of a Class of Dirichlet-Type Spaces and Related Operators

Authors: Xiaosong Liu; Songxiao Li
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, some characterizations are given in terms of the boundary value and Poisson extension for the Dirichlet-type spaceDμ. The multipliers of Dμ and Hankel-type operators from Dμ to L2PμdA are also investigated.


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## Body

## 1. Introduction

LetD be the unit disk of complex plane ℂ. For 0<p<∞, the Hardy space, denoted by Hp, is the space consists of all f∈HD such that(1)fHpp=sup0<r<112π∫02πfreiθpdθ<∞.Here,HD is the space of analytic functions on D.Let∂D denote the boundary of D and dA denote the normalized Lebesgue measure on D. Let μ be a positive Borel measure on ∂D. An f∈HD is said to belong to the space Dμ, called the Dirichlet-type space, if(2)∫Df′z2PμzdAz<∞,where(3)Pμz=∫02π1−z2eit−z2dμt2π.The spaceDμ was introduced by Richter in [1] for studying analytic two isometrics. It was shown in [1] that Dμ⊂H2. The norm on Dμ is defined as follows:(4)fDμ2=fH22+∫Df′z2PμzdAz.The spaceDμ is a Hilbert space with(5)f,gDμ=f,gH2+∫Df′zg′z¯PμzdAz,Dμ=H2 when μ=0. If dμ=dm, then Dμ coincides with the Dirichlet space D. By (Proposition 2.2 in [1]), we have(6)∫∂DDζfdμζ=∫Df′z2PμzdAz.Here,(7)Dζf=12π∫02πfeit−fζeit−ζ2dt.Letf∈L2∂D. We say that f∈L2μ if(8)∫∂D∫02πfeiθ−fζ2eiθ−ζ2dθdμζ<∞.The norm of the spaceL2μ is given by(9)fL2μ2=fL2∂D2+∫∂D∫02πfeiθ−fζ2eiθ−ζ2dθdμζ.The spaceDμ has been investigated by many authors. In [2], Richter and Sundberg studied the cyclic vectors of Dμ. Shimorin studied the reproducing kernels and extremal functions of Dμ in [3], see [4–6], for the study of Carleson measure for Dμ. The study of composition operators and Toeplitz operators on Dμ can be found in [7, 8], respectively, see [9–11], for more study of the space Dμ.In this paper, we provided some characterizations for the spaceDμ by the boundary value and Poisson extension. Moreover, we study the multipliers of Dμ and the Hankel-type operator from Dμ to L2PμdA.In this paper, we always assume thatμ is a positive Borel measure on ∂D and C is a positive constant that may differ from one occurrence to the other. The notation F≲G means that there exists a C such that F≤CG. The notation F≍G indicates that G≲F and also F≲G.


## 2. Characterizations of the SpaceDμ

Letf∈L1∂D. The Poisson extension of f, denoted by f^, is(10)f^z=∫02πfeit1−z2eit−z2dt2π,z∈D.It is well known thatf^ is a harmonic function on D.LetC1D denote the space of all functions on D with continuous partial derivatives. For f∈C1D, the gradient of f is defined by(11)∇f=∂f∂x,∂f∂y.First, we state some lemmas.Lemma 1 (see [6, 8]).
Letf∈L2∂D. Then,(12)∫∂DDζfdμζ<∞,if and only if(13)∫D∇f^z2PμzdAz<∞.Remark 1.
Letf∈L2∂D and F∈C1D such that limr⟶1Freiθ=feiθ(a.e.) for eiθ∈∂D. Then,(14)∫∂DDζfdμζ≲fL2∂D2+∫D∇Fz2PμzdAz.
Forf∈H2, let fb denote the boundary value of f.Corollary 1.
Letf∈H2. Then, f∈Dμ if and only if fb∈L2μ.Proof.
Sincef∈H2, then f=fb^. The desired result follows from Lemma 1.Lemma 2.
Letf∈L2∂D. Then, the following statements are equivalent:(a)
∫∂DDζfdμζ<∞.(b)
∫D∇f^z2PμzdAz<∞.(c)
limr⟶1−∫Df^2^z−f^z2dμrz<∞, where(15)dμrz=∫∂Dr21−r2ζ−rzdμζdAz.Proof.
a⟺b This implication follows by Lemma 1.Proof.
b⟺c For z∈D,r∈0,1, set(16)Pμrz=∫∂Dr21−z2ζ−rz2dμζ.
From [11], we see that Pμrz is subharmonic with(17)limr⟶1−Pμrz=Pμz.
By Green’s formula, we obtain(18)Pμrz=2π∫D∂2∂w∂¯wPμrwlog1−w¯zw−zdAw≍∫D∫∂Dr21−r2ζ−rwdμζlog1−w¯zw−zdAw.
According to (17) and (18) and Hardy-Littlewood’s identity (see page 238 in [12]), we have(19)∫D∇f^z2PμzdAz=limr⟶1−∫D∇f^z2PμrzdAz≍limr⟶1−∫D∇f^z2∫D∫∂Dr21−r2ζ−rwdμζlog1−w¯zw−zdAwdAz=limr⟶1−∫D∫D∇f^z2log1−w¯zw−zdAz∫∂Dr21−r2ζ−rwdμζdAw=limr⟶1−∫Df^2^w−f^w2dμrw.
The proof is complete.Theorem 1.
Letf∈H2. Then, the following statements are equivalent:(a)
f∈Dμ.(b)
limr⟶1−∫Df−f^z2zdμrz<∞.(c)
∫∂DDζfdμζ<∞ and(20)limr⟶1−∫Df^2z−fz2dμrz<∞.(d)
∫∂DDζfdμζ<∞ and there exists a harmonic function g such that f≤g on D and(21)limr⟶1−∫Dg2z−fz2dμrz<∞.Proof.
a⇔b This implication follows by Lemma 2 and(22)f2^z−fz2=f−f^z2z.
a⇒c If f∈Dμ, then ∫∂DDζfdμζ<∞. Since(23)f^z2=∫∂Dfζ1−z2ζ−z2dζ2π2≤∫∂Dfζ21−z2|ζ−z|2dζ2π∫∂D1−z2ζ−z2dζ2π=f2^z.
We getc from Lemma 2 and Corollary 1.
c⇒d Inequality (20) implies(24)limr⟶1−∫Df^z2^−fz2dμrz<∞.
Letg=f^. Then, g2≤f^2^. Thus,(25)limr⟶1−∫Dgz2−fz2dμrz<∞.
d⇒a By Lemma 2,(26)limr⟶1−∫Df2^z−f^z2dμrz<∞.
Assume thatg is a harmonic function such that f≤g. Note that f^ is the least harmonic function equal to or greater than f (see [12]); hence, f^≤g. By Lemmas 1 and 2 and Corollary 1, f∈Dμ. The proof is complete.


## 3. Multipliers ofDμ

LetI⊂∂D. The Carleson box SI is(27)SI=rζ∈D:1−I<r<1;ζ∈I.Assume thatν is a positive Borel measure on D. If supI⊂∂DνSI/I<∞, then we say that ν is a Carleson measure.If there exists a constantC>0 (see [4, 5])(28)∫Dfz2dνz≤CfDμ2,for allf∈Dμ,then we call that ν is a μ-Carleson measure.Letg∈L∞∂D and f∈L2μ. g is called the pointwise multipliers of L2μ if gf∈L2μ. We denote the space of all pointwise multipliers of L2μ by ML2μ.Lemma 3.
Letν be a positive Borel measure on D. Then, ν is a μ-Carleson measure if and only if(29)∫Dg^z2dνz≲gL2μ2,for all g∈L2μ.Proof.
First, we assume thatν is a μ-Carleson measure. Suppose that g∈L2μ. Without loss of generality, let g be a real-valued function. Suppose that g˜ is the harmonic conjugate of g^. Set f=g^+ig˜. Then, ∇f^z≍f′z by the Cauchy–Riemann equation. From Lemma 2.3 in [7] and Lemma 1, we obtain(30)∫Dg^z2dνz≤∫Dfz2dνz≲fDμ2=fH22+∫Df′z2PμzdAz≍f02+∫D∇g^z2PμzdAz≲gL2μ2.
Conversely, forf∈Dμ, by Corollary 1, fb∈L2μ and f=f^b. Then,(31)∫Dfz2dνz≲fbL2μ2≲fDμ2,which implies that ν is a μ-Carleson measure.Theorem 2.
g∈ML2μ if and only if g∈L∞∂D and ∇g^2PμdA is a μ-Carleson measure.Proof.
Assume thatg∈L∞∂D and ∇g^2PμdA is a μ-Carleson measure. Let f∈L2μ. By Remark 1, we obtain(32)fgL2μ2≲fgL2∂D2+∫D∇f^g^z2PμzdAz≤fgL2∂D2+∫Dg^z2∇f^z2PμzdAz+∫Df^z2∇g^z2PμzdAz.
By Lemma1 and Corollary 1, we obtain(33)∫Dg^z2∇f^z2PμzdAz≤Cg^L∞D2fL2μ2.
In addition, since∇g^2PμdA is a μ-Carleson measure, by Lemma 3, we have(34)∫Df^z2∇g^z2PμzdAz≤CfL2μ2.
Combining (32)–(34), we obtain that g∈ML2μ.
Conversely, assume thatg∈ML2μ. Then, by Theorem 2.7 in [6], we see that g∈L∞∂D. For f∈Dμ, by the Closed Graph Theorem, Lemma 1, and Corollary 1, we obtain(35)∫D∇fg^z2PμzdAz≤CfgL2μ2≤CfL2μ2≤CfDμ2.
Next, we show that∇g^2PμdA is a μ-Carleson measure. From the fact that ∇f≍f′z, we obtain(36)∫Dg^z2∇fz2PμzdAz≤C∫D∇fz2PμzdAz≍C∫Df′z2PμzdAz≤CfDμ2.
Then, by (35) and (36),(37)∫Dfz2∇g^z2PμzdAz≤C∫D∇fg^z2+g^z2∇fz2PμzdAz≤CfDμ2,which implies that ∇g^2PμdA is a μ-Carleson measure.
By Theorem2, we obtain the following result.Corollary 2.
Letf∈H2. Then, f∈MDμ if and only if fb∈ML2μ.


## 4. Hankel-Type Operators onDμ

LetP denote the set of all polynomials on D. From [1, 2], we see that P is dense in Dμ. Let(38)Pfz=∫Dfw1−w¯z2dAw.From Theorem 1.10 in [13], we see that P:L2D⟶A2 is a bounded projection. Here, A2 is the Bergman space which consists of all f∈HD such that ∫Dfz2dAz<∞. For f∈A2, we define a Hankel-type operator hf on P by(39)hfg=Pfg¯¯,g∈P.Lemma 4 (see Theorem 2.3 in [10]).
Letτ,σ>−1. Then, f∈Dμ if and only if(40)∫D∫Dfz−fw21−z¯w4+σ+τPμzdAσzdAτw<∞,where dAσz=1−z2σdAz.Lemma 5 (see Theorem 3.4 in [10]).
LetT be the operator defined by(41)Tgz=∫Dgw1−w¯z2dAw,g∈L2D.
Then,T:L2PμdA⟶L2PμdA is bounded.Theorem 3.
Letg∈L2D such that g2PμdA is a μ-Carleson measure. Then, Tg2PμdA is a μ-Carleson measure.Proof.
Suppose thatg2PμdA is a μ-Carleson measure. Then, by Lemma 5,(42)∫DTfgz2PμzdAz≤C∫Dfzgz2PμzdAz≤CfDμ2,for all f∈Dμ. So, it is enough to show that(43)∫DfzTgz−Tfgz2PμzdAz≤CfDμ2,for every f∈Dμ.
By Hölder’s inequality, we have(44)fzTgz−Tfgz2≤∫Dfz−fw1−w¯z2gwdAw2≲∫Dgw2dAw∫Dfz−fw21−w¯z4dAw=gL2D2∫Dfz−fw21−w¯z4dAw.
Consequently, by Lemma4, we obtain(45)∫DfzTgz−Tfgz2PμzdAz≤gL2D2∫D∫Dfz−fw21−w¯z4dAwPμzdAz≤gL2D2fDμ2.
The desired result follows.Theorem 4.
Letu∈A2. Then, the operator hu:Dμ⟶L2PμdA is bounded if and only if u2PμdA is a μ-Carlson measure.Proof.
Suppose thatu2PμdA is a μ-Carlson measure. Let g∈Dμ. Then, ug¯∈L2PμdA. By Lemma 4, we get that hug∈L2PμdA and(46)hugL2PμdA≤Tug¯L2PμdA≤Cug¯L2PμdA≤CgDμ.
So,hu:Dμ⟶L2PμdA is bounded.
Conversely, assume thathu:Dμ⟶L2PμdA is bounded. We need to prove that(47)ug¯L2PμdA≤CgDμ,for anyg∈Dμ.
By Hölder’s inequality we have(48)∫Duwgz−gw¯1−w¯z2dAw2≤∫Duw2dAw∫Dgz−gw21−w¯z4dAw=uA22∫Dgz−gw21−w¯z4dAw.
Since(49)uzgz¯−hugz¯=∫Duwgz−gw¯1−w¯z2dAw,by Lemma 4 and the fact that hu:Dμ⟶L2PμdA is bounded, we obtain(50)ug¯L2PμdA2≤uA22∫D∫D|gz−gw|2|1−w¯z|4dAwPμzdAz+hu2gDμ2≲uA22+hu2gDμ2.
The proof is complete.



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