# Predual ofQK Spaces

Authors: Jizhen Zhou
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

A necessary and sufficient condition is given for a positive measureμ on 𝔻 to be a K-Carleson measure. We give the predual of QK spaces in terms of this condition.


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## Body

## 1. Introduction

In study of a function space, it is of interest to study the dual and predual of function space. It is well known that Fefferman’s and Sarason’s theorems are(VMOA)*≅H1 and (H1)*≅BMOA. Anderson et al.  gave the similar results on the Bloch space in [1]. The reader can refer to [2, 3] about the predual of Qp spaces. We note that Qp spaces are a kind of QK spaces. Now our question is what is the predual of QK spaces. But the technique that is used to prove the predual of Qp spaces does not work for QK spaces. Enlightened by [4], we started from the characterizations of K-Carleson measure by an integral operator which contains the normalized nonnegative Borel measure on the unit disk. In this paper, we obtain a principal result that the predual of QK spaces is the analytic space 𝒞K, which is introduced in Section 3. We now recall a few fundamental definitions and establish some notation.Letg(a,z)=-log|φa(z)| be the Green function on the unit disk 𝔻={z:|z|<1} with logarithmic singularity at a∈𝔻, where φa(z)=(a-z)(1-a-z)-1 is the Möbius transformation of 𝔻. Denote by H(𝔻) the set of all analytic functions on 𝔻.LetK:[0,∞)→[0,∞) be a right-continuous and nondecreasing function. The space QK consists of all functions f∈H(𝔻) satisfying
(1)∥f∥QK2=supa∈𝔻∫𝔻‍|f′(z)|2K(g(a,z))dA(z)<∞,
where dA is an area measure on 𝔻 normalized so that A(𝔻)=1.Equipped with the norm|f(0)|+∥f∥QK, the space QK is Banach. It is easy to check that the space QK is Möbius invariant in the sense that ∥f∘φa∥QK=∥f∥QK for any f∈QK and a∈𝔻. See [5, 6] for a general theory of QK spaces. Note that the space QK gives Qp if we choose K(t)=tp for 0<p<∞. See [7, 8] for a summary of recent research for Qp spaces.Recall that a functionf∈H(𝔻) is said to belong to the Bloch space, denoted by ℬ, if
(2)∥f∥ℬ=supz∈𝔻(1-|z|2)|f′(z)|<∞.By [5], we know that
(3)QK⊂ℬ.The following two conditions have played a crucial role in the study ofQK spaces during the last years:
(4)∫1∞‍φK(t)dtt2<∞,(5)∫01‍φK(t)dtt<∞,
where
(6)φK(t)=sup0<s<1K(st)K(s),0<t<∞.Throughout the paper,K satisfies the following condition:
(7)∫01‍K(log1r)rdr<∞.Otherwise, the spaceQK only contains constant functions (cf. [5]). By  Theorem  2.1 in [5], we may assume that K is defined on [0,1] and extend its domain to [0,∞) by setting K(t)=K(1) for t>1. As the discussion in [6], we may assume that K(2t)≈K(t).For a subarcI⊂∂𝔻, the boundary of 𝔻, let
(8)S(I)={rζ∈𝔻:1-|I|<r<1,ζ∈I}.If|I|⩾1, then we set S(I)=𝔻. A positive measure μ on 𝔻 is said to be a K-Carleson measure if
(9)∥μ∥K=supI⊂∂𝔻∫S(I)‍K(1-|z|2|I|)dμ(z)<∞.By results in [6], we know that a function f∈H(𝔻) belongs to the space QK if and only if |f′(z)|2dA(z) is a K-Carleson measure.In the paper, we say thatK1≲K2 (for two functions K1 and K2) if there is a constant C>0 (independent of K1 and K2) such that K1≤CK2. We say K1≈K2 (i.e., K1 is comparable with K2) whenever K1≲K2≲K1.


## 2.K-Carleson Measure

For anyI⊂∂𝔻 and the nondecreasing K, denote
(10)ωI,K(z)=K(1-|z|2|I|)∫∂𝔻‍χI(ξ)Pz(ξ)|dξ|,z∈𝔻,
where
(11)Pz(ξ)=1-|z|2|ξ-z|2
is the poisson kernel and χI(ξ)=1 for ξ∈I, χI(ξ)=0 for ξ∈∂𝔻∖I.Lemma 1.
Letμ be a nonnegative measure on 𝔻. Let K satisfy condition (4). Then for any arcI⊂∂𝔻,
(12)∫𝔻‍ωI,K(z)dμ(z)<∞
holds if and only if μ is a K-Carleson measure.Proof.
First we assume that (12) holds. Now we show that μ is a K-Carleson measure. For any given z∈S(I), we have
(13)∫∂𝔻‍χI(ξ)Pz(ξ)|dξ|≥14.
This gives
(14)ωI,K(z)=K(1-|z|2|I|)∫∂𝔻‍χI(ξ)Pz(ξ)|dξ|≥14K(1-|z|2|I|).
Then
(15)∫S(I)‍K(1-|z|2|I|)dμ(z)≤4∫𝔻‍ωI,K(z)dμ(z)<∞.
The above inequality shows that μ is a K-Carleson measure.
Conversely, suppose thatμ is a K-Carleson measure. For a nonnegative integer n, we use In for the arc in ∂𝔻 which has the same center as I and length 2n|I|. For ξ∈In and n≥1, we have the following estimate:
(16)Pz(ξ)≲12n|I|,z∈S(In+1)∖S(In).
If z∈S(I1), we have
(17)ωI,K(z)===K(1-|z|2|I|)∫∂𝔻‍χI(ξ)Pz(ξ)|dξ|≲K(1-|z|2|I|).
If z∈S(In+1)∖S(In), we have
(18)ωI,K(z)===K(1-|z|2|I|)∫∂𝔻‍χI(ξ)Pz(ξ)|dξ|≲12nK(1-|z|2|I|).
Then
(19)∫𝔻‍ωI,K(z)dμ(z)=∫S(I1)‍ωI,K(z)dμ(z)=+∑n=1∞‍∫S(In+1)∖S(In)‍ωI,K(z)dμ(z)≲∥μ∥K=+∑n=1∞‍12n∫S(In+1)∖S(In)‍K(1-|z|2|I|)dμ(z)≲∥μ∥K=+∑n=1∞‍φK(2n)2n∫S(In+1)‍K(1-|z|22n+1|I|)dμ(z)≲∥μ∥K(1+∑n=1∞‍φK(2n)2n)≲∥μ∥K(1+∫1∞‍φK(s)s2ds).
We have the desired result by condition (4).Let𝕄 be the set of all nonnegative measure σ on 𝔻 with the normalized condition σ(𝔻)=1. For ξ∈∂𝔻, let
(20)Γ(ξ)={z∈𝔻,|z-ξ|<2(1-|z|)}.
For any z∈𝔻, denote
(21)ωσ,K,z(ξ)=∫Γ(ξ)‍K(1-|z|21-|w|)dσ(w),(22)ωσ,K(z)=∫∂𝔻‍ωσ,K,z(ξ)Pz(ξ)|dξ|.
The following estimate can be found in [9]:
(23)∫∂𝔻‍χΓ(ξ)(w)|dξ|===4arcsin(min(1,(1-|w|)4|w|))≈1-|w|2.
Then we have
(24)∫∂𝔻‍ωσ,K,z(ξ)|dξ|===∫∂𝔻‍∫Γ(ξ)‍K(1-|z|21-|w|)dσ(w)|dξ|===∫∂𝔻‍∫𝔻‍K(1-|z|21-|w|)χΓ(ξ)(w)dσ(w)|dξ|===∫𝔻‍K(1-|z|21-|w|)(∫∂𝔻‍χΓ(ξ)(w)|dξ|)dσ(w)==≲K(1)∫𝔻‍(1-|w|2)dσ(w)==≤K(1)σ(𝔻).
This shows that ωσ,K,z∈L1(∂𝔻). Hence, the definition of ωσ,K is logical.Theorem 2.
LetK satisfy condition (4). For all σ∈𝕄,
(25)∫𝔻‍ωσ,K(z)dμ(z)<∞,
if and only if μ is a K-Carleson measure.Proof.
Suppose thatμ is a K-Carleson measure. For any w∈𝔻, denote Iw={ξ∈∂𝔻:w∈Γ(ξ)}. Clearly, Iw is an arc on ∂𝔻 with the midpoint w/|w|. We obtain the following estimate by (23):
(26)|Iw|=∫∂𝔻‍χIw(ξ)|dξ|≈1-|w|.
Note thatK(t)≈K(2t) for any 0<t<∞. Then
(27)ωσ,K(z)=∫∂𝔻‍ωσ,K,z(ξ)Pz(ξ)|dξ|=∫∂𝔻‍∫Γ(ξ)‍K(1-|z|21-|w|)Pz(ξ)dσ(w)|dξ|=∫∂𝔻‍∫𝔻‍K(1-|z|21-|w|)χΓ(ξ)(w)Pz(ξ)dσ(w)|dξ|≈∫𝔻‍K(1-|z|21-|w|)(∫∂𝔻‍χIw(ξ)Pz(ξ)|dξ|)dσ(w)=∫𝔻‍ωIw,K(z)dσ(w).
Therefore, by Lemma 1 and Fubini’s theorem,
(28)∫𝔻‍ωσ,K(z)dμ(z)≈∫𝔻‍∫𝔻‍ωIw,K(z)dσ(w)dμ(z)=∫𝔻‍(∫𝔻‍ωIw,K(z)dμ(z))dσ(w)<∞.
Conversely, we assume that (25) holds. For any arc I, let w be the point in 𝔻 such that Iw=I. Let σ be the point mass at w. Then
(29)ωσ,K(z)=K(1-|z|21-|w|)∫∂𝔻‍χIw(ξ)Pz(ξ)|dξ|=ωIw,K(z).
This and Lemma 1 give that
(30)∫𝔻‍ωσ,K(z)dμ(z)=∫𝔻‍ωIw,K(z)dμ(z)<∞.
The proof is complete.


## 3. Predual ofQK Spaces

In this section, we will apply Theorem2 to get the predual of QK spaces.Definition 3.
LetK be a right-continuous and nondecreasing function. Denote by MK the set of all measurable functions G on 𝔻 such that the measure |G(z)|2dA(z) is a K-Carleson measure.By Theorem2, we can define
(31)∥G∥MK=supσ∈𝕄(∫𝔻‍|G(z)|2ωσ,K(z)dA(z))1/2.
It is easy to check that ∥·∥MK is a norm.Denote byMK the set of all measurable functions F on 𝔻 such that
(32)|||F|||MK=infσ∈𝕄(∫𝔻‍|F(z)|2(1-|z|2)2ωσ,K(z)dA(z))1/2<∞.Lemma 4.
For the spaceMK, define
(33)∥F∥MK=supG∈MK|〈F,G〉|∥G∥MK,
where
(34)〈F,G〉=∫𝔻‍F(z)G(z)¯(1-|z|2)dA(z).
Then ∥·∥MK is a norm.Proof.
It is obvious thatF=0a.e., then ∥F∥MK=0. Conversely, if we set
(35)G(z)=F(z)|F(z)|,F∈MK
then
(36)〈F,G〉=∫‍F(z)G(z)¯(1-|z|2)dA(z)=∫𝔻‍|F(z)|(1-|z|2)dA(z)=0.
This implies F=0a.e. We have
(37)∥F∥MK=0⟺F=0a.e.
For any givena∈ℂ, it is easy to see that
(38)∥aF∥MK=supG∈MK|〈aF,G〉|∥G∥MK=|a|supG∈MK|〈F,G〉|∥G∥MK=|a|∥F∥MK.
GivenF1,F2∈MK, we have
(39)∥F1+F2∥MK=supG∈MK|〈F1+F2,G〉|∥G∥MK≤supG∈MK|〈F1,G〉|∥G∥MK+supG∈MK|〈F2,G〉|∥G∥MK=∥F1∥MK+∥F2∥MK.
The proof is complete.Remark 5.
Note that∥F∥MK≤|||F|||MK. In fact, for any σ∈𝕄,F∈MK, and G∈MK, we have
(40)|〈F,G〉|=|∫𝔻‍F(z)G(z)¯(1-|z|2)dA(z)|≤(∫D‍|F(z)|2(1-|z|2)2ωσ,K(z)dA(z))1/2≤≤×(∫D‍|G(z)|2ωσ,K(z)dA(z))1/2≤(∫D‍|F(z)|2(1-|z|2)2ωσ,K(z)dA(z))1/2≤×∥G∥MK.
This shows that |〈F,G〉|≤|||F|||MK∥G∥MK. Hence, ∥F∥MK≤|||f|||MK.Theorem 6.
LetK satisfy condition (4). If MK is equipped with the norm
(41)∥F∥MK=supG∈MK|〈F,G〉|∥G∥MK,
then (MK)*≅MK under the pairing
(42)〈F,G〉=∫𝔻‍F(z)G(z)¯(1-|z|2)dA(z).Proof.
For any givenG∈MK, it is easy to see that |G(z)|2dA(z) is a K-Carleson measure. By Theorem 2, for any σ∈𝕄,
(43)∫𝔻‍|G(z)|2ωσ,K(z)dA(z)<∞.
By Lemma 4, we have
(44)|〈F,G〉|≤∥F∥MK∥G∥MK.
This shows that L(F)=〈F,G〉 is a bounded functional on MK for F∈MK. We have ∥L∥=∥G∥MK by the elementary knowledge of functional analysis, where ∥·∥ is norm of L. This gives G∈(MK)*.
Conversely, letL be a bounded linear functional on MK. For any given F∈MK, we have
(45)|L(F)|≤∥L∥∥F∥MK≤∥L∥·|||F|||MK=∥L∥infσ∈M∫𝔻‍|F(z)|2∣(1-|z|2)2ωσ,K(z)dA(z),
where ∥·∥ is norm of L. So for any fixed σ∈𝕄, we have
(46)|L(F)|≤∥L∥·∥F∥L2(σ,K,2),
where the space L2(σ,K,2) consists of all Lebesgue measure functions F on 𝔻 such that
(47)∥F∥L2(σ,K,2)=(∫𝔻‍|F(z)|2(1-|z|2)2ωσ,K(z)dA(z))1/2<∞.
Hence, L can be extended to Lσ as a bounded linear functional on L2(σ,K,2) such that L(F)=Lσ(F) for any F∈MK and ∥Lσ∥≤∥L∥. By the Hölder inequality, we obtain that the dual of L2(σ,K,2) is L2(ωσ,K(z)dA(z)) under the pair
(48)〈F,G〉=∫𝔻‍F(z)G(z)¯(1-|z|2)dA(z),
where the space L2(ωσ,K(z)dA(z)) consists of all Lebesgue measurable functions F on 𝔻 such that
(49)∫𝔻‍|F(z)|2ωσ,K(z)dA(z)<∞.
Then there exists a Gσ∈L2(ωσ,K(z)dA(z)) such that
(50)Lσ(h)=〈h,Gσ〉,h∈L2(σ,K,2).
Note that the functionGσ is independent of σ. In fact, for any given τ∈𝕄 which is different from σ, we have
(51)Lτ(h)=〈h,Gτ〉,h∈L2(τ,K,2).
Given any z0∈𝔻, consider the Bergman disk B(z0,r)={z∈𝔻,|φz0(z)|<r}. Define
(52)h(z)=eiarg{Gσ(z)-Gτ(z)}χB(z0,r)(z),
to be the test function. It is easy to see that h∈MK⊂L2(σ,K,2)∩L2(τ,K,2). Then we have
(53)〈h,Gσ〉=Lσ(h)=Lτ(h)=〈h,Gτ〉.
The above equalities show that
(54)0=〈h,Gσ-Gτ〉=∫B(z0,r)‍h(z)(Gσ(z)-Gτ(z))¯(1-|z|2)dA(z)=∫B(z0,r)‍|Gσ(z)-Gτ(z)|(1-|z|2)dA(z).
Hence, Gσ=Gτa.e. on B(z0,r) for any given z0∈𝔻. This implies that Gσ=Gτa.e. on 𝔻. We now have a G∈L2(ωσ,K(z)dA(z)) so that, for any σ∈𝕄,
(55)L(F)=〈F,G〉,F∈L2(σ,K,2),(56)∫𝔻‍|G(z)|2ωσ,K(z)dA(z)=∥Lσ∥≤∥L∥<∞.
Theorem 2 shows that |G(z)|2dA(z) is a K-Carleson measure. Hence, G∈MK.Definition 7.
LetK be a right-continuous and nondecreasing function. Let 𝕄 be the set of all nonnegative measure σ on 𝔻 with the normalized condition σ(𝔻)=1. Denote by 𝒞K the set of all analytic functions f∈H(𝔻) such that
(57)|||f|||CK=infσ∈𝕄∫𝔻‍|f′(z)|2(1-|z|2)2ωσ,K(z)dA(z)<∞,
where ωσ,K is defined as in (22).Remark 8.
In fact,f∈CK if and only if f′∈L2(σ,K,2) for any σ∈𝕄. Obviously, we have |||f|||𝒞K=infσ∈𝕄∥f′∥L2(σ,K,2). See (47) about the definition of ∥·∥L2(σ,K,2).We need the following result to proof the main theorem (cf. [10]). Let ψ∈H(𝔻). Define an operator on H(𝔻) as
(58)Tψ(z)=∫𝔻‍|ψ(w)|(1-|w|2)b-1|1-w¯z|b+1dA(w),b≥1.Lemma 9.
SupposeK satisfies conditions (4) and (5). If dμ(z)=|ψ(z)|2dA(z) is a K-Carleson measure, then |Tψ(z)|2dA(z) is a K-Carleson measure.Theorem 10.
LetK satisfy conditions (4) and (5). The dual of 𝒞K is QK under the pairing
(59)〈f,g〉C=∫𝔻‍f′(z)g′(z)¯(1-|z|2)dA(z).Proof.
Chooseg∈QK. Then |g′(z)|2dA(z) is a K-Carleson measure. Theorem 2 gives that, for any σ∈𝕄,
(60)∫𝔻‍|g′(z)|2ωσ,K(z)dA(z)<∞.
By the Hölder inequality, we have
(61)|〈f,g〉C|=|∫𝔻‍f′(z)g′(z)¯(1-|z|2)dA(z)|≤(∫𝔻‍|f′(z)|2(1-|z|2)2ωσ,K(z)dA(z))1/2≤×(∫𝔻‍|g′(z)|2ωσ,K(z)dA(z))1/2.
This and Theorem 2 give that
(62)|〈f,g〉C|≲infσ∈M(∫𝔻‍|f′(z)|2(1-|z|2)2ωσ,K(z)dA(z))1/2∥g′∥MK.
Therefore, g∈(𝒞K)*.
Conversely, letL be a bounded linear function on 𝒞K. Since f∈𝒞K⇔f′∈H(𝔻)∩MK, L can be viewed as a bounded linear functional L~ on MK; that is, L(f)=L~(f′). By Theorem 6, there exists a G∈MK such that L~(F)=〈F,G〉 for any F∈MK.
Consider the Bergman projectionP from L2(𝔻) to the Bergman space A2:
(63)P(G)(z)=∫𝔻‍G(w)(1-w¯z)2dA(w),G∈L2(𝔻).
It is easy to see that
(64)|P(G)(z)|≤∫𝔻‍|G(w)||1-w¯z|2dA(w).
Hence,|P(G)(z)|2dA(z) is a K-Carleson measure by Lemma 9. This shows that P(G) is analytic and in MK. Let g be the function satisfying g′=P(G). Then g∈QK.
Forf∈𝒞K, we have f′∈H(𝔻)∩MK. The Bergman projection P is self-adjoint. Hence,
(65)L(f)=L~(f′)=〈f′,G〉=〈f′,P(G)〉=〈f′,g′〉=〈f,g〉C.
We obtain
(66)L(f)=∫𝔻‍f′(z)g′(z)¯(1-|z|2)dA(z),f∈CK.
We complete the proof of the predual Theorem 10.



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