# On Characterizations of Weighted Harmonic Bloch Mappings and Its Carleson Measure Criteria

Authors: Munirah Aljuaid; M. A. Bakhit
Journal: Journal of Function Spaces
 (2023)

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## Abstract

Forα>0, several characterizations of the α-Bloch spaces of harmonic mappings are given. We also obtain several similar characterizations for the closed separable subspace. As an application, we give relations between BHα and Carleson’s measure.


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## Body

## 1. Introduction

LetΩ denote a simply connected region in the complex plane ℂ; a harmonic mapping is a complex-valued function h defined on Ω such that the Laplace equation satisfied
(1)Δh≔4hww¯≡0,where hww¯ is the complex second partial derivative of the harmonic mapping h.It is known that in the literature, a harmonic mappingh can be written in the form f+g¯, where f and g are analytic functions. This form is unique if we fix w0 such that gw0=0.LetD=w∈ℂ:w<1 be the well-known open unit disk in ℂ and HolD and HarD denote the class of analytic functions and harmonic mappings on D, respectively.In the last few decades, the Banach spaces of analytic functions onD have been gaining a great deal of attention, but for the harmonic extensions of analytic spaces, it is still limited. Besides [1] by F. Colonna, papers such as [2] for the study of the operator theory on some spaces of harmonic mappings, [3] for characterizations of Bloch-type spaces of harmonic mappings, [4] for composition operators on some Banach spaces of harmonic mappings, [5] for the study of harmonic Bloch and Besov spaces, [6] for the study harmonic Zygmund spaces, [7] for the study of harmonic ν-Bloch mappings and [8] for the study of harmonic Lipschitz-type spaces. For α>0, α-Bloch space for harmonic mapping is defined such that
(2)BHα=h∈HarD:Bhα<∞,where
(3)Bhα≔supw∈D1−w2αhww+hw¯w.The mappingh↦hBHα≔h0+Bhα defines a norm which yields a Banach space structure on BHα. This space is an extension to harmonic mappings of the classical α-Bloch space Bα introduced by Zhu in [9], see also [10]. We recall that f∈HolD belongs to Bα if and only if
(4)Bfα≔supw∈D1−w2αf′w<∞,with norm fBα=f0+Bfα. Thus, representing h∈HarD as f+g¯ with f,g∈HolD and g0=0, we see that hw=f′ and hw¯=g′¯. Therefore,
(5)Bhα=supw∈D1−w2αf′w+g′w,12Bfα+Bgα≤maxBfα,Bgα≤Bhα≤Bfα+Bgα.Consequently,h∈BHα if and only if the functions f,g∈HolD such that h=f+g¯ with g0=0 are in the classical α-Bloch space. When α=1, the space Bα is the (analytic) Bloch space B and the corresponding harmonic extension denoted by BH. The elements of BH were first introduced in [1].The little harmonicα-Bloch space BH,0α is defined such that
(6)BH,0α=h∈BHα:limw⟶11−w2αhww+hw¯w=0.BH,0α is a separable closed subspace of BHα (see [2]); for more information about BHα and BH,0α, see [2, 3, 11] and [1].Forb∈D, the conformal automorphism is given by
(7)φbw=b−w1−b¯w,forallw∈D,and Green’s function with logarithmic singularity at the fixed point b is defined by
(8)gw,b=log1φbw.Forb∈D and 0<δ<1, the pseudohyperbolic disk Db,δ with the pseudohyperbolic center b and pseudohyperbolic radius δ is given by
(9)Db,δ=w∈D:φbw<δ.The pseudohyperbolic diskDb,δ is a Euclidean disk with Euclidean center 1−b2δ/1−δ2b2 and Euclidean radius 1−δ2b/1−δ2b2 (see [12]). Now, we let λ denote the normalized Lebesgue area measure on D, since Db,δ⊂D is a Lebesgue measureable set; then, the Euclidean area of Db,δ is given by
(10)Db,δ=πb21−δ221−δ2b22.Thus, by directed computation, we have the following fact:Fact 1.
Letδ∈0,1; then, for all w∈Db,δ,
(11)1−w2≈Db,δ.
For anyw,b∈D, the hyperbolic distance between w and b is given by
(12)ϱb,w=12log1+φbw1−φbw.
Meanwhile, forR∈0,∞, the hyperbolic disk is given by
(13)Δb,R=w∈D:ϱb,w<R.
Throughout this paper, we say that two quantitiesQ1h and Q2h, depending on the harmonic mappings h, are equivalent denoted by Q1h≈Q2h, if there exists a constant C>0 such that
(14)1CQ2h≤Q1h≤CQ2h.
Moreover, ifQ1h≈Q2h, we have Q2h<∞⇔Q1h<∞.In this work, we expand the study carried out in [13, 14] and [15] for harmonic mappings.


## 2. Some Integral Criteria for Harmonicα-Bloch Mappings

The following lemma needed in the prove of the main theorem of this section (see Lemma 3 in [15]).Lemma 1.
Assume thatq,α∈0,∞ and q−α>−1; then,
(15)Jq,α=∬Dlog1wqdλw1−w2α<∞,where λ is the normalized Lebesgue area measure on D.The following theorem is the main theorem of this section.Theorem 2.
Assume thatα∈0,∞,δ∈0,1 and let p,q∈1,∞. Then for h∈HarD, the following quantities are equivalent:
(A)
hBHαp(B)
supb∈D1/Db,δ1−αp/2∬Db,δhww+hw¯wpdλw(C)
supb∈D∬Db,δhww+hw¯wp1−w2αp−2dλw(D)
supb∈D∬Dhww+hw¯wp1−w2αp−21−φbw2qdλw(E)
supb∈D∬Dhww+hw¯wpgqw,b1−w2αp−2dλwProof.
LetF∈HolD. Then, Fp is a subharmonic function. Thus,
(16)F0p≤1πδ2∬D0,δFwpdλw.
Now, assume thatF=hw∘φb; then,
(17)hwbp≤1πδ2∬D0,δhwφbwpdλw=1πδ2∬Db,δhwwpφb′w2dλw.
Also, suppose thatF=hw¯∘φb; then, we obtain
(18)hw¯bp≤1πδ2∬Db,δhw¯wpφb′w2dλw.
From [16], we know that
(19)φb′ζ≤41−b2b,ζ∈D.
Also, from [2], we know that hwζp+hw¯ζp≤hwζ+hw¯ζp.
Using the inequality,(20)hwζ+hw¯ζp≤2phwζp+hw¯ζp.
Therefore,(21)hwb+hw¯bp≤2phwbp+hw¯bp≤162pπδ21−b22∬Db,δhwwp+hw¯wpdλw.
Thus, we have(22)supb∈D1−b2αphwb+hw¯bp≤supb∈D162pπδ21−b22−αp∬Db,δhwwp+hw¯wpdλw≤supb∈DMδ,p,αDb,δ1−αp/2∬Db,δhww+hw¯wpdλw≲supb∈D1Db,δ1−αp/2∬Db,δhww+hw¯wpdλw,where Mδ,p,α=162p/πδαp1−δ22−αp is a constant. Therefore, A≲B.
The equivalenceB≈C follows directly from Fact 1, for all w∈Db,δ and δ∈0,1.
C⇒D. Since 1−δ2≤1−φbw2, so for every w∈Db,δ, we obtain
(23)supb∈D∬Db,δhww+hw¯wp1−w2αp−2dλw≤supb∈D11−δ2q∬Dhww+hw¯wp1−w2αp−21−φbw2qdλw≲supb∈D∬Dhww+hw¯wp1−w2αp−21−φbw2qdλw.
Hence,C≲D.
It is clear that1−φbw2≲gw,b for all w,b∈D; thus, we can infer that D≲E.
Now, making a change of variablesw=φbw in quantity E, we have
(24)∬Dhww+hw¯wpgqw,b1−w2αp−2dλw=∬Dhwφbw+hw¯φbwplog1wq1−φbw2αp−2φb′w2dλw=∬Dhwφbw+hw¯φbwp1−φbw2αplog1wqdλw1−w22≤hBHαp∬Dlog1wqdλw1−w22≲hBHαp.
From Lemma1, we know that
(25)Jq,2=∬Dlog1wqdλw1−w22<∞.
Thus, we deduce thatE≲A.
SupposeD1/4=w:w<1/4, for w∈D1/4; we have
(26)φb′w2=1−b21−b¯w4≤11−w4≤434,11−w2≤1615.
Therefore,(27)JD1/4=∬D1/4hww+hw¯wplog1wαpφb′w2dλw≤hBHαp∬D1/4log1wαpφb′w21−w2αpdλw≤4341615αpJαp,0hBHαp.
SinceJαp,0 is finite by Lemma 1, we have
(28)JD1/4≲hBHαp.
Next, sincelog1/s≤1/δ1−s2 for any 0<δ≤s≤1, so
(29)log1w≤41−w.
Then, forw∈D\D1/4, we obtain
(30)JD\D1/4=∬D\D1/4hww+hw¯wplog1wαpφb′w2dλw≤4αp∬D\D1/4hww+hw¯wp1−w2αpφb′w2dλw≤π4αphBHαp.
Thus,(31)JD\D1/4≲hBHαp.
Combining (28) and (31), we have F≲A. This completes the proof.Remark 3.
The above characterizations of harmonicα-Bloch functions is an extension of Theorem 1 proved by Zhao in [15]. We extend the known characterizations of α-Bloch space for analytic function to the harmonic setting using the conditions of the analytic functions f and g and their subharmonicity on the unit disk. Therefore, we used the proof technique as in the proof of Theorem 1 in [15].


## 3. Some Integral Criteria for Little Harmonicα-Bloch Mappings

We will give some concerned integral-type criteria for the little harmonicα-Bloch space BH,0α and the little harmonic Besov space.Theorem 4.
Suppose thatα∈0,∞,δ∈0,1 and let p,q∈1,∞. For any harmonic function h∈HarD, the following statements are equivalent:
(A˜)
h∈BH,0α(B˜)
limb⟶1−1/Db,δ1−αp/2∬Db,δhww+hw¯wpdλw=0(C˜)
limb⟶1−∬Db,δhww+hw¯wp1−w2αp−2dλw=0(D˜)
limb⟶1−∬Dhww+hw¯wp1−w2αp−21−φbw2qdλw=0(E˜)
limb⟶1−∬Dhww+hw¯wpgqw,b1−w2αp−2dλw=0Proof.
Suppose thath∈BH,0α; then,
(32)limw⟶1hww+hw¯w1−w2α=0.
By making the change of variablesζ=φbw, we have
(33)IDb=∬Dhww+hw¯wpgqw,b1−w2αp−2dλw=∬Dhwφbw+hw¯φbwp1−φbw2αplog1wqdλw1−w22.
Letδ∈0,1 such that π1−δ2q−1/δqq−1=ε, for ε>0. Set Dδ=w:w<δ; for any w∈D\Dδ, we have log1/w≤1−w2/δ. Then,
(34)ID\Dδb=∬D\Dδhwφbw+hw¯φbwp1−φbw2αplog1wqdλw1−w22≤hBHαp∬D\Dδlog1wqdλw1−w22≤1δqhBHαp∬D\Dδdλw1−w22−q=π1−δ2q−1δqq−1hBHαp.
This means that(35)ID\Dδb≤εhBHαp.
Sinceh∈BH,0α, then limb⟶1hwφbw+hw¯φbw1−φbw2α=0, and the convergence is uniform for any w∈Dδ. So for the given ε>0, there is δ0∈0,1 such that
(36)hwφbw+hw¯φbwp1−φbw2αp<ε,∀b∈D\Dδ0.
Thus, we obtain(37)ID\Dδ0b=∬D\Dδ0hwφbw+hw¯φbwp1−φbw2αplog1wqdλw1−w22≤ε∬D\Dδ0log1wqdλw1−w22.
Then, we have(38)ID\Dδ0b≤εJq,2.
By Lemma1, for all q∈1,∞, we know that Jq,2<∞.
Now combining (35) and (38), we have
(39)IDb≤εAh,p,α,∀b∈D\Dδ0,where Ah,p,α is a constant depending on p,α and the harmonic function h. That is, limb⟶1Ib=0. Thus, we deduce that A~⇒E~.
Since1−φbw2≲gw,b for all w,b∈D, we have E~⇒D~.
Next,D~⇒C~ follows immediately from the inequality
(40)∬Db,δhww+hw¯wp1−w2αp−2dλw≲∬Dhww+hw¯wp1−w2αp−21−φbw2qdλw.
Forw∈Db,δ and δ∈0,1, we have that 1−w2≈Db,δ; then, we get C~⇔B~.
By the inequality(41)1−b2αphwb+hw¯bp≲1Db,δ1−p/2∬Db,δhww+hw¯wpdλw,we obtain B~⇒A~.
Leth∈BH,0α; that is, limw⟶1−hww+hw¯w1−w2α=0. Then, for any ε>0, there is δ∈1/4,1 such that hww+hw¯wp1−w2αp<ε, for every w∈D\Dδ.
Now, we set(42)J~Db=∬Dhww+hw¯wplog1wαpφb′w2dλw.
For eachw∈D\Dδ, we know log1/w≤41−w2. Thus,
(43)J~D\Dδ=∬D\Dδhww+hw¯wplog1wαpφb′w2dλw≤22αpεπ.
Sincelimb⟶1−φb′w2=0 for every w∈D, and the uniformly convergence for w∈Dδ. Then, for any ε>0 and δ∈1/4,1, there is δ0∈0,1 such that φb′w2<ε, for every b∈D\Dδ0.
Hence,(44)J~D\Dδ0=∬D\Dδ0hww+hw¯wplog1wαpφb′wdλw≤hBHαp∬D\Dδ0log1wαpφb′w1−w2αpdλw≤εJαp,αphBHαp.
Again by Lemma1, for all p∈1,∞ and α∈0,∞ we know that Jαp,αp<∞. This means that
(45)J~D\Dδ0≲hBHαp.
By combining equations (43) and (45), we have
(46)J~D≤εAh,p,α,∀b∈D\Dδ0,where Ah,p,α is a constant depending on p,α and the harmonic function h. That is, limb⟶1−J~Db=0. Thus, we deduce that A~⇒F~. That ends the proof of Theorem 4.Remark 5.
Whenh is analytic function, BH,0α is the little α-Bloch space B0α, and Theorem 4 is proved by Zhao in [15].


## 4. Carleson’s Measures and Harmonicα-Bloch Mappings

Letμ be a positive measure on D. For a subarc J⊆∂D, we let SJ be the Carleson box based on J; that is,
(47)SJ≔ζ∈D:1−J≤ζ<1,ζ/ζ∈J.ForJ=∂D, we let SJ=D.Lets>0; then, a positive Borel measure μ on D is called a s-Carleson measure if
(48)μs=supJ⊆∂DμSJJs<∞.Note thats=1 gives the classical Carleson measure. We say that μ is a vanishing s-Carleson measure if
(49)limJ⟶1μSJJs=0.As is well known, the Berezin transform of a positive Borel measureμ on D is bounded if and only if μ is a Carleson measure (see, for example, [13, 14]). Then, for any δ∈0,1, we say dμ is a Carleson measure if
(50)supb∈DμDb,δλDb,δ<∞,where λ is the normalized Lebesgue area measure on D. Moreover, we say dμ is a compact Carleson measure if
(51)limb⟶1μDb,δλDb,δ=0.For allα∈0,∞, a positive measure μ on D is a bounded α-Carleson measure if and only if
(52)supb∈D∬D1−b21−b¯w2αdμζ<∞.Moreover,μ is a compact α-Carleson measure if and only if
(53)limb⟶1∬D1−b21−b¯w2αdμζ=0.For allα∈−1,∞,p∈0,∞, we denote by AHp,αD the weighted harmonic Bergman space, where AHp,αD is the set of all h∈HarD for which
(54)hAHp,αp=∬Dhwpdλαw<∞,where dλαw=1−w2αdλw.Theorem 6.
Assume thath∈HarD, and let α∈0,∞. Then, the following are equivalent:
(a)
h∈BHα(b)
supb,w∈Dhw−hb/ρb,w<∞(c)
supb∈D∬DexpChφbw−hbdλαw<∞,where C is a positive constantProof.
Firstly,a⇒b. Suppose that h∈BHα, and let Fb∈HarD defined as
(55)Fbζ=hφbζ−hb,∀b,ζ∈D.
Then,Fb0=0 and FbBHα≤hBHα<∞. By Theorem 3.6 in [2], we obtain
(56)Fbζ≤12hBHαlog1+ζ1−ζ.
Now, settingw=φbζ, we have
(57)hw−hb≤12hBHαlog1+φbw1−φbw,which means that
(58)supb,w∈Dhw−hbρb,w≤hBHα<∞.That is, b holds.
Secondly,b⇒c. Suppose that
(59)0<h∗=supb,w∈Dhw−hbρb,w<∞.
Then for anyr≥0,
(60)w:w∈D,Fbw>r⊂w:w>exp2r/h∗−1exp2r/h∗+1,when 0<C<2α+2/h∗, then
(61)∬DexpCFbwdλαw=C∫0∞expCr·λαw:w∈D,Fbw>rdr≤4πCα+1∫0∞expCrexp−2rα+1h∗dr=4πCα+1h∗2α+1−Ch∗=4πCh∗2α+12−Cα+1h∗.
Finally,c⇒a. For some C>0, let
(62)h∗∗=supb∈D∬DexpChφbw−hbdλαw=supb∈D∬DexpCFbwdλαw<∞.
Then,(63)FbwAHp,αp≤h∗∗C<∞.
For anyh∈HarD, since f,g∈HolD such that h=f+g¯ with g0=0, h has the Taylor series
(64)∑m=0∞amwm+∑m=1∞cmw¯m.
Hence, by a simple calculation for a Taylor series ofFbw, which converges uniformly on Dδ=w:w<δ,δ∈0,1, we have
(65)a1=Fbw0=2α+2α+11−δ2α+1+11−δ2α+1∬DδFbww¯dλαw.
Now, we letδ⟶1; then,
(66)Fbw0+Fbw¯0≤α2+3α+2∬DFbwdλαw.
Similarly, forδ⟶1, we see that
(67)Fbw¯0≤α2+3α+2∬DFbwdλαw,which means that
(68)1−w2αhww+hw¯w≤2Cα2+3α+2h∗∗.
So, we haveh∈BHα.Theorem 7.
Assume thath∈HarD, and let α∈0,∞ and p∈0,∞. Then, the following are equivalent:
(a˜)
h∈BHα(b˜)
dμ1=hww+hw¯wplog1/wαpdλw is p-Carleson measure(c˜)
dμ2=hww+hw¯wp1−w2αpdλw is p-Carleson measureProof.
First of all,a~⇒b~. Suppose that h∈BHα, and let the integral below:
(69)I=∬D1−b21−b¯w2pdμ1w=∬w>1/4+∬w≤1/41−b21−b¯w2pdμ1w=I1+I2.
Sincelog1/w≲1−w2 when w>1/4,
(70)I1=∬w>1/4hww+hw¯wp1−b21−b¯w2plog1wαpdλw≲∬w>1/4hww+hw¯wp1−b21−w2α1−b¯w2pdλw≲hBHαp∬D1−b21−b¯w2pdλw≲hBHαp.
At the same time,(71)I2=∬w≤1/4hww+hw¯wp1−b21−b¯w2plog1wαpdλw≤1615αphBHαp∬w≤1/41−b21−b¯w2plog1wαpdλw≤1615αp43phBHαp∬w≤1/4log1wαpdλwhBHαp.
Consequently,IhBHαp. So, we see that b~ holds.
Next,b~⇒c~. This is unmistakable, since 1−w2≤2log1/w.
Finally,c~⇒a~. Assuming that dμ2 is p-Carleson measure, then
(72)∞>I=∬Dhww+hw¯wp1−b21−b¯w2p1−w2αpdλw≥∬Dhww+hw¯wp1−w2αpdλw.
Moreover,(73)hwb+hw¯bp≤3π∬Dhww+hw¯wp1−b21−b¯w2pdλw.
Hence,(74)hwb+hw¯bp1−w2αp≤3π∬Dhww+hw¯wp1−w2αp1−b21−b¯w2pdλw≤I<∞.
So, we haveh∈BHα.Now, we give new characteristics of the little harmonicα-Bloch space BH,0α.Theorem 8.
Assume thath∈HarD, and let α∈0,∞. Then, the following are equivalent:
(i)
h∈BH,0α(ii)
limb⟶1supw∈Δb,Rhw−hb/ρb,w=0(iii)
limb⟶1∬DexpChφbw−hbdλαw=π/α+1,where C is a positive constantProof.
Firstly,i⇔ii. Suppose that h∈BH,0α, as in the proof of Theorem 6; let Fb∈HarD defined as
(75)Fbζ=hφbζ−hb,∀b,ζ∈D.
Then,Fb0=0 and FbBHα≤hBHα<∞. By Theorem 3.6 in [2], we obtain
(76)Fbζ≤12hBHαlog1+ζ1−ζ.
ForR∈0,∞ and δ=eR−1/eR+1, we obtain
(77)hw−hb≤12ρ0,φbwsupz∈D0,δ1−z2αFbzz+Fbz¯z.
Meanwhile, sincelimb⟶1φbz=1 for any z∈D0,δ and h∈BH,0α, we have
(78)limb⟶1supz∈D0,δ1−φbζ2αhζζ+hζ¯ζ=0.
By combining equations (77) and (78), we deduce (ii).
The other direction comes easily from inequality(79)1−w2αhww+hw¯w≤supw∈Δb,Rhw−hbρb,w.
Secondly,i⇔iii. For any f∈HolD, since B0α is the closure in Bα of the polynomials, there exist a polynomial p such that (see [17])
(80)f−pBα<2α+1C,C∈0,∞.
Also sinceBH,0α is the closure in BHα of the polynomials, there exist a polynomial P=p1+p2¯ where p1,p2∈Bα, such that
(81)h−PBHαp<2α+1C.
Hence,(82)supb∈D∬DexpCh−Pφbw−h−Pbdλαw<∞.
Furthermore,(83)M=supb∈D∬DexpChφbw−hbdλαw<∞.
Now, setδ=1−ε, where ε∈0,1 in (77); we get δ1∈0,1 so that for D\D0,δ1¯,
(84)πα+1≤∬DexpC2hφbw−hbdλαw≤∬D\D0,δexpC2hφbw−hbdλαw+∬D0,δexpC2hφbw−hbdλαw≤2πMε+π1−εα+12α+12εCε/2.
So, we see that (iii) holds.
On the other hand, suppose that (iii) is true. For anyC∈0,∞, we see that
(85)1−w2αhww+hw¯w≤α2+3α+2C∬DexpC2hφbw−hbdλαw.
So, we haveh∈BH,0α.Next, we give relation betweenBH,0α and the compact Carleson measure.Theorem 9.
Assume thath∈HarD, and let α∈0,∞ and p∈0,∞. Then, the following are equivalent:
(i)
h∈BH,0α(ii)
dμ1=hww+hw¯wplog1/wαpdλw is a compact p-Carleson measure(iii)
dμ2=hww+hw¯wp1−w2αpdλw is a compact p-Carleson measureProof.
First of all,i⇒ii. Suppose that h∈BH,0α, and let ε∈0,1. Then, there is δ∈1/4,1 such that
(86)1−w2αhww+hw¯w≤ε,for w∈D\D0,δ, and
(87)∬D\D0,δhww+hw¯wp1−b21−b¯w2plog1wαpdλw≤Cε.
Also,(88)∬D0,δhww+hw¯wp1−b21−b¯w2plog1wαpdλw≤C1−b2phBHαp.
By combining equations (87) and (88), we see that (ii) holds.
Next,ii⇒iii. This is very clear since 1−w2≤2log1/w.
After that,iii⇒i. Assuming that dμ2 is a compact p-Carleson measure, as in the proof of Theorem 7, we have
(89)hwb+hw¯bp1−w2αp≲3π∬Dhww+hw¯wp1−w2αp1−b21−b¯w2pdλw<∞.
So, we haveh∈BH,0α.



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