# Composition Operators Mapping Logarithmic Bloch Functions into Hardy Space

Authors: E. G. Kwon
Journal: Journal of Function Spaces
 (2018)

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## Abstract

Characterizing the hyperbolic Hardy classes, severalg-functions of hyperbolic type are introduced. Using this, necessary and sufficient conditions on the inducing self-maps are established for the boundedness of the composition operators from logarithmic Bloch spaces into Hardy spaces.


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## Body

## 1. Introduction

This paper is to characterize the class of holomorphic self-mapf of the open complex unit disc D for which the composition operator induced by f maps logarithmic Bloch space boundedly into Hardy space Hp. Main result of this paper is Theorem 2 whose primitive form is as follows.Theorem 1.
Iff is a holomorphic self-map of D and if 0<p<∞, -1/2<α<∞, then the following are equivalent:(i)
h∘f∈H2p/(1+2α) for all holomorphic h on D satisfying (1)supz∈Dh′z1-zloge1-z-α<∞;(ii)
sup0≤r<1∫02πlog⁡(1/(1-freiθ2))pdθ<∞;(iii)
∫02π∫01f′reiθ/(1-freiθ2)2log⁡(1/(1-freiθ2))2α(1-r)drp/(1+2α)dθ<∞.Whenα=0 the above equivalence is known. We refer to [1–3], wherein the results are investigated in D and in the ball of Cn, respectively.Also, the casep<1+2α was considered in [4], so the main case under consideration is p≥1+2α. In the latter case, a different approach, based on duality, is used.Note that Theorem1 not only characterizes the composition operators mapping logarithmic Bloch functions into the Hardy space but also introduces a kind of g-function. The result will be stated precisely and more extensively in Section 3.The restriction of the range ofα is essential. If α<-1/2, then it reduces to a trivial result. If α=-1/2, then it corresponds to another space instead of Hardy space. These will be treated separately in the last section.


## 2. Preliminaries

### 2.1. Hardy Space and Hyperbolic Hardy Class

LetD be the unit disc of the complex plane. For 0<p<∞ and for f subharmonic in D, we denote (2)fp≔limr→1∫02πfreiθpdθ2π1/p.Then the right side limit is monotone increasing. And by definition, the Hardy space Hp=Hp(D) consists of holomorphic f in D for which fp<∞, while the Yamashita hyperbolic Hardy class ϱHp consists of holomorphic self-map f of D for which ϱfp<∞. Here ϱ(z) denotes the hyperbolic distance of z and 0 in D, namely, (3)ϱz=12log⁡1+z1-z.Though ϱHp is not a linear space, it has, as hyperbolic counterparts, many properties analogous to those of Hp.For holomorphic self-mapsf of D, we let (4)λf=log⁡11-f2,f#=f′1-f2 following the notation of Yamashita and let (5)Mλfθ=sup⁡λfreiθ:0≤r<1. Then they have the following basic properties:(6)ϱfp,λfp,f#paresubharmonicfunctionsforanyp>0(7)ϱfandf#areautomorphisminvariantinthesensethatϱf,0=ϱφa∘f,aandφa∘f#=f#for any a∈D, where (8)φaz=a-z1-a¯z,z∈D.(9)λfp<∞⟺ϱfp<∞⟺f∈ϱHp(10)MλfLp≲λfpiff∈ϱHp(11)Δλfp=4pp-1f2+λff#2λfp-2≈f#2λfp-1Here and throughout, Lp means Lp([0,2π]), Δ denotes the Laplacian: Δ=4(∂2/∂z∂z¯), ψ≲ϕ means that ψ is bounded by a positive uniform constant times ϕ, and ψ≈ϕ means that either both sides are zero or the quotient ψ/ϕ lies between two positive uniform constants. We refer to [1, 5–7] for (6)~(11). For a general theory of Hp and ϱHp, we refer to [8–10] and [7, 11, 12], respectively.


### 2.2. Logarithmic Bloch Space

For-∞<α<∞, Blog⁡α denotes the weighted Bloch space consisting of holomorphic functions h in D satisfying (12)hBlog⁡α≔supz∈Dh′z1-z2loge1-z2-α<∞.Blog⁡α is a Banach space equipped with the norm h0+hBlog⁡α. Though the invariance “hBlog⁡α=h∘φaBlogα for all a∈D and h∈Blog⁡α” is satisfied only when α=0, it is not difficult to see that we still have(13)h∈Blogα⟺h∘φa∈Blogα once a∈D is fixed.


### 2.3. Bloch Pullback Problem

The composition operator induced by a holomorphic self-mapf of D will be denoted as usual by Cf, that is, Cfh=h∘f. For two function spaces X and Y, we denote C(X,Y) by the set of self-maps f for which Cf(X)⊂Y. In this notation, (i)⇔(ii) of Theorem 1 can be expressed as (14)CBlogα,Hp=ϱHp/21+2α,-12<α<∞.The problem of characterizing C(Blogα,Hp) is a kind of Bloch pullback problem. Bloch pullback problem was initiated by P. Ahern and W. Rudin. See [13–20] for Bloch-BMOA pullbacks and [1–3, 17, 21] for Bloch-Hardy pullbacks.


### 2.4. New HyperbolicG-Functions

Forh holomorphic in D, it is well known that G-function of Littlewood-Paley defined by(15)Ghθ≔∫01h′reiθ21-rdr1/2,0≤θ<2πsatisfies Gh∈Lp⇔h∈Hp and(16)GhLp≈h-h0p.See [10, 22, 23]. Analogously, for holomorphic self-map f of D, the hyperbolic version of G-function defined by(17)ϱGfθ≔∫01f#reiθ21-rdr,0≤θ<2πsatisfies ϱGf∈Lp⇔f∈ϱHp and(18)ϱGfLp≈fpprovided f(0)=0. See [17].We pay attention to the absence of the square root in the defining ofϱGf in (17) when we compare it to that of Gh in (15). The difference actually explains lots of known parallelism (see [11]) between H2 and ϱH1. Suggested by (11), we define for -1/2<α<∞(19)ϱGαfθ≔∫01f#reiθ2λfreiθ2α1-rdr1/1+2α,0≤θ<2π.Note that ϱG0f=ϱGf. Of course, main objective of introducing (19) is to establish an equivalence as (18).


## 2.1. Hardy Space and Hyperbolic Hardy Class

LetD be the unit disc of the complex plane. For 0<p<∞ and for f subharmonic in D, we denote (2)fp≔limr→1∫02πfreiθpdθ2π1/p.Then the right side limit is monotone increasing. And by definition, the Hardy space Hp=Hp(D) consists of holomorphic f in D for which fp<∞, while the Yamashita hyperbolic Hardy class ϱHp consists of holomorphic self-map f of D for which ϱfp<∞. Here ϱ(z) denotes the hyperbolic distance of z and 0 in D, namely, (3)ϱz=12log⁡1+z1-z.Though ϱHp is not a linear space, it has, as hyperbolic counterparts, many properties analogous to those of Hp.For holomorphic self-mapsf of D, we let (4)λf=log⁡11-f2,f#=f′1-f2 following the notation of Yamashita and let (5)Mλfθ=sup⁡λfreiθ:0≤r<1. Then they have the following basic properties:(6)ϱfp,λfp,f#paresubharmonicfunctionsforanyp>0(7)ϱfandf#areautomorphisminvariantinthesensethatϱf,0=ϱφa∘f,aandφa∘f#=f#for any a∈D, where (8)φaz=a-z1-a¯z,z∈D.(9)λfp<∞⟺ϱfp<∞⟺f∈ϱHp(10)MλfLp≲λfpiff∈ϱHp(11)Δλfp=4pp-1f2+λff#2λfp-2≈f#2λfp-1Here and throughout, Lp means Lp([0,2π]), Δ denotes the Laplacian: Δ=4(∂2/∂z∂z¯), ψ≲ϕ means that ψ is bounded by a positive uniform constant times ϕ, and ψ≈ϕ means that either both sides are zero or the quotient ψ/ϕ lies between two positive uniform constants. We refer to [1, 5–7] for (6)~(11). For a general theory of Hp and ϱHp, we refer to [8–10] and [7, 11, 12], respectively.


## 2.2. Logarithmic Bloch Space

For-∞<α<∞, Blog⁡α denotes the weighted Bloch space consisting of holomorphic functions h in D satisfying (12)hBlog⁡α≔supz∈Dh′z1-z2loge1-z2-α<∞.Blog⁡α is a Banach space equipped with the norm h0+hBlog⁡α. Though the invariance “hBlog⁡α=h∘φaBlogα for all a∈D and h∈Blog⁡α” is satisfied only when α=0, it is not difficult to see that we still have(13)h∈Blogα⟺h∘φa∈Blogα once a∈D is fixed.


## 2.3. Bloch Pullback Problem

The composition operator induced by a holomorphic self-mapf of D will be denoted as usual by Cf, that is, Cfh=h∘f. For two function spaces X and Y, we denote C(X,Y) by the set of self-maps f for which Cf(X)⊂Y. In this notation, (i)⇔(ii) of Theorem 1 can be expressed as (14)CBlogα,Hp=ϱHp/21+2α,-12<α<∞.The problem of characterizing C(Blogα,Hp) is a kind of Bloch pullback problem. Bloch pullback problem was initiated by P. Ahern and W. Rudin. See [13–20] for Bloch-BMOA pullbacks and [1–3, 17, 21] for Bloch-Hardy pullbacks.


## 2.4. New HyperbolicG-Functions

Forh holomorphic in D, it is well known that G-function of Littlewood-Paley defined by(15)Ghθ≔∫01h′reiθ21-rdr1/2,0≤θ<2πsatisfies Gh∈Lp⇔h∈Hp and(16)GhLp≈h-h0p.See [10, 22, 23]. Analogously, for holomorphic self-map f of D, the hyperbolic version of G-function defined by(17)ϱGfθ≔∫01f#reiθ21-rdr,0≤θ<2πsatisfies ϱGf∈Lp⇔f∈ϱHp and(18)ϱGfLp≈fpprovided f(0)=0. See [17].We pay attention to the absence of the square root in the defining ofϱGf in (17) when we compare it to that of Gh in (15). The difference actually explains lots of known parallelism (see [11]) between H2 and ϱH1. Suggested by (11), we define for -1/2<α<∞(19)ϱGαfθ≔∫01f#reiθ2λfreiθ2α1-rdr1/1+2α,0≤θ<2π.Note that ϱG0f=ϱGf. Of course, main objective of introducing (19) is to establish an equivalence as (18).


## 3. Equivalence between Norms

### 3.1. Main Result Revisited

Equipped with the notions introduced in Section2, Theorem 1 can be stated as the first part of the following.Theorem 2.
Let0<p<∞,-1/2<α<∞, and q=p/(1+2α). Let f be a holomorphic self-map of D. Then the following (i), (ii), and (iii) are mutually equivalent:(i)
f∈CBlogα,H2q(ii)
f∈ϱHp(iii)
ϱGαf∈Lp.
Moreover, if we assumef(0)=0 then(20)ϱGαfLp≈λfpand(21)Cf2q/p≈1+λfp,where Cf denotes (22)Cf≔suph∘f2q:h∈Blogα,h0+hBlogα≤1.It follows by (13) that the restriction f(0)=0 is not an essential one because Cf is bounded if and only if Cg is with g=φf(0)∘f. Note that we used the notation Cf also for the case 2q<1 when H2q are not normed spaces.


### 3.2. More onG-Function Equivalence

For (20), we in fact can prove more extensively the following: we define for -1/2<α<∞ and 0≤ϵ<∞ that (23)ϱGα,ϵfθ≔∫01f#reiθ2ϵ+λfreiθ2α1-rdr1/1+2α,0≤θ<2π.ThenTheorem 3.
Under the assumption of Theorem2 with f(0)=0,(24)ϱGα,ϵfLp≈ϵ+λfp.Immediately after this with (21) is the following.Corollary 4.
Under the assumption of Theorem2 with f(0)=0, (25)ϱGα,1fLp≈1+λfp≈Cf2q/p.To cover all of our results stated up to here, it is sufficient to prove (24) and (21). This will be done in Section 5 after stating preparatory lemmas in Section 4.


## 3.1. Main Result Revisited

Equipped with the notions introduced in Section2, Theorem 1 can be stated as the first part of the following.Theorem 2.
Let0<p<∞,-1/2<α<∞, and q=p/(1+2α). Let f be a holomorphic self-map of D. Then the following (i), (ii), and (iii) are mutually equivalent:(i)
f∈CBlogα,H2q(ii)
f∈ϱHp(iii)
ϱGαf∈Lp.
Moreover, if we assumef(0)=0 then(20)ϱGαfLp≈λfpand(21)Cf2q/p≈1+λfp,where Cf denotes (22)Cf≔suph∘f2q:h∈Blogα,h0+hBlogα≤1.It follows by (13) that the restriction f(0)=0 is not an essential one because Cf is bounded if and only if Cg is with g=φf(0)∘f. Note that we used the notation Cf also for the case 2q<1 when H2q are not normed spaces.


## 3.2. More onG-Function Equivalence

For (20), we in fact can prove more extensively the following: we define for -1/2<α<∞ and 0≤ϵ<∞ that (23)ϱGα,ϵfθ≔∫01f#reiθ2ϵ+λfreiθ2α1-rdr1/1+2α,0≤θ<2π.ThenTheorem 3.
Under the assumption of Theorem2 with f(0)=0,(24)ϱGα,ϵfLp≈ϵ+λfp.Immediately after this with (21) is the following.Corollary 4.
Under the assumption of Theorem2 with f(0)=0, (25)ϱGα,1fLp≈1+λfp≈Cf2q/p.To cover all of our results stated up to here, it is sufficient to prove (24) and (21). This will be done in Section 5 after stating preparatory lemmas in Section 4.


## 4. Preparatory Lemmas

We describe some lemmas, whose proof will be deferred to Section6, that will be used in proving our main theorem.Lemma 5.
Letf be a holomorphic self-map of D and -2≤α<∞. Then, for any positive ϵ the function (26)Fϵ≔f#2ϵ+λfαis subharmonic in D. Furthermore, for 0≤ϵ<∞, Mp(r,Fϵ) is an increasing function of r and(27)Fϵr2eiθ≤∫02πPr,θ-tFϵreitdt2π,0≤r<1,where P(r,θ) is the Poisson kernel: P(r,θ)=(1-r2)/(1-2rcos⁡θ+r2).The subharmonicity ofFϵ in Lemma 5 gives the following, where dA=dxdy.Lemma 6.
Let0<p<∞ and 0≤ϵ<∞. Then for holomorphic self-maps f of D,(28)ϵ+λfpp-ϵ+λf0p≈∫Df#z2ϵ+λfzp-1log1zdA≈∫01∫02πf#reiθ2ϵ+λfreiθp-11-rdrdθ.We need the following inequalities which are not difficult to guess. See, for example, [24] or [25] for (29) and (30), and see [26] for (31).Lemma 7.
Let0<β<∞ and -1<α<∞. Then, for 0≤r<1,(29)loge1-rβ≲1+∑k=0∞k+1β-1r2k;(30)∑k=0∞2kk+1αr2k-1≲11-rloge1-rα;(31)log⁡log⁡e1-r≲∑k=0∞k+1-1r2k.


## 5. Proof of Main Results

Letf be a holomorphic self-map of D with f(0)=0. We are sufficient to show (24) and (21). We assume f is not constant because there’s nothing left to prove when f is a constant. Let us denote for simplicity by S the boundary of D and σ the arc length measure on S normalized to be dσ(ζ)=dζ.


### 5.1. Proof of (24)

For notational clearance we prove only forϵ=0. But replacing ϱGαf and λf, respectively, by ϱGα,ϵf and ϵ+λf, it is easy to check that the proof below works for general ϵ in the same way.To showϱGαfLp≲λfp, we divide it into two cases: q≤1 and q>1.Ifq≤1, then by Hölder’s inequality, (10) and Lemma 6, it follows that (32)ϱGαfLpp=∫S∫01f#rζ2λfrζ2α+1-p+p-11-rdrqdσζ≤∫SMλfζp1-q∫01f#rζ2λfrζp-11-rdrqdσζ≲λfpp1-q∫S∫01f#rζ2λfrζp-11-rdrdσq≈λfpp.Next, supposeq>1. Set β=1+2α and let μ≔(f#)2λf2α.Note, by (11) and (27),(33)Δλfβ≈f#2λf2α=μand(34)μr2eiθ≤∫02πPr,θ-tμreitdt2π,0≤r<1.We make use of the representation(35)ϱGαfLpβ=ϱGαfβLq=suph∫ShζϱGαfζβdσζ,where the supremum is taken with respect to all nonnegative trigonometric polynomials h with hLs≤1, 1/s+1/q=1.If we setH the Poisson integral of h on S, then (34), a change of the order of the integration, and (33) give(36)∫ShζϱGαfζβdσζ=1π∫02πheiθdθ∫01μr2eiθ1-r2rdr≤1π∫02πheiθdθ∫011-r2rdr∫02πPr,θ-tμreitdt=2∫Sdσζ∫01Hrζμrζ1-r2rdr≈∫DHzμz1-z2dAz≲∫DHzΔλfβzlog1zdAzwhile straightforward calculation gives (37)ΔλfβH=4HΔλfβ+4∂λfβ∂¯H+∂¯λfβ∂H,where ∂=∂/∂z and ∂¯=∂/∂z¯. Whence by (36) we have(38)∫ShζϱGαfζβdσζ≲∫DΔλfβHzlog1zdA+4∫D∂λfβ∂¯H+∂¯λfβ∂Hzlog1zdA≔I+II.Now, using Green’s theorem with limiting process and Hölder’s inequality, we obtain(39)I≲λfβqHs≲λfpβ.On the other hand, direct differentiation gives(40)∂λfβ=∂¯λfβ=βfλf2αf#=βfλfαμ,and (41)∂H=∂¯H=12H′,where H denotes the holomorphic function in D whose real part is H with H(0)=0; whence (42)II≤4β∫DH′zfzλfαzμzlog⁡1zdA≲∫S∫01H′rζfrζλfαrζμrζ1-rdrdσζ.Since f≤λf, we have, by Schwarz inequality, (43)II≲∫SMλfζβ/2GHζϱGαfζβ/2dσζ.Applying Hölder’s inequality with triple (2q,s,2q) and applying (11) to the last integral, we arrive at (44)II≲λfpβ/2GHLsϱGαfLpβ/2.From (16), we know GHLs≲Hs, and it follows from the theorem of M. Riesz ([8, 23]) that Hs≲Hs≲1. Thus(45)II≲λfpβ/2ϱGαfLpβ/2.Gathering estimates (38), (39), and (45) up, we arrive at (46)∫ShζϱGαfζβdσζ≤I+II≲λfpβ+λfpβ/2ϱGαfLpβ/2 for all positive trigonometric polynomials h with hLs≤1. That is, (47)ϱGαfLpβ≲λfpβ+λfpβ/2ϱGαfLpβ/2. Therefore we obtain via the arithmetic-geometric mean inequality that(48)ϱGαfLp≲λfp.We next showλfp≲ϱGαfLp. Set β=1+2α again. Take N>1 such that p<βN. Hölder’s inequality with paring (βN/p,βN/(βN-p)) gives (49)λfpp≈∫Df#z2λfzp-11-z2dxdy=∫Df#z2λfzp/N+p1-1/N-11-z2dxdy≤∫S∫01f#2λf2α1-rdrp/βN·∫01f#2λf2δ1-rdr1-p/βNdσ, where 2δ=p(1-1/N)(βN/(βN-p))-1. Applying Hölder’s inequality one more time to the last quantity with paring (N,N/(N-1)), (50)λfpp≲∫SϱGαfpdσ1/N∫SϱGδfpdσ1-1/N=ϱGαfLpp/NϱGδfLpp1-1/N. Therefore by applying the arithmetic-geometric mean inequality and (48), we obtain the desired inequality. This proof is completed.


### 5.2. Proof of (21)

Ifh∈Blogα, then (16), the definition of Blogα, and (24) gives (51)h∘f-h∘f02q2q≈∫S∫011-rh′f·f′2rζdrqdσζ≲hBlogα2q∫S∫011-rf#rζ21+λfrζ2αdrqdσζ=hBlogα2qϱGα,1fLpp≈hBlogα2q1+λfpp. This with simple inequality (52)h∘f2q2q-h∘f02q≲h∘f-h∘f02q2q gives (53)h∘f2q2q≲h0+hBlogα2q1+λfpp; whence this verifies Cf2q/p≲1+λfp.We next show1+λfp≲Cf2q/p. For each nondyadic t∈[0,1], let (54)htz=1+∑k=0∞γktk+1αz2k,z∈D, where γk=γk(t) is the Rademacher function (see [8, 23]) defined by (55)γkt=signsin2k+1πt. Then ht is holomorphic in D and by (30) there is a positive constant Cα such that (56)1-zht′z≤Cαlog⁡e1-zα, so that ht∈Blogα with htBlogα≤Cα. Thus, the definition of Cf gives(57)∫Sht∘frζ2qdσζ≤1+Cα2qCf2q.On the other hand, it follows by (29) that(58)1+λfpp=limr→1∫Sloge1-frζ2pdσζ≲limr→1∫S1+∑k=0∞k+12αfrζ22kqdσζ,while Khinchin’s inequality (see [23] Theorem 8.4) gives(59)1+∑k=0∞k+12αfrζ22kq≲∫01ht∘frζ2qdt.Now (57), (58), and (59) together verify 1+λfp≲Cf2q/p. This proof is completed.


## 5.1. Proof of (24)

For notational clearance we prove only forϵ=0. But replacing ϱGαf and λf, respectively, by ϱGα,ϵf and ϵ+λf, it is easy to check that the proof below works for general ϵ in the same way.To showϱGαfLp≲λfp, we divide it into two cases: q≤1 and q>1.Ifq≤1, then by Hölder’s inequality, (10) and Lemma 6, it follows that (32)ϱGαfLpp=∫S∫01f#rζ2λfrζ2α+1-p+p-11-rdrqdσζ≤∫SMλfζp1-q∫01f#rζ2λfrζp-11-rdrqdσζ≲λfpp1-q∫S∫01f#rζ2λfrζp-11-rdrdσq≈λfpp.Next, supposeq>1. Set β=1+2α and let μ≔(f#)2λf2α.Note, by (11) and (27),(33)Δλfβ≈f#2λf2α=μand(34)μr2eiθ≤∫02πPr,θ-tμreitdt2π,0≤r<1.We make use of the representation(35)ϱGαfLpβ=ϱGαfβLq=suph∫ShζϱGαfζβdσζ,where the supremum is taken with respect to all nonnegative trigonometric polynomials h with hLs≤1, 1/s+1/q=1.If we setH the Poisson integral of h on S, then (34), a change of the order of the integration, and (33) give(36)∫ShζϱGαfζβdσζ=1π∫02πheiθdθ∫01μr2eiθ1-r2rdr≤1π∫02πheiθdθ∫011-r2rdr∫02πPr,θ-tμreitdt=2∫Sdσζ∫01Hrζμrζ1-r2rdr≈∫DHzμz1-z2dAz≲∫DHzΔλfβzlog1zdAzwhile straightforward calculation gives (37)ΔλfβH=4HΔλfβ+4∂λfβ∂¯H+∂¯λfβ∂H,where ∂=∂/∂z and ∂¯=∂/∂z¯. Whence by (36) we have(38)∫ShζϱGαfζβdσζ≲∫DΔλfβHzlog1zdA+4∫D∂λfβ∂¯H+∂¯λfβ∂Hzlog1zdA≔I+II.Now, using Green’s theorem with limiting process and Hölder’s inequality, we obtain(39)I≲λfβqHs≲λfpβ.On the other hand, direct differentiation gives(40)∂λfβ=∂¯λfβ=βfλf2αf#=βfλfαμ,and (41)∂H=∂¯H=12H′,where H denotes the holomorphic function in D whose real part is H with H(0)=0; whence (42)II≤4β∫DH′zfzλfαzμzlog⁡1zdA≲∫S∫01H′rζfrζλfαrζμrζ1-rdrdσζ.Since f≤λf, we have, by Schwarz inequality, (43)II≲∫SMλfζβ/2GHζϱGαfζβ/2dσζ.Applying Hölder’s inequality with triple (2q,s,2q) and applying (11) to the last integral, we arrive at (44)II≲λfpβ/2GHLsϱGαfLpβ/2.From (16), we know GHLs≲Hs, and it follows from the theorem of M. Riesz ([8, 23]) that Hs≲Hs≲1. Thus(45)II≲λfpβ/2ϱGαfLpβ/2.Gathering estimates (38), (39), and (45) up, we arrive at (46)∫ShζϱGαfζβdσζ≤I+II≲λfpβ+λfpβ/2ϱGαfLpβ/2 for all positive trigonometric polynomials h with hLs≤1. That is, (47)ϱGαfLpβ≲λfpβ+λfpβ/2ϱGαfLpβ/2. Therefore we obtain via the arithmetic-geometric mean inequality that(48)ϱGαfLp≲λfp.We next showλfp≲ϱGαfLp. Set β=1+2α again. Take N>1 such that p<βN. Hölder’s inequality with paring (βN/p,βN/(βN-p)) gives (49)λfpp≈∫Df#z2λfzp-11-z2dxdy=∫Df#z2λfzp/N+p1-1/N-11-z2dxdy≤∫S∫01f#2λf2α1-rdrp/βN·∫01f#2λf2δ1-rdr1-p/βNdσ, where 2δ=p(1-1/N)(βN/(βN-p))-1. Applying Hölder’s inequality one more time to the last quantity with paring (N,N/(N-1)), (50)λfpp≲∫SϱGαfpdσ1/N∫SϱGδfpdσ1-1/N=ϱGαfLpp/NϱGδfLpp1-1/N. Therefore by applying the arithmetic-geometric mean inequality and (48), we obtain the desired inequality. This proof is completed.


## 5.2. Proof of (21)

Ifh∈Blogα, then (16), the definition of Blogα, and (24) gives (51)h∘f-h∘f02q2q≈∫S∫011-rh′f·f′2rζdrqdσζ≲hBlogα2q∫S∫011-rf#rζ21+λfrζ2αdrqdσζ=hBlogα2qϱGα,1fLpp≈hBlogα2q1+λfpp. This with simple inequality (52)h∘f2q2q-h∘f02q≲h∘f-h∘f02q2q gives (53)h∘f2q2q≲h0+hBlogα2q1+λfpp; whence this verifies Cf2q/p≲1+λfp.We next show1+λfp≲Cf2q/p. For each nondyadic t∈[0,1], let (54)htz=1+∑k=0∞γktk+1αz2k,z∈D, where γk=γk(t) is the Rademacher function (see [8, 23]) defined by (55)γkt=signsin2k+1πt. Then ht is holomorphic in D and by (30) there is a positive constant Cα such that (56)1-zht′z≤Cαlog⁡e1-zα, so that ht∈Blogα with htBlogα≤Cα. Thus, the definition of Cf gives(57)∫Sht∘frζ2qdσζ≤1+Cα2qCf2q.On the other hand, it follows by (29) that(58)1+λfpp=limr→1∫Sloge1-frζ2pdσζ≲limr→1∫S1+∑k=0∞k+12αfrζ22kqdσζ,while Khinchin’s inequality (see [23] Theorem 8.4) gives(59)1+∑k=0∞k+12αfrζ22kq≲∫01ht∘frζ2qdt.Now (57), (58), and (59) together verify 1+λfp≲Cf2q/p. This proof is completed.


## 6. Proof of Lemmas

For measurable functionsf in D, let us denote as usual (60)Mpr,f=∫02πfreiθpdθ2π1/p,0≤r<1.


### 6.1. Proof of Lemma5

Let0<ϵ<∞ and eϵ=c. From (61)Fϵ≔f#2ϵ+λfα=f′21-f2-2log⁡c1-f2αwe get (62)∂∂zFϵ=ϕFϵ,∂∂z¯Fϵ=∂∂zFϵ¯,where (63)ϕ=f′′f′+2+αlogc1-f2-1f′f¯1-f2.Thus, (64)∂2∂z∂z¯Fϵ=ϕ2+∂ϕ∂z¯Fϵ.Noting that (65)∂ϕ∂z¯=f#22+αlog⁡c1-f2-1-αf2log⁡c1-f2-2and α≥-2, we have (66)∂2∂z∂z¯Fϵ≥0.Thus, Fϵ is subharmonic if 0<ϵ<∞.For0<ϵ<∞, subharmonicity of Fϵ implies that Mp(r,Fϵ) is an increasing function of r: (67)Mpr1,Fϵ≤Mpr2,Fϵifr1<r2<1.Also, by subharmonicity, (68)Fϵr2eiθ≤∫02πPr,θ-tFϵreitdt2π,0≤r<1.Letting ϵ→0, monotone convergence theorem guarantees (27) and (28) also for ϵ=0. This proof is completed.


### 6.2. Proof of Lemma6

First let0<ϵ<∞. Apply Green’s Theorem of the form (69)ddr∫02πψdθ=1r∬rDΔψdxdyvalid for ψ=(ϵ+λf)p∈C2(rD¯). Recalling (11), it follows by a limiting process after integrating with respect to dr that (70)ϵ+λfpp-ϵ+λf0p≈∫02πdθ2π∫01f#ρeiθ2ϵ+λfρeiθp-1log⁡1ρρdρ.Thus, the first equivalence follows.The second equivalence follows from Lemma5: setting Fϵ≔f#2ϵ+λfp-1, the inequality (71)rlog⁡1r≤1-r≤log⁡1rand the increasing property of M1(r,Fϵ) give (72)∫DFϵzlog⁡1zdA≤∫01∫02πFϵreiθ1-rdrdθ=∫01∫02πFϵr2eiθ1-r22rdrdθ≤4∫01∫02πFϵreiθlog⁡1rrdrdθ=4∫DFϵzlog1zdA.Next, lettingϵ→0 we obtain (28) for ϵ=0 also. This proof is completed.


## 6.1. Proof of Lemma5

Let0<ϵ<∞ and eϵ=c. From (61)Fϵ≔f#2ϵ+λfα=f′21-f2-2log⁡c1-f2αwe get (62)∂∂zFϵ=ϕFϵ,∂∂z¯Fϵ=∂∂zFϵ¯,where (63)ϕ=f′′f′+2+αlogc1-f2-1f′f¯1-f2.Thus, (64)∂2∂z∂z¯Fϵ=ϕ2+∂ϕ∂z¯Fϵ.Noting that (65)∂ϕ∂z¯=f#22+αlog⁡c1-f2-1-αf2log⁡c1-f2-2and α≥-2, we have (66)∂2∂z∂z¯Fϵ≥0.Thus, Fϵ is subharmonic if 0<ϵ<∞.For0<ϵ<∞, subharmonicity of Fϵ implies that Mp(r,Fϵ) is an increasing function of r: (67)Mpr1,Fϵ≤Mpr2,Fϵifr1<r2<1.Also, by subharmonicity, (68)Fϵr2eiθ≤∫02πPr,θ-tFϵreitdt2π,0≤r<1.Letting ϵ→0, monotone convergence theorem guarantees (27) and (28) also for ϵ=0. This proof is completed.


## 6.2. Proof of Lemma6

First let0<ϵ<∞. Apply Green’s Theorem of the form (69)ddr∫02πψdθ=1r∬rDΔψdxdyvalid for ψ=(ϵ+λf)p∈C2(rD¯). Recalling (11), it follows by a limiting process after integrating with respect to dr that (70)ϵ+λfpp-ϵ+λf0p≈∫02πdθ2π∫01f#ρeiθ2ϵ+λfρeiθp-1log⁡1ρρdρ.Thus, the first equivalence follows.The second equivalence follows from Lemma5: setting Fϵ≔f#2ϵ+λfp-1, the inequality (71)rlog⁡1r≤1-r≤log⁡1rand the increasing property of M1(r,Fϵ) give (72)∫DFϵzlog⁡1zdA≤∫01∫02πFϵreiθ1-rdrdθ=∫01∫02πFϵr2eiθ1-r22rdrdθ≤4∫01∫02πFϵreiθlog⁡1rrdrdθ=4∫DFϵzlog1zdA.Next, lettingϵ→0 we obtain (28) for ϵ=0 also. This proof is completed.


## 7. Remarks and Acknowledgement

In view of our main result, Theorem2, there arises a natural question: what is CBlogα,Hp when ≤-1/2 ?Letα<-1/2. If we denote the identity map of D by I, that is, I(z)=z,z∈D, then(73)I∈CBlogα,Hpforanypsimply because (74)∫01I′rζ1-Irζ22log11-Irζ22α1-rdr∈L∞S⊂LpSwhich gives G(h∘I)∈Lp(S), equivalently h∘I∈Hp, for all h∈Blogα.Letf be holomorphic self-map of D satisfying f(0)=0. Then f is subordinate to I. By (73) and Littlewood Subordination Theorem ([8]), we have f∈CBlogα,Hp for any p. But by (13) we can remove the condition f(0)=0. Therefore we have (75)CBlogα,Hp=holomorphicself-mapofDfor any p.The caseα=-1/2 was considered in [26] (partially) in connection with (31) and the concept of the hyperbolic Nevanlinna class (see [27]).



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*Source: 8150908-2018-07-04.xml*