# Univalent Functions in the Möbius InvariantQK Space

Authors: Fernando Pérez-González; Jouni Rättyä
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

It is shown that a univalent functionf belongs to QK if and only if supa∈𝔻∫01M∞2(r,f∘φa-f(a))K′(log(1/r))dr<∞, where φa(z)=(a-z)/(1-a¯z), provided K satisfies certain regularity conditions. It is also shown that under these conditions QK contains all univalent Bloch functions if and only if ∫01(log((1+r)/(1-r)))2K′(log(1/r))dr<∞.


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## Body

## 1. Introduction

The aim of this paper is to characterize univalent functionsf in the Möbius invariant QK space in terms of the maximum modulus of f∘φa-f(a), where φa(z)=(a-z)/(1-a¯z), provided K satisfies certain regularity conditions. We will begin with a brief overview of characterizations of univalent functions in classical function spaces of the unit disc together with necessary definitions. Then, we will state the above-mentioned characterization of univalent functions in QK and its consequences. The proofs are postponed to the end of the paper and will be presented in the sequential order.


## 2. Notation, Background, and Results

Letℋ(𝔻) denote the algebra of all analytic functions in the unit disc 𝔻:={z:|z|<1}. A function f∈ℋ(𝔻) is said to be univalent if it is one to one, and the class of all univalent functions is denoted by 𝒰. For the theory of univalent functions, see [1–3].For0<p≤∞, the Hardy spaceHp consists of those f∈ℋ(𝔻) for which (2.1)‖f‖Hp:=lim⁡r→1-Mp(r,f)<∞,
where (2.2)Mp(r,f):=(12π∫02π|f(reiθ)|pdθ)1/p,0<p<∞,
are the standard Lp-means of the restriction of f to the circle of radius r centered at the origin, and (2.3)M∞(r,f):=max⁡0≤θ≤2π|f(reiθ)|
is the maximum modulus function. For the theory of Hardy spaces, see [4, 5].Hardy and Littlewood [6], Pommerenke [7], and Prawitz [8] proved that if f∈𝒰, then f∈Hp, if and only if(2.4)∫01M∞p(r,f)dr<∞,0<p<∞.
For more information on univalent functions in Hardy spaces, see [2, 7–13]. A result due to Baernstein II [14] states that f∈BMOA, the space of analytic functions in the Hardy space H1 with boundary values of bounded mean oscillation, if and only if supa∈𝔻∥f∘φa-f(a)∥Hp<∞, where φa(z)=(a-z)/(1-a¯z) is the automorphism of 𝔻 which interchanges the origin and the point a. This, applied to (2.4), shows that f∈𝒰 belongs to BMOA if and only if(2.5)sup⁡a∈D∫01M∞p(r,f∘φa-f(a))dr<∞
for some (equivalently for all) 0<p<∞. However, Pommerenke [15] has shown that BMOA and the Bloch space ℬ contain the same univalent functions. The Bloch spaceℬ consists of those f∈ℋ(𝔻) for which (2.6)‖f‖B:=sup⁡z∈D|f′(z)|(1-|z|2)<∞.
If, in addition, f′(z)(1-|z|2)→0, as |z|→1-, then f∈ℬ0. For the theory of Bloch spaces, see [3, 16]. If Ω is a simply connected proper subdomain of the complex plane and f∈𝒰 such that f(𝔻)=Ω, then (2.7)dΩ(f(z))≤|f′(z)|(1-|z|2)≤4dΩ(f(z)),z∈D,
where dΩ(w) stands for the Euclidean distance from w to the boundary of Ω, see, for example, [3]. Therefore, univalent functions in the Bloch space can be characterized by the following well-known geometric condition; f∈ℬ if and only if supw∈ΩdΩ(w)<∞, that is, if the image of 𝔻 under f does not contain arbitrarily large discs. For other characterizations of univalent Bloch functions, see [11].Aulaskari et al. [17] improved the result by Pommerenke by showing that ℬ∩𝒰=Qp∩𝒰 for any p>0. Recall that Qp is a Möbius invariant subspace of ℬ and consists of those f∈ℋ(𝔻) for which (2.8)‖f‖Qp2:=sup⁡a∈D∫D|f′(z)|2gp(z,a)dA(z)<∞,
where dA(z) is the Euclidean area element on 𝔻. In particular, Q1=BMOA and Qp=ℬ for all p>1. Moreover, Q0 is the classical Dirichlet space𝒟 which consists of all f∈ℋ(𝔻) with finite area of image counting multiplicities. The Dirichlet space 𝒟 is a special case of classical Besov spaces. A geometric characterization of univalent functions in Besov spaces was found by Walsh [18], see also the related results by Donaire et al. [19]. For the theory of Qp spaces, see [20, 21].ForK:[0,∞)→[0,∞), the Möbius invariant space QK consists of those f∈ℋ(𝔻) for which (2.9)‖f‖QK2:=sup⁡a∈D∫D|f′(z)|2K(g(z,a))dA(z)<∞.
Moreover, if the integral above tends to zero as a approaches to the boundary of 𝔻, then f∈QK,0. If K(t)=tp, then QK=Qp, and therefore QK can be viewed as a generalized Qp space. For results on QK, see [22–24] and the references therein. From now on, the weight K is assumed to admit the following basic properties:  (1)
K(0)=0;(2)
∫01/eK(log(1/r))rdr<∞;(3)
K′(t)≥0 for all t∈(0,1);(4)
K′′(t)≤0 for all t∈(0,1).Requirements (1) and (2) are standard; the first one ensures thatK indeed plays an essential role in the definition, and the second one guarantees the nontriviality of QK as well as the inclusions QK⊂ℬ and QK,0⊂ℬ0. Conditions (3) and (4) are, of course, restrictions, yet, for example, K(t)=tp satisfies both of them for all 0<p≤1.Before proceeding further, we give an example related toQK spaces in order to illustrate the variety of spaces induced by different choices of K.Example 2.1.
For0<p<∞ and q∈ℝ, consider the weight
(2.10)K(t)=K(p,q)(t){:={tp(log⁡1t)q,0<t<e-q/p,e-q(qp)q,e-q/p≤t<1,0≤q<∞,:={tp(log⁡1t)q,0<t<12,2-p(log⁡2)q,12≤t<1,-∞<q<0.
Obviously, QK(p,0)=Qp for all 0<p<∞. Since
(2.11)K′(t)=tp-1(log⁡1t)q-1(plog⁡1t-q)
for either 0<t<e-q/p (the case q≥0) or 0<t<1/2 (the case q<0), the weight K is increasing. If 0<p≤1, then the integral
(2.12)∫01K(log⁡1r)(1-r2)-2rdr
diverges for all q∈ℝ, and therefore QK⊊ℬ by [22, Theorem 2.3]. Moreover, [22, Theorem 2.6] shows that
(2.13)Qp1⊊QK(p1,-q1)⊊QK(p1,-q2)⊊QK(p2,q2)⊊QK(p2,q1)⊊Qp2
for all 0<p1<p2≤1 and 0<q1<q2<∞.Let us turn back to univalent functions. A special case of [11, Theorem 4] shows that a univalent function f belongs to Qp if and only if(2.14)sup⁡a∈D∫01M∞2(r,f∘φa-f(a))(log⁡1r)p-1dr<∞,0<p≤1.
Of course, this is not a natural way to state the result because 𝒰∩Qp=𝒰∩ℬ for all p, but it appears to be useful for our purposes. The case p=1 of (2.14), which corresponds toBMOA, reduces to (2.5) with p=2. An appropriate interpretation of (2.14) allows us to conclude what happens with QK. Namely, since p(log(1/r))p-1=K′(log(1/r)) for K(t)=tp, condition (2.14), with (log(1/r))p-1 being replaced by K′(log(1/r)), gives a candidate for a characterization of univalent functions in QK. The main result of this paper is Theorem 2.2 which shows that this is indeed the case, provided(2.15)-(K′′(t)+K′(t))K(t)(K′(t))2≤C,0<t<∞,
for some positive constant C.Theorem 2.2.
Letf∈𝒰 and assume that K satisfies condition (2.15). Then, f∈QK if and only if
(2.16)sup⁡a∈D∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr<∞.
Moreover, f∈QK,0 if and only if
(2.17)lim⁡|a|→1-∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr=0.It is an immediate consequence of the proof that the assertions in Theorem2.2 remain valid for areally mean q-valent functions. In addition to standard techniques of univalent functions, the proof of Theorem 2.2 uses a result by Pavlović and Peláez [25] on weighted integrals of analytic functions and their derivatives. An application of this result yields the requirement (2.15). We will analyze the importance of (2.15) after discussing consequences of Theorem 2.2 and its proof.The first part of the proof of Theorem2.2 shows that f∈ℋ(𝔻) belongs to QK if and only if (2.18)sup⁡a∈D∫01A(f(Δ(a,r)))K′(log⁡1r)dr<∞,
where A(f(Δ(a,r))) denotes the area of image of the pseudohyperbolic disc Δ(a,r):={z:|φa(z)|<r} under f counting multiplicities. This, together with Theorem 2.2, shows that for f∈QK∩𝒰, the quantities A(f(Δ(a,r))) and M∞2(r,f∘φa-f(a)) are of the same growth (uniformly in a) when measured in terms of L1(K′(log(1/r))dr).Sinceℬ∩𝒰=Qp∩𝒰 for any p>0 by a result due to Aulaskari et al. [17], it is natural to ask when does the identity ℬ∩𝒰=QK∩𝒰 hold. An application of Theorem 2.2 yields Corollary 2.3 which answers this question provided K satisfies (2.15).Corollary 2.3.
Assume thatK satisfies condition (2.15). Then, QK∩𝒰=ℬ∩𝒰 if and only if
(2.19)∫01(log⁡1+r1-r)2K′(log⁡1r)dr<∞.
Moreover, if (2.19) is satisfied, then QK,0∩𝒰=ℬ0∩𝒰.Wulan [26] showed that every areally mean q-valent Bloch function belongs to QK whenever K satisfies(2.20)∫01(log⁡11-r)2K(log⁡(1/r))1-rrdr<∞.This result is more general than Corollary2.3 because the crucial condition (2.15) is not needed. However, K is concave whenever (4) is satisfied, and in that case, K′(t)≤K(t)/t, and then (2.20) implies (2.19).We next analyze the necessity of condition (2.15). This is needed in two consecutive steps in the proof of Theorem 2.2. These steps together establish the asymptotic inequality(2.21)∫01M∞2(r,f)1rK′(log⁡1r)dr≲∫01M∞2(r,f′)(∫r1K(log⁡1r)ds)dr
for all f∈ℋ(𝔻). As usual, we write A≲B, if there exists C>0 such that A≤CB, and the notation A≳B is understood in an analogous manner. In particular, if A≲B and A≳B, then we write A≍B. Set now v(r)=∫r1K(log(1/r)), so that (2.21) becomes(2.22)∫01M∞2(r,f)v′′(r)dr≲∫01M∞2(r,f′)v(r)dr.
We will show next that (2.22) does not necessarily remain true even for univalent functions f, if K, that induces v, does not satisfy (2.15). To see this, take (2.23)v(r)=1-rlog⁡(1/(1-r)),v′(r)≍-1log⁡(1/(1-r)),v′′(r)≍1(1-r)(log⁡(1/(1-r)))2,r⟶1-,
and f∈𝒰 with f2(z)=1-log(1-z), so that (2.24)M∞2(r,f)≍log⁡11-r,M∞2(r,f′)≍1(1-r)2log⁡(1/1-r),r⟶1-.
For these choices, the right-hand side of (2.22) is finite, whereas the left-hand side is not. If K(t)=-1/logt, then v(r)=∫r1K(log(1/r))≍(1-r)(log(1/(1-r)))-1, but (2.15) for this K is equivalent to -2-logt+tlogt≤C, which clearly fails as t→0+. Moreover, f∈𝒰∩ℬ, but (2.19) fails for K(t)=-1/logt. This justifies the assumption (2.15) in Theorem 2.2 and Corollary 2.3.


## 3. Proof of Theorem2.2

We will prove the first assertion, the second one, then immediately follows by the proof.


### 3.1. Sufficiency of (2.16)

Letf∈𝒰. Recall first that M2(r,h) is nondecreasing for any h∈ℋ(𝔻). Therefore we may use conditions (1) and (3) together with Fubini’s theorem to obtain(3.1)I(f,K,a):=∫D|(f∘φa)′(z)|2K(log⁡1|z|)dA(z)≍12π∫D|(f∘φa)′(z)|2|z|K(log⁡1|z|)dA(z)=∫01M22(r,(f∘φa-f(a))′)∫r11sK′(log⁡1s)dsr2dr≤∫01M22(r,(f∘φa-f(a))′)∫r1K′(log⁡1s)dsrdr=∫01∫Δ(0,s)|(f∘φa-f(a))′(z)|2dA(z)K′(log⁡1s)ds.
If h∈𝒰, then(3.2)1π∫Δ(0,s)|h′(z)|2dA(z)≤1π∫|w|≤M∞(s,h)dA(w)=2∫0M∞(s,h)tdt=M∞2(s,h),
and an application of this inequality to h=f∘φa-f(a), together, with (3.1), gives (3.3)I(f,K,a)≲∫01M∞2(s,f∘φa-f(a))K′(log⁡1s)ds.
Therefore a univalent function f belongs to QK if (2.16) is satisfied. Note that this part of the proof uses the univalence of f, but does not rely on (2.15). Moreover, if f is areally mean q-valent, then the reasoning in (3.2) remains valid as soon as the right-hand side of the inequality is multiplied by q.


### 3.2. Necessity of (2.16)

To prove this, we will need a special case of a result due to Pavlović and Peláez [25], which states that(3.4)∫01M∞2(r,h)ω(r)dr≍|h(0)|2+∫01M∞2(r,h′)(∫r1ω(s)ds)2ω(r)dr,h∈H(D),
provided ω:[0,1)→(0,∞) is differentiable and satisfies(3.5)ω′(r)ω(r)2∫r1ω(x)dx≤C,0<r<1,
for some constant C>0. Assumption (2.15) is equivalent to (3.5) for ω(r):=1/rK′(log1/r). Therefore, (3.4) yields (3.6)∫01M∞2(r,f∘φa-f(a))1rK′(log⁡1r)dr≲∫01M∞2(r,(f∘φa)′)r(K(log⁡(1/r)))2K′(log⁡(1/r))dr.
We next show that(3.7)r(K(log⁡(1/r)))2K′(log⁡(1/r))≤(C+2)∫r1K(log⁡1s)ds,
where C is the constant in (2.15). To prove (3.7), consider the function (3.8)g(r):=(C+2)∫r1K(log⁡1s)ds-r(K(log⁡(1/r)))2K′(log⁡(1/r)).
By the assumptions (1), (3), and (4) on K, g(r)→0 as r→1-. Moreover, (2.15) yields (3.9)g′(r)=-CK(log⁡1r)-(K′(log⁡(1/r))+(K′′(log⁡(1/r))))(K(log⁡(1/r)))2(K′(log⁡(1/r)))2≤0,
and thus g is decreasing, and therefore (3.7) holds. This, together with Fubini's theorem, shows that (3.10)∫01M∞2(r,f∘φa-f(a))1rK′(log⁡1r)dr≲∫01M∞2(r,(f∘φa)′)∫r1K(log⁡1s)dsdr=∫01K(log⁡1s)∫0sM∞2(r,(f∘φa)′)drds,
from which the inequality ∫0sM∞2(r,h)dr≤πM22(s,h), valid for all h∈ℋ(𝔻) (see [9, page 841] or [7, Hilfssatz 1]), yields (3.11)∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr≲∫01M22(s,(f∘φa)′)K(log⁡1s)ds≍I(f,K,a)≤‖f‖QK2.
Thus, (2.16) is satisfied if f∈QK.


## 3.1. Sufficiency of (2.16)

Letf∈𝒰. Recall first that M2(r,h) is nondecreasing for any h∈ℋ(𝔻). Therefore we may use conditions (1) and (3) together with Fubini’s theorem to obtain(3.1)I(f,K,a):=∫D|(f∘φa)′(z)|2K(log⁡1|z|)dA(z)≍12π∫D|(f∘φa)′(z)|2|z|K(log⁡1|z|)dA(z)=∫01M22(r,(f∘φa-f(a))′)∫r11sK′(log⁡1s)dsr2dr≤∫01M22(r,(f∘φa-f(a))′)∫r1K′(log⁡1s)dsrdr=∫01∫Δ(0,s)|(f∘φa-f(a))′(z)|2dA(z)K′(log⁡1s)ds.
If h∈𝒰, then(3.2)1π∫Δ(0,s)|h′(z)|2dA(z)≤1π∫|w|≤M∞(s,h)dA(w)=2∫0M∞(s,h)tdt=M∞2(s,h),
and an application of this inequality to h=f∘φa-f(a), together, with (3.1), gives (3.3)I(f,K,a)≲∫01M∞2(s,f∘φa-f(a))K′(log⁡1s)ds.
Therefore a univalent function f belongs to QK if (2.16) is satisfied. Note that this part of the proof uses the univalence of f, but does not rely on (2.15). Moreover, if f is areally mean q-valent, then the reasoning in (3.2) remains valid as soon as the right-hand side of the inequality is multiplied by q.


## 3.2. Necessity of (2.16)

To prove this, we will need a special case of a result due to Pavlović and Peláez [25], which states that(3.4)∫01M∞2(r,h)ω(r)dr≍|h(0)|2+∫01M∞2(r,h′)(∫r1ω(s)ds)2ω(r)dr,h∈H(D),
provided ω:[0,1)→(0,∞) is differentiable and satisfies(3.5)ω′(r)ω(r)2∫r1ω(x)dx≤C,0<r<1,
for some constant C>0. Assumption (2.15) is equivalent to (3.5) for ω(r):=1/rK′(log1/r). Therefore, (3.4) yields (3.6)∫01M∞2(r,f∘φa-f(a))1rK′(log⁡1r)dr≲∫01M∞2(r,(f∘φa)′)r(K(log⁡(1/r)))2K′(log⁡(1/r))dr.
We next show that(3.7)r(K(log⁡(1/r)))2K′(log⁡(1/r))≤(C+2)∫r1K(log⁡1s)ds,
where C is the constant in (2.15). To prove (3.7), consider the function (3.8)g(r):=(C+2)∫r1K(log⁡1s)ds-r(K(log⁡(1/r)))2K′(log⁡(1/r)).
By the assumptions (1), (3), and (4) on K, g(r)→0 as r→1-. Moreover, (2.15) yields (3.9)g′(r)=-CK(log⁡1r)-(K′(log⁡(1/r))+(K′′(log⁡(1/r))))(K(log⁡(1/r)))2(K′(log⁡(1/r)))2≤0,
and thus g is decreasing, and therefore (3.7) holds. This, together with Fubini's theorem, shows that (3.10)∫01M∞2(r,f∘φa-f(a))1rK′(log⁡1r)dr≲∫01M∞2(r,(f∘φa)′)∫r1K(log⁡1s)dsdr=∫01K(log⁡1s)∫0sM∞2(r,(f∘φa)′)drds,
from which the inequality ∫0sM∞2(r,h)dr≤πM22(s,h), valid for all h∈ℋ(𝔻) (see [9, page 841] or [7, Hilfssatz 1]), yields (3.11)∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr≲∫01M22(s,(f∘φa)′)K(log⁡1s)ds≍I(f,K,a)≤‖f‖QK2.
Thus, (2.16) is satisfied if f∈QK.


## 4. Proof of Corollary2.3

### 4.1. Sufficiency of (2.19)

SinceQK⊂ℬ, it suffices to show that 𝒰∩ℬ⊂QK. To see this, let f∈ℬ. An application of the inequality (4.1)M∞(r,g)≤‖g‖B12log⁡1+r1-r+|g(0)|,g∈B,
to the function g=f∘φa-f(a)∈ℬ gives(4.2)M∞(r,f∘φa-f(a))≤‖f∘φa-f(a)‖B12log⁡1+r1-r=‖f‖B12log⁡1+r1-r
for all a∈𝔻. This yields (4.3)sup⁡a∈D∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr≤14‖f‖B2∫01(log⁡1+r1-r)2K′(log⁡1r)dr<∞,
and thus f∈QK by Theorem 2.2.


### 4.2. Necessity of (2.19)

IfK satisfies condition (2.15) and QK∩𝒰=ℬ∩𝒰, then the univalent Bloch function log((1+z)/(1-z)), together with Theorem 2.2, shows that (2.19) holds.


### 4.3. The CaseQK,0∩𝒰=ℬ0∩𝒰

Assume thatK satisfies (2.19) and let ɛ>0. Since QK,0⊂ℬ0, it suffices to show that 𝒰∩ℬ0⊂QK,0. To see this, let f∈ℬ0. By applying (4.2) we obtain (4.4)∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr≤∫0RM∞2(r,f∘φa-f(a))K′(log⁡1r)dr+‖f‖B214∫R1(log⁡1+r1-r)2K′(log⁡1r)dr=I1(R)+I2(R).
Since f∈ℬ0⊂ℬ, the assumption (2.19) implies that I2(R)<ɛ/2 for all R sufficiently large. Fix such an R. The image of the circle of radius r centered at the origin under a univalent function h-h(0) is a Jordan curve with zero in its inner domain. The length of this image is 2πrM1(r,h′), and hence M∞(r,h)≤πrM1(r,h′)+|f(0)|. This estimate, applied to h=f∘φa-f(a), together with the assumption f∈ℬ0, yields (4.5)I1(R)≤π∫0R(12π∫02π|f′(φa(reiθ))φa′(reiθ)|dθ)2K′(log⁡1r)rdr=14π∫0R(∫02π|f′(φa(reiθ))|(1-|φa(reiθ)|2)dθ)2K′(log⁡(1/r))(1-r2)2rdr<ɛ2
for all sufficiently large |a|. Therefore, (4.6)∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr⟶0,|a|⟶1-,
and hence f∈QK,0 by Theorem 2.2.


## 4.1. Sufficiency of (2.19)

SinceQK⊂ℬ, it suffices to show that 𝒰∩ℬ⊂QK. To see this, let f∈ℬ. An application of the inequality (4.1)M∞(r,g)≤‖g‖B12log⁡1+r1-r+|g(0)|,g∈B,
to the function g=f∘φa-f(a)∈ℬ gives(4.2)M∞(r,f∘φa-f(a))≤‖f∘φa-f(a)‖B12log⁡1+r1-r=‖f‖B12log⁡1+r1-r
for all a∈𝔻. This yields (4.3)sup⁡a∈D∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr≤14‖f‖B2∫01(log⁡1+r1-r)2K′(log⁡1r)dr<∞,
and thus f∈QK by Theorem 2.2.


## 4.2. Necessity of (2.19)

IfK satisfies condition (2.15) and QK∩𝒰=ℬ∩𝒰, then the univalent Bloch function log((1+z)/(1-z)), together with Theorem 2.2, shows that (2.19) holds.


## 4.3. The CaseQK,0∩𝒰=ℬ0∩𝒰

Assume thatK satisfies (2.19) and let ɛ>0. Since QK,0⊂ℬ0, it suffices to show that 𝒰∩ℬ0⊂QK,0. To see this, let f∈ℬ0. By applying (4.2) we obtain (4.4)∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr≤∫0RM∞2(r,f∘φa-f(a))K′(log⁡1r)dr+‖f‖B214∫R1(log⁡1+r1-r)2K′(log⁡1r)dr=I1(R)+I2(R).
Since f∈ℬ0⊂ℬ, the assumption (2.19) implies that I2(R)<ɛ/2 for all R sufficiently large. Fix such an R. The image of the circle of radius r centered at the origin under a univalent function h-h(0) is a Jordan curve with zero in its inner domain. The length of this image is 2πrM1(r,h′), and hence M∞(r,h)≤πrM1(r,h′)+|f(0)|. This estimate, applied to h=f∘φa-f(a), together with the assumption f∈ℬ0, yields (4.5)I1(R)≤π∫0R(12π∫02π|f′(φa(reiθ))φa′(reiθ)|dθ)2K′(log⁡1r)rdr=14π∫0R(∫02π|f′(φa(reiθ))|(1-|φa(reiθ)|2)dθ)2K′(log⁡(1/r))(1-r2)2rdr<ɛ2
for all sufficiently large |a|. Therefore, (4.6)∫01M∞2(r,f∘φa-f(a))K′(log⁡1r)dr⟶0,|a|⟶1-,
and hence f∈QK,0 by Theorem 2.2.



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*Source: 259796-2011-11-14.xml*