# Random Power Series inQp,0 Spaces

Authors: Haiyin Li; Yan Wu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Aulaskari et al. proved if0<p<1 and εn is sequence of independent, identically distributed Rademacher random variables on a probability space, then the condition Σn=0∞n1−pan2<∞ implies that the random power series Rfz=∑n=0∞anεnzn∈Qp almost surely. In this paper, we improve this result showing that the condition Σn=0∞n1−pan2<∞ actually implies Rf∈Qp,0 almost surely.


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## Body

## 1. Introduction

LetD=z:∣z∣<1 be the unit disk in the extended complex plane ℂ^, S1=∂D and let HD be the space of all analytic functions on D.If0<r<1 and f∈HD, we set
(1)Mpr,f=12π∫02πfreiθpdθ1/p,0<p<∞,and M∞r,f=sup∣z∣=r∣fz∣. A function f∈HD is said to belong to the Hardy space Hp, if
(2)∥f∥Hp=sup0<r<1Mpr,f<∞.Fora∈D, let
(3)σaz=a−z1−a¯z,z∈D,then σa is a Möbius transformation of D which interchanges a and 0. For −1<p<∞, it is said that f belongs to the Qp space if f∈HD and
(4)∥f∥Qp2=supa∈D∫Df′z21−σaz2pdAz<∞.Furthermore, we say thatf belongs to the little Qp space, denoted by Qp,0, if f∈Qp and
(5)lim∣a∣⟶1−∫Df′z21−σaz2pdAz=0.Qp spaces were introduced by Aulaskari et al. in [1] and have been fully studied in recent years. It is known that Q0 is the classical Dirichlet space, Q1 is BMOA, and Q1,0 is VMOA (see [2]). It is proved in [3] that Qp is the Bloch space if 1<p<∞ and in [4] that Qp contains only constants if −1<p<0. The proper inclusions
(6)H∞⊂BMOA⊂⋂p<∞Hp,and Qp⊂BMOA0<p<1 are well known. We refer to [1, 5–7] for more information on Qp spaces.Recall that an analytic functionfz=Σk=0∞akznk is called a lacunary series if
(7)λ=infknk+1nk>1.For lacunary series, the following equivalent relations ([1]) hold for Qp,0<p<1:
(8)∑k=0∞nk1−pak2<∞⟺f∈Qp⟺f∈Qp,0,and for BMOA:
(9)∑k=0∞ak2<∞⟺f∈BMOA⟺f∈VMOA.Letεnn=0∞ be a sequence of independent, identically distributed (i.i.d.) Rademacher random variables on a probability space Ω,F,P; that is, Pεn=1=Pεn=−1=1/2 for all n≥0. For each fz=Σn=0∞anzn∈HD, we define the random power series:
(10)Rfz=∑n=0∞anεnzn.Usually, random power series are similar to lacunary series. For example, they are very well behaved if the coefficients are square-summable and very badly behaved if not.Aulaskari et al. [8] have shown that for 0<p<1, ∑n=0∞n1−pan2<∞ implies Rf∈Qp a.s., a result which is similar to the one for lacunary series. However, the condition Σn=0∞an2<∞ implies f∈⋂p<∞Hp a.s. [9], while there exists a sequence an with Σn=0∞an2<∞ and yet Rf∉BMOA for any choice of εn ([10]). Sledd [11] and Duren [12] have shown that the condition Σn=0∞an2logn<∞ implies that Rf∈BMOA a.s. Furthermore, this condition actually gives the stronger conclusion that Rf∈VMOA a.s.In this paper, we further discuss the random power series inQp,0 space for 0<p<1. We get the following:Theorem 1.
Suppose that0<p<1 and let fz=Σn=0∞anzn∈HD. If Σn=0∞n1−pan2<∞ for 0<p<1, then
(11)Rfz=∑n=0∞anεnzn∈Qp,0a.s.In what follows,C always denotes some positive constant which may differ at different situations.


## 2. Preliminaries and Basic Properties

The sequence of Rademacher functionsφnn=0∞ is defined as follows
(12)φ1t=1,0<t<12,−1,12<t<1,0,t=0,12,1.and for n≥2,
(13)φnt=sgnsin2nπt,0≤t≤1.In general,φnt vanishes at all multiples of 2−n and takes the values ±1 elsewhere. See [13, 14] for properties of these functions.The following Khintchine’s inequality (Appendix A of [13, 15]) for Rademacher functions will be used later.Lemma 2 (Khintchine’s inequality).
Letann=1∞ be a sequence of complex numbers with Σn=0∞an2<∞, then the series Σn=0∞anφnt converges a.s. Furthermore, let 0<p<∞, then for all N≥0,
(14)Ap∑n=0Nan21/2≤E∑n=0Nanφntp1/p≤Bp∑n=0Nan21/2,for some positive constants Ap and Bp depend only on p.Forα≥1 and β≥1, the following inequality holds for the beta function Bα,β.Lemma 3 (beta function inequality).
Letα≥1 and β≥1. Then there exist two positive constants C1 and C2 depending only on β such that
(15)C11αβ≤Bα,β=∫01tα−11−tβ−1dt≤C21αβ.Proof.
By Stirling’s formula, it is not difficult to get(16)∫01tα−11−tβ−1dt=ΓαΓβΓα+β≤C21αβ.
On the other hand, fora≥1,
(17)∫01tα−11−tβ−1dt≥∫1−1/a1tα−11−tβ−1dt≥1−1aα−1∫1−1/a11−tβ−1dt≥C11αβ.
The proof is complete.☐An important tool to study function spaces is Carleson type measure. LetI be any arc on S1 and ∣I∣ be the normalized length of I. The Carleson sector SI, based on I, is defined by
(18)SI=z=reiθ:1−∣I∣≤r<1,eiθ∈I.For0<p<∞, a positive Borel measure μ on D is called a bounded p-Carleson measure if there exists a positive constant C such that
(19)μSI≤CIp.We say thatμ is a vanishing p-Carleson measure if
(20)μSI=oIp,∣I∣⟶0.Whenp=1, we get the standard definition of the original Carleson measure.Letf∈HD. It is known that f∈Qp if and only if dμz=1−∣z∣pf′z2dxdy is a bounded p-Carleson measure and f∈Qp,0 if and only if dμz=1−∣z∣pf′z2dxdy is a vanishing p-Carleson measure, respectively [5]. Notice that a bounded p−Carleson measure is a finite measure; hence, for 0<p≤1, we have
(21)f∈Qp⇒∫011−rpMq2r,f′dr<∞,0<q≤2.Conversely, forf∈HD, the condition ∫011−rpM∞2r,f′dr<∞ implies f∈Qp for 0<p<1, and ∫011−rM∞2r,f′dr<∞ implies f∈BMOA (see [16]).The following proposition gives the stronger conclusion thatf∈Qp,0 and f∈VMOA when p=1, respectively.Proposition 4.
Letfz∈HD and 0<p≤1. If
(22)∫011−rpM∞2r,f′dr<∞,then f∈Qp,0.Proof.
For anyI⊂S1, we have
(23)∫If′reiθ2dθ≤M∞2r,f′I.
Letdμz=1−∣z∣pf′z2dxdy,
(24)μSI≤∫1−∣I∣1∫I1−rpf′reiθ2dθdr≤∣I∣∫1−∣I∣11−rpM∞2r,f′dr.
The integral(25)∫011−rpM∞2r,f′dr<∞,implies that for any ε>0, there exists a δ,0<δ<1, such that
(26)∫1−δ11−rpM∞2r,f′dr<ε.
Thus(27)μSI<∣I∣ε≤Ipε,holds for any subarc I on S1 with ∣I∣<δ. This shows that the measure dμz=1−∣z∣pf′z2dxdy is a vanishing p-Carleson measure, hence f∈Qp,0.☐Remark 5.
Aulaskari et al. showed in p. 426 of their paper [16] that the exponent p in the left-hand side of (17) can not be increased.SubstitutingM∞r,f′ by Mqr,f′ for some 2<q<∞, we can get the following result.Proposition 6.
Let0<p<1, 2<q<∞, and 1−2/q≥p. For fz∈HD, if
(28)∫011−rpMq2r,f′dr<∞,then f∈Qp,0.Proof.
For anyI⊂S1, by Ho¨lder’s inequality, we have
(29)∫If′reiθ2dθ≤∫If′reiθqdθ2/q∫Idθq−2/q≤Mq2r,f′I1−2/q,for 2<q<∞. Let dμz=1−∣z∣pf′z2dxdy,
(30)μSI≤∫1−∣I∣1∫I1−rpf′reiθ2dθdr≤I1−2/q∫1−∣I∣11−rpMq2r,f′dr.
The integral(31)∫011−rpMq2r,f′dr<∞,implies that for any ε>0, there exists a δ,0<δ<1, such that
(32)∫1−δ11−rpMq2r,f′dr<ε.
Thus, for1−2/q≥p, we have
(33)μSI<I1−2/qε≤Ipε,holds for any subarc I on S1 with ∣I∣<δ. This shows that the measure dμz=1−∣z∣pf′z2dxdy is a vanishing p-Carleson measure, hence f∈Qp,0.☐Remark 7.
It should be mentioned that the proof of Proposition4 and 6 are similar to that of Theorem 3.2 and 3.3 in [16]. Furthermore, for p=1 and 2<q<∞, it is unknown whether ∫011−rpMq2r,f′dr<∞ implies f∈VMOA.


## 3. Proof of Theorem1

Theorem 8.
Suppose thatfz=Σn=0∞anzn∈HD. If Σn=0∞n1−pan2<∞ for 0<p<1, then
(34)Rfz=∑n=0∞anεnzn∈Qp,0a.s.Proof.
Assume thatΣn=0∞n1−pan2<∞ for 0<p<1. For every 2<q<∞, by Fubini’s theorem, Jensen’s inequality, and Khintchine’s inequality, we have
(35)E∫01Mq2r,Rf′1−rpdr=∫01E∫02π∑n=0∞nanεnrn−1ein−1θqdθ2π2/q1−rpdr≤∫01E∫02π∑n=0∞nanεnrn−1ein−1θqdθ2π2/q1−rpdr≤Bp∫01∑n=0∞n2an2r2n−21−rpdr=Bp∑n=0∞n2an2∫01r2n−21−rpdr.
From the beta function inequality of Lemma3,
(36)∫01r2n−21−rpdr≤C212n−11+p≤C21n1+p,for n≥1. So we get
(37)E∫01Mq2r,Rf′1−rpdr≤C∑n=0∞n1−pan2<∞,which implies that
(38)∫011−rpMq2r,Rf′dr<∞,a.s.
Fixq≥2/1−p. By Proposition 6, we have Rf∈Qp,0 a.s.
This theorem is proved.☐Letq≥1 and 0<α≤1, we say that f∈HD belongs to the mean Lipschitz space Λq,α (see [13]), if
(39)Mqr,f′=O11−r1−a,asr⟶1.Furthermore, if(40)Mqr,f′=o11−r1−a,asr⟶1,we say that f∈λq,α.SinceMqr,f′ is an increasing function of r, it is easy to see that the condition
(41)∫01Mq2r,Rf′1−rpdr<∞,implies that
(42)Mqr,Rf′=o11−rp+1/2,asr⟶1.By the condition1−2/q≥p of Proposition 6, we get
(43)Mqr,Rf′=o11−r1−1/q,asr⟶1.So we get the following result.Theorem 9.
Letfz=Σn=0∞anzn∈HD. If Σn=0∞n1−pan2<∞ for 0<p<1, then
(44)Rfz=∑n=0∞anεnzn∈Λq,1qa.s.where q>2 and 1−2/q≥p. Furthermore, Rfz∈λq,1/q a.s.



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*Source: 1449080-2021-10-04.xml*