# Necessary and Sufficient Conditions for Inner Functions to Be inQK(p,p-2)-Spaces

Authors: Atte Reijonen
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Under some regularity conditions onK, inner functions in QK(p,p-2)-spaces are characterized in the following way: an inner function belongs to QK(p,p-2) if and only if it is a Blaschke product associated with {zn} satisfying supa∈D∑nK(1-|φa(zn)|)<∞, where φa(z)=(a-z)/(1-a¯z). The result generalizes earlier theorems in (Essén et al., 2006) and (Pérez-González and Rättyä, 2009).


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## Body

## 1. Introduction

Inner functions are bounded analytic functions in the unit discD={z∈C:|z|<1} such that their moduli are one almost everywhere on the boundary T={z∈C:|z|=1}. Each inner function can be represented as a product of a Blaschke product and a singular inner function [1]. Singular inner functions are of the form (1)Sz=exp⁡-∫Tz+wz-wdσw,z∈D,where σ is a positive measure on T which is singular with respect to the Lebesgue measure, and, for w∈T and m∈N∪{0}, (2)Bz=wzm∏nznznzn-z1-z¯nz,z∈D,is a Blaschke product associated with zn⊂D∖{0} if ∑n‍(1-|zn|)<∞ [2]. Information about inner functions can be found in [3]; see also [4–6], for example.An analytic functionf in D belongs to QK(p,q) for p∈(0,∞) and q∈(-2,∞) if (3)supa∈D⁡∫Df′zp1-zqKgz,adAz<∞.Here K:[0,∞)→[0,∞) is such that K(1)>0, g(z,a)=log(1-a¯z)/(a-z) is Green’s function, and dA(z) is the Lebesgue area measure. Information about QK(p,q)-spaces can be found in [7, 8]; see also the beginning of the next section.Our purpose is to study the behaviour of inner functions inQK(p,q)-spaces. In particular, we are interested in the Möbius invariant QK(p,p-2)-spaces. Related to QK(p,p-2), it is proved that only inner functions there are Blaschke products if K satisfies certain regularity conditions; see Proposition 7. Under corresponding assumptions, our main result, Theorem 1, gives a complete characterization of inner functions in the spaces. The results generalize [9, Theorems  1.3 and  1.4] and the essential content of [10, Theorem  5.1].Theorem 1.
Letp∈(1/2,∞), and assume that there exists α,β∈(1-min{p,1},min{p,1}) such that a nondecreasing K satisfies r-αK(r)↘ and r-βK(r)↗ for r∈(0,1). Then an inner function belongs to QK(p,p-2) if and only if it is a Blaschke product associated with a sequence {zn} satisfying(4)supa∈D⁡∑nK1-φazn<∞,where φa(z)=(a-z)/(1-a¯z).Note that an assertion similar to Theorem1 can be found in [11]. The proof there, however, contains some inaccuracies and does not seem to yield the claimed result. It is also worth noticing that the weight function K is essentially so-called normal one in Theorem 1 [12].The notationf↗ means that f is essentially increasing; that is, f(r1)≲f(r2) for r1≤r2. The term essentially decreasing, shortly f↘, is understood in an analogous manner. Here a≍b if a≲b and a≳b. The notation a≲b is used if there exists a constant C>0 such that a≤Cb, and a≳b is understood in a similar manner. Note that the constant C may depend on fixed parameters or functions.The remainder of the paper is organized as follows. In the next section, some auxiliary results are presented. Section3 contains necessary conditions for singular inner functions to be in QK(p,q)-spaces and the main purpose of the last section is to prove Theorem 1. Also Proposition 7 and an alternative version of Theorem 1 are stated and proved in the last section.


## 2. Auxiliary Results

In this section, we present auxiliary results related toQK(p,q)-spaces and inner functions. The weight function K plays a crucial role in these results.For simplicity, we denote the following conditions:(a)
K is continuous and nondecreasing.(b)
∫01‍(1-r)qKlog⁡(1/r)rdr<∞.(c)
K(r)≍K(2r) for r∈(0,∞).If (b) does not hold, then QK(p,q) contains constant functions only [7]. Hence it is natural to assume (b), even though it would not be necessary. Nevertheless, it will always be mentioned if an assumption is made.In the next lemma, we recall some basic properties ofQK(p,q)-spaces. It follows by combining results of [7]. For the lemma, we denote that f∈H(D) belongs to Bα for α∈(0,∞) if supz∈D|f′(z)|(1-|z|)α<∞. If α=1, then we use the notation B. Here f∈H(D) means that f is analytic in D. Moreover, write Aqp=QK(p,q) if K≡1.Lemma A.
Letp∈(0,∞) and q∈(-2,∞), and assume that K satisfies (a) and (b). Then the following statements are valid: (i)
QK(p,q)⊂B(q+2)/p. Moreover, QK(p,q)=B(q+2)/p if and only if (5)∫011-r-2Klog⁡1rrdr<∞.(ii)
IfK1(r)=K(r) for r∈(0,1) and K1(s)=K(1) for s∈[1,∞), then QK1(p,q)=QK(p,q).(iii)
f∈QK(p,q) if and only if f∈H(D) and (6)supa∈D⁡∫Df′zp1-zqK1-φazdAz<∞.(iv)
Aqp⊂QK(p,q). Moreover, Aqp=QK(p,q) if and only if K(0)>0.It is worth noticing that the statements (i)–(iii) of Lemma A are valid even ifK is discontinuous. This can be seen by looking the proofs of [7, Theorems  2.1 and  3.1].Denote thatQ(I)={rξ:1-|I|<r<1,ξ∈I}, where I is a subarc of T such that |I|<1. Then a positive measure μ on D is a K-Carleson measure if (7)supI⁡∫QIK1-zIdμz<∞.Now we can characterizeQK(p,q)-spaces by using K-Carleson measures. This result has been earlier presented in [11, Theorem  1], but practically, it follows by modifying the proof of [10, Theorem  3.1].Lemma B.
Letp∈(0,∞) and q∈(-2,∞), and assume that K satisfies (a)–(c) and(8)∫01φKrdrr<∞,whereφKr=supt∈0,1⁡KrtKt.Then f∈QK(p,q) if and only if f∈H(D) and |f′(z)|p(1-|z|)qdA(z) is a K-Carleson measure.The next lemma shows that under certain conditions one may apply the Schwarz-Pick theorem without any essential loss of information. Before the lemma, we underline that the assumptions (a)–(c) are not necessary therein. In fact, it suffices to assume thatK:(0,1)→[0,∞) but, of course, if (b) does not hold, then the statements of the lemma are trivial.Lemma 2.
Letp∈[1,∞) and q∈(-2,∞), and let ϕ be an inner function. If (i)
p=1 and (9)esssupr∈δ,1⁡1-r-qK1-r1-δ-1∫δr1-sq-1K1-s1-δds≲1,δ∈0,1;(ii)
p∈(1,∞) and (10)supr∈δ,1⁡∫δr1-sq-pK1-s1-δds∫r11-sq/1-pK1-s1-δ1/1-pdsp-1≲1,δ∈0,1;then(11)∫δ11-ϕreiθp1-rq-pK1-r1-δdr≲∫δ1ϕ′reiθp1-rqK1-r1-δdris satisfied for all δ∈(0,1) and almost all θ∈[0,2π).Proof.
Assume first that (i) holds, and denoteωδ,q(r)=(1-r)qK(1-r)/(1-δ). Since 1-|ϕ(reiθ)|≤∫r1‍|ϕ′(seiθ)|ds for almost all θ∈[0,2π), by Fubini’s theorem and (i), we obtain (12)∫δ11-ϕreiθωδ,qr1-rdr≤∫δ1∫r1ϕ′seiθdsωδ,qr1-rdr=∫δ1ϕ′seiθ∫δsωδ,qr1-rdr≲∫δ1ϕ′seiθωδ,qsdsfor almost all θ∈[0,2π). If (ii) holds, then an application of [13, Theorem  2] yields (13)∫δ11-ϕreiθpωδ,qr1-rpdr≤∫δ1ωδ,qr1/p1-rp∫r1ϕ′seiθdspdr≲∫δ1ϕ′reiθpωδ,qrdrfor almost all θ∈[0,2π). This completes the proof.Corollary 3.
Letp∈[1,∞), q∈(-2,∞), and α∈[0,∞) such that p>q+α+1, and let ϕ be an inner function. If K satisfies r-αK(r)↘ for r∈(0,1), then (11) holds for all δ∈(0,1) and almost all θ∈[0,2π).Proof.
By the assumption, we have(14)∫δr1-sq-pK1-s1-δds=∫δr1-sq-pK1-s/1-δ1-s/1-δα1-s1-δαds≲K1-r/1-δ1-rα∫0r1-sα+q-pds≍1-rq+1-pK1-r1-δ,r∈δ,1.Similarly, for p>1, we obtain (15)∫r11-sq/1-pK1-s1-δ1/1-pdsp-1≲1-rp-q-1K1-r1-δ-1,r∈δ,1.Now the assertion follows from Lemma 2 applying the formulas above.It is worth noticing that [9, Lemma  2.1] is a special case of Lemma 2. This is easy to see by choosing K≡1 and α=0 in Corollary 3.Denote thatK satisfies (A), if (8) and (a)–(c) are satisfied, and (B) if (9) holds for p∈(0,1] and (10) holds for p∈(1,∞). Related to (A) and (B), we end this section by proving the following consequence of Lemma 2.Corollary 4.
Letp∈[1,∞) and q∈(-2,∞), and assume that K satisfies (A) and (B). Let ϕ=∏n=1k‍ϕn, where ϕn is an inner function for all n=1,…,k. Then ϕ∈QK(p,q) if and only if ϕn∈QK(p,q) for all n=1,…,k.Proof.
Hölder’s inequality yields|ϕ′(z)|p≤kp-1∑n=1k‍|ϕn′(z)|p. Hence ϕ∈QK(p,q) if ϕn∈QK(p,q) for all n=1,…,k. The other implication follows by applying Lemmas B and 2 together with the fact that |ϕ|≤|ϕn| for all n=1,…,k.


## 3. Singular Inner Functions andQK(p,q)-Spaces

This section contains necessary conditions for singular inner functions to be inQK(p,q)-spaces.WriteIξ=Iξ(|I|)={eiθ:θ∈(ξ-I/2,ξ+|I|/2)}, where ξ∈[0,2π) and |I|∈(0,1).Lemma 5.
Letp∈[1,∞) and q∈(-2,∞), and assume that K satisfies (A) and (B). If S is the singular inner function associated with a measure σ and there exists ξ∈[0,2π) such that lim|I|→0+σ(Iξ)/|I|=∞ and either (16)limI→0+⁡Iq+2-p∫I/σIξ1rq-pKrdr=∞or (17)limI→0+⁡Iq+2-pσIξIp∫0I/σIξrqKrdr=∞,then S∉QK(p,q).Proof.
Sincelim|I|→0+σ(Iξ)/|I|=∞, we may assume that |I|/σ(Iξ)<1. Then, for θ∈(ξ-(I/2),ξ+(|I|/2)), Lemma 2 yields (18)∫1-I1S′reiθp1-rqK1-rIrdr≍∫1-I11-Sreiθp1-rq-pK1-rIdr≳∫1-I11-exp⁡-∫θ-Iθ+I1-rreiθ-eit2dσtp1-rq-pK1-rIdr≳∫1-I11-exp⁡-σIξ1-rI2p1-rq-pK1-rIdr=Iq+1-p∫011-exp-σIξsIpsq-pKsds≍Iq+1-p∫0I/σIξ1-exp⁡-σIξsIpsq-pKsds+Iq+1-p∫I/σIξ1sq-pKsds≕J1Iξ+J2Iξ.Since 1-e-x≍x for all x∈(0,M) with fixed M∈(0,∞), we obtain (19)J1Iξ≍Iq+1-pσIξIp∫0I/σIξsqKsds.Hence the assertion follows by Lemma B.Ifσ(Iξ)>0 for all |I|>0, then lim|I|→0+σ(Iξ)/|I|=∞ by [14, Theorem  7.15]. In particular, there exists always ξ∈[0,2π) such that lim|I|→0+σ(Iξ)/|I|=∞. Hence the following result is a direct consequence of Lemma 5.Corollary 6.
Letp∈[1,∞) and q∈(-2,∞), and assume that K satisfies (A) and (B). If S is the singular inner function associated with a measure σ and (20)supξ∈0,2π⁡Iq+2-p∫I/σIξ1rq-pKrdr=∞,I⟶0+,then S∉QK(p,q).


## 4. Blaschke Products andQK(p,p-2)-Spaces

The main purpose of this section is to prove Theorem1. We begin by showing that all inner functions in QK(p,p-2) are Blaschke products if K satisfies certain regularity conditions. After that an alternative version of Theorem 1 is stated and proved; and finally, Theorem 1 follows by applying this result.Proposition 7.
Letp∈(0,∞) and q∈(-2,∞), and assume that a nondecreasing K satisfies (b). Then the following statements are valid: (i)
Ifp>q+2, then only inner functions in QK(p,q) are finite Blaschke products.(ii)
If∫01‍φK(r)(dr/r)<∞ and ∫1∞‍φK(r)(dr/r2)<∞, then only inner functions in QK(p,p-2) are Blaschke products.Proof.
Since only inner functions in VMOA are finite Blaschke products [15], the statement (i) follows from the inclusions QK(p,q)⊂B(q+2)/p⊂VMOA for p>q+2.
By using [16, Lemma  2] twice together with [16, Corollary  3], [10, Lemma  2.3], and Lemma A(ii), we can assume that K is second differentiable, K(r)≍K(2r) for r∈(0,∞), K(s)/s→∞ as s→0+, and K(t)=K(1) for t∈[1,∞). Hence it is clear that (A) holds. Moreover, the condition ∫01‍r-2K(r)dr=∞ follows from [11, Lemma  3] and (B) with q=p-2 follows from the proof of Corollary 3 using [10, Lemma  2.2].
We may assume thatp∈[1,∞) because the inclusion QK(p,p-2)⊂B yields QK(p1,p1-2)⊂QK(p2,p2-2) for 0<p1≤p2<∞. Hence Corollary 6 implies that QK(p,p-2) does not contain any singular inner functions. Now, since each inner function can be presented as a product of a Blaschke product and a singular inner function, the statement (ii) follows by Corollary 4.Remark 8.
In many cases,QK(p,p-2) contains also nontrivial Blaschke products unlike QK(p,p-2-ɛ) for ɛ∈(0,p). The contrast between these spaces is strong also in the general case. More precisely, using the inclusion QK(p,q)⊂B(q+2)/p and [1, Theorem  5.1], we find that if f∈QK(p,p-2-ɛ), then f(eiθ)∈Λɛ/p; that is, the boundary function satisfies the Lipschitz condition of order ɛ/p [1]. In particular, f belongs to the disc algebra A. On the other hand, by [10, Corollary  3.1], it is easy to find K such that log⁡(1-z) belongs to QK(2,0).Next we prove an alternative version of Theorem1. The proof uses some ideas from [17, 18]. For the result, denote p^=min⁡{p,1}.Theorem 9.
Letp∈(1/2,∞), and assume that K satisfies the following conditions: (i)
K is nondecreasing.(ii)
∫01‍(1-r)p-2Klog⁡(1/r)rdr<∞.(iii)
∫01‍φK(r)rp^-2dr<∞.(iv)
∫1∞‍φK(r)r-p^-1dr<∞.Then an inner function belongs to QK(p,p-2) if and only if it is a Blaschke product associated with a sequence {zn} satisfying (4).Proof.
By a similar manner as in the proof of Proposition7, we may assume that K satisfies the basic assumption of [11], which means that, in addition to (i) and (ii), K is continuous, K(r)≍K(2r) for r∈(0,∞) and K(s)=K(1) for s∈[1,∞).
Assume first that an inner functionB belongs to QK(p,p-2) for some p∈(1/2,∞). Then Proposition 7 implies that B is a Blaschke product. Hence, using [11, Theorem  10], we obtain that the zero sequence of B satisfies (4).
IfB is a Blaschke product associated with a sequence {zn} satisfying (4), then [11, Lemma  5] with parameters t=p^-2 and c=2(p^-1) yields (21)supa∈D⁡∑n1-φaznp^∫DK1-z1-φazn¯z2p^1-z2-p^dAz<∞.Therefore B∈QK(p,p-2) by [11, Proposition  8], and hence the assertion follows.We proceed to prove Theorem1. First, it is known that K satisfies (i) of Theorem 9. On the other hand, since log⁡(1/r)≍1-r for r∈(1/e,1), Lemma A(ii) together with the assumption yields (22)∫011-rp-2Klog⁡1rrdr≍K1+∫1/e11-rp-2Klog⁡1/rlog⁡1/rβlog1rβdr≲K1∫1/e11-rβ+p-2dr<∞.Therefore the condition (ii) holds. Moreover, we obtain (23)∫01φKrrp^-2dr=∫01supt∈0,1⁡KrtrtβKtrtβrp^-2dr≲∫01rβ+p^-2dr<∞,and consequently (iii) is satisfied. Regarding (iv), we may assume, by Lemma A(ii), that there exists α∈[β,p^) such that r-αK(r)↘ for all r∈(0,∞). Thus (24)∫1∞φKrr-p^-1dr=∫1∞supt∈0,1⁡KrtrtαKtrtαr-p^-1dr≲∫1∞rα-p^-1dr<∞;and hence, the assertion finally follows by Theorem 9.We end this paper with the following remark.Remark 10.
We make the following observations about Theorem1 and Proposition 7: (i)
IfK(r)=rs with s∈[0,∞), then QK(p,q)=F(p,q,s) [19, 20]. Related to F(p,q,s), Theorem 1 and Proposition 7 generalize [9, Theorems  1.3 and  1.4]. Moreover, by [10, Lemmas  2.1,  2.2 and Corollary  3.1], it is easy to see that Theorem 1 generalizes the essential content of [10, Theorem  5.1].(ii)
The Möbius invariance ofQK(p,p-2) plays an important role in the proof of [11, Theorem  10]. Hence, if one want to characterize Blaschke products in QK(p,q)-spaces where p<q+2, using a similar technique as in this paper, it is necessary to assume that QK(p,q) is Möbius invariant.



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