# Second-Order Linear Differential Equations with Solutions in Analytic Function Spaces

Authors: Jianren Long; Yu Sun; Shimei Zhang; Guangming Hu
Journal: Journal of Function Spaces
 (2019)

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## Abstract

This research is concerned with second-order linear differential equationf′′+A(z)f=0, where A(z) is an analytic function in the unit disc. On the one hand, some sufficient conditions for the solutions to be in α-Bloch (little α-Bloch) space are found by using exponential type weighted Bergman reproducing kernel formula. On the other hand, we find also some sufficient conditions for the solutions to be in analytic Morrey (little analytic Morrey) space by using the representation formula.


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## Body

## 1. Introduction

We shall assume that the reader is familiar with the definitions of classical function spaces, for example, Hardy space, Bergman space, and Bloch space. One of main objectives in the research of complex linear differential equations with analytic coefficients in unit disc is to consider the relationship between the growth of coefficients and the growth of solutions. Many results concerning fast growing solutions have been obtained by Nevanlinna and Wiman-Valiron theories; for example, see [1–5] and reference therein. However, it is very difficult to study the slowly growing solutions of complex linear differential equations by using Nevanlinna and Wiman-Valiron theories; hence, the different approaches are employed, for example, Herold’s comparison theorem [6], Gronwall’s lemma [7], Picard’s successive approximations [8], and some methods based on Carleson measures [9]. There are many results concerning the slowly growing solutions that have been obtained; for example, see [6, 7, 9–11] and reference therein.In [9], Pommerenke considered the second-order equation: (1)f′′+Azf=0,where A(z) is an analytic function in the unit disc D=z:z<1 and obtained a sharp sufficient conditions for A(z) which guarantee all solutions of (1) are in the classical Hardy space H2(D). The coefficient condition was given in terms of Carleson measures by Green’s formula and Carleson’s theorem for Hardy space. The leading method of this approach has been extended to study, for example, solutions in the Hardy space and Bergman space [6] and Dirichlet type space [10].Recently, Gröhn-Huusko-Rättyä [12] found sufficient conditions for the coefficient A(z) guaranteeing all solutions of (1) are in Bloch (little Bloch) space by taking advantage of the reproducing formula in small weighted Bergman space [13]. Motivated from [12], now we use the reproducing formula in Bergman space with exponential type weights to get the conditions for the coefficient A(z) such that all solutions of (1) are in α-Bloch (little α-Bloch) space. In the same paper, they obtained also some sufficient conditions on the coefficient A(z) which guarantee all solutions of (1) belong to BMOA (VMOA) space, in which some properties of Bloch space were used. It follows from [14–16] that BMOA space is a special kind of analytic Morrey space, and the analytic Morrey space and Bloch space are different. Therefore, we obtain also some sufficient conditions on A(z) which guarantee all solutions of (1) to be in analytic Morrey (little analytic Morrey) space by using different ways in [12]. The definition of BMOA space and analytic Morrey space are recalled in Section 2 below.Note. If A≲B, then there exists a positive constant C such that A≤CB. Similarly, if A≳B, then there exists a positive constant C such that A≥CB. If A and B satisfy A≳B and A≲B, then we say that A and B are comparable, by A≈B. Note that σa(z)=(a-z)/(1-a¯z) is the Möbius transformation of D for a,z∈D, and dm(z) denotes the normalized Lebesgue area measure.The paper is organized as follows: some definitions of related function spaces are recalled in Section2. Some results in which all solutions of (1) are in α-Bloch spaces are discussed in Section 3. We obtain some results in which all solutions of (1) are in analytic Morrey spaces in Section 4.


## 2. Some Related Function Spaces and Notations

Let∂D be the boundary of D and A(D) be the space of analytic functions in D. For 1≤p<∞, Hardy space Hp(D) consists of f∈A(D) with(2)fHpp=sup0<r<1⁡12π∫02πfreiθpdθ<∞.For a subarc I⊂∂D, the length of I is defined as(3)I=12π∫Idζ,and let(4)SI=rζ∈D:1-I≤r<1,ζ∈Idenote the Carleson square in D.Denote(5)fI=1I∫Ifζdζ2πas the average of f∈A(D) over I. BMOA space consists of those functions f∈H2(D) such that(6)fBMOA=supI⊂∂D⁡1I∫Ifζ-fI2dζ2π1/2<∞.VMOA space consists of those functions f∈BMOA such that(7)limI→0⁡1I∫Ifζ-fI2dζ2π=0.Morrey spaces were introduced in the 1930s [17] in connection to partial differential equations and were subsequently studied as function spaces in harmonic analysis on Euclidean spaces. The analytic Morrey space L2,λ(D)(0≤λ≤1) was introduced recently by Wu and Xie [15], and then many researchers pay attention to the spaces; for example, see [16, 18, 19] and reference therein. The analytic Morrey space L2,λ(D) consists of those functions f∈H2(D) such that(8)fL2,λ=supI⊂∂D⁡1Iλ∫Ifζ-fI2dζ2π1/2<∞.Little analytic Morrey space L02,λ(D) consists of those functions f∈L2,λ(D) and(9)limI→0⁡1Iλ∫Ifζ-fI2dζ2π=0.Clearly, for λ=0 or λ=1, L2,λ(D) reduces to H2(D) and BMOA, respectively; hence the case of 0<λ<1 is marked, and then BMOA⊂L2,λ(D)⊂H2(D).The following lemma gives some equivalent conditions ofL2,λ(D) by [19, Theorem 3.1] or [20, Theorem 3.21].Lemma 1.
Suppose that0<λ≤1 and f∈A(D). Then the following statements are equivalent: (i)
f∈L2,λ(D).(ii)
supa∈D⁡1-a21-λ∫Df′z21-σaz2dm(z)<∞.(iii)
supI⊂∂D⁡1/Iλ∫SIf′z21-z2dm(z)<∞.(iv)
supa∈D⁡1-a2(1-λ)/2f∘σa-f(a)H2<∞.From Lemma1, we see the following.Lemma 2.
Letf and λ be defined as Lemma 1. Then f∈L02,λ(D) if and only if(10)lima→1-⁡1-a21-λ∫Df′z21-σaz2dmz=0or(11)lima→1-⁡1-a21-λ/2f∘σa-faH2=0.Here we use the reproducing formula in Bergman space with exponential type weights to study (1). To this end, let 0<p<∞; the exponential type weighted Bergman space AP(ωγ,α) is the space of functions f∈A(D) such that(12)fAPωγ,αp≔∫Dfzpωγ,αzdmz<∞,where ωγ,α is the exponential type weight as(13)ωγ,αρ=1-ργexp⁡-c1-ρα,γ≥0,c>0,α>0.In [21], it is proved that the point evaluation Lζfz=f(ζ) is bounded linear function on AP(ωγ,α). Hence there exists a reproducing kernel Kζ∈AP(ωγ,α) with KζA2(ωγ,α)=Lζ such that(14)fζ=Lζfz≔∫DfzKζz¯ωγ,αzdmz,f∈APωγ,α,where(15)Kζz=∑n=0∞zζ¯n2∫01ρ2n+1ωγ,αρdρ.Finally, we recall the definition ofα-Bloch space Bα, which can be found in [22]. Let α∈(0,∞); it is defined as(16)Bα≔h∈AD:hBα≔supz∈Dh′z1-z2α<∞.Little α-Bloch space B0α is the subspace of Bα consisting of functions h with(17)limz→1-⁡h′z1-z2α=0.Clearly, Bα (B0α) is the Bloch (little Bloch) space for α=1.


## 3. Sufficient Conditions of Solutions inBα(B0α)

In the section, we consider the estimation of coefficient similarly toSection 7 in [12], in which the exponential type weighted Bergman reproducing kernel formula is used. Some sufficient conditions on A(z) guaranteeing all solutions of (1) to be in Bα (B0α) are obtained by these estimates as follows.Theorem 3.
Letωγ,α(ρ) be an exponential type weight. If(18)limρ→1-¯supz∈D⁡1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmηis sufficiently small, then all solutions of (1) belong to B2(1+α).Theorem 4.
Letωγ,α(ρ) be an exponential type weight. Suppose that the condition of Theorem 3 is satisfied, and(19)limz→1-¯limρ→1-¯1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmη=0.Then all solutions of (1) belong to B02(1+α).In order to prove Theorems3 and 4, we need the following lemma.Lemma 5 (see [21]).
Iff,g∈A2(ωγ,α), then(20)∫Dfzgz¯ωγ,αzdmz=∫Df′zg′z¯1-z21+αωγ,αzdmz+f0g0¯.Proof of Theorem3.
Letf be any solution of (1), then fρ is analytic in D¯ and satisfies fρ′′(z)+ρ2A(ρz)fρ(z)=0. Therefore,(21)fρ′z=-∫0zρ2fρζAρζdζ+fρ′0,z∈D.By the reproducing formula and Fubini’s theorem, (22)fρ′z=-∫0z∫DfρηKζη¯ωγ,αηdmηρ2Aρζdζ+fρ′0=-∫Dfρηωγ,αη∫0zKζη¯ρ2Aρζdζdmη+fρ′0,z∈D.Applying Lemma 5, (23)fρ′z=-∫Dfρ′η1-η21+αωγ,αη∫0zKζ′η¯ρ2Aρζdζdmη+fρ0∫0zρ2Aρζ2∫01ρωγ,αρdρdζ+fρ′0.Therefore, (24)fρB21+α≲fρB21+αsupz∈D⁡1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmη+fρ0supz∈D1-z21+α∫0zAρζ2∫01ρωγ,αρdρdζ+fρ′0,z∈D.This implies that (25)fρB21+α1-Csupz∈D⁡1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmη≲fρ0supz∈D⁡1-z21+α∫0zAρζdζ+fρ′0,where C is a positive constant and 0<ρ<1. It follows from Fubini’s theorem and the reproducing formula that (26)supz∈D⁡1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmη=supz∈D⁡1-z21+α∫0z∫Dωγ,αηKζ′η¯dmηAρζdζ≳supz∈D⁡1-z21+α∫0zAρζdζ.Therefore, we deduce f∈B2(1+α) by the above formula and (25) and letting ρ→1-.Proof of Theorem4.
By Theorem3, we know that f∈B2(1+α). Fubini’s theorem and the reproducing formula yield(27)limz→1-¯limρ→1-¯1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmη≳limz→1-¯limρ→1-¯1-z21+α∫0zAρζdζ.The condition of Theorem 4 implies (28)limz→1-¯limρ→1-¯1-z21+α∫0zAρζdζ=0.By using the similar reason as in the proof of Theorem 3, we have (29)1-z21+αfρ′z≲fρB21+α1-z21+α∫Dωγ,αη∫0zKζ′η¯Aρζdζdmη+fρ01-z21+α∫0zAρζdζ+1-z21+αfρ′0,z∈D.Then by the condition of Theorem 4,(30)limz→1-¯limρ→1-¯1-z21+αfρ′z=0.Therefore, the assertion follows.


## 4. Sufficient Conditions of Solutions inL2,λ(D)(L02,λ(D))

In the section, some sufficient conditions onA(z) which guarantee all solutions of (1) belong to analytic Morrey space are obtained by using two ideas. On the one hand, a sufficient condition on A(z) guaranteeing that all solutions of (1) belong to BMOA is obtained in [12, Theorem 3]; here we study the condition on A(z) which guarantee that all solutions of (1) are in analytic Morrey space by using similar idea in [12, Theorem 3]. On the other hand, a sufficient condition on A(z) guaranteeing that all solutions of (1) are in analytic Morrey space is shown by using representation formula.It is also known that BMOA space is a subspace of Bloch space, and then the properties of Bloch space were used in the proof of [12, Theorem 3]. However, it was proved that analytic Morrey space and Bloch space are different in [16]. Therefore, we need different methods from Section 6 of [12] to deal with the case of analytic Morrey space; the following results are proved.Theorem 6.
LetA(z)∈A(D) and 0<λ<1. Suppose that (31)supa∈D∫DAz21-z221-σaz2dmzis sufficiently small and(32)supz∈DAz1-z5-λ/2∫0zdr1-r3-λ/2<3-λ2.Then all solutions of (1) are in L2,λ(D).Theorem 7.
LetA(z)∈A(D) and 0<λ<1. Suppose that the conditions of Theorem 6 are satisfied, and(33)lima→1-∫DAz21-z221-σaz2dmz=0and(34)limz→1-Az1-z5-λ/2∫0zdr1-r3-λ/2=0.Then all solutions of (1) are in L02,λ(D).In order to prove Theorems6 and 7, we recall the definition of Carleson measure. For a non-negative measure μ on D, it is called a Carleson measure if(35)μ≕supI⊂∂DμSII1/2<∞.Moreover, if in addition (36)limI→0μSII=0,then μ is called a compact Carleson measure.For the proof of Theorems6 and 7, the following lemmas are needed.Lemma 8 (sse [14]).
Suppose thatμ is a Carleson measure. Then(37)μ≈supa∈D∫Dσa′zdμz.Lemma 9 (see [23]).
Let0<p,q<∞. If f∈A(D), then (38)∫Dfzp1-z2qdmz≲f0p+∫Df′zp1-z2p+qdmz.Lemma 10 (see [24]).
Let0<p,q<∞. If f∈A(D), then(39)fnap1-a2q+2≲∫Dfnzp1-z2q1-σaz2dmz.Lemma 11 (see [22]).
Suppose that0<p,q<∞. Then, for f∈A(D),(40)supa∈D⁡fnap1-a2qand(41)supa∈D⁡fn+1ap1-a2q+pare comparable.Lemma 12.
Let0<λ<1 and f∈L2,λ. Then (42)fL2,λ2≲supa∈Df′a21-a23-λ+supa∈D1-a21-λ∫Df′′z21-z221-σaz2dmz.Proof.
An auxiliary function is constructed as follows:(43)gz=f′σaz1-a21-a¯z2.By Lemmas 1 and 9,(44)fL2,λ2≈1-a21-λ∫Df′z21-σaz2dmz=1-a21-λ∫Df′σaz21-σa∘σaz2σa′z2dmz=1-a21-λ∫Df′σaz21-σaz22dmz1-z2=1-a21-λ∫Dgz21-z2dmz≲g021-a21-λ+1-a21-λ∫Dg′z21-z23dmz,where g02=f′a21-a22.
Note that(45)1-a21-λ∫Dg′z21-z23dmz≤I1+I2,where(46)I1=1-a21-λ∫Df′σaz′21-a¯z41-a221-z23dmzand(47)I2=1-a21-λ∫Df′σaz21-a221-a¯z61-z23dmz.Clearly,(48)I1≲1-a21-λ∫Df′′σaz21-σaz241-z2dmz=1-a21-λ∫Df′′z21-z221-σaz2dmz.It follows from Lemmas 10 and 11 that (49)I2=1-a21-λ∫Df′σaz21-σaz221-a¯z21-z2dmz≤1-a21-λsupa∈D⁡f′a21-a22∫D1-z21-a¯z2dmz≲1-a21-λsupa∈Df′′a21-a24∫D1-z21-a¯z2dmz≲1-a21-λ∫Df′′z21-z221-σaz2dmz∫D1-z21-a¯z2dmz.It is easy to get that supa∈D∫D((1-z2)/1-a¯z2)dm(z) is finite. Then the lemma is completely proved.Lemma 13 (see [25]).
Suppose thatμ is a positive Borel measure and 0<p<∞. Then μ is a Carleson measure if and only if Hp(D)⊂Lp(dμ). Namely, if there exists a continuous including mapping i:Hp(D)→Lp(dμ), then(50)fLpdμ≲μfHpD.Lemma 14 (see [18]).
Let0<λ<1. If f∈L2,λ, then(51)fz≲fL2,λ1-z21-λ/2,z∈D.Proof of Theorem6.
We can obtain that all solutionsf of (1) belong to B(3-λ)/2 by [11, Corollary 4] and (32). It follows from Lemma 12 that (52)fρL2,λ2≲supa∈D⁡1-a23-λf′ρz2ρ2+1-a21-λ∫Dρ4f′′ρz21-z221-σaz2dmz≲fρB3-λ/22+supa∈D⁡1-a21-λ·∫Dfρz-fρa2Aρz21-z221-σaz2dmz+supa∈D⁡1-a21-λfρa2∫DAρz21-z221-σaz2dmz≲fρB3-λ/22+J1+J2,where fρ(z)=f(ρz). By Lemmas 1, 8, and 13, (53)J1≲supa∈D⁡1-a21-λ∫Dfρσz-fρa2Aρσaz2·1-σaz23σa′zdmz≲supa∈D⁡1-a21-λfρσz-fρaH22·supb∈D⁡∫DAρσaz21-σaz23σa′zσb′zdmz≲fρL2,λ2supa∈D⁡supb∈D⁡∫DAρσaz21-σaz23σa′zσb′zdmz≲fρL2,λ2supa∈D⁡supb∈D∫DAρz21-z221-σb∘σa-1z2dmz≲fρL2,λ2supc∈D∫DAρz21-z221-σcz2dmz,where dμ(z)=Aρσaz21-σaz23σa′zdm(z) is a Carleson measure.
It follows from Lemma14 that (54)
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.Denote(55)Ja,ρ=∫DAρz21-z221-σaz2dmz,0<ρ<1.For |a|≤1/2, the estimate is trivial. Let 1/2≤a≤1/(2-ρ). Since 1-a¯z≤21-a¯z/ρ for z≤ρ, (56)Ja,ρ=∫D0,ρAz21-zρ221-a21-a¯z/ρdmzρ2≤4ρ2∫DAz21-z221-σaz2dmz≤16∫DAz21-z221-σaz2dmzfor any 1/2<ρ<1. Let 1/(2-ρ)≤a≤1. Now (57)Ja,ρ≤supa∈D⁡Az21-z24∫D1-z221-σaz21-ρz24dmz.By Lemma 10, we get (58)Ja,ρ≲supa∈D⁡∫DAz21-z221-σaz2dmz·∫D1-z221-σaz21-ρz24dmz.By [12, Theorem 3], (59)Ja,ρ≲supa∈D⁡∫DAz21-z221-σaz2dmz·∫011-s31-a1-ρs41-asds≲supa∈D⁡∫DAz21-z221-σaz2dmz·∫011-a1-ρs1-asds=supa∈D⁡∫DAz21-z221-σaz2dmz·1-aa-ρlog⁡1+a-ρ1-a.Since ρ≤2-1/a, 1-a/a-ρlog⁡(1+(a-ρ)/(1-a)) is bounded. Therefore, (60)Ja,ρ≲supa∈D⁡∫DAz21-z221-σaz2dmz.If (31) is sufficiently small, then the norm fρL2,λ2 is uniformly bounded for 1/2<ρ<1. By letting ρ→1-, we reduce f∈L2,λ.Proof of Theorem7.
We can get that all solutions of (1) belong to B0(3-λ)/2 by the similar reason as in the proof of [11, Theorem 1]; here it is omitted. The next proof is also similar to the proof of Theorem 6; here it is also omitted.Next we consider the estimate of coefficient similarly toSection 8 in [12], in which the well-known representation formula(61)hz=12π∫02πheis1-ze-isds,z∈D,is used, where h∈H1(D); it is found in [26, Theorem 3.6]. Here, we prove the following results.Theorem 15.
LetA∈A(D) and 0<λ≤1. Suppose that(62)limρ→1-¯supa∈D⁡1-a21-λ∫D1-σaz2∫0zAρζdζ2dmzis bounded and(63)limρ→1-¯supa∈D⁡∫D1-σaz2∫Dhρ,z′s2log⁡s21-σas2dmsdmzis sufficiently small, where hρ,z(s)=∫0z(A(ρς)/(1-s¯ς))dς¯. Then all solutions of (1) are in L2,λ(D).Theorem 16.
LetA∈A(D) and 0<λ≤1. Suppose that the conditions of Theorem 15 are satisfied, and(64)lima→1-¯limρ→1-¯1-a21-λ∫D1-σaz2∫0zAρζdζ2dmz=0,lima→1-¯limρ→1-¯∫D1-σaz2∫Dhρ,z′s2log⁡s21-σas2dmsdmz=0.Then all solutions of (1) are in L02,λ(D).In order to prove Theorems15 and 16, the following lemma is needed.Lemma 17 (see [12]).
Iff,g∈H2(D), then(65)12π∫02πfeitgeit¯dt=2∫Df′sg′s¯log⁡1sdms+f0g0¯.Proof of Theorem15.
Letf be any solution of (1). Then fρ is analytic in D¯ and satisfies fρ′′(z)+ρ2A(ρz)fρ(z)=0. Therefore,(66)fρ′z=-∫0zρ2fρςAρςdς+fρ′0,z∈D.By the representation formula and Fubini’s theorem, (67)fρ′z=-12π∫02πfρeis∫0zρ2Aρς1-e-isςdςds+fρ′0=-ρ22π∫02πfρeishρ,zeis¯ds+fρ′0,z∈D,where(68)hρ,zs=∫0zAρς1-s¯ςdς¯.Since fρ,hρ,z∈H2(D), Lemma 17 implies(69)12π∫02πfρeishρ,zeis¯dt=2∫Dfρ′shρ,z′s¯log⁡1sdms+fρ0hρ,z0¯.So,(70)fρ′z2≤8∫Dfρ′shρ,z′s¯log⁡1sdms2+2fρ0hρ,z0¯-fρ′02≲∫Dfρ′s21-σas2dms∫Dhρ,z′s2log⁡s21-σas2dms+4fρ0hρ,z0¯2+4fρ′02.Therefore, by Lemma 1, (71)fρL2,λ2≈supa∈D⁡1-a21-λ∫Dfρ′z21-σaz2dmz≲fρL2,λ2supa∈D⁡∫D1-σaz2∫Dhρ,z′s2log⁡s21-σas2dmsdmz+4fρ02supa∈D1-a21-λ∫D1-σaz2hρ,z02dmz+4fρ′02supa∈D⁡1-a21-λ∫D1-σaz2dmz.Then(72)fρL2,λ21-Csupa∈D⁡∫D1-σaz2∫Dhρ,z′s2log⁡s21-σas2dmsdmz≲4fρ02supa∈D⁡1-a21-λ∫D1-σaz2∫0zAρζdζ2dmz+4fρ′02,where C is a positive constant. By letting ρ→1-, (62) and (63), the assertion follows.Proof of Theorem16.
By the assumption and Theorem15, we get f∈L2,λ. By using the similar reason as the proof of Theorem 15 and the condition of Theorem 16, we have (73)lima→1-¯limρ→1-¯1-a21-λ∫Dfρ′z21-σaz2dmz≲fρL2,λ2lima→1-¯limρ→1-¯∫D1-σaz2∫Dhρ,z′s2log⁡s21-σas2dmsdmz+4fρ02lima→1-¯limρ→1-¯1-a21-λ∫D1-σaz2∫0zAρζdζ2dmz+4fρ′02lima→1-¯1-a21-λ∫D1-σaz2dmz=0.The assertion follows by Lemmas 1 and 2.



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