Second-Order Linear Differential Equations with Solutions in Analytic Function Spaces
This research is concerned with second-order linear differential equation where is an analytic function in the unit disc. On the one hand, some sufficient conditions for the solutions to be in -Bloch (little -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.
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 , Gronwall’s lemma , Picard’s successive approximations , and some methods based on Carleson measures . There are many results concerning the slowly growing solutions that have been obtained; for example, see [6, 7, 9–11] and reference therein.
In , Pommerenke considered the second-order equation: where is an analytic function in the unit disc and obtained a sharp sufficient conditions for which guarantee all solutions of (1) are in the classical Hardy space . 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  and Dirichlet type space .
Recently, Gröhn-Huusko-Rättyä  found sufficient conditions for the coefficient guaranteeing all solutions of (1) are in Bloch (little Bloch) space by taking advantage of the reproducing formula in small weighted Bergman space . Motivated from , now we use the reproducing formula in Bergman space with exponential type weights to get the conditions for the coefficient such that all solutions of (1) are in -Bloch (little -Bloch) space. In the same paper, they obtained also some sufficient conditions on the coefficient 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 which guarantee all solutions of (1) to be in analytic Morrey (little analytic Morrey) space by using different ways in . The definition of BMOA space and analytic Morrey space are recalled in Section 2 below.
Note. If , then there exists a positive constant such that . Similarly, if , then there exists a positive constant such that . If and satisfy and , then we say that A and B are comparable, by . Note that is the Möbius transformation of for , and denotes the normalized Lebesgue area measure.
The paper is organized as follows: some definitions of related function spaces are recalled in Section 2. Some results in which all solutions of (1) are in -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 be the boundary of and be the space of analytic functions in . For , Hardy space consists of withFor a subarc , the length of is defined asand letdenote the Carleson square in .
Denoteas the average of over . BMOA space consists of those functions such thatVMOA space consists of those functions such that
Morrey spaces were introduced in the 1930s  in connection to partial differential equations and were subsequently studied as function spaces in harmonic analysis on Euclidean spaces. The analytic Morrey space was introduced recently by Wu and Xie , and then many researchers pay attention to the spaces; for example, see [16, 18, 19] and reference therein. The analytic Morrey space consists of those functions such thatLittle analytic Morrey space consists of those functions andClearly, for or , reduces to and BMOA, respectively; hence the case of is marked, and then BMOA.
Lemma 1. Suppose that and . Then the following statements are equivalent: (i).(ii).(iii).(iv).
From Lemma 1, we see the following.
Lemma 2. Let and be defined as Lemma 1. Then if and only ifor
Here we use the reproducing formula in Bergman space with exponential type weights to study (1). To this end, let ; the exponential type weighted Bergman space is the space of functions such thatwhere is the exponential type weight asIn , it is proved that the point evaluation is bounded linear function on Hence there exists a reproducing kernel with such thatwhere
Finally, we recall the definition of -Bloch space , which can be found in . Let ; it is defined asLittle -Bloch space is the subspace of consisting of functions withClearly, () is the Bloch (little Bloch) space for .
3. Sufficient Conditions of Solutions in ()
In the section, we consider the estimation of coefficient similarly to ection in , in which the exponential type weighted Bergman reproducing kernel formula is used. Some sufficient conditions on guaranteeing all solutions of (1) to be in () are obtained by these estimates as follows.
Theorem 3. Let be an exponential type weight. Ifis sufficiently small, then all solutions of (1) belong to .
Lemma 5 (see ). If , then
Proof of Theorem 3. Let be any solution of (1), then is analytic in and satisfies . Therefore,By the reproducing formula and Fubini’s theorem, Applying Lemma 5, Therefore, This implies that where is a positive constant and . It follows from Fubini’s theorem and the reproducing formula that Therefore, we deduce by the above formula and (25) and letting .
Proof of Theorem 4. By Theorem 3, we know that . Fubini’s theorem and the reproducing formula yieldThe condition of Theorem 4 implies By using the similar reason as in the proof of Theorem 3, we have Then by the condition of Theorem 4,Therefore, the assertion follows.
4. Sufficient Conditions of Solutions in ()
In the section, some sufficient conditions on 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 guaranteeing that all solutions of (1) belong to BMOA is obtained in [12, heorem 3]; here we study the condition on which guarantee that all solutions of (1) are in analytic Morrey space by using similar idea in [12, heorem 3]. On the other hand, a sufficient condition on 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, heorem 3]. However, it was proved that analytic Morrey space and Bloch space are different in . Therefore, we need different methods from ection 6 of  to deal with the case of analytic Morrey space; the following results are proved.
Theorem 6. Let and . Suppose that is sufficiently small andThen all solutions of (1) are in .
In order to prove Theorems 6 and 7, we recall the definition of Carleson measure. For a non-negative measure on , it is called a Carleson measure ifMoreover, if in addition then is called a compact Carleson measure.
Lemma 8 (sse ). Suppose that is a Carleson measure. Then
Lemma 9 (see ). Let . If , then
Lemma 10 (see ). Let . If , then
Lemma 11 (see ). Suppose that . Then, for ,andare comparable.
Lemma 12. Let and . Then
Proof. An auxiliary function is constructed as follows:By Lemmas 1 and 9,where .
Note thatwhereandClearly,It follows from Lemmas 10 and 11 that It is easy to get that is finite. Then the lemma is completely proved.
Lemma 13 (see ). Suppose that is a positive Borel measure and . Then is a Carleson measure if and only if . Namely, if there exists a continuous including mapping , then
Lemma 14 (see ). Let . If , then
Proof of Theorem 6. We can obtain that all solutions of (1) belong to by [11, orollary 4] and (32). It follows from Lemma 12 that where . By Lemmas 1, 8, and 13, where is a Carleson measure.
It follows from Lemma 14 that DenoteFor , the estimate is trivial. Let . Since for , for any . Let . Now By Lemma 10, we get By [12, heorem 3], Since , is bounded. Therefore, If (31) is sufficiently small, then the norm is uniformly bounded for . By letting , we reduce .
Proof of Theorem 7. We can get that all solutions of (1) belong to by the similar reason as in the proof of [11, heorem 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 to ection in , in which the well-known representation formulais used, where ; it is found in [26, heorem 3.6]. Here, we prove the following results.
Theorem 15. Let and . Suppose thatis bounded andis sufficiently small, where . Then all solutions of (1) are in .
Lemma 17 (see ). If , then