disc D, for which |f(z)|(1 â |z|2)q is uniformly bounded in D, and f â Dp if the integral /D |f ... + ··· + A1(z)f + A0(z)f = 0 with entire coefficients is relatively well known in the complex plane [15], ... f(k) + A(z)f = 0, where A(z) is analytic in D and k â N. .... Three different proofs for Theorem 3.1 are given in Section 5. Corollary 3.2.
J. Heittokangas, R. Korhonen and J. R¨ atty¨ a Nagoya Math. J. Vol. 187 (2007), 91–113
LINEAR DIFFERENTIAL EQUATIONS WITH SOLUTIONS IN THE DIRICHLET TYPE SUBSPACE OF THE HARDY SPACE ¨ ¨ J. HEITTOKANGAS, R. KORHONEN and J. R ATTY A Dedicated to Nikolaos Danikas Abstract. Sufficient conditions for the analytic coefficients of the linear differential equation f (k) + Ak−1 (z)f (k−1) + · · · + A1 (z)f 0 + A0 (z)f = 0 are found such that all solutions belong to a given Hq∞ -space, or to the Dirichlet type subspace D p of the classical Hardy space H p , where 0 < p ≤ 2. For 0 < q < ∞, the space Hq∞ consists of those functions f , analytic in the unit disc D, for which |f (z)|(1 − |z|2 )q is uniformly bounded in D, and f ∈ D p if R the integral D |f 0 (z)|p (1 − |z|2 )p−1 dσz converges.
§1. Introduction The growth of entire solutions of the linear differential equation (1.1)
f (k) + Ak−1 (z)f (k−1) + · · · + A1 (z)f 0 + A0 (z)f = 0
with entire coefficients is relatively well known in the complex plane [15], [20], [23], [32], where efficient tools, such as Wiman-Valiron and Nevanlinna theories, are available. As for local considerations, Nevanlinna theory has been applied to fast growing analytic solutions [3], [4], [7], [8], [16], [18], [21], [22], but the analysis of slowly growing solutions seems to require a different approach [16], [17], [19], [26], [30]. Chr. Pommerenke [26] studied the second-order equation (1.2)
f 00 + A(z)f = 0,
where A(z) is an analytic function in the unit disc D = {z : |z| < 1}. Received August 9, 2005. 2000 Mathematics Subject Classification: Primary 34M10; Secondary 30D50, 30D55.
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¨ ¨ J. HEITTOKANGAS, R. KORHONEN AND J. RATTY A
Theorem A. ([26, Theorem 2]) Let A(z) be an analytic function in D. There is an absolute constant α > 0 with the following property: If Z 1 − |a|2 dσz ≤ α, (1.3) sup |A(z)|2 (1 − |z|2 )3 |1 − a ¯z|2 a∈D D then all solutions of (1.2) belong to the Hardy space H 2 . The definitions of the Hardy spaces and other relevant concepts are postponed to Section 2 below. Theorem B. ([26, Theorem 3]) Let A(z) be an analytic function in D, and let 0 < δ0 < 1. There is an absolute constant β > 0 with the following property: If (1.4)
1 sup sup 0≤θ≤2π 0 0, there exists an α, depending only on q and k, such that whenever |A(z)|(1 − |z|) k ≤ α, then all solutions of (1.6) belong to Hq∞ . The purpose of this study is to find sufficient conditions for the analytic coefficients of the linear differential equation (1.1) such that all solutions belong to a given weighted H ∞ -space, or to the Dirichlet type subspace D p of the Hardy space H p , where 0 < p ≤ 2. In particular, the results obtained generalize Theorems A, B and D and Corollary C to equation (1.1). A number of related results are also presented including, for instance, boundary versions of the generalizations of Theorems A and D. The remainder of the paper is organized as follows. The notation is fixed and the required function spaces are defined in Section 2. The results are presented and analyzed in Section 3, where some examples are also given. The necessary auxiliary results, which will be repeatedly used in the proofs of the results in Section 5, are listed in Section 4. §2. Notation Throughout the paper, D(0, R) denotes the Euclidian disc of radius R centered at the origin, so D(0, 1) = D. For 0 < p ≤ ∞, the Hardy space H p consists of those functions f , analytic in D, for which (2.1)
kf kH p = sup Mp (r, f ) < ∞, 0≤r 0}. Theorem 3.3. Let 0 < δ < 1. For every q > 0 there exists a constant α = α(q, k) > 0 such that if the coefficients A j (z) of (1.1) satisfy (3.2)
sup |Aj (z)|(1 − |z|2 )k−j ≤ α,
|z|≥δ
j = 0, . . . , k − 1,
then all solutions of (1.1) belong to H q∞ . Example 3.4. The functions (3.3)
1
fn (z) = (1 − z) 2 (−a1 +1+(−1)
n
√
(a1 −1)2 +4a0 )
,
n = 1, 2,
are linearly independent solutions of the differential equation a1 0 a0 f = 0, (3.4) f 00 − f − 1−z (1 − z)2 where a0 , a1 ∈ R such that (a1 − 1)2 + 4a0 > 0. If a1 > q + 1 − a0 /q then f1 6∈ Hq∞ , and therefore the constant α in Theorems 3.1 and 3.3 satisfies α ≤ 2 min max{|q + 1 − a0 /q|, 2|a0 |} = a0 ∈R
4q(q + 1) 2q + 1
LINEAR ODE’S WITH SOLUTIONS IN DIRICHLET TYPE SPACES
97
in the case k = 2. This clearly implies that α tends to zero as q tends to zero. The functions in (3.3) and f3 (z) = (1 − z)2 are linearly independent solutions of f 000 −
a1 00 a0 + a 1 0 2a0 f − f − f = 0, 2 1−z (1 − z) (1 − z)3
where a0 , a1 ∈ R. Therefore the constant α in Theorems 3.1 and 3.3 satisfies α ≤ 2 min max{|q + 1 − a0 /q|, 2|q + 1 + a0 (1 − 1/q)|, 8|a0 |} a0 ∈R 16q(q + 1) 3q + 1 , 0 < q ≤ 1, = 16q(q + 1) , 1 ≤ q < ∞, 5q − 1
in the case k = 3. Once again this implies that α tends to zero as q tends to zero. As an immediate consequence of Theorem 3.3 it is deduced that if |Aj (z)|(1 − |z|2 )k−j tends to zero as z approaches the boundary ∂D, then all solutions of (1.1) belong to Hq∞ for all q > 0. ∞ Corollary 3.5. If Aj ∈ Hk−j,0 for j = 0, . . . , k − 1, then all solutions T ∞ of (1.1) belong to 0 0 be small enough so that 1 + q0 < (q + s + 1)/p. By Theorem 3.1, there exists a constant α > 0 such that if (3.1) holds, then all solutions of (1.1) belong to H q∞ . Now, by Proposition E 0 q0 +1 ⊂ F (p, q, s), which = B and [33, Proposition 5.7], it follows that H q∞ 0 0 yields the assertion. Proof of Theorem 3.3. Without loss of generality, assume that α ≤ 1 and δ < r < 1. By [17, Theorem 5.1] there is a constant C 1 > 0, depending only on the initial values of f , such that ! Z rX k−1 iθ iθ 1/(k−j) |f (re )| ≤ C1 exp k |Aj (se )| ds 0 j=0
for all θ ∈ [0, 2π], and so the assumption (3.2) yields Z δX k−1 iθ |f (re )| ≤ C1 exp k (5.2) |Aj (seiθ )|1/(k−j) ds 0 j=0
+k
Z
δ
= C2 exp k
Z
δ
k−1 rX j=0
≤ C2 exp α1/k k 2
Z
k−1 rX j=0
iθ 1/(k−j)
|Aj (se )|
iθ 1/(k−j)
|Aj (se )|
δ
r
ds 1−s
!
≤ C2
ds
ds
!
!
1 , 1/k (1 − r)α k2
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¨ ¨ J. HEITTOKANGAS, R. KORHONEN AND J. RATTY A
from which the assertion follows by choosing α = (q/k 2 )k . Proof of Theorem 3.9. The assertion follows by Theorem 3.10. Proof of Theorem 3.10. By Lemma 4.2 and the reasoning in [28, pp. 42– 43] it follows that, for all n = 0, . . . , k − 1 and j = 0, . . . , k − 1, Z 1 (n) sup |Aj (z)|p (1 − |z|2 )p(k+n−j)−1 dσz |I| |I|≤1−δ S(I) Z (n) ≤ 10 sup |ϕ0a (z)||Aj (z)|p (1 − |z|2 )p(k+n−j)−1 dσz D
|a|≥δ
≤ C sup
|a|≥δ
Z
D
|Aj (z)|p (1 − |z|2 )p(k−j)−1
1 − |a|2 dσz , |1 − a ¯z|2
where the constant C > 0 depends only on p and k. Hence, for α small enough, all solutions belong to D p ∩ Hp∞ by Theorem 3.12. Proof of Corollary 3.11. Theorem 3.10 implies that the solutions of (1.1) belong to D p . The assumption (3.9) together with [33, Lemma 2.9] ∞ show that Aj ∈ Hk−j,0 for j = 0, . . . , k − 1. Therefore, by Corollary 3.5, all T solutions of (1.1) belong to 0
