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Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space. (English) Zbl 1161.34060
The authors study the growth relation between the coefficients and the solutions of the $$k(\geq 2)$$th-order linear differential equation $f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0.\tag{$$*$$}$ In the paper [Trans. Am. Math. Soc. 360, No. 2, 1035–1055 (2008; Zbl 1133.34045)], they pose the following direct problem and the inverse problem: (1) Suppose that, for every $$j = 0, \dots , k-1$$, the coefficient $$A_j(z)$$ of $$(\ast)$$ belongs to a certain analytic function space depending on $$j$$. Find the function space or spaces to which all solutions of $$(\ast)$$ belong. (2) Suppose that all solutions of $$(\ast)$$ belong to a certain analytic function space. Find the function space or spaces to which the coefficient $$A_j(z)$$, $$j = 0, \dots , k-1$$, of $$(\ast)$$ belongs.
In this paper under review, they give some sufficient conditions on the analytic coefficients of $$(\ast)$$ for all solutions to belong to a weighted $$H^{\infty}$$-space, $$H^{\infty}_q$$ $$(0<q<\infty)$$, or to the Dirichlet-type subspace $$\mathcal{D}^p$$ of the Hardy space $$H^p$$ $$(0<p\leq 2)$$, that is, the family of analytic functions $$f$$ in the unit disk $$D$$ for which $$|f(z)|(1-|z|^2)^q$$ is uniformly bounded in $$D$$ or the integral $$\int_D|f'(z)|^p(1-|z|^2)^{p-1}\,d\sigma_z$$ converges, respectively. Here $$d\sigma_z$$ denotes the element of the Lebesgue area measure on $$D$$.
The authors generalize the preceding results concerning special forms of $$(\ast)$$ by Ch. Pommerenke [Complex Variables, Theory Appl. 1, 23–38 (1982; Zbl 0464.34010)] and by the first author [Annales Academiae Scientiarum Fennicae. Mathematica. Dissertationes. 122. Helsinki: Suomalainen Tiedeakatemia. Joensuu: Univ. Joensuu, Department of Mathematics (2000; Zbl 0965.34075)].
As candidates for analytic function spaces in problems (1) and (2), they study also the spaces $$H^{\infty}_{q,0}$$ $$(0 < q < \infty)$$, the $$\alpha$$-Bloch spaces $$\mathcal{B}^{\alpha}$$ and the little $$\alpha$$-Bloch spaces $$\mathcal{B}^{\alpha}_0$$ $$(0 < \alpha < \infty)$$ consisting of analytic functions $$f$$ in $$D$$, for which $$\lim_{r\to 1-} (1-r^2)^q M(r,f) = 0$$, $$\sup_{0\leq r<1} M(r, f') (1- r^2)^{\alpha} < \infty$$ and $$\lim_{r\to 1-}(1-r^2)^{\alpha} M(r, f') = 0$$ with $$M(r,g)=\max_{|z|=r}|g(z)|$$, respectively. They are all Banach spaces with corresponding norms.
Further, the authors observe a general family of function spaces $$F(p, q, s)$$ and $$F_0(p, q, s)$$ for $$0 < p < \infty$$, $$-2 < q < \infty$$ and $$0\leq s < \infty$$, which were introduced by R. Zhao [Annales Academiae Scientiarum Fennicae. Mathematica. Dissertationes. 105. Helsinki: Suomalainen Tiedeakatemia (1996; Zbl 0851.30017)], since for appropriate parameter values $$p$$, $$q$$ and $$s$$, these spaces coincide with several classical function spaces such as the weighted Bergman space $$A_q^p$$, the $$Q_p$$ or $$Q_{p,0}$$ spaces and the spaces of analytic functions with bounded or vanishing mean oscillation as well as $$\mathcal{D}^p$$ above.
For example, Theorem 3.1 states the following: For every $$q > 0$$ there exists a constant $$\alpha = \alpha (q, k) > 0$$ such that if the coefficients $$A_j(z)$$ of $$(\ast)$$ satisfy $$\|A_j\|_{H^{\infty}_{k-j}}:=\sup_{z\in D} |A_j(z)| (1-|z|^2)^{k-j} \leq \alpha$$ $$(j = 0, \dots, k-1)$$, then all solutions of $$(\ast)$$ belong to $$H^{\infty}_q$$.
They give three different proofs for this theorem with application of their growth estimates given in [Ann. Acad. Sci. Fenn., Math. 29, No. 1, 233–246 (2004; Zbl 1057.34111)]. Theorem 3.3 refines this norm condition on the $$A_j$$ by replacing $$\sup_{z\in D}$$ with $$\sup_{1>|z|\geq \delta}$$ for $$0<\delta<1$$. Corollaries to Theorem 3.3 show that if $$A_j(z) \in A^1_{k-j-2}$$ and thus if $$A_j \in H^{\infty}_{k-j,0}$$ for $$j = 0, \dots , k-1$$, then all solutions of $$(\ast)$$ belong to $$\bigcap_{0<q<1} H^{\infty}_q$$.
Examples of linearly independent solutions to $$(\ast)$$ with some restricted coefficients $$A_j$$ are provided to give some upper bound on the constant $$\alpha$$ in Theorems 3.1 and 3.3 and to show the sharpness of these corollaries. Then such conditions that all solutions of $$(\ast)$$ should belong to $$\mathcal{D}^p \cap H^{\infty}_p$$ are lined up from Theorem 3.9 to Corollary 3.17.

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain 30D50 Blaschke products, etc. (MSC2000) 30D55 $$H^p$$-classes (MSC2000)
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