## The Growth and Conditions of Solutions in Some Function Spaces for Complex Linear Differential Equations

Yang Weifeng,

 基金资助: 湖南省自然科学基金.  2016JJ6029湖南省教育厅科研基金.  16B061

 Fund supported: the Hunan Provincial Natural Science Foundation.  2016JJ6029the Scientific Research Fund of Hunan Provincial Education Department.  16B061

Abstract

In this paper, the relationship between the solutions and coefficients of the complex linear differential equation $f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_1(z)f'+A_0(z)f=0, z\in {\mathbb D}$ is investigated. A new growth estimate of the solutions is provided. Some sufficient conditions on the coefficients are also obtained such that all solutions belong to the $H_q^\infty$ space, the $F(p, q, s)$ space and the ${\cal N}^p_\alpha$ space.

Keywords： Complex linear differential equation ; Weighted Hardy space ; F(p, q, s) space ; Growth of solution

Yang Weifeng. The Growth and Conditions of Solutions in Some Function Spaces for Complex Linear Differential Equations. Acta Mathematica Scientia[J], 2020, 40(6): 1481-1491 doi:

## 1 引言

${\mathbb D}$为复平面上的单位圆盘, $H({\mathbb D}) $${\mathbb D} 内解析函数的全体, {\mathbb D} 内任意阶齐次复线性微分方程具有如下一般形式 $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_1(z)f'+A_0(z)f = 0, z\in {\mathbb D},$$ 其中 A_j(z)\in H({\mathbb D})$$ (j = 0, 1, 2, \cdots, k-1)$.

1982年, Pommerenke在文献[3]中获得了$k = 2$时方程(1.1)的解属于Nevanlinna类和Hardy空间$H^2$的充分条件.在文献[3]的基础上, Heittokangas在文献[4]中研究了单位圆盘内任意阶线性微分方程, 证明了系数函数是$H$-函数是解为有限级的解析函数的充分必要条件, 并获得了方程$f^{(k)}+A_0(z)f = 0$的解属于Nevanlinna类, 加权Hardy空间$H^p_q$, $\alpha$-Bloch空间, $F(p, q, s)$空间的条件.李叶舟[5], 曹廷彬和仪洪勋[6]先后利用Nevanlinna值分布理论研究了复线性微分方程的解增长性与系数函数增长性之间的关系.后来Heittokangas, Korhonen和Rättyä在文献[7-10]中研究了方程(1.1)的解属于Hardy空间、增长空间、Dirichlet型空间、Nevanlinna类的条件.

2004年, Heittokangas等在文献[7]中获得了方程(1.1)解的下列增长估计.

2011年, 李浩和乌兰哈斯在文献[11]中获得了方程(1.1)的解属于$Q_K$空间, Dirichlet空间的条件. 2013年, 李浩和李颂孝在文献[12]中进而给出了方程(1.1)解属于$Q_K$型空间和Dirichlet型空间的条件, 其中有关方程(1.1)解属于$Q_K$型空间的结果如下:

(ⅰ)当$p>q+2$时, $f(z)\in Q_K(p, q)$;

(ⅱ)当$p\leq q+2$时, $f(z)\in Q_K(p, q)\cap H^\infty$.

$0<p, s<\infty$, $-2<q<\infty$, $F(p, q, s)$空间是由满足下列条件的${\mathbb D}$内解析函数$f(z)$构成的集合

$\begin{eqnarray} f(z) = \sum\limits^{k-1}_{j = 0}g_jz^{j}-\sum\limits^{k-1}_{j = 0}\sum\limits^j_{i = 0}\frac{(-1)^i \big({}^j_i\big)}{(k-j+i-1)!}\int^{z}_{0}(z-\xi)^{k-j+i-1}A_{j}^{(i)}(\xi)f(\xi){\rm d}\xi, \end{eqnarray}$

(ⅰ) $f(z)\in F(p, q, s)$;

(ⅱ) $\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D} |f^{(n)}(z)|^p(1 -|z|^2)^{(n-1)p+q}(1-|\varphi_a(z)|^2)^s {\rm d}\sigma (z) < \infty$;

(ⅲ) $\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D} |f^{(n)}(z)|^p(1 -|z|^2)^{(n-1)p+q}g^s(z, a){\rm d}\sigma (z) < \infty$;

(ⅳ) ${\rm d}\mu(z) = |f^{(n)}(z)|^p(1 -|z|^2)^{(n-1)p+q+s}{\rm d}\sigma(z)$是有界$s$-Carleson测度.

(1) 当$c<0$时, $I_{c, t}(z) $${\mathbb D} 内有界; (2) 当 c>0 时, I_{c, t}(z)$$ {\mathbb D}$内无界, 且当$|z|\rightarrow 1^-$时, $I_{c, t}(z)\sim \frac{1}{(1-|z|^2)^c}$;

(3) 当$c = 0$时, $I_{c, t}(z) $${\mathbb D} 内无界, 且当 |z|\rightarrow 1^- 时, I_{c, t}(z)\sim \log\frac{1}{(1-|z|^2)} . ## 3 主要结果 定理3.1 设 f(z) 是方程(1.1)的解, 则存在正常数 C , 使得 其中 C 仅与 f^{(j)}(0)$$ A^{(j)}(0) $$(j = 0, 1, 2, \cdots, k-1) 有关. 设 f(z) 是方程(1.1)的解.由引理2.1, 有 其中 g_j 仅与 f^{(j)}(0)$$ A^{(j)}(0) $$(j = 0, 1, 2, \cdots, k-1) 有关. z = r{\rm e}^{{\rm i}\theta} , \xi = t{\rm e}^{{\rm i}\theta} , 则有 利用Gronwall引理(参见文献[2, 引理5.10]), 可得到 又因为, 对于 i = 1, 2, \cdots, j , 有 那么 \int^{r}_{0}(r-t)^{k-j+i-1}|A_{j}^{(i)}(\xi)|{\rm d}t\leq c_{i, j}+\frac{(k-j+i-1)!}{(k-j-1)!}\int^{r}_{0}(r-t)^{k-j-1}|A_{j}(t{\rm e}^{{\rm i}\theta})|{\rm d}t , 其中 c_{i, j} 均为正常数, 只与 A^{(j)}(0)$$ (j = 0, 1, 2, \cdots, k-1)$有关.因此有

(ⅰ)当$0<\beta \leq 1$时, $f(z)\in H^\infty$;

(ⅱ)当$\beta = 0$时, $f(z)\in H^\infty_{q}$;

(ⅲ)当$\beta <0$时, $f(z)\in \varepsilon^{-\beta }$.

由条件(3.1)可知$|A_j(z)|\leq\frac{\alpha_j}{(1-|z|^2)^{k-j-\beta_j}} $$(j = 0, 1, 2, \cdots, k-1) .由定理3.1, 有 \begin{eqnarray} |f(r{\rm e}^{{\rm i}\theta})|&\leq& C_1\exp\bigg(\sum\limits^{k-1}_{j = 0}\frac{2^j}{(k-j-1)!}\int^{r}_{0}(r-t)^{k-j-1}|A_{j}(t{\rm e}^{{\rm i}\theta})|{\rm d}t\bigg)\\ &\leq& C_1\exp\bigg(\sum\limits^{k-1}_{j = 0}\frac{2^j}{(k-j-1)!}\int^{r}_{0}(1-t)^{k-j-1}|A_{j}(t{\rm e}^{{\rm i}\theta})|{\rm d}t\bigg)\\ &\leq& C_1\exp\bigg(\sum\limits^{k-1}_{j = 0}\frac{2^j\alpha_j}{(k-j-1)!}\int^{r}_{0}(1-t)^{\beta_j-1}{\rm d}t\bigg), \end{eqnarray} 其中 C_1 为正常数, 仅与 f^{(j)}(0)$$ A^{(j)}(0) $$(j = 0, 1, 2, \cdots, k-1) 有关, 后面的 C_2 , C_3$$ C_4$也是如此.

$\beta<0$时, 由(3.2)式可得

$\beta = 0$时, 由(3.2)式有

$1\geq \beta>0$时, 由(3.2)式可知$|f(r{\rm e}^{{\rm i}\theta})|\leq C_1\exp\bigg(\sum\limits^{k-1}_{j = 0}\frac{2^j\alpha_j}{\beta_j(k-j-1)!}\bigg)$, 从而$f(z)\in H^\infty$.定理3.2证毕.

$$$\|A_j\|_{H^\infty_{k-j}} = \sup\limits_{z\in {\mathbb D}}|A_j(z)|(1-|z|^2)^{k-j}\leq \alpha, \quad j = 1, 2, \cdots, k-1,$$$

$$$\|A_{0} \|_{H^\infty_{k-t}} = \sup\limits_{z\in {\mathbb D}}|A_0(z)|(1-|z|^2)^{k-t}\leq \alpha.$$$

(ⅰ)当$p<q+s+1$, $t = 0$时, $f(z)\in F(p, q, s)$;

(ⅱ)当$p\geq q+s+1$, $p> q+2$, $\max\{\frac{p-q-s-1}{p}, \frac{p-q-2}{p}\}<t<1$时, $f(z)\in F(p, q, s)$;

(ⅱ)当$p\geq q+s+1$, $p\leq q+2$, $\frac{1-s}{p}<t<1$时, $f(z)\in F(p, q, s)$.

令${\rm d}\mu_a(z) = (1-|\varphi_a(z)|^2)^s {\rm d}\sigma (z)$, $f^{(j)}_\rho (z) = f^{(j)}(\rho z)$, $A_{\rho, j}(z) = A_j(\rho z) $$(j = 0, 1, \cdots, k) , 其中 \frac{1}{2} \leq \rho < 1 . f(z) 是方程(1.1)的解, 则 f_\rho^{(k)}(z) = -A_{\rho, k-1}(z)f_\rho^{(k-1)}(z)-\cdots -A_{\rho, 1}(z)f_\rho'(z)-A_{\rho, 0}(z)f_\rho(z). 由引理2.2, 可得 \begin{eqnarray} \|f_\rho \|^p_{F(p, q, s)}&\leq &C_1 \sup\limits_{a\in {\mathbb D}} \int_{\mathbb D} |f_\rho ^{(k)}(z)|^p(1 -|z|^2)^{(k-1)p+q}{\rm d}\mu_a(z)+\sum\limits^{k-1}_{j = 0}|f_\rho^{(j)}(0)|\\ &\leq & C_1\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}\sum\limits^{k-1}_{j = 0}|A_{\rho, j}(z)f_\rho^{(j)}(z)|^p(1 -|z|^2)^{(k-1)p+q}{\rm d}\mu_a(z)+C_2, \end{eqnarray} 其中 C_1 = C_1(k, p, q, s)$$ C_2 = \sum\limits^{k-1}_{j = 0}|f^{(j)}(0)|$均为常数.

$p<q+s+1$, $t = 0$时, 由(3.3), (3.4)和(3.5)式以及引理2.2, 可得

$p\geq q+s+1$, $p>q+2$, $1>t>\max\{\frac{p-q-s-1}{p}, \frac{p-q-2}{p}\}$时.利用引理2.2和引理2.3, 由(3.3), (3.4)和(3.5)式, 可得

$\begin{eqnarray} \|f_\rho \|^p_{F(p, q, s)}&\leq & C_1\bigg(\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}\sum\limits^{k-1}_{j = 1}|A_{\rho, j}(z)f_\rho^{(j)}(z)|^p(1 -|z|^2)^{(k-1)p+q}{\rm d}\mu_a(z)\\ &&+\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}|A_{\rho, 0}(z)f_\rho(z)|^p(1 -|z|^2)^{(k-1)p+q}{\rm d}\mu_a(z)\bigg)+C_2\\ &\leq & C_1C_4\alpha^p\|f_\rho \|^p_{F(p, q, s)}+C_2+C_1 \alpha^p\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}|f_\rho(z)|^p(1 -|z|^2)^{(t-1)p+q}{\rm d}\mu_a(z)\\ &\leq & C_1C_4\alpha^p\|f_\rho \|^p_{F(p, q, s)}+C_2\\ & &+C_5 \alpha^p\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}(|f(0)|+\|f_\rho\|_{F(p, q, s)})^p(1 -|z|^2)^{(t-1)p+q}{\rm d}\mu_a(z)\\ &\leq & C_1C_4\alpha^p\|f_\rho \|^p_{F(p, q, s)}+C_2{}\\ & &+C_6 \alpha^p\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}(|f(0)|^p+\|f_\rho\|^p_{F(p, q, s)})(1 -|z|^2)^{(t-1)p+q}{\rm d}\mu_a(z). \end{eqnarray}$

$\begin{eqnarray} (1-C_1C_4\alpha^p-C_6C_7\alpha^p)\|f_\rho \|^p_{F(p, q, s)}\leq C_2+C_6C_7|f(0)|^p, \end{eqnarray}$

$q+2>p$, 由(3.5)式可得

$\begin{eqnarray} \|f_\rho \|^p_{F(p, q, s)}&\leq & C_1C_4\alpha^p\|f_\rho \|^p_{F(p, q, s)}+C_2+C_1 \alpha^p\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}|f_\rho(z)|^p(1 -|z|^2)^{(t-1)p+q}{\rm d}\mu_a(z)\\ &\leq & C_1C_4\alpha^p\|f_\rho \|^p_{F(p, q, s)}+C_2\\ & &+C_8 \alpha^p\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}\big(|f(0)|^p+\frac{\|f_\rho\|^p_{F(p, q, s)}}{(1-|z|^2)^{q+2-p}}\big)(1 -|z|^2)^{(t-1)p+q}{\rm d}\mu_a(z)\\ &\leq & C_1C_4\alpha^p\|f_\rho \|^p_{F(p, q, s)}+C_8\alpha^p \bigg(|f(0)|^p\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}(1 -|z|^2)^{(t-1)p+q}{\rm d}\mu_a(z)\\ & & +\|f_\rho\|^p_{F(p, q, s)}\sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}(1 -|z|^2)^{tp-2}{\rm d}\mu_a(z)\bigg)+C_2. \end{eqnarray}$

$C_{10} = \sup\limits_{a\in {\mathbb D}} \int_{\mathbb D}\big(\ln\frac{e}{1-|z|^2}\big)^p(1 -|z|^2)^{(t-1)p+q}(1-|\varphi_a(z)|^2)^s {\rm d}\sigma (z).$则(3.10)式等价于

$\begin{eqnarray} (1-C_1C_4\alpha^p-C_8C_{10}\alpha^p)\|f_\rho \|^p_{F(p, q, s)}\leq C_2+C_7 C_8\alpha^p|f(0)|^p. \end{eqnarray}$

$\begin{eqnarray} \|A_j\|_{H^\infty_{k-j-t}} = \sup\limits_{z\in {\mathbb D}}|A_j(z)|(1-|z|^2)^{k-j-t}\leq \alpha, \quad j = 0, 1, 2, \cdots, k-1, \end{eqnarray}$

(ⅰ)当$p<q+s+1$, $0<t<1$时, $f(z)\in F(p, q, s)\cap H^\infty$;

(ⅱ)当$p\geq q+s+1$, $p> q+2$, $\max\{\frac{p-q-s-1}{p}, \frac{p-q-2}{p}\}<t<1$时, $f(z)\in F(p, q, s)\cap H^\infty$;

(ⅲ)当$p\geq q+s+1$, $p\leq q+2$, $\frac{1-s}{p}<t<1$时, $f(z)\in F(p, q, s)\cap H^\infty$.

由定理3.3可知三种情形下均有$f(z)\in F(p, q, s)$, 再由定理3.2可知三种情形下均有$f(z)\in H^\infty$.因此三种情形下均有$f(z)\in F(p, q, s)\cap H^\infty$.证毕.

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