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In this article, by the use of nth derivative characterization, we obtain several some sufficient conditions for all solutions of the complex linear differential equation to lie in weighted Dirichlet spaces and derivative Hardy spaces, respectively, where 
$$ \begin{align*}f^{(n)}+A_{n-1}(z)f^{(n-1)}+\ldots+A_1(z)f'+A_0(z)f=A_n(z) \end{align*} $$
$A_i(z) (i=0,1,\ldots ,n)$ are analytic functions defined in the unit disc. This work continues the lines of the investigations by Heittokangas, et al. for growth estimates about the solutions of the above equation.
In this paper, we will show the Carleson measure characterizations for the boundedness and compactness of the Cesàro-type operator
\begin{equation*}\mathcal{C}_{\mu}(f)(z)=\sum^{\infty}_{n=0}\left( \int_{[0,1)}t^nd\mu(t)\right) \left(\sum^{n}_{k=0}a_k \right)z^n, \quad z\in \mathbb{D},\end{equation*}
acting on a number of important analytic function spaces on 
$\mathbb{D}$, where µ is a positive finite Borel measure. The function spaces are some newly appeared analytic function spaces (e.g., Bergman–Morrey spaces 
$A^{p,\lambda}$ and Dirichlet–Morrey spaces 
$\mathcal{D}_p^{\lambda}$) . This work continues the lines of the previous characterizations by Blasco and Galanopoulos et al. for classical Hardy spaces and weighted Bergman spaces and so forth.
In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, 
$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$
$\mu $ is a positive Radon measure, K is a 
$\mu $-measurable function on the open unit disk 
$\mathbb {D}$, and 
$\sigma _w(z)$ is the classical Möbius transform of 
$\mathbb {D}$.
In this note, we mainly study operator-theoretic properties on the Besov space 
$B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.
