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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 
$\mathbb {D}$.
$\beta$-plane zonal jets
Zonal jets and inertia–gravity waves are ubiquitous on planets such as Earth, Jupiter and Saturn. Motivated by the modification of energy flux of balanced flow by inertia–gravity waves, this paper studies the impact of near-inertial waves (NIWs) on zonal jets on a 
$\beta$-plane. Using a two-dimensional quasi-geostrophic and NIW coupled system on a 
$\beta$-plane (Xie & Vanneste, J. Fluid Mech., vol. 774, 2015, pp. 143–169), we find NIWs catalytically impact several features of zonal jets. The NIWs inhibit jet formation due to the waves’ catalytic induction of downscale mean energy flux. As the strength of NIWs increases, a critical point exists beyond which zonal jets are annihilated. The jet spacing is captured by the Rhines scale 
$L\sim \sqrt {U/\beta }$ with 
$U$ estimated from the upscale energy flux induced by the mean flow alone, which again shows that the NIWs’ impact is catalytic. Also, the temporal asymmetry of NIWs leads to the spatial asymmetry of jet dynamics. The jet profiles are asymmetric with a stronger shear on the left flank. And similar to the left turning of vortex dipole under the impact of NIWs, the NIW-modified jets migrate poleward. The NIWs also show a catalytic role in jet migration: the net momentum flux directly induced by NIWs is of secondary importance in the zonal mean momentum dynamics and impedes jet migration, while the advective effect of NIW-modified mean flow dominates the jet migration velocity.
