New characterizations of Bloch spaces, Bers-type and Zygmund-type spaces and Related Questions

In terms of Berezin symbols, we give new characterizations of the Bloch spaces $\mathcal{B}$ and $\mathcal{B}_{0}$б Bers-type and the Zygmund-type spaces of analytic functions on the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$ю We discuss some properties of Toeplitz operators on the Bergman space $L_{a}^{2}(\mathbb{D})$. We provide a new characterization of certain function space with variable exponents. Namely, given a function $f(z)= {\displaystyle\sum\limits_{n=0}^{\infty}} \widehat{f}(n)z^{n}\in \mathrm{Hol}(\mathbb{D})$ with a bounded sequence $\left\{ \widehat{f}(n)\right\} _{n\geq0}$ of Taylor coefficients $\widehat{f}(n)=\frac{f^{(n)}(0)}{n!},$ $\left( n=0,1,2,\dots\right) $, we have $f\in H_{p(\cdot),q(\cdot),\gamma(\cdot)}$ if and only if
$$ \int\limits_{0}^{1} \left( \frac{1}{2\pi} {\displaystyle\int\limits_{0}^{2\pi}} \left\vert \widetilde{D}_{(\widehat{f}(n)e^{int})}(\sqrt{r})\right\vert ^{p(t)}dt\right) ^{\frac{q(t)}{p(t)}}(1-r)^{\frac{\gamma(t)p(t)-q(t)}{p(t)} }dr<+\infty. $$
Here $D_{(a_{n})}$ denotes the associate diagonal operator on the Hardy–Hilbert space $H^{2}$ defined by the formula $D_{(a_{n})}z^{n}=a_{n}z^{n}\text{ }(n=0,1,2,\dots)$.