Toeplitz operators and Bergman projections on weighted spaces of holomorphic functions

For an open subset $O \subset \mathbb{C}$, $1 \leq p < \infty$ and a continuous function $v: O \rightarrow ]0,\infty[$ put $||f||_{v,p}=\left(\int_O |f(z)|^pv(z)dm(z)\right)^{1/p}$ and $||f||_{v,\infty}= \text{ ess } \sup_{z\in O}|f(z)|v(z)$, where $dm$ is the area measure on $O$.
Consider the spaces $L_v^p=\{f:O \rightarrow \mathbb{C} \text{ measurable}: ||f||_{v,p} < \infty \}$, $H_v^p=\{h \in L_v^p: h \text{ holomorphic}\}$.
Let $P_v:L_v^2 \rightarrow H_v^2$ be the orthogonal projection (Bergman projection). For measurable $f$ and holomorphic $h$ on $O$ put $T_f(h) = P_v(fh)$ (Toeplitz operator).
If $O$ is the unit disc $\mathbb{D}$ or the upper half plane we present conditions on $v$ and $f$ such that $T_f$ is a well-defined and bounded operator $H_v^{\infty}\rightarrow H_v^{\infty}$.
Moreover, if $v(z)= \exp\left(\frac{-\alpha}{(1-|z|^l)^{\beta}}\right) \ \text{ and } \ \tilde{v}(z)= \exp\left(\frac{-\tilde{\alpha}}{(1-|z|^l)^{\tilde{\beta}}}\right), \ \ z \in \mathbb{D}$, we determine all $l,\alpha, \tilde{\alpha},\beta, \tilde{\beta} >0$ and $1 \leq p \leq \infty$ such that $P_{\tilde{v}}$ is also a bounded operator $L_v^p \rightarrow H_v^p$.
This is a joint work with J.Bonet and J.Taskinen. $\sup z$