## Norms of some operators from Bergman spaces to weighted and Bloch-type spaces.(English)Zbl 1160.47027

Let $$B$$ be the open unit ball in $$\mathbb C^n$$, $$n \in \mathbb N$$. The weighted Bergman spaces $$A_{\alpha}^p$$ are defined by
$A_{\alpha}^p:= \{ f \in H(B); \; \| f\| _{A_{\alpha}^p}:= \int_B | f(z)| c_{\alpha} (1-| z| ^2)^{\alpha} \; dV(z) < \infty \},\quad p>1,\quad \alpha > -1,$
where $$dV(z)$$ is the Lebesgue measure and $$c_{\alpha}$$ chosen so that $$\int_B c_{\alpha} (1-| z| ^2)^{\alpha} \; dV(z) =1$$. For $$u \in H(B)$$ and an analytic self-map $$\varphi$$ of $$B$$, the corresponding weighted composition operator is defined by
$u C_{\varphi}: H(B) \to H(B), \; f \to u (f \circ \varphi).$
In this paper, the authors determines the norm of such a weighted composition operator acting between the weighted Bergman space $$A_{\alpha}^p$$ and the weighted Banach space of holomorphic functions
$H_{\mu}^{\infty}:= \{ f \in H(B) \mid \| f\| _{\mu}:= \sup_{z \in B} \mu(z) | f(z)| < \infty \}.$
In the second part of the article, the author restricts himself to the setting $$n=1$$. Let $$\varphi$$ be an analytic self-map of the open unit disk $$\mathbb D$$, and $$g \in H(\mathbb D)$$. For $$f \in H(\mathbb D)$$, the products of Volterra and composition operators are given by $C_{\varphi} J_g f(z):= \int_{0}^{\varphi(z)} f(\xi) g(\xi) \; d \xi \text{ and } J_g C_{\varphi} f(z):= \int_0^z f(\varphi(\xi)) g(\xi) \; d \xi.$ The norms of such operators acting between weighted Bergman spaces $$A_{\alpha}^p$$ and weighted Bloch type spaces $$B_{\mu}$$ are computed.

### MSC:

 47B33 Linear composition operators 47B38 Linear operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions