On an Integral-Type Operator Acting between Bloch-Type Spaces on the Unit Ball

Abstract

Let $𝔹$ denote the open unit ball of ${ℂ}^n$. For a holomorphic self-map $\phi$ of $𝔹$ and a holomorphic function $g$ in $𝔹$ with $g\left(0\right)=0$, we define the following integral-type operator: $I_{\phi }^{g}f\left(z\right)={\int }_0^1\Re f\left(\phi \right(tz\left)\right)g\left(tz\right)(dt/t)$, $z\in 𝔹$. Here $\Re f$ denotes the radial derivative of a holomorphic function $f$ in $𝔹$. We study the boundedness and compactness of the operator between Bloch-type spaces ${ℬ}_{\omega }$ and ${ℬ}_{\mu }$, where $\omega$ is a normal weight function and $\mu$ is a weight function. Also we consider the operator between the little Bloch-type spaces ${ℬ}_{\omega ,0}$ and ${ℬ}_{\mu ,0}$.