Notes on the Bergman projection type operator in $\Bbb C^n$

Abstract

n this paper, we will define the Bergman projection type operator $P_r$ and find conditions on which the operator $P_r$ is bound-ed on $L^p (B, d \nu)$. By using the properties of the Bergman projection type operator $P_r$, we will show that if $f \in L^p_a (B, d \nu)$, then $(1 -$ $\parallel w \parallel^2 ) \nabla f(w) \cdot z \in L^p (B, d \nu)$. We will also show that if $(1 -\parallel w \parallel^2)$ $ \frac{ \nabla f(w) \cdot z}{\langle z, w\rangle } \in L^p (B, d \nu)$, then $f \in L^p_a (B, d\nu)$.