  Suppose $f\in L^p(\mathbb{D})$, where $p\geq1$ and $\mathbb{D}$ is the unit
disk. Let $\mathfrak{J}_0$ be the integral operator defined as follows:
$\mathfrak{J}_0[f](z)=\int_{\mathbb{D}}\frac{z}{1-\bar{w}z}f(w)\mathrm{d}A(w)$,
where $z$, $w\in\mathbb{D}$ and
$\mathrm{d}A(w)=\frac{1}{\pi}\mathrm{d}x\mathrm{d}y$ is the normalized area
measure on $\mathbb{D}$. Suppose $\mathfrak{J}_0^*$ is the adjoint operator of
$\mathfrak{J}_0$. Then $\mathfrak{J}^*_0=\mathfrak{B}\mathfrak{C}$, where
$\mathfrak{B}$ and $\mathfrak{C}$ are the operators induced by the Bergman
projection and Cauchy transform, respectively. In this paper, we obtain the
$L^1$, $L^2$ and $L^{\infty}$ norm of the operator $\mathfrak{J}_0^*$.
Moreover, we obtain the $L^p(\mathbb{D})\rightarrow L^\infty(\mathbb{D})$ norm
of the operators $\mathfrak{C}$ and $\mathfrak{J}_0^*$, provided that $p>2$.
This study is a continuation of the investigations carried out in [4] and [11].
