Norm estimates of the Cauchy transform and related operators

Suppose $f\in L^p(\mathbb{D})$, where $p\ge 1$ 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 ${\mathfrak{J}}_0^{\ast }$ is the adjoint operator of ${\mathfrak{J}}_0$. Then ${\mathfrak{J}}_0^{\ast }=\mathfrak{B}\mathfrak{C}$, 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 $L^{\mathrm{\infty }}$ norm of the operator ${\mathfrak{J}}_0^{\ast }$. Moreover, we obtain the $L^p(\mathbb{D})\to L^{\mathrm{\infty }}(\mathbb{D})$ norm of the operators $\mathfrak{C}$ and ${\mathfrak{J}}_0^{\ast }$, provided that $p>2$. This study is a continuation of the investigations carried out in [4] and [11].