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].
Categories: math.CV