Norm estimates of the Cauchy transform and related operators

Abstract

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].