$L^p$-theory for Cauchy-transform on the unit disk

Let $\mathbb{D}$ be the unit disk and $\varphi\in L^p($\mathbb{D}$, $\mathrm{d}$A)$,where$1\leq p\leq\infty$\mathrm{.}For$z\in$\mathbb{D}$$, the Cauchy-transform on $\mathbb{D}$, denote by $\mathcal{P}$, is defined as follows: $$\mathcal{P}[\phi ](z)=-{\int }_{\mathbb{D}}(\frac{\phi (w)}{w-z}+\frac{z\overline{\phi (w)}}{1-\bar{w}z})\mathrm{d}A(w)\mathrm{.}$$ The Beurling transform on $\mathbb{D}$, denote by $\mathcal{H}$, is now defined as the $z$-derivative of $\mathcal{P}$. In this paper, by using Hardy’s type inequalities and Bessel functions, we show that $|$\mathcal{P}$|{L^2\to L^2}=\alpha\approx1.086$,where$\alpha$isasolutiontotheequation:$2J_0(2/\alpha)-\alpha J_1(2/\alpha)=0$,and$J_0$,$J_1$ are Bessel functions. Moreover, for $p>2$, by using Taylor expansion, Parseval’s formula and hypergeometric functions, we also prove that $|$\mathcal{P}$|{L^p\to L^{\infty}}=2(\Gamma(2-q)/\Gamma^2(2-\frac{q}{2}))^{1/q}$,where$q=p/(p-1)$ is the conjugate exponent of $p$, and $\mathrm{\Gamma }$ is the Gamma function. Finally, applying the same techniques developed in this paper, we show that the Beurling transform $\mathcal{H}$ acts as an isometry of $L^2(\mathbb{D},\mathrm{d}A)$.