$L^p$-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 $1\leq p\leq\infty$. For $z\in\mathbb{D}$, the Cauchy-transform on $\mathbb{D}$, denote by $\mathcal{P}$, is defined as follows: $$\mathcal{P}[\varphi](z)=-\int_{\mathbb{D}}\left(\frac{\varphi(w)}{w-z}+\frac{z\overline{\varphi(w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ 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 $\alpha$ is a solution to the equation: $2J_0(2/\alpha)-\alpha J_1(2/\alpha)=0$, 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 $\Gamma$ 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 $L^2(\mathbb{D}, \mathrm{d}A)$.