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

Abstract

Let $\mathbb{D}$ be the unit disk and $\phi \in L^p(\mathbb{D},\mathrm{d}A)$, where $1\le p\le \infty$. For $z\in \mathbb{D}$, the Cauchy-transform on $\mathbb{D}$, denote by $\mathcal{P}$, is defined as follows: $$\mathcal{P}[\phi ](z)=-{\int }_{\mathbb{D}}\left(\frac{\phi (w)}{w-z}+\frac{z\overline{\phi (w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ 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 $\Vert \mathcal{P}{\Vert }_{L^2\to L^2}=\alpha \approx 1.086$, 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 $\Vert \mathcal{P}{\Vert }_{L^p\to L^{\infty }}=2(\Gamma (2-q)/{\Gamma }^2(2-\frac{q}{2}){)}^{1/q}$, 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 $L^2(\mathbb{D},\mathrm{d}A)$.

29 pages

References in corpus (1)

• Norm estimates of the Cauchy transform and related operators