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arXiv 2008.03068math.CV
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unit disk上柯西变换的$L^p$ -理论

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

David Kalaj, Petar Melentijević, Jian-Feng Zhu

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29 pages
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摘要：

令$\mathbb{D}$为单位盘，$φ\in L^p(\mathbb{D}, \mathrm{d}A)$，其中$1\leq p\leq\infty$。对于$z\in\mathbb{D}$，$\mathbb{D}$上的柯西变换（用$\mathcal{P}$表示）定义如下：$$\mathcal{P}[φ](z)=-\int_{\mathbb{D}}\left(\frac{φ(w)}{w-z}+\frac{z\overline{φ(w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ $\mathbb{D}$上的Beurling变换（用$\mathcal{H}$表示）现在被定义为$\mathcal{P}$的$z$ -衍生物。本文利用Hardy型不等式和Bessel函数证明了方程$2J_0(2/α)-αJ_1(2/α)=0$的解为$\|\mathcal{P}\|_{L^2\to L^2}=α\approx1.086$，$J_0$，$J_1$为Bessel函数.此外，对于$p>2$，利用Taylor展开式、Parseval公式和超几何函数，我们还证明了$\|\mathcal{P}\|_{L^p\to L^{\infty}}=2(Γ(2-q)/Γ^2(2-\frac{q}{2}))^{1/q}$，其中$q=p/(p-1)$是$p$的共轭指数，$Γ$是Gamma函数.最后，应用本文中开发的相同的技术，我们表明，Beurling变换$\mathcal{H}$作为等距的$L^2(\mathbb{D}, \mathrm{d}A)$。

英文摘要：

Let $\mathbb{D}$ be the unit disk and $φ\in L^p(\mathbb{D}, \mathrm{d}A)$, 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: $$\mathcal{P}[φ](z)=-\int_{\mathbb{D}}\left(\frac{φ(w)}{w-z}+\frac{z\overline{φ(w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ 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 $\|\mathcal{P}\|_{L^2\to L^2}=α\approx1.086$, where $α$ is a solution to the equation: $2J_0(2/α)-αJ_1(2/α)=0$, 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 $\|\mathcal{P}\|_{L^p\to L^{\infty}}=2(Γ(2-q)/Γ^2(2-\frac{q}{2}))^{1/q}$, where $q=p/(p-1)$ is the conjugate exponent of $p$, and $Γ$ 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 $L^2(\mathbb{D}, \mathrm{d}A)$.

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