论文标题
$ l^p $ - 单元磁盘上凯奇 - 转换的理论
$L^p$-theory for Cauchy-transform on the unit disk
论文作者
论文摘要
令$ \ mathbb {d} $为单位磁盘，$φ\ in l^p（\ mathbb {d}，\ mathrm {d} a）$，其中$ 1 \ leq p \ leq \ leq \ infty $。对于$ z \ in \ mathbb {d} $，$ \ mathbb {d} $上的cauchy-transform，用$ \ mathcal {p} $表示，定义如下： $$ \ MATHCAL {p} [φ]（z）= - \ int _ {\ Mathbb {d}}} \ left（\ frac {φ（w）} {w-z} +\ frac {z \ +edline {φ（w）}} {1- \ bar {w} z} \ right）\ mathrm {d} a（w）a（w）。$$ $ \ mathbb {d} $上的beurling变换，用$ \ mathcal {h} $表示，现在被定义为$ z $ - $ \ mathcal {p} $的$ z $。 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 $是Bessel功能。此外，对于$ p> 2 $，通过使用Taylor扩展，Parseval的公式和超几何功能，我们还证明了$ \ | \ | \ Mathcal {p} \ | _ _ {l^p \ to l^{\ infty}} = 2（γ（2- q）/γ^2（2- \ frac {q} {2}）））^{1/q} $，其中$ q = p/q/（p/p/q/q} $是$ p $的共轭指数，$γ$是gamma函数。最后，应用本文开发的相同技术时，我们表明beurling变换$ \ mathcal {h} $充当$ l^2（\ Mathbb {d}，\ mathrm {d} a）$的等轴测图。
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 $L^2(\mathbb{D}, \mathrm{d}A)$.
