论文标题
Cauchy转换和相关运营商的规范估计值
Norm estimates of the Cauchy transform and related operators
论文作者
论文摘要
假设$ f \在l^p（\ mathbb {d}）$中，其中$ p \ geq1 $和$ \ mathbb {d} $是单位磁盘。令$ \ Mathfrak {J} _0 $为积分运算符，定义如下：$ \ Mathfrak {J} _0 [f]（z）= \ int _ {\ Mathbb {d}}} \ frac {z} $ w \ in \ Mathbb {d} $和$ \ mathrm {d} a（w）= \ frac {1}π\ mathrm {d} x \ mathrm {d} y $是$ \ m athbb {d} $的归一化区域。假设$ \ mathfrak {j} _0^*$是$ \ mathfrak {j} _0 $的伴随运营商。然后$ \ mathfrak {J}^*_ 0 = \ Mathfrak {b} \ Mathfrak {C} $，其中$ \ Mathfrak {b} $和$ \ Mathfrak {C {C} $是伯格曼投资和Cauchy Tronsports诱导的操作员。在本文中，我们获得了运算符$ \ mathfrak {J} _0^*$的$ l^1 $，$ l^2 $和$ l^{\ infty} $ norm。此外，我们获得了$ l^p（\ mathbb {d}）\ rightArrow l^\ infty（\ mathbb {d}）$ norm of操作员的$ \ mathfrak {c} $和$ \ mathfrak {c} $和$ \ mathfrak {j} _0^*$，提供了$ p> 2 $。这项研究是在[4]和[11]中进行的研究的延续。
Suppose $f\in L^p(\mathbb{D})$, where $p\geq1$ and $\mathbb{D}$ is the unit disk. Let $\mathfrak{J}_0$ be the integral operator defined as follows: $\mathfrak{J}_0[f](z)=\int_{\mathbb{D}}\frac{z}{1-\bar{w}z}f(w)\mathrm{d}A(w)$, where $z$, $w\in\mathbb{D}$ and $\mathrm{d}A(w)=\frac{1}π\mathrm{d}x\mathrm{d}y$ is the normalized area measure on $\mathbb{D}$. Suppose $\mathfrak{J}_0^*$ is the adjoint operator of $\mathfrak{J}_0$. Then $\mathfrak{J}^*_0=\mathfrak{B}\mathfrak{C}$, where $\mathfrak{B}$ and $\mathfrak{C}$ are the operators induced by the Bergman projection and Cauchy transform, respectively. In this paper, we obtain the $L^1$, $L^2$ and $L^{\infty}$ norm of the operator $\mathfrak{J}_0^*$. Moreover, we obtain the $L^p(\mathbb{D})\rightarrow L^\infty(\mathbb{D})$ norm of the operators $\mathfrak{C}$ and $\mathfrak{J}_0^*$, provided that $p>2$. This study is a continuation of the investigations carried out in [4] and [11].
