Suppose $0<\alpha <\mathrm{\infty }$, ${\mathfrak{B}}_{\alpha }$ is an integral operator of $L^p(\text{ID},\mathrm{d}A)$ which is defined as follows: ${\mathfrak{B}}_{\alpha }\left[f\right]\left(z\right)={\int }_{\text{ID}}\frac{1}{(1-\overline{w}z{)}^{\alpha }}f\left(w\right)\mathrm{d}A\left(w\right)$, where $\mathbb{D}$ is the unit disk and $\mathrm{d}A\left(w\right)$ is the normalized area measure. For $0<\alpha <2$, we obtain the norm estimates of $\parallel {\mathfrak{B}}_{\alpha }{\parallel }_p$, where $1\le p\le \mathrm{\infty }$. Our results are sharp for $p=1,2,\mathrm{\infty }$. Moreover, we show that ${\mathfrak{B}}_{\alpha }$ is a compact operator and of Schatten p-class, where $p>\frac{1}{2-\alpha }$. For $2<\alpha <3$, we show that ${\mathfrak{B}}_{\alpha }\left(L^p\right(\text{ID}\left)\right)\subseteq L^q\left(\text{ID}\right)$, where $p>\frac{3}{3-\alpha }$ and q is the conjugate exponent of p. This result is sharp (see Remark 1.2) and the norm estimate of $\parallel {\mathfrak{B}}_{\alpha }{\parallel }_{L^p(\text{ID},\mathrm{d}A)\to L^q(\text{ID},\mathrm{d}A)}$ is also obtained.

认为$0<\alpha <\mathrm{\infty }$,${\mathfrak{乙}}_{\alpha }$是一个积分算子${\mathrm{大号}}^p(\text{ID},\mathrm{d}\mathrm{一个})$定义如下：${\mathfrak{乙}}_{\alpha }\left[F\right]\left(z\right)={\int }_{\text{ID}}\frac{1}{(1-\overline{w}z{)}^{\alpha }}F\left(w\right)\mathrm{d}\mathrm{一个}\left(w\right)$， 在哪里$\mathbb{D}$是单位磁盘和$\mathrm{d}\mathrm{一个}\left(w\right)$是归一化面积度量。为了$0<\alpha <2$，我们得到范数估计$\parallel {\mathfrak{乙}}_{\alpha }{\parallel }_p$， 在哪里$1\le p\le \mathrm{\infty }$. 我们的结果是尖锐的$p=1,2,\mathrm{\infty }$. 此外，我们表明${\mathfrak{乙}}_{\alpha }$是紧算子和 Schatten p类，其中$p>\frac{1}{2-\alpha }$. 为了$2<\alpha <3$, 我们证明${\mathfrak{乙}}_{\alpha }\left({\mathrm{大号}}^p\right(\text{ID}\left)\right)\subseteq {\mathrm{大号}}^q\left(\text{ID}\right)$， 在哪里$p>\frac{3}{3-\alpha }$q是p的共轭指数。这个结果是尖锐的（见备注 1.2）和范数估计$\parallel {\mathfrak{乙}}_{\alpha }{\parallel }_{{\mathrm{大号}}^p(\text{ID},\mathrm{d}\mathrm{一个})\to {\mathrm{大号}}^q(\text{ID},\mathrm{d}\mathrm{一个})}$也获得。