Constructive 条件 for the bounded multiple integral operator with quasi-homogeneous kernel in Lebesgue space with exponential weights and computation of the operator norm

在加指数权的Lebesgue中, 应用权函数方法和各种分析技巧, 讨论拟齐次核的重积分算子$T$: \begin{align*} & \ \ \ \ T(f_{1},f_{2},\cdots,f_{n-1})(x_{n})\\ &=\int_{\mathbb{R}^{n-1}}K(e^{\lambda_{1}x_{1}},\cdots,e^{\lambda_{n-1}x_{n-1}},e^{\lambda_{n}x_{n}}) \prod_{i=1}^{n-1}f_{i}(x_{i})\textbf{d}x_{1}\cdots\textbf{d}x_{n-1}, \end{align*} 其中$K(u_{1},\cdots,u_{n-1},u_{n})$是$\lambda$阶齐次函数, 得到算子 $$ T: \prod_{i=1}^{n-1}L_{p_{i}}^{\exp\{\alpha_{i}x_{i}\}}(\mathbb{R})\rightarrow L_{q_{n}}^{\exp\{\alpha_{n}(1-q_{n})x_{n}\}}(\mathbb{R}) $$ 有界的充分必要条件, 且在$T$有界时, 得到$T$的算子范数计算公式, 解决了构建有界算子$T$的一个理论问题, 这对于Fourier变换 $F(\omega)=\int_{\mathbb{R}^{n}}f(x)e^{-i\omega x}\textbf{d}x$等算子的研究有一定意义.