Equivalence condition for the best matching parameters of multiple integral operator with generalized homogeneous kernel and applications

Abstract

By introducing the matching parameters $a_{1},a_{2},\ldots,a_{n}$ and using the weight function method, the inequality $~\|T(f_{1},f_{2},\ldots,f_{n-1})\|_{q_{n},\alpha_{n}(1-q_{n})}\!\leq\!~M(a_{1},a_{2},\ldots,a_{n})~\prod_{i=1}^{n-1}\|f_{i}\|_{p_{i},\alpha_{i}}$ can be obtained for the operator $T$: <disp-formula><tex-math><![CDATA[$$ T(f_{1},f_{2},\ldots,f_{n-1})(x_{n})=\int_{\mathbb{R}_{+}^{n-1}}K(x_{1},\ldots,x_{n-1},x_{n})\prod_{i=1}^{n-1}f_{i}(x_{i}) {d}x_{1}\cdots{d}x_{n-1}.$$]]></tex-math></disp-formula> If the constant factor $M(a_{1},a_{2},\ldots,a_{n})$ is the operator norm $\|T\|$ of $T:\prod_{i=1}^{n-1}L_{p_{i}}^{\alpha_{i}}(0,+\infty)\rightarrow~L_{q_{n}}^{\alpha_{n}(1-q_{n})}(0,+\infty)$, then $a_{1},a_{2},\ldots,a_{n}$ are called the best matching parameters of the operator $T$. In this paper, for the generalized homogeneous kernel, the necessary and sufficient conditions for the best matching parameters of the operator $T$ are discussed, as the method of determining that the operator $T$ is bounded and the calculation formula for the operator norm of $T$ are obtained. Finally, some special cases are given as applications.