A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications

[摘要] Let $x=(x_{1},x_{2},\ldots,x_{n})$, and let $K(u(x),v(y))$ satisfy $u(rx)=ru(x)$, $v(ry)=rv(y)$, $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$, and $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$. In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel $K(u(x),v(y))$ and discuss its applications in the theory of operators.

[学科分类] 电力

[关键词] Generalized homogeneous kernel;Hilbert-type multiple integral inequality;Necessary and sufficient condition;The best constant factor;Bounded operator;Operator norm