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 $u(rx)=ru(x)$, $v(ry)=rv(y)$, $K(ru,v)=r^{lambdalambda_{1}}K(u, r^{-rac{lambda_{1}}{lambda_{2}}}v)$, and $K(u,rv)=r^{lambdalambda_{2}}K(r^{-rac{lambda_{2}}{lambda_{1}}}u, v)$. 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.