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

Let $x=(x_{1},x_{2},\ldots,x_{n})$x=(x1,x2,…,xn), and let $K(u(x),v(y))$K(u(x),v(y)) satisfy $u(rx)=ru(x)$u(rx)=ru(x), $v(ry)=rv(y)$v(ry)=rv(y), $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$K(ru,v)=rλλ1K(u,r−λ1λ2v), and $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$K(u,rv)=rλλ2K(r−λ2λ1u,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 $K(u(x),v(y))$K(u(x),v(y)) and discuss its applications in the theory of operators.