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.

中文翻译：

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一类具有广义齐次函数核的希尔伯特型多重积分不等式及其应用

设$ x =（x_ {1}，x_ {2}，\ ldots，x_ {n}）$，并令$ K（u（x），v（y））$满足$ u（rx）= ru（ x）$，$ v（ry）= rv（y）$，$ K（ru，v）= r ^ {\ lambda \ lambda_ {1}} K（u，r ^ {-\ frac {\ lambda_ {1 }} {\ lambda_ {2}}} v）$和$ K（u，rv）= r ^ {\ lambda \ lambda_ {2}} K（r ^ {-\ frac {\ lambda_ {2}} { \ lambda_ {1}}} u，v）$。在本文中，我们获得了具有内核$ K（u（x），v（y））$的Hilbert型多重积分不等式的充要条件和最佳常数因子，并讨论了其在算符理论中的应用。