| Summary: | Abstract Let x = ( x 1 , x 2 , … , x n ) $x=(x_{1},x_{2},\ldots,x_{n})$ , and let K ( u ( x ) , v ( y ) ) $K(u(x),v(y))$ satisfy u ( r x ) = r u ( x ) $u(rx)=ru(x)$ , v ( r y ) = r v ( y ) $v(ry)=rv(y)$ , K ( r u , v ) = r λ λ 1 K ( u , r − λ 1 λ 2 v ) $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$ , and K ( u , r v ) = r λ λ 2 K ( r − λ 2 λ 1 u , v ) $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$ . 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 ) ) $K(u(x),v(y))$ and discuss its applications in the theory of operators.
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