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Let    $x=(x_{1},x_{2},\ldots,x_{n})$   x  =  (   x  1   ,   x  2   ,  …  ,   x  n   )    , and let    $K(u(x),v(y))$   K  (  u  (  x  )  ,  v  (  y  )  )satisfy    $u(rx)=ru(x)$   u  (  r  x  )  =  r  u  (  x  )    ,    $v(ry)=rv(y)$   v  (  r  y  )  =  r  v  (  y  )    ,    $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$   K  (  r  u  ,  v  )  =   r   λ   λ  1K  (  u  ,   r   −    λ  1    λ  2v  )    , and    $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$   K  (  u  ,  r  v  )  =   r   λ   λ  2K  (   r   −    λ  2    λ  1u  ,  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.

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