The parameter conditions for the existence of the Hilbert-type multiple integral inequality and its best constant factor

By means of the weight function, the following results are given. The Hilbert-type multiple integral inequality with the lambda -order homogeneous kernel integral R+n</mml:msubsup>integral R+m</mml:msubsup>K(<mml:mfenced close="<parallel>" open="parallel to">x</mml:mfenced>m,rho,<mml:mfenced close="<parallel>" open="parallel to">y</mml:mfenced>n,rho )f(x)g(y)dxdy <= M<mml:msub><mml:mfenced close="<parallel>" open="parallel to">f</mml:mfenced>p,alpha <mml:msub><mml:mfenced close="<parallel>" open="parallel to">g</mml:mfenced>q,beta is true if and only if <mml:mfrac>alpha +mp</mml:mfrac>+<mml:mfrac>beta +nq</mml:mfrac>=lambda +m+n, and the expression of the best possible constant factor is obtained. Furthermore, its application in the operator theory is discussed.