The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications

For x= ( x₁,…, x_n ) , u ( x ) = ( ∑_i=1^n a_i x_iρ )1/ρ , v ( y ) = ( ∑_i=1^n b_i y_iρ )1/ρ , by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K ( u ( x ),v ( y ) ) =G ( u^λ₁ ( x ) v^λ₂ (y) ) and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered.