Equivalent Parameter Conditions for the Construction of Hilbert-type Integral Inequalities with a Class of Non-homogeneous Kernels

Using the weight function methods, the equivalent parameter conditions for the validity of Hilbert-type integral inequalities with non-homogeneous kernel K(x, y)=φ^λ(x^λ₁y^λ₂)φ′(x^λ₁y^λ₂) and the best constant factor are discussed. The necessary and sufficient conditions for constructing such Hilbert-type inequalities and the formula for expressing the best constant factor are obtained. Many new Hilbert-type integral inequalities with some specific non-homogeneous kernels and the best constant factors are also obtained. Finally, the norm and boundedness of corresponding singular integral operators are discussed.