The Adaptation Parameter Conditions for Hilbert-type Integral Inequalities with Quasi-homogeneous Kernels

The weighting coefficient method and real analysis techniques are used to discuss how to select the adaptation parameters to obtain Hilbert-type integral inequalities with quasi-homogeneous kernel and the best constant factor. The necessary and sufficient conditions for the adaptation parameters for constructing the best Hilbert-type integral inequality with quasi-homogeneous kernel and the expression formula of the best constant factor are obtained. This solves a fundamental theoretical problem in the study of constructing optimal Hilbert-type integral inequalities. Finally, its applications to finding the norm of integration operators are discussed.