n-Variable pan-homogeneous kernel multiple integral operator: equivalent conditions for optimal matching parameters and applications

In this paper, we introduce the concept of n-variable pan-homogeneous functions, to unify homogeneous functions, generalized homogeneous functions and a class of non-homogeneous functions. With this weight function method, we investigate optimal matching parameters for Hilbert-type multiple integral inequalities with n-variable pan-homogeneous kernels, and establish both necessary and sufficient conditions as well as expressions for best constant factor. We further provide equivalent conditions for optimal matching parameters of corresponding multiple integral operators T and expressions for operator norm. We discuss boundedness and norm calculation for non-homogeneous kernel multiple integral operator in the form \textexp(-x_1^\lambda _1\displaystyle\sum\limits_i=2^nx_i^\lambda _i) .