<i>n</i>元泛齐次核重积分算子最佳搭配参数的等价条件及应用

n元泛齐次核重积分算子最佳搭配参数的等价条件及应用

摘要: 提出n元泛齐次函数的概念．统一了齐次函数、广义齐次函数，以及一类非齐次函数；探讨了基于权函数法的n元泛齐次核Hilbert型重积分不等式的最佳搭配参数，并得到其充分必要条件与最佳常数因子的表达式；给出对应重积分算子T最佳搭配参数的等价条件算子范数的表达式；讨论了形如 \texte^-x_1^\lambda _1\sum\limits_i=2^nx_i^\lambda _i 的非齐次核重积分算子的有界性及其范数求解问题．

Abstract: 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 \texte^-x_1^\lambda _1\sum\limits_i=2^nx_i^\lambda _i. .