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

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

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及算子范数的表达式；讨论了形如exp(- xλ1 1∑ n i=2 xλi 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 exp(- x λ 1 1∑ n i=2 x λ i i ).

关键词

n元泛齐次核 / Hilbert型重积分不等式 / 加权Lebesgue空间 / 最佳搭配参数 / 算子范数 / n元指数型耦合核

Key words

n-variable pan-homogeneous kernel / Hilbert-type multiple integral inequality / weighted Lebesgue space / optimal matching parameters / operator norm / n-variable exponential coupling kernel

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基金资助

广东省教育厅科研资助项目(2023KTSCX034)

广东省教育厅科研资助项目(2024ZDZX2088)

广东省本科高校“2025年人工智能赋能基础课程教学改革研究”资助项目(25GGYB03)

高等学校大学数学教学研究与发展中心资助项目(CMC20240605)

广东省教育厅创新团队资助项目(2022WCXTD009)

广东省研究生教育创新计划资助项目(2024SFKC_042)