Hardy-Littlewood积分算子范数不等式及其推广

摘要

著名的Hardy-Littlewood不等式在分析数学及其应用中均起着重要的作用.但要求出该不等式中的最佳常数的值,却是一个困难的问题.为此,笔者在《常用不等式》(第3版)中曾将该问题作为未解决问题中的第109题.在笔者论文"关于Hardy-Littlewood不等式中的最佳常数"的基础上,通过将求最佳常数问题转化为求相应的算子范数等新的分析技巧,得到了Hardy-Littlewood积分算子的范数不等式.作为它的推广,得到n维向量空间上具有径向核的新的积分算子范数不等式.%How to obtain the sharp constant of the Hardy-Littlewood inequality remains unsolved.In this paper, by means of the new analysis technique of the sharp constant factor is changed into the corresponding operator norm, Hardy-Littlewood integral operator norm inequalities are proved.As its generalizations, some new integral operator norm inequalities with radial kernel on n-dimensional vector spaces are established.