调和分析中的几何不等式

Geometric inequalities in harmonic analysis

几何不等式一直是分析、几何、方程、概率和组合学研究的热门内容之一,而分数次积分不等式又在分析学中扮演重要角色.因其在Fourier变换限制性猜想、Radon变换和k平面变换等问题中发挥重要作用,多年来一直备受分析学家们的高度关注.本文简要回顾一些分数次积分不等式,介绍经典几何极值不等式,以及研究最优化问题的有用工具重排不等式;重点介绍结合对称重排思想和竞争对称性方法在证明分数次积分不等式最优化函数中的应用.本文还将回顾混合范数空间的基本性质,并介绍其上的一些分数次积分不等式.

Geometric inequalities have been hot topics in analysis, geometry, PDE, probability and combinatorial theory. Among so many geometric inequalities, fractional integral inequalities exceptionally attract attention of analysts, which play an important role in Analysis. The reason is due to their diverse connections to questions regarding the restriction of the Fourier transform, Radon transform and the k-plane transform. In this paper,we simply review a series of fractional integral inequalities, some geometric extremal inequalities and symmetric decreasing rearrangement inequalities which are extremely useful analytic tools to deal with many classic extremal problems. Together with competing symmetries, we focus on presenting rearrangement method to determine the optimisers of some fractional integral inequalities. We also introduce some properties of Lebesgue spaces with mixed norms and consider some fractional integral inequalities in mixed norm spaces as well.