On geometric inequalities related to fractional integration
The first part of this thesis establishes a series of geometric ineqalities related to fractional integration in some geometric settings, including bilinear and multilinear forms. In the second part of this thesis, we study some kinds of rearrangement inequalities. In particular, some applications of rearrangement inequalities will be given, for instance, the determination of the extremals of some geometric problems. By competing symmetries and rearrangement inequalities, we prove the sharp versions of geometric inequalities introduced in the first part in Euclidean spaces. Meanwhile, there are the corresponding conformally equivalent formulations in unit sphere and in hyperbolic space. The last part is about collaborative work on the regularity of the Hardy-Littlewood maximal functions. We give a simple proof to improve Tanaka's result of the paper entitled "A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function". Our proof is based on the behaviour of the local maximum of the non-centered Hardy-Littlewood maximal function.