TY - JOUR T1 - Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$ AU - Zhang , Shutao AU - Han , Yazhou JO - Analysis in Theory and Applications VL - 2 SP - 178 EP - 203 PY - 2022 DA - 2022/07 SN - 38 DO - http://doi.org/10.4208/ata.OA-2021-0025 UR - https://global-sci.org/intro/article_detail/ata/20798.html KW - Hardy-Littlewood-Sobolev inequality, reversed Hardy-Littlewood-Sobolev inequality, rearrangement free method. AB -

For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on $\mathbb{S}^n,$ a new proof is given for the attainability of their sharp constants. Classical methods used in [22] and [8] depends on rearrangement inequalities. Here, we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method. The merit of the method is that it does not rely on rearrangement inequalities.