(untitled) (Analysis in Theory and Applications, 2022)
For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on 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.