Extremal functions for the Caffarelli–Kohn–Nirenberg inequalities: A simple proof of the symmetry

In this article, we give a simple proof of the result due to Lin and Wang ensuring the foliated Schwarz symmetry of the extremal functions for the Caffarelli–Kohn–Nirenberg inequalities. This new proof uses a direct and powerful method due to Bartsch, Weth and Willem using polarizations.