# symmetrized rearrangement on sphere.

I am trying to undestand the Corollary 2.2 from Osgood, Phillips and Sarnak (see http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.558&rep=rep1&type=pdf), that is, if $$u \in W^{1}(S^{2})'$$ with mean value zero and antipodal symmetry then $$\int_{S^{2}}|\nabla_0 u_s|^{2}\frac{dA_0}{4\pi} \le \frac{1}{2}\int_{S^{2}}|\nabla_0 u|^{2}\frac{dA_0}{4\pi},$$ where $$u_s$$ is the symmetric decreasing rearrangement of $$u$$.

They suggest to use the usual symmetrization procedure with the isoperimetric inequality for $$S^{2}$$ ( that is, $$L^{2} \ge A(4\pi-A)$$), replaced by a sharper form holding for domains with antipodal symmetry: $$L^{2} \ge 2A(4\pi-A).$$

From Pólya-Szego there is a following result: If $$D$$ is a open domain in $$\mathbb{R}^{n}$$ then $$\int_{D^{*}} |\nabla u^{*}|dx \le \int_{D}|\nabla u|dx,$$ where $$D^{*}$$ is the symmetric rearrangement of $$D$$, that is, $$D^{*}$$ is the open centered ball whose volume agrees with $$D$$.

How can I match the Polya-Szego inequality with the sharp isoperimetric inequality to prove the first inequality?