One of the equalities that we will frequently use to estimate \(u_*\) pointwise is the following: \(u_*(s)-u_*(t)=\displaystyle \int _t^su^{\prime }_*(\sigma )d\sigma \) . But such an equality is true only if \(u_*\) is absolutely continuous on \([t,s]\) . Thus in this chapter, we will show how the relative rearrangement will allow us to answer the following natural question: if u is regular (say \(C^k(\overline {\Omega })\) ), what can we say about the regularity of \(u_{*}\) ? Through this study, we will deduce inequalities that will link the gradient of u and the derivative of \(u_{*}\) . These inequalities will give the classical Polyà–Szegö inequalities but also various other extensions.