Stability theorems for GNS inequalities: a reduction principle to the radial case

A symmetrization techique, introduced by Cianchi, Fusco, Maggi and Pratelli concerning the Sobolev inequality, is adapted to the Gagliardo-Nirenberg-Sobolev inequality (GNS) to obtain a reduction step of the problem of showing its quantitative version. More precisely we prove a stability result for the GNS inequality under the hypothesis that it holds, in turn, in the smaller class of radial symmetric decreasing functions.