  We provide a scheme of a recent stability result for a family of
Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an
improved entropy - entropy production inequality associated with an appropriate
fast diffusion equation (FDE) written in self-similar variables. This result
can be rephrased as an improved decay rate of the entropy of the solution of
(FDE) for well prepared initial data. There is a family of
Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar
structure. When the exponents are in a range for which the optimal functions
for (CKN) are radially symmetric, we investigate how the methods for (GNS) can
be extended to (CKN). In particular, we prove that the solutions of the
evolution equation associated to (CKN) also satisfy an improved decay rate of
the entropy, after an explicit delay. However, the improved rate is obtained
without assuming that initial data are well prepared, which is a major
difference with the (GNS) case.
