  We let U=SU(2) and K=SO(2) and denote N_{U}(K) the normalizer of K in U. For
a an element of U\ N_{U} (K), we let \mu_{a} be the normalized singular measure
supported in KaK. For p a positive integer, it was proved that \mu_{a}^{( p)},
the convolution of p copies of \mu_{a}, is absolutely continuous with respect
to the Haar measure of the group U as soon as p>=2. The aim of this paper is to
go a step further by proving the following two results : (i) for every a in U\
N_{U} (K) and every integer p >=3, the Radon-Nikodym derivative of
\mu_{a}^{(p)} with respect to the Haar measure m_{U} on U, namely
d\mu_{a}^{(p)}/d m_{U}, is in L^{2}(U), and (ii) there exist a in U\ N_{U} (K)
for which d\mu_{a}^{(2)}/ dm_{U} is not in L^{2}(U), hence a counter example to
the dichotomy conjecture. Since L^{2} (G) \subseteq L^{1} (G), our result gives
in particular a new proof of the result when p>2.
