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.
Categories: math.CA