Orbital measures on SU(2)/SO(2)

TL;DR

This paper investigates the regularity of orbital measures on the group SU(2)/SO(2), proving that for p≥3 their convolution derivatives are in L^2, and providing a counterexample for p=2 to a previous conjecture.

Contribution

It establishes L^2 integrability of the Radon-Nikodym derivatives for convolutions of orbital measures on SU(2)/SO(2), and refutes the dichotomy conjecture with a counterexample.

Findings

For p≥3, the derivatives are in L^2(U).

Counterexample shows for some a, the derivative for p=2 is not in L^2.

Provides a new proof for the case p>2.

Abstract

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.