Pseudo-differential Operators on SU(2)

Abstract

In this chapter we carry out the analysis of operators on SU(2) with an application to the operators on the three-dimensional sphere $\mathbb{S}^3$. In particular, we derive a much simpler symbolic characterisation of pseudo-differential operators on SU(2) than the one given in Definition 10.9.5. In turn, this will also yield a characterisation of full symbols of pseudo-differential operators on the 3-sphere $\mathbb{S}^3$. We note that this approach works globally on the whole sphere, since the version of the Fourier analysis that we use is different from the one in, e.g., [110, 122, 111] which covers only a hemisphere, with singularities at the equator. For a general introduction and motivation for the analysis on SU(2) we refer the reader to the introduction in Part IV where the cases of SU(2) and $\mathbb{S}^3$ were put in a perspective.