Some Open Problems on the Analysis of the Cauchy-Fueter System in Several Variables

In the last few years there has been a resurgence of interest for the analysis of regular functions of a quaternionic variable and, more generally, for the analysis of monogenic functions of Clifford variables [28], of which quaternions are one of the simplest examples. There are many reasons for such a rebirth of a theory whose origin goes back to Fueter’s works in the first half of this century (actually, as Professor Shapiro pointed out [16], the first works in this direction are due to Moisil and Theodoresco [19], [20] $)$ . On one hand, Clifford analysis has acquired particular relevance in theoretical physics (see e.g. [5] $)$ , and, in particular, quaternionic quantum field theory is now being actively developed (see [5]); on the other hand, Clifford analysis has become increasily relevant to questions in classical harmonic analysis, as it has been clearly demonstratcd by thc work of McIntosh[18]$ (see also [28]) and more source of interest is the very recent novel approach due to Kravchenko and Shapiro, [16]