Null-sets of \(H\)-solutions (Q1431039)

The authors study the problem of null-sets of Leutwiler' s system of partial differential equations of first order which is the basis of a modified quaternionic analysis. This system reads as follows: \[ \begin{gathered} t(u_x- u_y- w_t)+ w= 0,\\ u_y= -v_x,\;u_t= -w_x,\;v_t= w_y.\end{gathered} \] Quaternionic functions \(f= u+ iv+ jw\) with \(u\), \(v\), \(w\) are real-valued and \(i\), \(j\) are quaternionic units are called \(H\)-solutions. This name is given in honor of Hodge. There are constructed non-trivial examples of null-sets of \(H\)-solutions. Moreover there were found subsets \(\Omega_0\subset\Omega\) with \(f|_{\Omega_0}= 0\) leads to \(f= 0\) on \(\Omega\). The results formulated in this little article convey considerable the understanding of the nature of the modified quaternionic analysis.