Generalized Moisil-Théodoresco systems and Cauchy integral decompositions
The authors study generalized Moisil-Théodoresco systems. Let \(\mathbb{R}^{(s)}_{0,m+1}\) be the space of \(s\)-vectors \((0\leq s\leq m+ 1)\) in the Clifford algebra \(\mathbb{R}_{0,m+1}\) constructed over the quadratic vector space \(\mathbb{R}^{0,m+1}\), let \(r,p,q\in\mathbb{N}\) with \(0\leq r\leq m+1\), \(0\leq p\leq q\) and \(r+ 2q\leq m+1\) and let \[ \mathbb{R}^{(r,p,q)}_{0,m+1}= \sum^q_{j= p}\bigoplus \mathbb{R}^{(r+ 2j)}_{0,m+1}. \] The Dirac operator \(\partial_x\) is defined by \(\partial_x= \sum^m_{i=0} e_i\partial_{x_i}\), where \(x= (x_0,x_1,\dots, x_m)\in \mathbb{R}^{m+1}\) and \(e= (e_i: i= 0,\dots, m)\) is an orthogonal basis for the quadratic space \(\mathbb{R}^{0,m+1}\). Then an \(\mathbb{R}^{(r,p,q)}_{0,m+1}\)-valued smooth function \(W\) defined in an open subset \(\Omega\subset\mathbb{R}^{m+1}\) is said to satisfy the generalized Moisil-Théodoresco system of type \((r,p,q)\) if \(\partial_x w= 0\) in \(\Omega\), where \(\partial_x\) is the Dirac operator in \(\mathbb{R}^{m+ 1}\). The authors study and characterize the structure of the solutions to Moisil-Théodoresco systems. A structure theorem is proved in Section 3 for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if \(\Omega\) is bounded with boundary \(\Gamma\), where \(\Gamma\) is an Ahlfors-David regular surface, and if \(w\) is a \(\mathbb{R}^{(r,p,q)}_{0,m+1}\)-valued Hölder continuous function on \(\Gamma\), then necessary and sufficient conditions are given under which \(w\) admits on \(\Gamma\) a Cauchy integral decomposition \(w= w_++ w_-\).
scientific article; zbMATH DE number 5313295
scientific article