Intrinsic Dirac operators in \(\mathbb{C}^ n\)
A compact, real, \(n\)-dimensional (i.e. dimension over \(\mathbb{R})\), \(C^1\) submanifold-with-boundary \(M\) of \(\mathbb{C}^n\) is of type one if it does not intersect the light cone on any of its points, and has no isotropic tangent vectors. These manifolds share many properties concerning monogenic functions (i.e. functions satisfying \(D_f = 0\), where \(D\) is the Dirac operator of \(\mathbb{C}^n\); these functions have values in the Clifford algebra over \(\mathbb{C}^n)\) with \(\mathbb{R}^n\), and in this paper a number of these properties, and their consequences, are considered. First of all, a function monogenic (alternative name: left regular) in a neighbourhood of \(M\) can be extended to the cell of harmonicity of \(M\), and this extension can be described by a Cauchy-like integral on \(\partial M\). Using this, the author proves that the module \(B^p_l\) of \( L_p\) functions of \(M\) which are restrictions of monogenic functions is a complete submodule of \(L^p(M)\). Using the weight \(\exp (B(x))\), it is possible to prove similar results for the solutions of the equation \(D_f - (D_B) f = 0\), where \(B\) is a scalar valued potential. The intrinsic Dirac operator \(D_M\) of \(M\) is defined in such a way that, if \(g \in C^1(M)\) has a holomorphic extension \(g^+\), in a neighbourhood of \(M\), then \(D_Mg\) has the holomorphic extension \(D^+_g\). For a suitable function \(f\), convolution of \(f\) with the Cauchy kernel, mapping \(f\) to \(T_mf\) gives a solution to the equation \(D_M g = - f\). Continuity properties of \(T_M\) are studied, and using \(T_M\) it is proved that, in \(L_2 (M)\), the complement of \(B^2_l\) is the image of the set of traceless functions (i.e. with boundary value zero) under \(D_M\). As a consequence, under certain conditions, traceless functions, harmonic on the cell of harmonicity, are identically zero.
scientific article; zbMATH DE number 889812