Generalized Hermitean Clifford Analysis

Complex Hermitean Clifford analysis, emerged presently as a refinement of the theory of several complex variables in the non-commutative field, have ob-tained full development in recent five years. As the classical Clifford analysis has important applications in the representation theory of Lie groups, the Atiya-Singer index theorem, the theory of singular integral operators and the theory of water wave equations, Hermitean Clifford analysis has potential importance.The complex Hermitean Clifford analysis has its center around the simulta-neous null solutions of a2x2matrix operator. This operator is a refinement of classical Clifford Dirac operator in the complex Hermitean setting. It is an impor-tant fact that, when the complex plane as the range of holomorphic functions is identified with a spinor space, holomorphic functions in several complex variables are exactly the complex valued functions which are null solutions of the2×2matrix operator.In this article, we generalize Hermitean Clifford analysis in two essential ways.First, we incorporate the2×2matrix operator into a single operator by means of algebraic approach. It has such an advantage that the null space of this operator is naturally increased, compared with that of the2×2matrix operator. The important point lies at the parallelism between the complex Hermitean Clifford analysis and the classical Clifford analysis arose from this single operator. This makes a solid foundation for the development of the complex Hermitean Clifford analysis.Secondly, we generalize the complex Hermitean Clifford analysis to other al-gebras, such as hypercomplex algebra, paracomplex algebra, bicomplex algebra, split quaternion algebra. These generalizations initiate from the Minkowski time-space theory and the bicomplex quantum mechanics. Our results provide new operators for analysis, and new algebraic structures for geometry. These geome-tries corresponding to these algebras is now becoming best tools in the field such as the relativity theory, special geometry, black hole, and supersymmetry.The context of each chapter is stated separatively as follow.In chapter two, we invent paracomplex Witt basis. With it we can introduce the paracomplex Hermitean Dirac operator, and then establish the integral theory in paracomplex Hermitean Clifford analysis. In particular, we obtain the Cauchy type integral formula which generalizes the Matinelli-Bochner integral formula in several complex variables. We unify the theory of paracomplex Hermitean Clifford analysis with that of complex Hermitean Clifford analysis by means of a parameter and by introducing a single Dirac type operator D.In chapter three, we establish the bicomplex Hermitean Clifford analysis. The bicomplex Hermitean Clifford analysis studies the bicomplex Dirac operator: D:C∞(R4n,W4n)â†'C∞(R4n,W4n), where W4n is the tensor product of the three algebras, i.e., the hyperbolic quater-nion B, the bicomplex number B, and the Clifford algebra R0,4n.The operator D is a square root of the Laplacian in R4n, introduced by the formula D=∑j=03Kj(?)zj with Kj the basis of B, and (?)zj the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B(?)R R0,4nWe initiate the study of a single operator D, which overturn the prevailing opinion in the literature that monogenic functions in Hermitean Clifford analysis in the complex or quaternionic setting are null solutions of a system of equations, instead of by a single equation like classical monogenic functions. In contrast to the Hermitean Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting.For the operator D, we establish the Cauchy integral formula, which gener-alizes the Matinelli-Bochner formula in the theory of several complex variables.In chapter four we extend the quaternionic Hermitean Clifford analysis to split quaternions. We construct the split quaternion Witt basis and introduce the split quaternionic Hermitean Dirac operator D which is a self-map of smooth functions in domains of R4n with values in the tensor product of the hyperbolic quaternions B, the split quaternions H, and the Clifford algebra R0,4n.D is defined by D=∑j=03Kj(?)zj with Kj the basis of B, and (?)zj the twisted Hermitian Dirac operators in the split quaternionic Clifford algebra H(?)R0,4n.We establish the counterpart of the Cauchy integral formula.In chapter five, we characterize the dual of the (Frechet) space of germs of left Hermitean monogenic matrix functions in a compact set M(?)R2n, which has important applications in the theory of the generalized functions and hyper-functions.