Narrow Results

In the Clifford analysis context a specific type of solution for the higher spin Dirac operators is studied; these higher spin Dirac operators can be seen as generalizations of the classical Rarita–Schwinger operator. To that end subspaces of the space of triple monogenic polynomials are introduced and their algebraic structure is investigated. Also a dimensional analysis is carried out.

The idea of multiplexing motivates us to develop the theory on the Fourier transform (FT) of multivector-valued functions. In this paper, in the framework of Clifford analysis, we establish a Heisenberg–Pauli–Weyl type uncertainty principle for the FT of multivector-valued functions.

The spectrum of masses of the colorless states of DW Hamiltonian operator of quantum SU(2) Yang–Mills field theory on ${ℝ}^D$ obtained via the precanonical quantization is shown to be purely discrete and bounded from below. The scale of the mass gap is estimated to be of the order of magnitude of the scale of the ultra-violet parameter $\varkappa$ which is introduced by precanonical quantization on dimensional grounds.

New higher dimensional distributions are introduced in the framework of Clifford analysis. They complete the picture already established in previous work, offering unity and structural clarity. Amongst them are the building blocks of the principal value distribution, involving spherical harmonics, considered by Horváth and Stein.

Around the central theme of "square root" of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.

We study a system of equations modeling the stationary motion of incompressible electrical conducting fluid. Based on methods of Clifford analysis, we rewrite the system of magnetohydrodynamics fluid in the hypercomplex formulation and represent its solution in Clifford operator terms.

In this paper, we generalize the results of¹, on theory of hyperholomorphic analysis to the case for several variables, that is, Clifford analysis. These results are not only of theoretical importance but also parctical interest. In fact, we also generalize³, the Dirac operator to . When the base i ={i₁, …,i_n} is given, the Dirac operator D is determined, while ^ψD may consider as the rotation transform on the vector ψ = {ψ¹, … , ψⁿ}. Such a generalization is similar to Descartes coordinate system to the moving coordinate system, therefore, it is very convenient to build some basic formulas on Clifford analysis and to find some analytic representations of differential equations. From this idea, we obtain a very important representation of Bergman kernel function:

The main purpose of our article is to prove a quantitative Hartogs-Rosenthal theorem concerning uniform approximation on compact sets by solutions of elliptic first-order differential operators with coefficients in a Banach algebra. This theorem, as well as some other related results and consequences, points out that certain properties of the standard Euclidean Dirac operators originally established in the settings of Clifford analysis, or single-variable complex analysis, persist under more general circumstances.

In Clifford analysis there are two well known generalizations of the concept of holomorphic functions issued from the idea of monogenic functions: the hypermonogenic functions introduced by H. Leutwiler and the holomorphic Cliffordian functions introduced by G. Laville and I. Ramadanoff. When these functions are of type with h scalar valued, they are called respectively H-solutions and p-holomorphic Cliffordian functions. The parity of the dimension d of the vector space ℝ^0,d generating the Clifford algebra ℝ_0,d induces quite different behaviours of these functions. We summarize the properties of the analytic Cliffordian monomials and classify the properties of H-solutions and of p-holomorphic Cliffordian functions with respect to the parity of d. We show that the restriction of an H-solution on ℝ⊕ℝ^0,d to x_d = 0 is a p-holomorphic Cliffordian function. By an explicit construction of the homogeneous polynomials which are H-solutions, we show the converse. Finally, a paravector-valued function of paravectors is an H-solution if and only if it is the restriction of a p-holomorphic Cliffordian function.

A differential and integral criterion for monogenicity is presented within the framework of Clifford analysis. The results obtained provide characterizations for the holomorphic and the two-sided biregular functions.

The aim of this paper is to make an overview on three generalizations to higher dimensions of the function's theory of a complex variable. The first one concerns the so-called monogenic functions which were introduced by F. Brackx, R. Delanghe and F. Sommen, the second is the theory of hyper-monogenic functions developed by H. Leutwiler, and the last one studies the theory of holomorphic Cliffordian functions due to G. Laville and I. Ramadanoff. The basic notions in this three theories will be given. Their links and differences will also be commented.

In this paper, we list some holomorphic function spaces in Clifford analysis and give their properties. Firstly, we give some properties of regular function space. Next we study hypermonogenic function space and its properties. Finally we study k-holomorphic function and give some of its properties in unbounded domain.

This paper has two parts. In the first part, we study the boundedness for a kind of singular integral operator, which is related to the Cauchy-type integral of k-monogenic function in Clifford Analysis. In the second part, we show the Hölder continuous of the singular integral operator.