Search

In this paper we study the Dirichlet problem in the ball for the so-called inframonogenic functions, i.e. the solutions of the sandwich equation ∂ x _ f ∂ x _ = 0 , where ∂ x _ stands for the Dirac operator in R m. The main steps in deriving our results are the establishment of some interior estimates for the first order derivatives of harmonic Hölder continuous functions and the proof of certain invariance property of the higher order Lipschitz class under the action of the Poisson integral. Using Mathematica we also implement an algorithm to find explicitly the solution of such a Dirichlet problem for a much wider class of partial differential equations in the ball of R 3 with polynomial boundary data. [ABSTRACT FROM AUTHOR]

Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, first, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases. [ABSTRACT FROM AUTHOR]

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in ℝm. The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain Ω⊂ℝm can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used. [ABSTRACT FROM AUTHOR]

Solutions of the sandwich equation ϕ ∂ ̲ [ f ] ψ ∂ ̲ = 0 , where ϕ ∂ ̲ stands for the Dirac operator with respect to a structural set ϕ , are referred to as (ϕ , ψ) -inframonogenic functions and capture the standard inframonogenic ones as special case. We derive a new integral representation formula for such functions as well as for multidimensional Ahlfors–Beurling transforms closely connected to the use of two different orthogonal basis in R m . Moreover, we also establish sufficient conditions for the solvability of a jump problem for the system ϕ ∂ ̲ [ f ] ψ ∂ ̲ = 0 in domains with fractal boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study a refinement of the Szegö-Radon transform in the hypermonogenic setting. Hypermonogenic functions form a subclass of monogenic functions arising in the study of a modified Dirac operator, which allows for weaker symmetries and also has a strong connection to the hyperbolic metric. In particular, we construct a projection operator from a module of hypermonogenic functions in R p + q onto a suitable submodule of plane waves parameterized by a vector on the unit sphere of R q. Moreover, we study the interaction of this Szegö-Radon transform with the generalized Cauchy-Kovalevskaya extension operator. Finally, we develop a reconstruction (inversion) method for this transform. [ABSTRACT FROM AUTHOR]