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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]

In this paper, we study the Bargmann–Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann–Radon transform for axially monogenic functions, by making use of the Funk–Hecke theorem. Then we present the explicit form of the general Cauchy–Kowalewski extension for radial function. Finally, by making use of the results we obtained, we give an application of the Bargmann–Radon transform for Cauchy–Kowalewski extension. [ABSTRACT FROM AUTHOR]

In this paper, we consider functions satisfying the sandwich equation ∂x_f∂x_=0, where ∂x_ stands for the Dirac operator in Rm. Such functions are referred as inframonogenic and represent an extension of the monogenic functions, ie, null solutions of ∂x_. In particular, for odd m, we prove that a C2‐function is both inframonogenic and harmonic in Ω⊂Rm if and only if it can be represented in Ω as f=f1+f2+f3x_+x_f4,where f1 and f2 are, respectively, left and right monogenic functions in Ω, while f3 and f4 are two‐sided monogenic functions there. Finally, in deriving some applications of our results, we have made use of the deep connection between the class of inframonogenic vector fields and the universal solutions of the Lamé‐Navier system in R3. [ABSTRACT FROM AUTHOR]

Solutions of the sandwich equation ∂ x _ f ∂ x _ = 0, where ∂ x _ stands for the first‐order differential operator (called Dirac operator) in the Euclidean space R m, are known as inframonogenic functions. These functions generalize in a natural way the theory of kernels associated with ∂ x _, the nowadays well‐known monogenic functions, and can be viewed also as a refinement of the biharmonic ones. In this paper we deepen study the connections between inframonogenic functions and the solutions of the homogeneous Lamé‐Navier system in R 3. Our findings allow to shed some new light on the structure of the solutions of this fundamental system in 3‐dimensional elasticity theory. [ABSTRACT FROM AUTHOR]

In this note we consider certain jump problem for poly-monogenic functions in fractal domains with higher order Lipschitz boundary data. This is accomplished by using a higher order Teodorescu operator which replaces the expected surface integral. Also, we give out the uniqueness of solutions basing the work on the method of removable singularities for monogenic functions making use of a Dolzhenko type theorem. [ABSTRACT FROM AUTHOR]

A conformal mapping in a plane domain locally maps circles to circles. More generally, quasiconformal mappings locally map circles to ellipses of bounded distortion. In this work, we study the corresponding situation for solutions to Stein–Weiss systems in the ( n + 1 ) D Euclidean space. This class of solutions is a transformation of a subset of monogenic locally quasiconformal mappings with nonvanishing Jacobian. In the theoretical part of this work, we prove that an M-conformal mapping locally maps the unit hypersphere onto explicitly characterized hyperellipsoids and vice versa. Then we discuss quasiconformal radial mappings and their relations with the Cauchy kernel and p -monogenic mappings. This is followed by the consideration of quadratic M-conformal mappings. In the applications part of this work, we provide the reader with some plot examples that demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space [ABSTRACT FROM AUTHOR]

In this article, we solve the Dirichlet boundary value problemwhereis a cylindrical domain,Dis the Cauchy-Riemann operator,is a Clifford algebra-valued function,,,is ak-dimensional submanifold of the boundary,,is a fixed point inside the domain. If the boundary data are Hölder continuously differentiable functions, then the unique solution is Hölder continuous. We shall prove an estimate of the solution by its boundary data. [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]