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In the framework of Clifford analysis, we study a second‐order elliptic (generally nonstrongly elliptic) system of partial differential equations of the form: ν∂x_fϑ∂x_=0$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}}{f}^{\vartheta}\kern-0.1em {\partial}_{\underset{\_}{x}}=0 $$, where ν∂x_$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}} $$ stands for the Dirac operator with respect to a structural set ν$$ \nu $$. The solutions of this system are known as (ν,ϑ)$$ \left(\nu, \vartheta \right) $$‐inframonogenic functions. Our main purpose is to describe necessary and sufficient conditions for the solvability of a ∂¯$$ \overline{\partial} $$‐problem associated with the sandwich operator ν∂x_(·)ϑ∂x_$$ {}^{\nu}\kern-0.1em {\partial}_{\underset{\_}{x}}{\left(\cdotp \right)}^{\vartheta}\kern-0.1em {\partial}_{\underset{\_}{x}} $$. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems. [ABSTRACT FROM AUTHOR]

Clifford analysis is a higher dimensional function theory and a refinement of harmonic analysis. We will extend Clifford analysis to the infinite‐dimensional setting. To do that, we have to introduce infinite‐dimensional Clifford algebras and the infinite‐dimensional Dirac operator. The paper's main result is the Wiener chaos decomposition in Clifford analysis. We apply the decomposition to white noise and Brownian motion. [ABSTRACT FROM AUTHOR]

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with, but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)‐module. The main aim of the paper is to explain that a good possibility is to take the so‐called generalized Cauchy–Riemann equations proposed a long time ago by Stein and Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)‐modules. This is an important step in developing the corresponding function theory. [ABSTRACT FROM AUTHOR]

In this article, we will give an affirmative answer to Gilbert’s conjecture on Hardy spaces of Clifford analytic functions in upper half-space of R 8 . It is based on an explicit construction of Clifford algebra C l 8 and Spinor space R 8 by octonion algebra. Furthermore, it provides an associative approach to the theory of octonionic analytic functions. Additionally, certain classical results about octonionic analytic functions have been reformulated, and a related subject has been treated in Octonionic Hardy space in upper half-space. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we study higher order Dirac operators constructed with k-vectors. We find a necessary and sufficient condition to determine whether a function cancels them. [ABSTRACT FROM AUTHOR]

Power series expansion is used to study the growth properties of left-regular function in Clifford setting, which generalize some classical results in complex analysis. [ABSTRACT FROM AUTHOR]

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions—called monogenic functions—are defined by means of the Dirac operators that factor a certain wave operator. One of the fundamental features of quaternionic analysis is the invariance of quaternionic analogues of holomorphic function under conformal (or Möbius) transformations. A similar invariance property is known to hold in the context of Clifford algebras associated to positive definite quadratic forms. We generalize these results to the case of Clifford algebras associated to all non-degenerate quadratic forms. This result puts the indefinite signature case on the same footing as the classical positive definite case. [ABSTRACT FROM AUTHOR]

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F ˜ λ , we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F ˜ λ. [ABSTRACT FROM AUTHOR]

This work introduces fractional d-bar derivatives in the setting of quaternionic analysis, by giving meaning to fractional powers of the quaternionic d-bar derivative. The definition is motivated by starting from nth-order d-bar derivatives for n ∈ N , and further justified by various natural properties such as composition laws and its action on special functions such as Fueter polynomials. [ABSTRACT FROM AUTHOR]

This paper aims to study the jump problem for monogenic functions in fractal hypersurfaces of Euclidean spaces. The notion of the Marcinkiewicz exponent has been taken into consideration. A new solvability condition is obtained, basing the work on specific properties of the Teodorescu transform in Clifford analysis. It is shown that this condition improves those involving the Minkowski dimension. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to solve a kind of Riemann–Hilbert problem for Ψ -hyperholomorphic functions, which are linked with the use of non-standard orthogonal basis of the Euclidean space R m . We approach this problem using the language of Clifford analysis for obtaining the explicit solution of the problem in a Jordan domain Ω ⊂ R m . Since our study is concerned with either a smooth or fractal boundary, the data of the problem involve Lipschitz class Lip (k - 1 + α , Γ) . [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of characterizing the sets of uniqueness for the solutions of the sandwich equation ∂ x ̲ 3 f ∂ x ̲ = 0, where ∂ x ̲ stands for the Dirac operator in R m. These solutions are referred to as infrabimonogenic functions and can be viewed as a non-commutative version of biharmonic functions. Our main result states that a pair of distinct spheres is a set of uniqueness for infrabimonogenic functions in a convex domain of an odd-dimensional space. [ABSTRACT FROM AUTHOR]

We give estimates of the Cauchy transform in Lebesgue integral norms in Clifford analysis framework which are the generalizations of Cauchy transform in complex plane, and mainly establish the (L p , L q) -boundedness of the Clifford Cauchy transform in Euclidean space R n + 1 using the Clifford algebra and the Hardy–Littlewood maximal function. Furthermore, we prove Hedberg estimate and Kolmogorov's inequality related to Clifford Cauchy transform. As applications, some respective results in complex plane are directly obtained. Based on the properties of the Clifford Cauchy transform and the principle of uniform boundedness, we solve existence of solutions to integral equations with Cauchy kernel in quaternionic analysis. [ABSTRACT FROM AUTHOR]

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F˜λ.

Let Ω⊂ℝm$$ \Omega \subset {\mathrm{\mathbb{R}}}^m $$ be a bounded regular domain, let ∂x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in ℝm$$ {\mathrm{\mathbb{R}}}^m $$, and let ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space ℝ0,m$$ {\mathrm{\mathbb{R}}}^{0,m} $$. For k∈{1,...,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, ℝ0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}^{(k)} $$ denotes the space of k$$ k $$‐vectors in ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form ∂x_Fk∂x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in Ω$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$‐vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to ∂Ω$$ \mathrm{\partial \Omega } $$, ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator ∂x_(.)∂x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of ℝm$$ {\mathrm{\mathbb{R}}}^m $$. [ABSTRACT FROM AUTHOR]

We study local and global properties of solutions of the boundary value problem for the Dirac wave operator. This includes explicit formulas, domain of dependence, range of influence, finite propagation speed and energy estimates. We also introduce the space of Dirac wave homogeneous polynomials and the corresponding Dirac-wave conjugate functions. [ABSTRACT FROM AUTHOR]

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]

We generalize Beurling's theorem for the Clifford‐Fourier transform and give some of its applications. We deduce analogs of Hardy, Cowling‐Price, and Gelfand‐Shilov theorems in the Clifford analysis setting. [ABSTRACT FROM AUTHOR]

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems ( for short RHBVPs ) for axially monogenic functions on Euclidean space R n + 1 , n ∈ N . With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to (D - α) ϕ = 0 , α ∈ R . [ABSTRACT FROM AUTHOR]

In this paper, we prove the sharp Pitt's inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. As an application, the Beckner's logarithmic uncertainty principle for the Clifford-Fourier transform is established. [ABSTRACT FROM AUTHOR]

We prove that k distinct concentric spheres are a set of uniqueness for infrapolymonogenic functions in Euclidean spaces of odd dimension. In even dimension, we provide examples of non-zero infrapolymonogenic functions which vanish in k distinct concentric spheres. [ABSTRACT FROM AUTHOR]

This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. We are interested in finding some structures in the solutions of these generalized Lamé-Navier systems. Using MATLAB we also implement algorithms to compute with such partial differential operators as well as to verify some theoretical results obtained in the paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. We are interested in finding some structures in the solutions of these generalized Lamé-Navier systems. Using MATLAB we also implement algorithms to compute with such partial differential operators as well as to verify some theoretical results obtained in the paper. [ABSTRACT FROM AUTHOR]

In this paper, the two-sided fractional Clifford-Fourier transformation (FrCFT) is defined and some of its properties are given. First, we give the definition of the two-sided FrCFT and discuss the relationship between the two-sided Clifford-Fourier transformation (CFT) and the two-sided FrCFT. Then, the inverse transformation of the two-sided FrCFT is given and the time-frequency properties are discussed. Finally, we study differential properties and the Plancherel identity of the two-sided FrCFT. [ABSTRACT FROM AUTHOR]

The purpose of this article is to define a new general weighted class of hyperholomorphic functions, the so called B α , ω q (G) Spaces. For this class we obtain characterizations by weighted Bloch B ω α spaces. Moreover, we characterize the hyperholomorphic B α , ω q (G) functions by the coefficients of certain lacunary series expansions in Clifford analysis. [ABSTRACT FROM AUTHOR]

Iterated Dirichlet problem, also called as Riquier or Navier problem, and polyharmonic Dirichlet problem are studied for n-Poisson equation in Clifford analysis using iterated polyharmonic Green function and polyharmonic Green-Almansi type function appropriate for the boundary conditions of the problems. [ABSTRACT FROM AUTHOR]

In this paper, we present the algebraic properties of dual quaternions and define a slice regularity of a dual quaternionic function. Since the product of dual quaternions is non-commutative, slice regularity is derived in two ways. Thereafter, we propose the Cauchy-Riemann equations and a power series corresponding to dual quaternions.

The aim of this paper is to establish a representation formula for the solutions of the Lamé-Navier system in linear elasticity theory. We also study boundary value problems for such a system in a bounded domain $ \Omega\subset {\mathbb R}^3 $, allowing a very general geometric behavior of its boundary. Our method exploits the connections between this system and some classes of second order partial differential equations arising in Clifford analysis.

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.

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]

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]

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for (φ , ψ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ (1 + α , Γ) . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for (φ , ψ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ (1 + α , Γ) . [ABSTRACT FROM AUTHOR]

In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in R m . Being defined as the solutions of elliptic (genera-lly non-strongly elliptic) partial differential equations, (φ , ψ) -inframonogenic and (φ , ψ) -harmonic functions do not share the good structure and properties of the harmonic ones. The aim of this paper it to show and clarified the relationship between these classes of functions. [ABSTRACT FROM AUTHOR]

As an analogue of the massless field equations in Euclidean space, we consider the so-called generalized Cauchy–Riemann equations introduced by E. Stein and G. Weiss. In the spin 1/2 case these equations reduce to the Dirac equation for spin 1/2 fields, which was thoroughly and intensively studied in Clifford analysis. For general spin it was recently shown that, in dimension 4, homogenous solutions form irreducible Spin modules. The next step then is to describe the corresponding Fischer decomposition, i.e. an irreducible decomposition of the space of spinor fields, which is well-known for spin 1/2 and for spin 1. The main aim of the present paper is to describe, still in dimension 4, the Fischer decomposition for spin 3/2. [ABSTRACT FROM AUTHOR]

Clifford-Legendre and Clifford–Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on m-dimensional euclidean space R m and taking values in the associated Clifford algebra R m . New recurrence and Bonnet-type formulae for these polynomials are provided, and their Fourier transforms are computed. Explicit representations in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of zeros. In the case m = 2 we describe a degeneracy between the even- and odd-indexed polynomials. [ABSTRACT FROM AUTHOR]

This paper studies some systems of second order partial differential equations associated to a Dirac type operator φ ∂ ̲ in R 2 and R 3 , with respect to an arbitrary structural set φ . We develop a method by which we can transform these general systems into those connected to the standard Dirac operator ∂ ̲ . One unexpected result is that there exists an isomorphism relating the above generalized equations to the original ones. [ABSTRACT FROM AUTHOR]

Harmonic functions are the solutions of the second-order partial differential equation ∂ x ̲ ∂ x ̲ u = 0 , where ∂ x ̲ stands for the Dirac operator factorizing the Laplacian in R m . In this paper, we consider functions satisfying the sandwich equation ∂ x ̲ u ∂ x ̲ = 0 , the so-called inframonogenic functions. It is easily seen that the real-valued solutions of both previous equations will be identical. However, the situation is quite different when Clifford algebra valued solutions are considered. This leads to different classes of functions, which appear together in some topics of linear elasticity theory. The main purpose of this paper is to deepen the understanding of inframonogenic functions as well as to contrast its behavior with the more traditional harmonic functions. [ABSTRACT FROM AUTHOR]