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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 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]

Firstly, this paper gives the Cauchy-type integral formula and the Plemelj formula of hypermonogenic functions on unbounded domains and discusses the Hölder continuity of the quasi-Cauchy-type singular integral operator for hypermonogenic functions on unbounded domains. Secondly, this paper studies the relation betweenandand introduces the modified quasi-Cauchy-type singular integral operator. Finally, this paper proves the existence and uniqueness of the fixed point of the operatorby the Banach’s Contraction Mapping Principle and gives a Mann iterative sequence which strongly converges to the fixed point of the operator. [ABSTRACT FROM AUTHOR]

In this article, we obtain an explicit expression of the difference and sum of the unit base in the Clifford analysis case by using the analogy with the complex one. These formulae then lead to the introduction of a class of Bernoulli polynomials in the Clifford analysis setting which are linked to the sum base. The newly introduced Bernoulli polynomials have properties which are very similar to those of the classical ones. Then we prove that each of the sum and difference of the unit base is of order 1. Finally, certain convergence properties of the difference and sum of simple monic bases of special monogenic polynomials are investigated. The obtained results generalize the complex case proved by Mikhail and Nassif to the Clifford setting. [ABSTRACT FROM PUBLISHER]

Abstract: The classical Hadamard three-circles theorem (1896) gives a relation between the maximum absolute values of an analytic function on three concentric circles. More precisely, it asserts that if f is an analytic function in the annulus , , and if , , and are the maxima of f on the three circles corresponding, respectively, to , , and r then In this paper we introduce a Hadamardʼs three-hyperballs type theorem in the framework of Clifford analysis. As a concrete application, we obtain an overconvergence property of special monogenic simple series. [Copyright &y& Elsevier]

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]

Clifford analysis may be regarded as a direct and elegant generalization to higher dimensions of the theory of holomorphic functions in the complex plane, centred around the notion of monogenic function, i.e. a null solution of the Dirac operator. This paper dealswith axial and biaxialmonogenic functions containing sphericalmonogenics. They are constructed bymeans of two fundamentalmethods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves. [ABSTRACT FROM AUTHOR]

Suppose that Ω is a bounded domain of ℝ with a fractal boundary Γ and let ℝ be the real Clifford algebra constructed over the quadratic space ℝ. Replacing the fractal dimensions of Γ with conditions of approximating character we will characterize the monogenicity of a ℝ-valued function F in the interior and exterior of Ω, in terms of its boundary value f = F|. Moreover, our geometric perspective allows for generalizations of certain two-sided monogenic extension results to a wide class of domains. [ABSTRACT FROM AUTHOR]

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation invariant differential operator $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial }$$ called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators $$\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }$$ and $$\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} ^\dag }$$ which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial }$$. In this paper, Teodorescu operators for the Hermitian Dirac operators $$\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }$$ and $$\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} ^\dag }$$ are constructed. Moreover, the structure of the Euclidean and Hermitian Teodorescu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators issuing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed. [ABSTRACT FROM AUTHOR]

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space $${{\mathbb R}^m}$$. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on $${{\mathbb R}^m}$$. Fix the direct sum $${{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}$$. In this article, we will study the decomposition of the space $${{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}$$ of spherical monogenics of order n under the action of Spin( p) × Spin( q). As a result, we obtain a Spin( p) × Spin( q)-invariant orthonormal basis for $${{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}$$. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space $${{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}$$. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a class of Fourier multipliers whose symbols are controlled by a polynomial on starlike Lipschitz surfaces and get the boundedness of these operators on Sobolev spaces and their endpoint estimates. [Copyright &y& Elsevier]

In the Clifford analysis context a specific type of solution for the higher spin Dirac operators $\mathcal{Q}_{k,l} (k \geq l \in \mathbb{N})$ 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. [ABSTRACT FROM AUTHOR]

The main goal of this article is to study special polynomial power series expansions for entire axially monogenic functions. In relation to the difference and sum bases of monogenic polynomials, which have link with Bernoulli polynomials, the best possible bounds for the orders of such bases are determined. Moreover the Tρ-property of such differenced base and the sum base is discussed. The results obtained contain previous ones of complex setting as special cases. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ ℝ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum ( d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface. [ABSTRACT FROM AUTHOR]

Abstract: Segal–Bargmann space and monogenic Fock space are introduced first. Then, with the help of exponential functions in Clifford analysis, two integral operators are defined to connect and together. The corresponding integral properties are studied in detail. [Copyright &y& Elsevier]

In this article we present a closed formula for the CK-extension of the Gaussian distribution in m, and the monogenic version of the holomorphic function [image omitted] which is a fundamental solution of the generalized Cauchy-Riemann operator. [ABSTRACT FROM AUTHOR]

In this article, half Robin problems for powers of Dirac operator in the unit ball B of Rm (m x≥ 3) are investigated. By using a certain kind of decomposition of k-monogenic functions an explicit formula for the solution and the necessary and sufficient conditions for the solvability of the problems are obtained. [ABSTRACT FROM AUTHOR]

New higher–dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co–ordinates, the "finite part" distribution $$ Fpx^{\mu }_{ + } $$ on the real line and the generalized spherical means involving vector–valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector–valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self–contained, it offers an overview of the results found so far. [ABSTRACT FROM AUTHOR]