Polyanalytic Hardy decomposition of higher order Lipschitz functions

This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve $\Gamma$ into a sum of two polyanalytic functions in each open domain defined by $\Gamma$. Our basic tools are the Hardy projections related to a singular integral operator arising in polyanalytic function theory, which, as it is proved here, represents an involution operator on the higher order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on the boundary of a domain

Hermitian clifford analysis

This paper gives an overview of some basic results on Hermitian Clifford analysis, a refinement of classical Clifford analysis dealing with functions in the kernel of two mutually adjoint Dirac operators invariant under the action of the unitary group. The set of these functions, called Hermitian monogenic, contains the set of holomorphic functions in several complex variables. The paper discusses, among other results, the Fischer decomposition, the Cauchy–Kovalevskaya extension problem, the axiomatic radial algebra, and also some algebraic analysis of the system associated with Hermitian monogenic functions. While the Cauchy–Kovalevskaya extension problem can be carried out for the Hermitian monogenic system, this system imposes severe constraints on the initial Cauchy data. There exists a subsystem of the Hermitian monogenic system in which these constraints can be avoided. This subsystem, called submonogenic system, will also be discussed in the paper

On a Generalized Lam\'e-Navier system in R3\mathbb{R}^3

This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lam\'e-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 Lam\'e-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.Comment: 19 pages 0 figure

Duality for Hermitean systems in R2n

In this paper, using the algebraic structure of the space of circulant (2 × 2) matrix, we characterize the dual of the (Frechet) space of germs of left Hermitean monogenic matrix functions in a compact set of Euclidean space with even di;ension. As an application we describe the dual space of the so-called h-monogenic functions satisfying simultaneously two Dirac type equations

Generalized Moisil-Théodoresco systems and Cauchy integral decompositions

Let ℝ0,m+1(s) be the space of s-vectors (0≤s≤m+1) in the Clifford algebra ℝ0,m+1 constructed over the quadratic vector space ℝ0,m+1, let r,p,q∈ℕ with 0≤r≤m+1, 0≤p≤q, and r+2q≤m+1, and let ℝ0,m+1(r,p,q)=∑j=pq⨁ ℝ0,m+1(r+2j). Then, an ℝ0,m+1(r,p,q)-valued smooth function W defined in an open subset Ω⊂ℝm+1 is said to satisfy the generalized Moisil-Théodoresco system of type (r,p,q) if ∂xW=0 in Ω, where ∂x is the Dirac operator in ℝm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if Ω is bounded with boundary Γ, where Γ is an Ahlfors-David regular surface, and if W is a ℝ0,m+1(r,p,q)-valued Hölder continuous function on Γ, then necessary and sufficient conditions are given under which W admits on Γ a Cauchy integral decomposition W=W++W−

Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis

Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article

A Hilbert transform for hermitean matrix functions on fractal domains

We consider Holder continuous circulant (2 × 2) matrix functions defined on the fractal boundary of a Jordan domain in R2n. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitean Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitean Dirac operators. In a previous paper a Hermitean Cauchy integral was constructed by means of a matrix approach using circulant (2×2) matrix functions, from which a Hilbert transform was derived, all of this for the case of domains with smooth boundary. However, crucial parts of the method used are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitean Hilbert transform in that case. As a consequence, we are able to give necessary and sufficient conditions for the Hermitean monogenicity of a circulant matrix function in the interior and exterior of the domain considered, in terms of its boundary value, where the boundary is required to be Ahlfors-David regular

Hölder norm estimate for a Hilbert transform in Hermitian Clifford analysis

A Hilbert transform for Holder continuous circulant (2 x 2) matrix functions, on the d-summable (or fractal) boundary I" of a Jordan domain Omega in a"e(2n) , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Holder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Holder exponents, the diameter of I" and a specific d-sum (d > d) of the Whitney decomposition of Omega. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary