Generalized Cauchy-Riemann Equations Research Articles

On a Statement of the Boundary Value Problem for a Generalized Cauchy–Riemann Equation with Nonisolated Singularities in a Lower-Order Coefficient

On a Statement of the Boundary Value Problem for a Generalized Cauchy–Riemann Equation with Nonisolated Singularities in a Lower-Order Coefficient

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

О постановке краевой задачи для обобщенного уравнения Коши-Римана с неизолированными особенностями в младшем коэффициенте

В статье выявлен эффект влияния попарно непересекающихся и не проходящих через начоло координат неизолированных особенностей в младшем коэффициенте обобщенного уравнения Коши-Римана на постановку краевых задач. Условие на границе области оказалось недостаточным для решения таких задач. Поэтому рассмотрен случай, объединяющий элементы задач Римана-Гильберта на границе области и линейного сопряжения на окружностях-носителях сингулярностей младшего коэффициента, лежащих внутри области. Библиография: 19 названий.

Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields

Abstract We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 𝔸 {\mathbb{A}} for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 𝔸 {\mathbb{A}} , which is also known as the Lorch derivative with respect to 𝔸 {\mathbb{A}} , and recently simply called 𝔸 {\mathbb{A}} -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.

A Riemann–Hilbert Type Problem for the Generalized Cauchy–Riemann Equation with Supersingular Points on a Half-Plane

For an equation with the Cauchy–Riemann operator having strong isolated point singularities in the lowest coefficient, an integral representation of the solution is found in the class of bounded functions at infinity, and a Riemann–Hilbert type problem on a half-plane is investigated. Specialists are well aware about the important role played in applications by I.N. Vekua’s theory of generalized analytic functions of an equation in the case when the coefficients and the right-hand side of the generalized Cauchy–Riemann system belong to the class of summable functions. It is deeply linked with many areas of analysis, geometry and mechanics, including quasi-conformal maps, surface theory, shell theory, and gas dynamics. In particular, it is widely used in modeling transonic gas flows, states of momentless stressed equilibrium of convex shells, and many other processes. The Vekua–Pompeiu integral plays a key role in this theory. When the lowest coefficient has strong singularities, some researchers believe that the Vekua–Pompeiu integral, which is main research engine in the theory of generalized analytic functions, cannot be applied. By using the Vekua–Pompeiu integral, we have succeeded in constructing solutions of the Cauchy–Riemann system the lowest coefficient of which contains strong isolated point singularities in an infinite domain. A Riemann–Hilbert type problem on a half-plane is investigated for the equation under consideration by using an explicit integral representation.

Fischer Decomposition of Massless Fields for Spin 3/2 in Dimension 4

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.

General massless field equations for higher spin in dimension 4

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.

On Classes of Solutions to Generalized Cauchy–Riemann Equations

We study classes of functions satisfying the generalized Cauchy–Riemann system and construct solutions by successively applying quaternionic differential operators. We extract some classes of solutions to this system.

Hilbert boundary value problem for generalized analytic functions with a singular line

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.

Dirichlet Problem for a Generalized Cauchy–Riemann Equation with a Supersingular Point on a Half-Plane

For equations with a Cauchy–Riemann operator involving a strong point singularity in the lower coefficient on a half-plane, an integral representation of the solution is obtained in the class of bounded functions and a Dirichlet-type problem is studied. The calculation of the Vekua–Pompeiu integral is examined in the case when the density of the integral has strong singularities in a set of points or lines.

Интегральные представления для обобщенного уравнения Коши-Римана с надсингулярной точкой в ​​полуплоскости

Интегральные представления для обобщенного уравнения Коши-Римана с надсингулярной точкой в ​​полуплоскости

Задачи типа Гильберта для уравнения Коши–Римана с сингулярной окружностью в младших коэффициентах

The Cauchy-Riemann system of equations occupies a special place in the class of first-order elliptic systems. Such a system with lowest terms and a right-hand side is called the generalized Cauchy-Riemann system (GCRS), which can be conveniently studied by making a transition from a real space into a complex space. There are several different mathematical theories of equations that generalize the methods of the theory of functions of a complex variable. In this regard, the work of L. Bers, in which the integration operations with respect to a complex variable for a generalized system of Cauchy-Riemann type are generalized, should primarily be mentioned. This approach has received a well-known finalization in the theory of pseudo-analytic functions. In the works of G.N. Polozhiy, the theory of p-analytic functions was developed, which is close in its ideas to the works of L. Bers. Another, more progressive research area, called «generalized analytical functions», was developed by the school of I.N. Vekua and his followers (B.V. Boyarsky and others). Here, the idea of correspondence between the functions of a complex variable and the solutions of the generalized Cauchy–Riemann equation using the functional analysis techniques is developed. The Vekua theory is constructed on the assumption that the coefficients of the function’s lowest terms belong to the space of summable functions with degree р > 2. The coefficients of such systems may admit «weak» singularities limited by the requirement of p-integrability. Thus, the Vekua theory does not cover even equations with coefficients having first-order singularities. However, other problems, the lowest coefficients of which admit first- order singularities or «strong singularities», boil down to GCRS. In this study, Hilbert type problems are solved for the GCRS the lowest coefficients of which admit strong singularity in a circle. &nbsp

Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equation on upper half ball

We study Riemann–Hilbert boundary value problems with variable coefficients for axially symmetric null-solutions to the iterated generalized Cauchy–Riemann equation, defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. First, we prove an Almansi-type decomposition theorem for axially symmetric null-solutions to the iterated generalized Cauchy–Riemann equation. Then, we give integral representation solutions to the Riemann–Hilbert problems for axially symmetric null-solutions to iterated generalized Cauchy–Riemann equation over the upper half unit ball centred at the origin in four-dimensional Euclidean space. In particular, we derive solutions to the Schwarz problem for axially symmetric null-solutions to iterated generalized Cauchy–Riemann equation over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Finally, we further extend the results to axially symmetric null-solutions to over the upper half unit ball centred at the origin in four-dimensional Euclidean space.

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of $${\mathbb {R}}^4$$ R 4

In this paper we are interested in finding solutions to Riemann–Hilbert boundary value problems, for short Riemann–Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann–Hilbert problems for axially monogenic functions defined over the upper half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann–Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy–Riemann equations.

Boundary value problem for a generalized Cauchy–Riemann equation with singular coefficients

For a generalized Cauchy–Riemann system whose coefficients admit higher-order singularities on a segment, we obtain an integral representation of the general solution and study a boundary value problem combining the properties of the linear conjugation problem and the Riemann–Hilbert problem in function theory.

Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equations in axially symmetric domains

We consider Riemann–Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric poly-monogenic functions, i.e., null-solutions to iterated generalized Cauchy–Riemann equations, defined in axially symmetric domains. This extends our recent results about RHBVPs with variable coefficients for axially symmetric monogenic functions defined in four-dimensional axially symmetric domains. First, we construct the Almansi-type decomposition theorems for poly-monogenic functions of axial type. Then, making full use of them, we give the integral representation solutions to the RHBVP considered. As a special case, we derive solutions to the corresponding Schwarz problem. Finally, we generalize the result obtained to functions of axial type which are null-solutions to perturbed iterated generalized Cauchy–Riemann equations Dαkϕ=0,k⩾2(k∈N),α∈R.

The Radon transform between monogenic and generalized slice monogenic functions

In Bures et al. (Elements of quaternionic analysis and Radon transform, 2009), the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on \({\mathbb {R}}^{n+1}\) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra \({\mathbb {R}}_n\) is mapping solutions of the generalized Cauchy–Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in \({\mathbb {R}}_n\) and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

О поведении решения краевой задачи для обобщенного уравнения Коши–Римана

О поведении решения краевой задачи для обобщенного уравнения Коши–Римана

Harmonic Analysis Associated with the One-Dimensional Dunkl Transform

This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy–Riemann equations D x u−∂ y v=0,∂ y u+D x v=0, where D x is the Dunkl operator (D x f)(x)=f′(x)+(λ/x)(f(x)−f(−x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform , a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces $H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})$ on the half-plane ${\mathbb{R}}^{2}_{+}$ for p≥p 0=2λ/(2λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform is shown to be a bounded mapping from $H_{\lambda}^{1}({\mathbb{R}})$ to $L^{1}_{\lambda}({\mathbb{R}})$ ; and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for $f\in H^{1}_{\lambda}({\mathbb{R}})$ .

Gelfand–Tsetlin bases for spherical monogenics in dimension 3

The main aim of this paper is to recall the notion of Gelfand–Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac or the generalized Cauchy–Riemann equation, respectively) in dimension 3. In the paper, using the GT construction, we obtain explicit orthogonal bases for spherical monogenics in dimension 3. We compare them with those constructed by the first and the second author recently (by a direct analytic approach) and we show in addition that the GT basis has the Appell property with respect to all three variables. The last fact is quite important for future applications.