On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical

We consider a certain analog of Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical. We establish sufficient conditions for an existence of limiting values of this integral on the curve of integration.


Introduction.
Let Γ be a closed Jordan rectifiable curve in the complex plane C. By D + and D − we denote, respectively, the interior and the exterior domains bounded by the curve Γ.
N. Davydov [1] established sufficient conditions for an existence of limiting values of the Cauchy type integral (1) on Γ from the domains D + and D − .This result stimulated a development of the theory of Cauchy type integral on curves which are not piecewisesmooth.
In particular, using the mentioned result of the paper [1], the following result was proved: if the curve Γ satisfies the condition (see [2]) (2) θ(ε) := sup ξ∈Γ θ ξ (ε) = O(ε), ε → 0 (here θ ξ (ε) := mes {t ∈ Γ : |t − ξ| ≤ ε}, where mes denotes the linear Lebesgue measure on Γ), and the modulus of continuity ω g (ε) := sup then the integral (1) has limiting values in every point of Γ from the domains D + and D − (see [3]).The condition (2) means that the measure of a part of the curve Γ in every disk centered at a point of the curve is commensurable with the radius of the disk.
In this paper we consider a certain analogue of Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical and study the question about an existence of its limiting values on the curve of integration.
We say that a continuous function For monogenic functions Φ : Ω ζ → A 3 we established basic properties analogous to properties of analytic functions of the complex variable: the Cauchy integral theorem, the Cauchy integral formula, the Morera theorem, the Taylor expansion (see [11]).

On existence of limiting values of a hypercomplex analogue of the Cauchy type integral.
In what follows, t 1 , t 2 , x, y, z ∈ R and the variables x, y, z with subscripts are real.For example, x 0 and x 1 are real, etc. Let and a singular integral For the Euclidean norm in A 3 the following inequalities are fulfilled: To estimate I 1 we choose a point Using the inequalities ( 7) and ( 8), we obtain It follows from Lemma 1.1 [9] that (10) , where ξ := x + iy and t := t 1 + it 2 .The following inequality follows from the relations ( 9) and ( 10): where the constant c(m) depends only on m.
To estimate the last integral we use Proposition 1 [10] (see also the proof of Theorem 1 [4]) and the condition (2).So, we have where the constant c does not depend on ε.
To estimate I 1 we introduce the domain Using the inequalities ( 8) and (11), we obtain: Estimating I 2 , by analogy with the estimation of I 1 , we obtain: where the constant c does not depend on ε.
Using the inequality |t − ξ| ≥ |t − ξ 0 |/2, where the point ξ 0 := x 0 + iy 0 corresponds to the point ζ 0 = x 0 + y 0 e 2 , and using the relations (7), (8), (11) and (2), by analogy with the estimation of I 1 , we obtain: The theorem follows from the Lemma 1 and the equalities In comparison with Theorem 1, note that additional assumptions about the function ϕ are required for an existence of limiting values of the function (6) be the curve congruent to the curve Γ ⊂ C. Consider the domain Π ± ζ := {ζ = x + ye 2 + ze 3 : x + iy ∈ D ± , z ∈ R} in E 3 .By Σ ζ we denote the common boundary of domains Π + ζ and Π − ζ .Consider the integral (6) Φ(ζ) = 1 2πi Γ ζ ϕ(τ )(τ − ζ) −1 dτ with a continuous density ϕ : Γ ζ → R. The function (6) is monogenic in the domains Π + ζ and Π − ζ , but the integral (6) is not defined for ζ ∈ Σ ζ .For the function ϕ : Γ ζ → R consider the modulus of continuity Theorem 1 in the case where the curve Γ satisfies the condition (2) and the modulus of continuity of the function ϕ satisfies a condition of the type (3), we establish the existence of certain limiting values of the integral (6) in points ζ 0 ∈ Γ ζ when ζ tends to ζ 0 from Π + ζ or Π − ζ along a curve that is not tangential to the surface Σ ζ outside of the plane of curve Γ ζ .

2 , e 2 e 3 , e 2 3 } 3 . 1 .
) dτ with the constant M := max{1, e 2 for any measurable set Γ ζ ⊂ Γ ζ and all continuous functions ψ : Γ ζ → A Lemma Let Γ be a closed Jordan rectifiable curve satisfying the condition (2) and the modulus of continuity of a function ϕ : Γ ζ → R satisfies the condition of the type (3).If a point ζ tends to ζ 0 ∈ Γ ζ along a curve γ ζ for which there exists a constant m < 1 such that the inequality
from Π + ζ or Π − ζ on the boundary Σ ζ .We are going to state these results in next papers.