Finite-dimensional smoothness of Cauchy-type integrals

An estimate for the modulus of continuity of a quaternion singular Cauchy integral

“…In particular, it was established that the broadest class of curves (see in [6,9]) for which it has the same form as on a circle is the class of regular curves (for which the measure of the part of a curve that enters the disk does not exceed a constant multiplied by the radius of the disk). For more general curves (see [6,[9][10][11][12][13]), the majorant worsens and depends on a curve.…”

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

“…(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…”

“…Theorem 1. 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). Then the integral (6) has boundary values Φ ± (ζ 0 ) for all ζ 0 ∈ Γ ζ that are expressed by the formulas:…”

“…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.…”