E. Gordadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE PROBLEM OF LINEAR CONJUGATION IN THE CASE OF NON-SMOOTH LINES

1. Let S_{Γ} be a singular operator, S_{Γ}φ = (πi)^{−1} ∫_{Γ} φ(t)(t − τ)^{−1} dt. If

S_{Γ} is bounded in L_{p}(Γ), then we write Γ ∈ R. If the operator ρS_{ρ} is also bounded in L_{p}(Γ), where ρ(t) is a positive measurable function, then we write ρ ∈ W_{p}(Γ). Denote by Γ_{t} a continuous on Γ arc (which open if t is an exterior point, and closed if Γ is unclosed, and t is one of the end points), such that t ∈ Γ_{t}.

Theorem. If Γ is the Jordan line, Γ ∈ R, and for every t ∈ \overline{Γ} there exists Γ_{t} such that ρ ∈ W_{p}(Γ_{t}), then ρ ∈ W_{p}(Γ).

2. Consider the boundary value problem of linear conjugation in the Hardy–Smirnov class E_{p}(D_{Γ}^{±}), p > 1. Suppose that Γ ∈ R, Γ is the Jordan line and Γ = ∪^{n}_{k=1}Γ_{a_{k}a_{k+1}}, a_{n+1} = a_{1} (Γ_{ab} denotes the arc with the

ends a and b which is continuous from a to b). Assume that Γ_{a_{k}a_{k+1}} can be supplimented up to the closed contour Γ_{k}^{0} ∈ R for which the problem under Simonenko’s conditions with the zero index has the unique solution. Contours of the type Γ_{k}^{0} have been considered by many authors, however, the existence of cusps was exclued everywhere. It is evident that the cusps and more complicated cases may occur for the lines Γ = ∪^{n}_{k=1}Γ_{a_{k}a_{k+1}} at the

points a_{k}. We introduce obvious enough additional restrictions on G(t) in the neighborhood of a_{k} which allow one to obtain the classical result for the above-mentioned lines. As an example, we can name both the piecewise smooth lines and the Radon lines.