ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 30, 45 Litvinchuk, G. S., Universidade da Madeira, Portugal On Solvability theory of Hilbert boundary value problem with a Carleman shift. We consider the following problem. Find a function Φ^{+}(z) analytic in the circle |z| < 1 satisfying the condition

Re{A(t)Φ^{+}(t) + B(t)Φ^{+}(α(t))} = h(t) on |t| = 1, (1)

where A, B, h ∈ H_{μ}(Γ) (Hólder class), α(t) is a linear fractional diffeomorphism of |t| = 1 onto itself preserving ou changing the orientation on |t| = 1 and satisfying the Carleman condition α(α(t)) ≡ t. The numbers of linear independent solutions and solvability conditions of problem (1) are calculated, and the very solutions are constructed in the terms of a factorization of (2 × 2) - matrix function determined by coefficients A, B. The particular cases are analysed when this factorization is realized effectively. Reference: G. S. Litvinchuk, Boundary value problems and singular integral operators with shift, Nauka, Moscow, 1977 (in Russian), pp. 1 - 448. Chinese transl. : Beijing, 1982.