Hilbert boundary value problem in the weighted spaces L 1(ρ)

Abstract

The paper studies a Hilbert boundary value problem in L 1(ρ), where ρ(t) = |1–t|α and α is a real number. For α > −1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > −1 and n+κ < 0. For α ≤ −1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.