Hilbert Boundary Value Problem in the Weighted Spaces L~1(p)

摘要

The paper studies a Hilbert boundary value problem in L~1(p), where p(t) = |1 - t|~α and α is a real number. For α > -1, it is proved that the homogeneous problem has n + k linearly independent solutions when n + k ≥ 0, where a(t) is the coefficient of the problem, besides, k-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 + k < 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.