Riemann and Hilbert boundary value problems in BMO classes for holomorphic functions

The classical Riemann and Hilbert boundary value problems for analytic functions are under consideration. We search the solution in BMOA class under assumption that the coefficient of the boundary condition belongs to the set of pointwise multipliers of BMO. We constract examples when the problem with non-negative index in the such natural setting has not solution in BMOA. Sufficient conditions on the coefficient are given when we have usual pattern of solvability in BMOA class.