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.