ON RIEMANN PROBLEM IN WEIGHTED SMIRNOV CLASSES WITH POWER WEIGHT

Weighted Smirnov classes with power weight in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problem with a measurable coefficient whose argument is a piecewise continuous function is considered in these classes. A Muckenhoupt type condition is imposed on the power type weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered, and conditions on degeneration order are found.