On the Solvability of Riemann Problems in Grand Hardy Classes

Abstract

The grand Hardy classes $$H_{p)}^{+}$$ and $${}_{m}H_{p)}^{-}$$ , $$p>1$$ , of functions analytic inside and outside the unit disk, which are generated by the norms of the grand Lebesgue spaces, are defined. Riemann problems of the theory of analytic functions with piecewise continuous coefficient are considered in these spaces. For these problems in grand Hardy classes, a sufficient solvability condition on the coefficient of the problem is found and a general solution is constructed. It should be noted that grand Lebesgue spaces are nonseparable and, therefore, certain classical facts (for example, part of the Riesz theorem) do not hold in these spaces, as well as in the Hardy spaces generated by them. Therefore, one must find a suitable subspace associated with differential equations and study the problems in these subspaces.