ABSTRACT.

The Riemann problem is stated as follows: find an analytic function in a domain \( D_{+} \cup D_{-} \) such that \( (*) \ \phi_{+}(t) = G(t)\phi_{-}(t) + g(t), \ t \in S \). Here \( S \) is the boundary of \( D_{+} \), \( D_{-} \) complements the complex plane to \( D_{+} \cup S \), the functions \( G = G(t) \) and \( g = g(t) \) belong to \( H^{\mu}(S) \), the space of Hölder-continuous functions. The theory of problem \( (*) \) is developed also for continuous \( G \). If \( G = 1 \), \( S \in C^{\infty} \), and \( g \) is a tempered distribution, then problem \( (*) \) has a solution in tempered distributions. It is proved that problem \( (*) \) for \( G \in L_{p}(S) \) and \( g \) a tempered distribution does not make sense. It is proved that if \( G \in C^{\infty}(S) \), \( G \ne 0 \) on \( S \), and \( g \) is a distribution of the class \( \mathcal{D}’ \), then the Riemann problem makes sense and a method for solving this problem is given. It is proved that if \( S = \mathbb{R} = (-\infty,\infty) \) and \( G \in L_{p}(\mathbb{R}) \), where \( p \ge 1 \) is a fixed number, then \( \ln |G| \) does not belong to \( L_{q}(\mathbb{R}) \) for any \( q \ge 1 \).