On a Carleman problem in the case of a doubly periodic group

Let $\Gamma$ be a doubly periodic group whose fundamental region $D$ is a rectangle, in which the ratio of the largest side to the shortest one does not exceed $3$. The generating transformations of the group and their inverse transformations induce, on the boundary, an involutive inverse shift, discontinuous at the vertices. We consider a particular case of the Carleman problem for functions that are analytic in $D$ (the so-called jump problem). We show that the regularization of the unknown function suggested by Torsten Carleman leads to an equivalent regularization of the problem. For this, we rely on the contraction mapping principle for Banach spaces and use the theory of Weierstrass elliptic functions. The integral representation was first introduced by Carleman during his talk at the International Congress of Mathematicians in Zürich in 1932. However, he did not investigate the Fredholm integral equation obtained by regularizing the jump problem. In particular, the question of equivalence of the jump problem and the corresponding Fredholm equation obtained through the given representation remained open.