On the regularization of one class of sum-difference equations with periodic coefficients

Abstract

Let $D$ be a square with the boundary $\Gamma$. A four-element linear sum-difference equation is considered in the class of functions that are holomorphic outside $D$ and vanish at infinity. The coefficients of the equation and the free term are holomorphic in $D$. The solution is sought in the form of a Cauchy-type integral over $\Gamma$ with unknown density. Its boundary values satisfy the Hölder condition on any compact set in $\Gamma$ that does not contain vertices. At most, logarithmic singularities are allowed at the vertices. To regularize the equation on $\Gamma$, a piecewise linear Carleman shift is introduced, which changes the orientation and still has fixed points. It is continuous at the vertices, but its derivatives are discontinuous at them. The regularization of uranium was carried out and the condition for its equivalence was found. Various applications and generalizations are indicated.