On the Existence of Solutions of a Two-DimensionalHypersingular Integral Equation in the Class of Functions with a Singularity on the Boundaryof the Domain

We consider a two-dimensional hypersingular integral equation in a convex bounded domain whose boundary is a smooth curve. The equation contains an integral operator with integral understood in the sense of Hadamard finite part. We study the existence of solutions having a power-law singularity on the boundary of the domain: the solution is sought in the class of functions represented as the ratio of a smooth function to the root of the distance from a point to the boundary. We prove that the action of an integral operator with a power-law pole singularity of the third order on a function in the class in which the solution is sought gives a function that is Hölder continuous on the entire domain. Further, we prove that the hypersingular equation has a solution with the above-mentioned power-law singularity on the boundary of the domain and indicate a boundary condition under which such a solution is unique.