A boundary value problem with degeneration on the boundary along the manifold of codimension $k > 2$

The article considers the boundary value problem for elliptic equations of arbitrary order $2m$ with degeneracy on the boundary of the domain along manifolds of codimension $k > 2$. The study uses methods of functional analysis and geometry of smooth manifolds proposed by Y. V. Egorov and V. A. Kondratiev. These methods allow us to investigate the boundary value problem in more general formulation. Aprioristic estimates for the solution of a task in the generalized spaces of Sobolev – Slobodetsky are obtained and the theorem of smoothness of solutions of a task is formulated.