Potentials and correspondence of boundaries

A theorem on the absolute continuity of $ACL_p$-functions on the level sets of $ACL_q$-functions is proved.
For $ACL_1$-homeomorphisms of plane domains the problem of boundary correspondence along Carathéodory prime ends is studied in connection with the singularities of the potentials of the gradient of these mappings.
Horneomorphisms with a bounded potential of the gradient are introduced – new classes of plane topological mappings for which estimates of the distortion of the relative distance in closed domains are established, and, in particular, the $C$-correspondence of boundaries.
Bibliography: 22 titles.