Siberian Federal University

The purpose of this paper is to introduce and study strongly m-subharmonic (shm)functions on complex manifolds X ⊂ CN , dimX = n, n 6 N. There are different ways to define shm-functions on complex manifolds: using local coordinates, using retraction π : CN → X or using Jensen measures (see for example [1, 8, 13]). In this paper we use the local coordinates. In Section 1 we present the definition and simplest properties of shm-functions in Cn. In Section 2, we provide the definition of shm-functions in the domains D ⊂ X of the complex manifold X and prove several of their potential properties. Section 3 introduces maximal functions and their properties, while Section 4 presents the main result of the work (Theorem 4.1) concerning the solvability of the Dirichlet problem in regular domains