The Dirichlet Problem in the Class of shm-functions on a Stein Manifold X

The purpose of this paper is to introduce and study strongly m-subharmonic (shm)functions
\non complex manifolds X ⊂ CN , dimX = n, n 6 N. There are different ways to define shm-functions on
\ncomplex manifolds: using local coordinates, using retraction π : CN → X or using Jensen measures (see
\nfor example [1, 8, 13]). In this paper we use the local coordinates. In Section 1 we present the definition
\nand simplest properties of shm-functions in Cn. In Section 2, we provide the definition of shm-functions
\nin the domains D ⊂ X of the complex manifold X and prove several of their potential properties. Section
\n3 introduces maximal functions and their properties, while Section 4 presents the main result of the work
\n(Theorem 4.1) concerning the solvability of the Dirichlet problem in regular domains