The Dirichlet problem in the class of $\mathrm{sh_m}$-functions on a Stein manifold $X$

The purpose of this paper is to introduce and study strongly $m$-subharmonic ($sh_m$) functions on complex manifolds $X\subset \mathbb{C}^N, dim X=n, n\leqslant N.$ There are different ways to define $sh_m$-functions on complex manifolds: using local coordinates, using retraction $\pi : {{\mathbb{C}}^{N}}\to X$ 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 $sh_m$-functions in ${{\mathbb{C}}^{n}}.$ In Section 2, we provide the definition of $sh_m$-functions in the domains $D\subset X$ 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.