Extension and approximation ofm-subharmonic functions

Abstract

Abstract.Let Ω ⊂ C n be a bounded domain, and let f be a real-valued function defined on the whole topological boundary ∂Ω. The aim of this paper is to find a characterization of the functions f which can be extended to the inside to a m-subharmonic function under suitable assumptions on Ω. We shall do so by using a function algebraic approach with focus on m-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of m-subharmonic functions.

The Dirichlet problem for $m$-subharmonic functions on compact sets

“…With the help of the Jensen measures defined in Definition 2.1 we can now define m-subharmonic functions defined on compact sets. For more results about these functions, see [3].…”

“…We shall need the following version of Edwards' celebrated duality theorem (see Theorem 2.8 in [3]).…”

“…It was proved in [3] that every m-harmonic function defined on an open set is pluriharmonic. The situation is different for the function theory on compact sets.…”

The classification of holomorphic (m, n)-subharmonic morphisms

“…Let Ω N ⊆ C N be a connected, and open set, and 1 ≤ k ≤ N . By C (1,1) we denote the space of (1, 1)-forms with constant coefficients, and then we define…”

“…For a similar phenomena in the Caffarelli-Nirenberg-Spruck model see e.g. Section 5 in [1]. On the other hand, there has been some progress obtaining a Poisson type formula for SH k using Choquet theory and representing measures, instead of Poletsky disks (see e.g.…”