A note on approximation of plurisubharmonic functions (Q1692361)

The classical Mergelyan property concerns approximating of holomorphic functions on a subdomain by global holomorphic functions. The PSH-Mergelyan property concerns the analogous property where holomorphic functions are replaced by plurisubharmonic functions. For a bounded pseudoconvex domain in \(\mathbb{C}^n\) with smooth boundary, \textit{N. Sibony} [Duke Math. J. 55, 299--319 (1987; Zbl 0622.32016)] proved the PSH-Mergelyan property for plurisubharmonic functions which extend continuously to the boundary. \textit{J. E. Fornæss} and \textit{J. Wiegerinck} [Ark. Mat. 27, No. 2, 257--272 (1989; Zbl 0693.32009)] extended this to domains with \(C^1\)-boundaries only. By more recent work of \textit{B. Avelin} et al. [Complex Var. Elliptic Equ. 61, No. 1, 23--28 (2016; Zbl 1338.32029)], the PSH-Mergelyan property is also valid for domains with \(C^0\)-boundaries only. There are also classical precise Mergelyan-type criteria on harmonic and subharmonic functions on \(\mathbb{C}\) and more generally \(\mathbb{R}^n\), involving the thinness of the complement of the concerned domains, see Section 4 in the paper. If we do not assume \(C^0\) boundaries, then \textit{L. Hed} [The Plurisubharmonic Mergelyan Property. Umeå (Thesis) (2012)] showed that the PSH-Mergelyan property fails, for domains as simple as \(\mathbb{D}\backslash [-1/2,1/2]\subset \mathbb{C}\). (Here \(\mathbb{D}\) is the unit disk.) In this paper, the authors show on the other hand that if only countably many boundary points are not \(C^0\), then the PSH-Mergelyan property still holds. The authors also provide some extra assumptions under which it is allowed that uncountably many boundary points are not \(C^0\). They also discuss approximation with stronger regularity (Hölder continuity) of the involved functions. The main assumptions and arguments in the paper are based on the modulus of continuity \(\omega\) of a function \(u:X\rightarrow \mathbb{R}\), which is required to satisfy \(|u(z)-u(w)|\leq \omega (|z-w|)\) for all \(z,w\in X\). Using the modulus of continuity and standard constructions for plurisubharmonic functions defined on intersecting domains, the authors prove the key Lemma 2.2, which approximates a given function \(u\) by another function \(v\) defined on a slightly bigger domain containing a small neighbourhood around a given boundary point. This is then applied to the non-\(C^0\)-boundary points. Here the assumption that there are only countably non-\(C^0\)-boundary points is used when the authors apply Gauthier's localisation theorem for checking that the constructed function satisfies the conclusion of the main theorem. In the case where all boundary points are \(C^0\), the argument gives a new proof of the mentioned result by Avelin et al. [loc. cit.]. To extend the result to the case of uncountably many non-\(C^0\)-boundary points, the authors introduce a condition \(PS(C,\varphi)\) (see Definition 3.1). In Theorem 3.5, this \(PS(C,\varphi )\) condition is required for the set of non-\(C^0\)-boundary points, where \(\varphi(t)=-\omega (t) \log (t)\) and \(\omega \) is the modulus of continuity which is required to be concave. In these results, some constraints on the modulus of continuity of the functions \(u\) are required. The authors give some examples showing that indeed some conditions on the modulus of continuity must be imposed, otherwise approximation is not possible.