ANNALES POLONICI MATHEMATICI 94.3 (2008) Subextension of plurisubharmonic functions without increasing the total Monge–Ampère mass by Rafał Czyż (Kraków) and Lisa Hed (Umeå) Abstract. We prove that subextension of certain plurisubharmonic functions is al- ways possible without increasing the total Monge–Ampère mass. 1. Introduction. Bedford and Burns [5] (see also [9]) proved that any smooth bounded domain in C n satisfying a certain non-degeneracy condition on the Levi form on the boundary is the domain of existence for plurisub- harmonic functions, and El Mir [19] constructed an example of a plurisub- harmonic function defined on the unit bidisc in C 2 for which the restriction to any smaller bidisc admits no subextension to C 2 . Bedford and Taylor [6] improved an example by Fornæss and Sibony [20] by constructing a smooth negative plurisubharmonic function on an arbitrary bounded domain in C n with C 2 -boundary that does not subextend. In this article we are interested in subextension without increasing the total Monge–Ampère mass. Before proceeding we need some background and notation. Let PSH(Ω) denote the set of all plurisubharmonic functions defined on a domain Ω ⊂ C n . Recall that a bounded domain Ω ⊆ C n is called hyperconvex if there exists a bounded plurisubharmonic function ϕ : Ω → (−∞, 0) such that the closure of the set {z ∈ Ω : ϕ(z ) <c} is compact in Ω for every c ∈ (−∞, 0). Let E 0 (Ω) be the class of bounded plurisubharmonic functions u such that lim z→ξ u(z )=0 for all ξ ∈ ∂Ω and  Ω (dd c u) n < ∞. We say that a negative function u ∈ PSH(Ω) is in the class F (Ω) if there is a decreasing sequence [u j ] of functions u j ∈E 0 (Ω) which converges pointwise to u on Ω and sup j  (dd c u j ) n < ∞. The class E (Ω) contains the functions in PSH(Ω) that are locally in F (Ω), and Theorem 4.2 2000 Mathematics Subject Classification : Primary 32U15; Secondary 32W20. Key words and phrases : complex Monge–Ampère operator, plurisubharmonic function, subextension. The first-named author was partially supported by ministerial grant number N N201 3679 33. [275] c  Instytut Matematyczny PAN, 2008