ANNALES POLONICI MATHEMATICI 100.2 (2011) A unicity theorem for plurisubharmonic functions by Nguyen Quang Dieu (Hanoi) Abstract. We give sufficient conditions for unicity of plurisubharmonic functions in Cegrell classes. 1. Introduction. Let Ω be an open subset of C n . An upper semicon- tinuous function u : Ω → [−∞, ∞) is said to be plurisubharmonic if the restriction of u to each complex line is subharmonic (we allow the function identically −∞ to be plurisubharmonic). We write PSH(Ω) (resp. PSH − (Ω)) for the cone of plurisubharmonic (resp. negative plurisubharmonic) functions on Ω. The domain Ω is said to be hyperconvex if there exists a continuous negative plurisubharmonic exhaustion function for Ω. Let u,v ∈ PSH − (Ω) be such that lim z→∂Ω u(z ) = lim z→∂Ω v(z )=0. In this note, we are aiming at sufficient conditions to ensure that u = v near the boundary ∂Ω. Before formulating the main result, it is convenient to recall the following concept. A compact subset K of Ω is said to be holomorphically convex if for every z ∈ Ω \ K, there exists a holomorphic function f on Ω such that ‖f ‖ K < |f (z )|. We will prove the following. Theorem A. Let Ω be a bounded hyperconvex domain in C n . Let K ⊂ Ω be a compact holomorphically convex subset of Ω. Let u 1 ,u 2 ∈ PSH − (Ω) be such that the following conditions hold: (a) lim z→∂Ω u 1 (z ) = lim z→∂Ω u 2 (z )=0. (b) (dd c u 1 ) n ≤ (dd c u 2 ) n on Ω \ K and  Ω (dd c u 2 ) n < ∞. (c) u 1 ≤ u 2 on Ω \ K. (d)  K (dd c u 1 ) n ≤  K (dd c u 2 ) n . Then u 1 = u 2 on Ω \ K. 2010 Mathematics Subject Classification : 31C15, 32F05. Key words and phrases : plurisubharmonic function, Monge–Amp` ere operator, hypercon- vex domain. DOI: 10.4064/ap100-2-5 [159] c  Instytut Matematyczny PAN, 2011