Capacity and stability on some Cegrell classes of $$m-$$subharmonic functions

Solutions to weighted complex m-Hessian equations on domains in Cn

“…Now subharmonicity of u and v forces v ≤ u + 2ε entirely on Ω. The proof is complete by letting ε → 0.Using the basic properties of m−subharmonic functions in Proposition 2.2 and the comparison principle Lemma 3.2, as in the plurisubharmonic case (see[BT82]), we have the following quasicontinuity property of m−subharmonic functions (see Theorem 2.9 in[Ch12] and Theorem 4.1 in[SA12]). Proposition 3.6.…”

“…From Theorem 3.14 in [Ch12] it follows that if u ∈ E m (Ω), the complex m-Hessian H m (u) = (dd c u) m ∧ β n−m is well defined and it is a Radon measure on Ω. On the other hand, by Remark 3.6 in [Ch12] the following description of E m (Ω) may be given…”

“…2.2 Next, we recall the classes E 0 m (Ω), F m (Ω) and E m (Ω) introduced and investigated in [Ch12]. Let Ω be a bounded m-hyperconvex domain in C n , which mean there exists an m− subharmonic function ρ : Ω → (−∞, 0) such that the closure of the set {z ∈ Ω : ρ(z) < c} is compact in Ω for every c ∈ (−∞, 0).…”

“…n . In [Ch12], Chinh introduced the Cegrell classes F m (Ω) and E m (Ω) which are not necessarily locally bounded and the complex m-Hessian operator is well defined in these classes. On the other hand, solving the Monge -Ampère equation in the class of plurisubharmonic functions is important problem in pluripotential theory.…”

“…In Section 2 we recall the definitions and results concerning the m-subharmonic functions which were introduced and investigated intensively in recent years by many authors (see [B l05], [SA12]). We also recall the Cegrell classes of m-subharmonic functions F m (Ω), N m (Ω) and E m (Ω) which were introduced and studied in [Ch12] and [T19]. In Section 3, we present a version of the comparison principle for the weighted m− Hessian operator H χ,m .…”