Maximal m-subharmonic functions and the Cegrell class Nm

On a Family of Quasimetric Spaces in Generalized Potential Theory

“…By Proposition 4.2 we know that convergence in (X , J p ) implies convergence in capacity, but by Example 4.3, we have that the converse statement is false. Hence, Theorem 6.3 is a generalization of [34,Theorem 7.2]. Note that this also implies improved results in the pluricomplex case, m = n, and therefore, Theorem 6.3 also generalizes the stability result by Cegrell and Kołodziej [16].…”

“…Let ψ ∈ E 0,m ( ) be such that −1 ≤ ψ ≤ 0, and K . Then by Błocki's inequality (see, e.g., Lemma 3.4 in [34]) and by (4.2) we get…”

“…Now assume that p ≤ m. Then we can continue our estimate by using the Hölder inequality. By [34], we get…”

“…In this section, we shall prove some new stability results for the complex Hessian operator. For previous results concerning stability of the complex Monge-Ampère equation or the complex Hessian equation, see, e.g., [16,19,34]. In those papers, the authors proved that under some assumption if μ j converges to μ, then the corresponding solutions U (μ j ) converges to U (μ) in capacity.…”

Geodesics in the space of $m$-subharmonic functions with bounded energy

“…Note that if u ∈ K, then F (u(z), z) dµ ≤ H m (ψ). By [50] there exists a uniquely determined function v ∈ E 0,m such that H m (v) = F (u(z), z) dµ, and by the comparison principle we have that v ≥ ψ. Thus, v ∈ K, i.e.…”