On a Family of Quasimetric Spaces in Generalized Potential Theory

Abstract

We construct a family of quasimetric spaces in generalized potential theory containing m-subharmonic functions with finite (p, m)-energy. These quasimetric spaces will be viewed both in $${\mathbb {C}}^n$$ C n and in compact Kähler manifolds, and their convergence will be used to improve known stability results for the complex Hessian equations.

On the regularity of the complex Hessian equation

“…, then it follows from [6] that (E p,m (Ω), J p ) is a complete quasimetric space. We shall as well need the following comparison principle from [26].…”

“…, where u, v ∈ E p,m (Ω), p > 0. In [6], it was proved that (E p,m (Ω), J p ) is a complete quasimetric space. Furthermore, convergence in (X, J p ) implies convergence in capacity, and in L q -norm, but the converse is not true.…”

Poincaré- and Sobolev- Type Inequalities for Complex m-Hessian Equations

“…This section introduces the necessary definitions and summarizes some basic facts. For additional details, we refer the reader to [6,41,47,48].…”

On quaternionic pluripotential theory associated to quaternionic $m$-subharmonic functions

“…Pluripotential theory for m-subharmonic functions developed rapidly in last two decades, and there are vast literatures (cf. [1,2,8,10,12,13,15,17,19,20,22,23,25,26,30] and references therein).…”