Degenerate quarternionic Monge-Ampère equations in weighted energy classes

2025/04/28 by Lin, Genglong
#31C10 #32U15 #32U40 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.19619

Abstract

In this paper, we consider degenerate quaternionic Monge-Ampère equations in weighted energy class E_χ(Ω) where Ω is a quarternionic domain in ℍⁿ and χ is a weight function which satisfies some natural conditions. Firstly we prove that the quaternionic Monge-Ampère operator is well-defined for functions in E_χ(Ω), in particular E_p(Ω),p>0. Secondly, we prove that fine property holds in the Cegrell type class E(Ω). As an application, we prove a mass concentration theorem for the quarternionic plurisubharmonic envelope. In the study of complex Monge-Ampère equation, characterization of finite energy range of complex Monge-Ampère operator was a central problem which aroused the interest of experts in the subject. As a quaternionic analogue, we prove a theorem which explicitly characterizes the finite energy range of quaternionic Monge-Ampère operator in the end.

Related

• Cegrell classes and a variational approach for the quaternionic Monge-Ampere equation
• On the finite energy classes of quaternionic plurisubharmonic functions
• The weigthed Monge-Ampère energy of quasiplurisubharmonic functions
• On some weighted energy classes of plurisubharmonic functions
• Weighted pluricomplex energy