Cegrell classes

The aim of the present paper "The Topology on the Space of δ-psh Functions in the Cegrell Classes" is to introduce a metric locally convex topology on the space δE = E−E of δ-psh functions in the Cegrell class E. We prove that with this topology δE is a non-separable and non-reflexive Fr´echet space. At the same time, we extend the Monge–Amp`ere operator from the class E to δE.

14p runordie5 04-07-2022 6 1   Download

In the paper, we prove the existence of solutions of the complex Monge–Ampère type equation −χ(u)(ddcu)n = µ in the class Eχ(Ω) if there exist subsolutions in this class. As an application, we prove that the complex Monge–Ampère equation (ddcu)n = µ is solvable in the class E(Ω) if there exist subsolutions locally. Moreover, by an example we show that the conditions in our above result are sharp.

17p runordie3 27-06-2022 18 3   Download

The aim of the paper "Uniqueness properties of m-subharmonic functions in Cegrell classes" is to provide sufficient conditions for unicity of m-subharmonic functions. Since two plurisubharmonic functions may equal on an open subset of a domain without being identical (e.g. u ≡ 0 and v(z) = max(log |z|, 0)), it seems natural to put additional assumptions on Hessian measures of u and v to guarantee that u ≡ v on the whole domain Ω.

15p runordie3 27-06-2022 23 2   Download