Modulability and duality of certain cones in pluripotential theory

Abstract

ModulabilityOrdered vector space Quasi-Banach space Topological dual Let p > 0, and let E p denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δE p = E p − E p , with the vector ordering induced by the cone E p is σ -Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone E p in the quasi-normed space δE p is closed, generating, and has empty interior.

On Dirichlet's principle and problem

“…It was proved in [3] (see also [4]) that for p = 1 the constant D(n, p) in Theorem 2.1 is strictly great than 1. For this reason we can not use similar variational method to prove the Dirichlet principle in the class E p when p = 1.…”

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

“…We start with the definition of msubharmonic functions and the m-Hessian operator. Let Ω ⊂ C n , n ≥ 2, be a bounded domain, 1 ≤ m ≤ n, and define C (1,1) to be the set of (1, 1)-forms with constant coefficients. With this set…”

“…It was proved in [1,34] that (δE p,k , · ) is a quasi-Banach space for p = 1, and a Banach space for p = 1. Furthermore, the cone E p,k (Ω) is closed in δE p,k (Ω).…”

“…Then it was proved in [1,34] that (δM p,m , | · | p,m ) is a quasi-Banach space, and for p = 1 a Banach space.…”

On a Family of Quasimetric Spaces in Generalized Potential Theory

“…Proposition 2. 4 Let n ≥ 2, 1 ≤ m ≤ n, and assume that ⊂ C n is an m-hyperconvex domain, and u, v ∈ E p,m ( ).…”