The Range of the Monge-Ampère operator (ω+ dd^c .)ⁿ in bounded domains

2025/09/28 by Omar Alehyane, Alehyane, Omar, Fatima Zahra Assila +3
Mathematics · #32Q15 #32U05 #32W20 #35A23 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.23944

openalex publication_date 2025/09/28 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

Let Ω be a bounded strictly pseudoconvex domain of ℂⁿ. We solve degenerate complex Monge-Ampère equations of the form (ω+ dd^c φ)ⁿ = μ in the generalized Cegrell classes K(Ω,ω,H), where H ∈ E(Ω) is maximal, ω is a smooth real (1,1)-form defined in a neighborhood of Ω and μ is a positive Radon measure. This generalizes the previous work of the last author \citeSal25 to the case of non-continuous functions H and also to the case of measures μ which do not vanish on pluripolar sets.

Citations

• Geodesic connectivity and rooftop envelopes in the Cegrell classes
• Plurisubharmonic envelopes and supersolutions
• Monge-Ampere measures on pluripolar sets

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